David L Dickinson
email: dickinsondl@appstate.edu

David Masclet
email: david.masclet@univ-rennes1.fr.

Laurent Denant Boemont

Romain Espinoza

Olivier L'

Using Ethical Dilemmas to predict Antisocial Choices with Real Payoff Consequences: an Experimental Study

Keywords: experiments, money burning, ethical dilemmas, anti-social behavioral, trolley problem JEL Codes: C90, C91, Z10

Anti-social behaviours are costly to organizations, and the ability to identify predictors of such behaviours can be valuable. In this paper, we used a within-subjects laboratory design to study choices in the well-known (hypothetical) Trolley problem as well as in a real payoff moneyburning experiment that can inform our understanding of moral preferences and antisocial behavior. Choices in both environments respond to incentives (i.e., the relative price of the ethical decision). Trolley problem decisions are consistent with previously known resultsindividuals prefer no action over action, and they prefer to avoid direct over indirect responsibility when negative consequences would be similar in either instance. In analyzing the determinants of anti-social money burning, our data identify money burning due to inequality aversion, but we also find evidence of pure nastiness (burning money of others to increase one's advantageous inequality). Importantly, we find that willingness to commit ethically dubious acts in the Trolley problem significantly predicts money burning and, more specifically, nastiness. We conclude that choices in hypothetical environments can predict consequential and inefficient antisocial behaviours. Also, utilitarian behaviour in the Trolley dilemma is not linked to antisocial money burning, which contrasts with conclusions in the literature.

Introduction

Economists have long challenged the assumption of homo economicus and recognize that people are not always own-payoff maximizing. Rather, they may be altruistic, fairness-minded, cooperative, or perhaps even anti-social. Using laboratory methods with real payoff, experimental economics has shown that participants in dictator games often share their endowment [START_REF] Forsythe | Fairness in simple bargaining experiments[END_REF][START_REF] Hoffman | Preferences, property rights, and anonymity in bargaining games[END_REF], they reciprocate in gift exchange or trust environments [START_REF] Berg | Trust, reciprocity, and social history[END_REF][START_REF] Fehr | Gift exchange and reciprocity in competitive experimental markets[END_REF], and they contribute positive amounts in public goods games (see surveys in [START_REF] Ledyard | Public goods: A survey of experimental research[END_REF][START_REF] Chaudhuri | Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature[END_REF]. Studies focusing on the darker side of human behaviour are surprisingly more limited in economics. This is all the more surprising given that unethical behavior within organizations is not rare and often results in high costs for the entire society. Anti-social behaviours can result in relational, workplace, or other costs to society that are nontrivial. Cyber-sabotage is now a growing concern, for example (see [START_REF] Line | Targeted attacks against industrial control systems: Is the power industry prepared?[END_REF], and survey data from the U.S. and Europe document antisocial workplace behaviours that include mistreatment, verbal abuse, and sabotage, with estimates indicating that such antisocial behaviours may impact 10%-35% of people in the workplace (see [START_REF] Charness | The dark side of competition for status[END_REF]. 1 Field data examples often pose difficulties in our attempt to understand the core determinants of antisocial tendencies given that they may be confounded with self-interest, hidden from view, or contaminated by reputational concerns. Behavioural metrics that help identify the likelihood that someone may engage in antisocial behaviours can therefore be a useful way to prevent antisocial behavioural costs and improve overall welfare.

Moral philosophers have examined choices in ethical dilemmas, such as the famous Trolley problem [START_REF] Foot | The problem of abortion and the doctrine of the double effect[END_REF], while in economics there has been some focus on the antisocial dimension of behaviour through experimental research on money burning [START_REF] Zizzo | Are People Willing to Pay to Reduce Others' Incomes?[END_REF][START_REF] Zizzo | Inequality and Procedural Fairness in a Money Burning and Stealing Experiment[END_REF] and nasty behavior (Abbink and[START_REF] Abbink | The pleasure of being nasty[END_REF]Herrman, 2009). An investigation into the components of moral preferences that predict antisocial behaviour, to our knowledge, does not yet exist. Our objective is to construct moral identifiers derived from the Trolley problem to predict antisocial (and consequential) behaviour in a money burning game.

The Trolley dilemma has captivated moral philosophers for decades [START_REF] Foot | The problem of abortion and the doctrine of the double effect[END_REF][START_REF] Thomson | The Trolley Problem[END_REF][START_REF] Spranca | Omission and commission in judgment and choice[END_REF][START_REF] Petrinovich | An empirical study of moral intuitions: Toward an evolutionary ethics[END_REF]. The dilemma describes a runaway trolley that, unless an action is taken, will run over one or more persons on a train track unable to escape. Various versions of the problem exist (see [START_REF] Shallow | Trolley problems in context[END_REF]), but we focus on perhaps two of the most classic scenarios. The first assumes a runaway trolley is bound to kill several individuals on a main set of tracks unless one pulls a lever to divert the trolley onto a side track where it will kill anyone who may be on the side track. Such a decision scenario is considered an "indirect" (or impersonal) moral choice in the sense that pulling the lever to save lives indirectly but intentionally results in the death of those on the side track. A second version is considered a "direct" (or personal) moral choice scenario where instead of pulling a lever one must push an individual onto the main track (and that person will die) in order to save the one or more individuals on the main track [START_REF] Thomson | The Trolley Problem[END_REF].

The Trolley dilemma has come under fire for its lack of realism and external validity [START_REF] Rai | Moral principles or consumer preferences? Alternative framings of the trolley problem[END_REF][START_REF] Bauman | Revisiting external validity: Concerns about trolley problems and other sacrificial dilemmas in moral psychology[END_REF][START_REF] Kahane | Sidetracked by trolleys: Why sacrificial moral dilemmas tell us little (or nothing) about utilitarian judgment[END_REF]. Nevertheless, others have found it useful for studying various components of moral reasoning (e.g., [START_REF] Cushman | The role of reasoning in moral judgments: Testing three principles of harm[END_REF][START_REF] Greene | An fMRI investigation of emotional engagement in moral Judgment[END_REF]Greene et al, 2011), such as the identification of behavioural norms or highlighting that certain moral dilemmas preferentially engage emotional centers in a way that may be important in predicting choice (e.g., [START_REF] Greene | An fMRI investigation of emotional engagement in moral Judgment[END_REF]. Still others have noted how the Trolley dilemma can highlight the difference between acts of omission versus commission [START_REF] Spranca | Omission and commission in judgment and choice[END_REF][START_REF] Cox | Status quo effects in fairness games: reciprocal responses to acts of commission versus acts of omission[END_REF], which is a relevant distinction in courts of law. And, while past criticism of the Trolley dilemma may have seemed justified due to the unrealistic nature of the decision it presents, the relevance of the Trolley dilemma is at a higher level than perhaps ever before with the recent rise in ethical concerns surrounding self-driving vehicles. [START_REF] Bonnefon | The social dilemma of autonomous vehicles[END_REF] highlight how the moral dilemma relates to the social dilemma of Autonomous Vehicle (AV) adoption, whereby most survey respondents agreed an AV should be programmed to sacrifice its passenger(s) if more pedestrians are saved as a result, but these same individuals thought it much less appropriate to program the AV as such if one's own life were at stake.

In economics, studies focusing on the antisocial dimension of behaviour include the seminal studies by [START_REF] Zizzo | Are People Willing to Pay to Reduce Others' Incomes?[END_REF] and [START_REF] Zizzo | Inequality and Procedural Fairness in a Money Burning and Stealing Experiment[END_REF], whose results show that many subjects are willing to incur a real cost in order to reduce other's payoffs-"money burning". Money burning may be explained by inequality aversion [START_REF] Zizzo | Are People Willing to Pay to Reduce Others' Incomes?[END_REF], but it may also result from a pure pleasure of being nasty (Abbink and Sadrieh, 2009). 2 Our goal is to contribute to the literature in the following ways. First, we attempt to investigate the determinants of antisocial behaviours using measures derived from both a realpayoff money burning experiment and a hypothetical Trolley dilemma. Are money burning decisions driven by inequality aversion concerns, pure nastiness, pre-emptive retaliation? In conjunction with our exploration of the determinants of money burning, we hope to further our understanding of some key determinants of (un)ethical choices. Indeed, specific Trolley dilemmas can identify more ethically dubious choices, thus allowing us to classify one's morality. We are also able to distinguish an immoral act of commission from an immoral act of omission, which yields a more rich set of morality variables to consider as predictors of money burning decisions.

Secondly, our paper will also contribute to the literature by investigating the extent to which costly money burning decisions and Trolley choices obey the law of demand. Responses to ethical dilemmas surrounding the adoption of autonomous vehicle technologies, which bear resemblance to the Trolley dilemma, were recently shown to be sensitive to the relative number of lives saved in the scenario [START_REF] Bonnefon | The social dilemma of autonomous vehicles[END_REF]. Within the context of demand for costly punishment. [START_REF] Nikiforakis | A comparative statics analysis of punishment in public-good experiments[END_REF] showed that voluntary contributions to provide a public good increase monotonically in punishment effectiveness, and Anderson and Putterman (2007) found that the price of punishment is a significant determinant of punishment demand. These previous studies suggest that even the moral domain of choice should obey the law of demand. To our knowledge, no previous study has attempted to investigate the role played by relative cost in the context of money burning decisions, and our set of Trolley dilemmas allows us to explore efficiency (utilitarian outcomes), which implies that the dominant concern should be minimizing the number of lives lost.

Finally, we investigate the relationship between decisions in the money burning experiment and choices in the Trolley experiment. Specifically, by exploiting our withinsubjects strategy method design, we examine the ability of moral identifiers derived from Trolley decisions to predict consequential choices in the money burning game. The predictive validity of ethical dilemma responses has been of interest in the recent literature although there is no clear cut evidence regarding the relationship. Some authors have recently argued that hypothetical ethical dilemmas are not useful for predicting behaviour in real dilemmas [START_REF] Bostyn | Of Mice, Men, and Trolleys: Hypothetical Judgment Versus Real-Life Behavior in Trolley-Style Moral Dilemmas[END_REF]. 3 Other studies suggest that antisocial personality types may be more willing to make a difficult moral choice that produces a utilitarian outcome [START_REF] Koenigs | Damage to the prefrontal cortex increases utilitarian moral judgments[END_REF][START_REF] Bartels | The mismeasure of morals: Antisocial personality traits predict utilitarian responses to moral dilemmas[END_REF][START_REF] Bracht | Moral judgments, gender, and antisocial preferences: An experimental study[END_REF]. 4 In this paper we attempt to contribute to this existing literature that attempts to explore the connection between some version of antisocial behavior and choice in hypothetical moral dilemma. The main originality of our work is that in the typical ethical dilemma studied it is always utilitarian to sacrifice the life because even more are saved. Thus, existing studies cannot separate utilitarian behaviour from less savoury preferences that may perversely desire to cause a death through action. Our design can separate utilitarian from immoral preferences by including Trolley dilemmas not typically considered, which then allows us to construct moral identifiers. 5 These moral identifiers are shown to have power predicting antisocial behaviour in the consequential (i.e., non-hypothetical) money burning game. If choices in hypothetical moral thought experiments can help identify those likely to make antisocial choices, it may be possible to improve overall welfare.

To preview our main findings, we observe that choices in both environments respond to incentives (i.e., the relative price of the ethical decision. In analyzing the determinants of antisocial money burning, our data identify money burning due to inequality aversion, but we also find evidence of pure nastiness (burning money of others to increase one's advantageous inequality). Trolley problem decisions are consistent with previously known resultsindividuals prefer no action over action, and prefer to avoid direct over indirect responsibility when negative consequences would be similar in either instance. Importantly, we find that willingness to commit ethically dubious acts in the Trolley problem significantly predicts money burning and, more specifically, nastiness. For example, those willing to commit an immoral act of omission in the Trolley dilemma are significantly more likely to display nasty money burning behavior (i.e., destruction of another's payoff even when one is already at a payoff advantage). We conclude that ethically questionable choices in hypothetical environments can predict consequential and inefficient antisocial behaviours. Also, utilitarian behaviour in the Trolley dilemma is not linked to antisocial money burning, which contrasts with conclusions in the literature.

Experimental design

Overview

Both experiments were administered in strategy method format, where decisions were elicited on multiple decisions prior to a randomized draw of one (in the incentivized money burning task) for real payoff. The Trolley dilemma was always administered first to the subjects, and we allowed the option to opt out and not complete the Trolley dilemma if one preferred to avoid the dilemma altogether. Table 1 describes the menu of dilemmas administered in our version of the Trolley dilemma. Importantly, we highlight that our choice menu allows us to examine how the likelihood of taking action responds to the number of people saved (X) relative to killed (Y). We are also able to examine preferences for inaction over action when the number of lives lost would be unaffected (i.e., X=Y dilemmas). And finally, we can examine how one's likelihood to take action differs if action is indirect (i.e., pull a lever to divert the runaway trolley) versus a more direct act (i.e., push an individual(s) onto the track to stop the runaway trolley), which we call INDIRECT versus DIRECT decision scenarios.

In what follows, we scored immorality as derived from the Trolley dilemma choices as follows: Immoral Omission is an indicator variable equal to one if a subject chose to not take action in the DIRECT and INDIRECT (X,Y)=(6,0) scenarios, where action would save 6 individuals without any lives being lost as a result. Another dichotomous variable, Immoral Commission, is equal to one if the subject chose action in both the (X,Y)=(6,6) and (1,1) scenarios of both the INDIRECT and DIRECT choice dilemmas. In the case of Immoral Commission, the subject prefers to be responsible (via action) for a given number of deaths rather than simply allow that same number of deaths to occur as a bystander. We created a final variable by taking a subject's average propensity to act in the remaining scenarios not used in the construction of the Immoral Omission or Immoral Commission variables. Such a variable, Action Propensity, represents one's willingness to take action, though it also describes utilitarian preferences in our dilemmas.

For the money burning game, a key treatment variable is whether only one in the pair (the "decider") or both could burn money. 6 Specifically, in the Bilateral Burn treatment, both players could mutually and simultaneously destroy a portion of each other's payoffs. That is, each of the two subjects in a randomly matched pair made money burning decisions and two random decisions were selected such that each subject was both a decider and passive recipient (i.e., potential money burn victim) in a consequential money burning choice. In Unilateral Burn, subjects were randomly assigned as decider or passive recipient before decision making, and only the deciders made decisions. After decisions were made in all 9 money burning scenarios, deciders and recipients were randomly matched and one scenario was selected at random to determine the payoff of both players in the money burning game. 7 This process was common knowledge.

Experimental procedures

The experiment was computerized and administered using the Z-tree platform (Fischbacher, 1999). We recruited 150 subjects at the University of Rennes 1 (France), each subject participated in only session, and none of the subjects had participated in a similar economic experiment. A total of 9 sessions were conducted (with 14 to 18 subjects per session), where in each session subjects were administered the Trolley dilemma and the Money Burning game. Table 3 contains summary information about number of participants in each treatment of the Money Burning game, which is identified by the order of presentation of the scenarios (see footnote 4) and whether the Unilateral or Bilateral Burn treatment. Importantly, subjects were given the choice to opt out of the Trolley Dilemma for whatever reason. A total of 12 subjects (8%) chose to opt out of the Trolley dilemma task, and we use this "opt-out" in the analysis of money burning choices below.

A session lasted approximately one hour (this includes the time spent to read the instructions). At the end of the experiment, one task was randomly selected for each pair of randomly matched subjects (and random role assignments, in the case of Unilateral Burn 6 We also varied the ordering of the Money Burning (x,y) pairs in the menu received (presenting the decision maker's endowment in Increasing, Decreasing, or Random order. Each subject saw only one ordering). We did not have a formal hypothesis regarding the ordering of the money burning scenarios, and later analysis documents that the varied ordering does not significantly impact outcomes in the task. We considered variation in the order more exploratory. Of course, there is no theoretical reason to believe that the ordering should matter, but this possibility has been investigated on the more well-known risky choice task [START_REF] Holt | Risk aversion and incentive effects[END_REF] lottery choice menu (see [START_REF] Bruner | Changing the probability versus changing the reward[END_REF]. 7 Our design made use of the strategy method, as opposed to direct elicitation method, in order to generate multiple observations from each subject in each decision experiment (other than the passive recipients in the money burning game, which were randomly selected prior to decision making in that game). [START_REF] Brandts | The strategy versus the direct-response method: a first survey of experimental comparisons[END_REF] survey experimental results comparing strategy method versus direct elicitation and conclude that the strategy method for response elicitation, in general, provides a conservative estimate of what choice would be using direct response elicitation-in our case, money burning choices may therefore be a conservative estimate of outcomes one would find using direct elicitation of just a single response in a single scenario. treatments). Payments were made anonymously at the end of the session and the average earnings were 25.52 Euros per subject.

Behavioural Predictions

Trolley Problem

The literature identifies two clear predictions we can make regarding outcomes in the Trolley dilemma. First, a widely reported result is that individuals are more willing to take action and save lives in the INDIRECT frame where a level is pulled, as compared to the DIRECT frame where an individual is pushed onto the track, holding constant the relative number of lives saved. This is related to the distinction between personal and impersonal moral dilemmas [START_REF] Greene | An fMRI investigation of emotional engagement in moral Judgment[END_REF]. Thus, our first hypothesis stems from this "contact principle" [START_REF] Cushman | The role of reasoning in moral judgments: Testing three principles of harm[END_REF]. This hypothesis implies that for each pair (X,Y) of lives (saved, lost), we predict an individual is more likely to take action in the Indirect frame.

H1 (Trolley):

For each (X,Y) dilemma, action is more likely in the INDIRECT Frame Secondly, both individuals and courts of law consider an act of omission to be a lesser "sin" than an act of commission that results in similar consequences (see [START_REF] Cox | Status quo effects in fairness games: reciprocal responses to acts of commission versus acts of omission[END_REF]. This principle with respect to the Trolley dilemma has been labelled the "action principle". In our set of Trolley dilemma choices we can then focus on (X,Y) pairs (6,6) and (1,1), where an equal number of individuals would perish whether or not action is taken. In these cases, we hypothesize a lesser likelihood to act given a preference to not be responsible (via action) for the deaths. To take action in such cases could be considered an immoral act of commission. Also of interest would be the (0,6) Trolley dilemma, where action costs no lives. In such a dilemma, to not take action would be consider an immoral act of omission.

H2 (Trolley):

When lives lost are unaffected (X=Y dilemmas), inaction is preferred over action (moral omission). Also, action is preferred over inaction when action is costless (any (X,0) dilemma) (moral commission).

Rejection of H2 implies acts of Immoral Omission or Commission.

This final Trolley Problem H3 hypothesis posits that, independent of one's moral tendencies, a decreased relative cost of action will increase action likelihood. In other words, we predict a downward sloping demand curve for lives saved in this moral dilemma.

H3 (Trolley):

The likelihood of action will increase in the relative number of lives saved.

Money Burning

First consider the theoretical predictions in the money burning game. Purely selfish decision makers are completely indifferent towards the other person's well-being and would not incur costs to affect others' payoffs, and Utilitarian preferences would never choose to reduce total welfare. Thus, under the assumption of either pure selfishness or Utilitarianism, there should be no money burning. Behavioral considerations, however, may be invoked to generate alternative hypotheses of interest. For example, individuals may be inequality averse such that utility depends not only on one's own payoff but also on the equality of the income distribution (see [START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF]. 8 In our design, advantageous inequality (i.e., x ≥ y) cannot be reduced by burning money, and so we are only able to identify disadvantageous inequality aversion if money is burned when x < y). Alternatively, individuals may have so-called nasty preferences, which we define as a desire to lower others' payoff to gain or increase one's advantageous inequality (i.e., money burning scenarios where x ≥ y). Because decisions in the Bilateral Burn treatment are impacted by any (unmeasured) expectations of others' money burning choices, the pure effect unconfounded by expectations is measured from the Unilateral Burn treatments. Thus, we have our first money burning hypothesis: H4 (Money Burning): Money will not be burned in Unilateral burn treatments (homo economicus or Utilitarian preferences assumed).

Rejection of H4 when x < y (tasks A1-A4 in Table 2) is evidence of disadvantageous inequality aversion, and when x ≥ y is evidence of nasty preferences.

If individuals burn others' money and it is common knowledge that money burning is Bilateral, then observed money burning may be due to inequality aversion (for x < y) or nastiness (for x ≥ y) or anticipatory negative reciprocity. This type of "pre-emptive retaliation" may be exist even in the presence of disadvantageous inequality if both players in a pair make a simultaneous choice to burn money. In other words, in the simultaneous choice Bilateral.treatments one may burn the counterpart's money on the expectation that the counterpart may burn some of one's payoff (see [START_REF] Abbink | The pleasure of being nasty[END_REF]. H5 (Money Burning): Money burning will be similar in Unilateral and Bilateral Burn treatments.

Rejection of H5 is evidence that people believe others will burn money.

Just as in the case of the Trolley dilemma, we expect that antisocial tendencies to burn resources of others will nevertheless respond to the price of doing so. Previous studies have shown that punishment decisions in a VCM context obey the law of demand. [START_REF] Nikiforakis | A comparative statics analysis of punishment in public-good experiments[END_REF][START_REF] Anderson | Do non-strategic sanctions obey the law of demand? The demand for punishment in the voluntary contribution mechanism[END_REF]. Based on these papers' findings one may reasonably conjecture that money burning decisions also obey the law of demand and, though the cost of burning is fixed in our design, the amount burnt varies. This implies that the cost of burning money relative to one's payoff in the chosen payoff distribution varies and we can expect an increase in money burning when the relative cost of burning money is low. This leads to H6. H6 (Money Burning): Burning money will be negatively related to its relative cost.

Finally, a unique contribution we offer in the paper is to consider moral descriptors of one's choices in the Trolley dilemma as an explanatory variable regarding one's choice to burn money. Immoral acts of commission and omission are defined in H2 based on a subset of our Trolley dilemmas. Someone who takes action in the (X,Y) Trolley dilemmas not implicated in H2 has a higher Action Propensity (or is more Utilitarian). The morality of those with higher Action Propensity is difficult to assess given that one may be willing to sacrifice one or more lives to save others for more than one reason. Such reasons may include both ethically dubious reasons (i.e., I prefer to push someone to save others) as well as Utilitarian reasons (e.g., I will do whatever leads to the most lives saved (least lives lost)). For this reason, clean moral descriptors of immorality for our final hypothesis focus on metrics derived from a subset of the Trolley dilemmas: H7 (Money Burning): Moral descriptors derived from the Trolley dilemmas-(X=Y) and (X,0) dilemmas-will predict increased money burning.

Results

Trolley Results

We start by showing summary data from the subjects who made Trolley dilemma choices in Figures 1 and2 (9 Trolley dilemma choices per subject). Of the 150 participants in our experiment, n=12 subjects opted out of the Trolley dilemma, leaving us with n=138 Trolley subject decision makers (we code these "Trolley opt-out" subjects for later use as a regressor in the money burning estimations in the next section).9 Figure 1 shows the proportion of choices in each treatment (Direct and Indirect dilemmas) for the subset of dilemmas that hold constant the number of individuals saved. Left to right on the horizontal axis shows dilemmas that increase the number of individuals sacrificed for a constant X=6 individuals saved. Two things stand out in Figure 1: the proportion of individuals who take action decreases as the relative cost, Y/X, increases; more surprisingly, greater than 20% of subjects did not choose to take action in the (6,0) dilemma where 6 individuals could be saved at zero cost, and some chose (indirect) action in the (6,6) dilemma where the same number of individuals would perish even if nothing were done. Both represent instances of what we call "Trolley immorality". Figure 2 organized the remaining subset of Trolley choices to hold constant the number of individuals who perish at Y=1. Left to right in Figure 2 organizes the Trolley dilemmas by a decreasing number of lives saved while holding constant the number who perish.

We again see that action in the Trolley dilemma is responsive to the relative cost (or effectiveness) of the action--subjects are less likely to take action when the relative cost, Y/X, increases (or, as the relative benefit X/Y decreases). In Figure 2, we also see that a nonzero number of subjects choose an immoral act of commission in Trolley dilemma (1,1) where action was preferred even though an individual would perish even with inaction.

As noted in Section 3 (Experimental Design), we elicit choices in the strategy method to maximize data generated per subject. Due to multiple decisions per subject, all models in Table 4 include standard errors clustering at the individual subject level. The model structure is a Probit estimation where the dependent variable is equal to one if that subject chooses to take "action" (i.e., pull the lever or push the individual(s) in that particular dilemma scenario). The different columns of Table 4 show estimations using different sets of independent variables. The first two columns use a dummy variable for each (X,Y) pair of lives saved (X) and sacrificed or killed (Y) compared to the omitted baseline scenario of (X,Y)=(6,0). Columns 3-5 replace the dummy variables with continuous variables measuring the number of lives sacrificed and saved.

The dummy variable identifying the DIRECT version of each Trolley dilemma has a consistently negative and significant coefficient estimate across all models, which supports Hypothesis H1. Individuals are significantly less likely to take action when it is a more personal moral dilemma (action would be direct) compared to impersonal (action would be indirect). Interestingly, this effect is somewhat muted for male subjects as seen by the significant and positive coefficient on Male*DIRECT in model 5. 10 Because many of the dilemmas confound morality of choice with utilitarian actions, we next examine hypothesis H2 using only the subset of Trolley dilemmas (X,Y)=(6,6), (1,1), and (6,0). The comparison of coefficients in our Table 4 estimations are not a transparent way to assess whether a statistically significant number of subjects chose action in the (6,6) and (1,1) dilemmas, or inaction in the (6,0) dilemma. Rather, we can test the null hypothesis that the proportion of subjects choosing action in the (X=Y) dilemmas is equal to zero against the alternative that it is greater than zero. For the test of immoral action, we test the null hypothesis that the proportion of subjects choosing action in the (6,0) dilemma is equal to 100% against the alternative hypothesis that it is less than 100%. For the case of n=138 observations, the observed proportions in both the case of DIRECT and INDIRECT framing of the Trolley dilemmas lie outside of the 95% confidence interval. This evidence implies rejection of H2 in favor of the existence of immoral acts of omission and commission being greater than zero. 11 Finally, we show support for Hypothesis H3 by using estimates in models (4) and ( 5) of Table 4. Here the marginal effect on # Lives Sacrificed (Y) holds constant the # Lives Saved (X), and vice versa. Thus, the negative and positive, respectively, effects of these variables on the likelihood of taking action confirm that Trolley choices respond to the relative number of lives saved to lost, which supports Hypothesis 3. In short, action in the Trolley dilemma responds to incentives and displays a downward sloping demand curve for lives saved. Nevertheless, a nontrivial number of subjects make choices that can be classified as immoral acts of commission or omission in the Trolley dilemma choices designed to highlight such dubious moral viewpoints. Having established the results from our Trolley Dilemma, shown them consistent with the extant literature, and also documented our morality metrics as revealing, we next turn to the results from the Money Burning game.

Money Burning Results

Summary results from the Money Burning game are shown in Figures 3 and4, and in Table 5. Figures 3 and4 summarize the frequency of money burning choices for categories of (x,y) pairs. Figure 3 shows money burning choices in each possible scenario, for both instances of Unilateral Burn and Bilateral Burn. Figure 4 highlights the apparent downward trend in money burning as the cost of burning money is larger relative to the recipient's budget-money burning also obeys the law of demand (Hypothesis 6). Table 5 shows the total number of instances (out of 9 scenarios) in which the subject burned money, on average (top row), along with summary information on the proportion of money burning choices for the different possible types of money burners. Depending on the relationship between the decider's payoff, x, and the passive recipient's payoff, y, one can consider decisions to burn money are reflective of disadvantageous inequality aversion (burning money when x<y) or nasty preferences (burning money when x > y). Others may never burn money (Homo Economicus or Utilitarian preferences), and some burn money in all 9 scenarios and so reflect an unconditional desire to behave antisocially (i.e., destroy resources and reduce total welfare of the pair). Though we have only limited data from deciders in the Unilateral Burn treatments, the Bilateral Burn data in Table 5 reflect similar proportions of burn choices in both Bilateral and Unilateral Burn treatments. This most likely indicates that Bilateral Burn choices are not driven primarily by expectations that others will burn money.

Tables 6 and7 show results from Probit estimations of the probability that someone makes the dichotomous choice to burn money and select the End Distribution over the Start Distribution in the Table 2 scenarios. Errors in both tables are clustered at the level of the individual subject, and we report marginal effects in the tables. The set of independent variables 11 One sample proportions tests, the Z statistic cannot be calculated for the boundary hypothesized proportions of 0% and 100%, and so we rather calculate our tests using null hypothesis proportions of 1% and 99%, respectively. Our conclusions remain intact even if allowing for a 5% « error » in decision making (at the p <.10 level for the (6,6) DIRECT and (6,0) INDIRECT dilemmas, but at the p < .01 level in all other cases). That is, if assuming that a small percentage of subject may make mistaken choices in ours ample, our conclusions are H2 result is largely unchanged.

in Table 6 includes: controls for the presentation order of the (x,y) distributions in the Money Burning menu set (Random, Increasing, or Decreasing); an indicator variable for the scenarios where burning was Unilateral (Bilateral is the reference group); indicator variables capturing payoff equality/inequality in the different (x,y) payoff distributions; a variable measuring the relative cost of money burning compared to one's own payoff; a variables measuring the simple preferences in the Trolley dilemma to take action); a set of subject-specific controls. Importantly, model (3) in Table 6 and the models in Table 7 include indicator variables to identify whether the subject committed Immoral Commission or Immoral Omission in the Trolley problem (6,6), (1,1), and (6,0) dilemmas. So, these two indicator variables capture a sense of the moral preferences of the subject as derived from the Trolley choices, and the test of significance on their coefficients is a test of whether such measures from hypothetical decision scenarios may yet hold power to predict decision in consequential decision tasks than contain at least some type of moral element. Table 7 focuses on estimates separating the subsamples of the data for the (x,y) distributions where x < y (disadvantageous payoff inequality) versus x ≥ y (advantageous inequality).

We first focus on the results in Table 6, and note that there is no evidence that the ordering of the (x,y) options matters. Our data support for hypothesis H5-money burning is equally likely in Bilateral and Unilateral Burn, which suggests that beliefs that others will burn money do not impact money burning decisions in our data. Statistically significant positive coefficients on Income<other in all three models support rejection of our hypothesis H4 in favor of the alterative hypothesis of disadvantageous inequality aversion [START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF] when x > y in the Start Distribution (Table 2). The marginally significant (p < .10) coefficient on the Relative Cost of burning money indicates that money burning is responsive to how much of one's payoff the burning choice will cost-a lower relative budget impact of burning marginally increases the likelihood that one burns money, which supports hypothesis H6. Model (2) includes an indicator variable for those who opted out of the Trolley dilemma, and we find a marginally significant impact of Opt-Out on the probability that one will burn money. This variable is absent is model (3) where we include the Trolley immorality measures as regressors, which necessarily implies we focus on the money burning data from those who also completed the Trolley dilemma choices. Importantly, in model (3) of Table 6, we find evidence that making a morally dubious choice(s) in the Trolley dilemma predicts a significantly increased likelihood of money burning.12 This is support for hypothesis H7. Thus, we offer first evidence in the literature, to our knowledge, that moral indicators from a hypothetical dilemma can predict significant increases in anti-social money burning choices with real payoff consequences. 13Table 7 shows results of related estimations where the subsample of x < y versus x ≥ y are used as a way to identify general money burning from "nastiness", which would be defined as a willingness to burn money for x ≥ y payoff distributions (i.e., a willingness to pay to burn money even when my payoff is at least as higher my counterpart's). The results from Table 7 show that the Diff Income (= |y -x|) represented in the Start Distribution only predicts a significantly higher probability of money burning when a subject's payoff is lower than the counterpart's, which again implies rejection of H4 in favor of disadvantageous inequality aversion that is sensitive to the size of inequality. Looking at the advantageous inequality subset of data in model ( 2), we see that the relative cost of burning money matters in terms of anti-social "nasty" choices. The higher the advantageous payoff inequality is in our design, the lower the relative cost to make the money burning choice. For this reason, we see the predicted marginally higher nastiness in those scenarios where the decider is at the largest payoff advantage (see also right-half of Figure 3). This offers some evidence of nasty preferences as the alternative upon rejection of the Utilitarian or Homo Economicus H4 hypothesis.

Interestingly, the immorality measures from the Trolley dilemma are significant predictors of the probability one burns money. Model (1) shows that both immoral acts of commission and omission in the Trolley dilemma predict a 36%-38% increase in the likelihood one burns money (p < .01 in both cases. The difference between these two effects is statistically insignificant, p > .10). We identify predictors of nastiness in model (2) of Table 7 and a key result is that we find that the immoral act of omission in Trolley dilemma #12 (i.e., not acting when 6 lives could be saved at the expense of zero lost lives) predicts a 30% increased likelihood of making a "nasty" money burning choice (p < .01). In a sense, our strongest way to judge morality from the Trolley dilemma is whether someone chose the immoral act of omission. In sum, we find strong support for hypothesis H7 and conclude that Trolley morality, though hypothetical, can be a significant predictor of consequential antisocial decisions.

Discussion.

In this paper we attempt to explain some key determinants of costly antisocial behaviours using measures derived from both a money burning game and a moral thought experiment. While ethical dilemmas and thought experiments have been of significant interest to moral philosophers for decades, we believe our study to be unique. Our particular innovation has been to use responses in the iconic Trolley dilemma to generate immorality indicators that have predictive power regarding one's decision making in consequential environments. The consequential environment we explore allows for costly antisocial choice and may be considered a type of behavioural marker for the likelihood of costly behaviours in field settings.

Our results highlight the importance of the relative cost of the ethical behaviour across the domains of both the hypothetical Trolley dilemma and the consequential Money Burning game. Subjects are more likely to make an ethically dubious choice if the costs of doing so are lower. Aside from identifying typical response patterns in the Trolley dilemma, we identified choices made from our set of Trolley dilemmas that would constitute morally questionable acts of omission or commission. We then estimated a significant increase in the likelihood of burning money for those subjects identified as willing to commit an immoral act of omission or commission in the Trolley dilemma. Upon further investigation, we found that immoral Trolley respondents' increased willingness to burn money was linked more strongly to disadvantageous inequality aversion than to nastiness. Nevertheless, we identified that choice in one Trolley scenario (not typically considered in the existing literature) is a highly significant predictor of the probability of nasty money burning. These results call into question some recent conclusions in the literature regarding increased Utilitarianism among those with antisocial personality traits [START_REF] Bartels | The mismeasure of morals: Antisocial personality traits predict utilitarian responses to moral dilemmas[END_REF]. Specifically, our research connects immoral preferences and actual anti-social behaviour in a stylized game.

As always, there are limitations to our study. First, it is likely the case that reputational concerns may be important if one is aware that selection based on Trolley dilemma responses may be at stake. And of course, the validity of a hypothetical ethical dilemma may always be a point of concern. For this reason, one of our main purposes is to highlight that response patterns in such hypothetical dilemmas may be instructive towards an understanding of consequential behavioural tendencies. At some level, the criticism of selection bias would apply to any number of hypothetical or self-response instruments used to screen individuals or assess situational risk. We believe the key is that we first understand the link between hypothetical responses and consequential behaviours, because researchers often have no alternative approach to study high stakes moral dilemmas.

Our hope is that this research will stimulate further investigations into the value of hypothetical choices towards predicting outcomes in other non-hypothetical but related decision domains. These findings may have interesting implications for how hypothetical scenario instruments could be used to screen individuals for antisocial tendencies that could be costly to an organization. Because the type of anti-social decision making we studied involves resource destruction when outcome inequality is present, it is intriguing to consider that the markers for such behavioural tendencies may already exist in well-known hypothetical thought scenarios. Imagine that an employer could use responses to the Trolley dilemma as a way to identify workers who may be more willing to engage in antisocial resource destruction. While this may seem like the type of worker one may wish to avoid (i.e., avoid such individuals in designing self-driving auto accident avoidance algorithms), those willing to destroy resources may have value to the employer in certain specialized roles (e.g., lead negotiator who must credibly be willing to walk away from a contractual arrangement or wage negotiations). Of course, such implications of our findings are themselves only a thought experiment, but we hope them to be useful at motivating why this may be a fruitful area for research extensions. If choices in hypothetical dilemmas can serve as behavioural markers that predict real world ethical choice, then we feel this is a useful step forward in an important area of behavioural research.

Current technological developments, such as the increased experimentation and utilization of drones and self-driving vehicles, render hypothetical moral dilemmas like the Trolley dilemma increasingly relevant to policy-makers as society attempts to understand barriers to technology adoption and implementation (e.g., [START_REF] Crockett | The trolley problem: Would you kill one person to save many others?[END_REF]. Our results have implications for how choices in hypothetical moral dilemmas may be used to understand or even possibly forecast certain types of behaviour in real world environments. -1921.8550 -1889.6102 -1998.7432 -1968.3391 -1954.1628 Notes: *. Notes: ratio along horizontal axis reflect the size of the burning cost, 10, relative to the decider's payoff level prior to burning the recipient's payoff. Left to right along the axis reflects an increasing cost of burning the recipient's payoff, relative to one's own payoff in the allocation.

Figure 1 .Figure 2 .

 12 Figure 1. Frequencies of taking action in the Trolley dilemma by treatment (number saved unchanged)

Figure 3 .

 3 Figure 3. Frequencies of money burning decision by treatment

Figure 4 .

 4 Figure 4. Money burning per relative cost

Table 1 .

 1 Trolley Dilemmas. Trolley dilemmas numbered here for discussion in the text (dilemmas were not numbered for subjects)

			INDIRECT		DIRECT
		Are you willing to pull a lever to	Are you willing to kill Y people
		divert the trolley to a different	by pushing them onto the track
		track to save X people, where Y	to save X people?
		on that side track will die.
	Trolley	Total	Total		Total	Total
	Dilemma	people	people		people	people
	#	killed	saved		killed	saved
	1	Y=6	X=6	Yes No	Y=6	X=6	Yes No
	2	Y=5	X=6	Yes No	Y=5	X=6	Yes No
	3	Y=4	X=6	Yes No	Y=4	X=6	Yes No
	4	Y=3	X=6	Yes No	Y=3	X=6	Yes No
	5	Y=2	X=6	Yes No	Y=2	X=6	Yes No
	6	Y=1	X=6	Yes No	Y=1	X=6	Yes No
	7	Y=1	X=5	Yes No	Y=1	X=5	Yes No
	8	Y=1	X=4	Yes No	Y=1	X=4	Yes No
	9	Y=1	X=3	Yes No	Y=1	X=3	Yes No
	10	Y=1	X=2	Yes No	Y=1	X=2	Yes No
	11	Y=1	X=1	Yes No	Y=1	X=1	Yes No
	12	Y=0	X=6	Yes No	Y=0	X=6	Yes No
	Note:

Table 2 :

 2 Money Burning choice tasks (Increasing treatment) Subjects chose the Start or End Distribution for each of the 9 tasks

	Task	Start	Damage	Burning	End
	#	Distribution		costs	Distribution
	A1	(50 , 250)	50	10	(40 , 200)
	A2	(50 , 200)	50	10	(40 , 150)
	A3	(50 , 150)	50	10	(40 , 100)
	A4	(50 , 100)	50	10	(40 , 50)
	A5	(50 , 50)	50	10	(40 , 0)
	A6	(100 , 50)	50	10	(90 , 0)
	A7	(150 , 50)	50	10	(140 , 0)
	A8	(200 , 50)	50	10	(190 , 0)
	A9	(250 , 50)	50	10	(240 , 0)

Table 3 .

 3 Summary of the Money Burning Treatments

	Session Participants	Treatment Description
	1	18	Increasing-Unilateral burn
	2	16	Increasing-Bilateral burn
	3	18	Increasing-Bilateral burn
	4	18	Decreasing-Unilateral burn
	5	16	Decreasing-Bilateral burn
	6	14	Decreasing-Bilateral burn
	7	18	Random-Unilateral burn
	8	16	Random-Bilateral burn
	9	16	Random-Bilateral burn
	Total	150	(n=96 Bilateral Burn, n=54 Unilateral burn)

Table 4 .

 4 Probability of Action (Pull level or Push person) in Trolley Dilemma

	Marginal Effects Reported (robust st errors in parenthesis)
	Independent variable	(1)	(2)	(3)	(4)	(5)

Table 5 :

 5 Descriptive statistics of money burning decisions

	All	Unilateral Burn Bilateral Burn

Notes: # subjects in bold, % subjects in parenthesis ( )

Table 6 .

 6 Probability of burning money *.10, **.05, ***.001 for the 2-tailed test. Standard errors clustered at the individual subject level. Increasing, Decreasing, Random (reference group) control for the differing order of the money burning allocation scenarios. Relative Cost takes on the value of the 10 experimental monetary units (EMU) cost divided by the payoff in EMU if choosing not to burn money. Trolley Opt-Out is an indicator for those subjects who chose not to complete the Trolley dilemma task. ^reduced as a result of those opting out of the Trolley dilemma choice, which is used to score morality variables.

	Marginal Effect (st. error) displayed	(1)	(2)	(3)
		Marg. Effect	Marg. Effect	Marg. Effect
	Independent Variable	(st. error)	(st. error)	(st. error)
	Increasing (x,y) order (=1)	-0.0077	-0.0047	-0.0196
		(0.0502)	(0.0498)	(0.0510)
	Decreasing (x,y) order (=1)	-0.0186	-0.0173	-0.0147
		(0.0467)	(0.0460)	(0.0493)
	Unilateral Burn (=1)	-0.0146	-0.0110	-0.0156
		(0.0456)	(0.0454)	(0.0440)
	|Income < other (x < y)|	0.0005***	0.0005***	0.0005***
		(0.0001)	(0.0003)	(.0002)
	Income > other (x > y)	-0.0002	-0.0002	-0.0002
		(0.0003)	(0.0003)	(0.0003)
	Income = other ( = 1)	0.0525	0.0513	0.0597
		(0.0524)	(0.0519)	(0.0583)
	Relative cost	-0.9551*	-0.9398*	-1.0741*
		(0.5447)	(0.5370)	(0.5787)
	Trolley Opt-Out (=1)	---	-0.1171*	---
			(0.0409)
	Action Propensity (Trolley dilemmas	---	---	0.0675
	2-10)			(0.0546)
	Immoral Commission (=1)	---	---	0.2655***
	(action in Trolley 1&11)			(0.1154)
	Immoral Omission (=1)	---	---	0.3428***
	(inaction in Trolley 12)			(0.1282)
	Male (=1)	---	---	-0.0157
				(0.0433)
	Happiness ∈ [1 ,10]	---	---	-0.0278*
	(10=highest current life happiness)			(0.0161)
	Religion ∈ [1 ,10]	---	---	0.0053
	(10=very important)			(0.0078)
	Age	---	---	-0.0135
				(0.0133)
	Observations	1107	1107	1026
	# Participants^	123	123	114^
	Log likelihood	-458.2919	-452.741	-406.183

Notes:

Table 7 .

 7 Probability of burning money *.10, **.05, ***.001 for the 2-tailed test. Standard errors clustered at the individual subject level. ^reduced as a result of those opting out of the Trolley dilemma choice, which is used to score morality variables.

	Marginal Effect (st. error) displayed
		(1)	(2)
	Independent Variable	Income ≤ other's	Income ≥ other's
	Increasing (x,y) order (=1)	0.0546	-0.0665
		(0.0705)	(0.0574)
	Decreasing (x,y) order (=1)	0.0312	-0.0510
		(0.0649)	(0.0597)
	Unilateral Burn (=1)	-0.0326	-0.0123
		(0.0527)	(0.0569)
	|Diff Income|	0.0004***	0.0007
		(0.0001)	(0.0005)
	Equal Income (x = y)	0.0513	0.2075
		(0.0509)	(0.1820)
	Relative cost	---	-2.3937*
			(1.3978)
	Action Propensity	0.0379	0.1121
	(Trolley dilemmas 2-10)	(0.0572)	(0.0717)
	Immoral Commission (=1)	0.3669***	0.1326
	(action in Trolley 1&11)	(0.1797)	(0.1146)
	Immoral Omission (=1)	0.3823***	0.3043***
	(inaction in Trolley 12)	(0.1774)	(0.1387)
	Male (=1)	0.0084	-0.0441
		(0.0527)	(0.0541)
	Happiness ∈ [1 ,10]	-0.0203	-0.0357*
	(10=highest current life happiness)	(0.0172)	(0.0195)
	Religion ∈ [1 ,10]	0.0085	-0.0071
	(10=very important)	(0.0092)	(0.0102)
	Age	-0.0211	-0.0105
		(0.0151)	(0.0176)
	Observations	570	570
	# Participants^	114	114
	Log likelihood	-196.646	-239.281

Notes:

Individual characteristics may be yet another factor that explains money burning decisions. For instance, some previous studies have shown that high basal testosterone is associated with an increased threshold for conflict (seeCarney and Mason, 2010, and references therein).

[START_REF] Bostyn | Of Mice, Men, and Trolleys: Hypothetical Judgment Versus Real-Life Behavior in Trolley-Style Moral Dilemmas[END_REF] examine whether Trolley dilemma responses predict propensity to deliver electric shocks to mice in dilemmas with similar but nonfatal scenarios.

Using self-report measure of antisocial personality tendencies[START_REF] Bartels | The mismeasure of morals: Antisocial personality traits predict utilitarian responses to moral dilemmas[END_REF] or patients with brain damage in regions important to emotion generation[START_REF] Koenigs | Damage to the prefrontal cortex increases utilitarian moral judgments[END_REF], these studies find tendencies towards increased utilitarianism in individuals with antisocial personality. Another recent study[START_REF] Bracht | Moral judgments, gender, and antisocial preferences: An experimental study[END_REF] is quite related to ours in that it also examines ethical choice in hypothetical dilemmas as well as in a consequential money transfer game. The differences in our study are notable, however. First, we do not pool data across indirect versus direct moral choices as they do, which is important given we identify a highly significant (p < .01) impact of this factor on one's willingness to take action (we also show that one of their key results is qualified in our findings by conditioning on the direct versus indirect nature of the dilemma). Secondly, both our hypothetical and consequential choice experiments vary the relative efficiency or cost of one's action, thus allowing a more thorough examination of ethical and antisocial choice. Finally, we use a morality measure derived from the Trolley dilemma to predict behavior in the consequential money burning game, while[START_REF] Bracht | Moral judgments, gender, and antisocial preferences: An experimental study[END_REF] use the outcome in their consequential game as a regressor in the ethical dilemma choice. While of potential interest, we find the causation of hypothetical-to-consequential choice more valuable in terms of implications and use as a potential screening or identification mechanism (e.g., job application/interview screener).

It is therefore important to note that many moral dilemmas confound the utilitarian choice from the choice one might make for non-utilitarian reasons. For example in the typical Trolley dilemma, it is utilitarian to pull the switch or push the individual, and yet one may be willing to act not because more lives are saved than lost, but rather because one prefers or perversely enjoys being responsible for someone's death.

Indeed, a very appealing hypothesis about distributional preference is inequality aversion (seeLoewenstein et al. 1989;[START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF] Bolton and Ockenfels, 2000; Charness and Rabin, 2002). These approaches assume that individual utility depends not only on one's own payoff but also on the equality of the income distribution.

We conducted a probit estimation of the determinants of the decision to opt out of the Trolley dilemma. Though few subjects opted out, we found one variable, "happiness" (self-reported current level of happiness in life) was a marginally significant determinant of the opt-out choice (p < .10). Specifically, those self-reporting higher levels of life happiness were marginally more likely to opt out of the Trolley dilemma.

This result is somewhat related to the gender result found in[START_REF] Bracht | Moral judgments, gender, and antisocial preferences: An experimental study[END_REF], who find males more likely to take action in a set of moral dilemmas. The study includes a variety of dilemmas in addition to a limited number of Trolley dilemmas, but they do not distinguish dilemmas in their set that involve a direct versus indirect action in the moral choice. As such, our result is an important qualification of what they report given our evidence suggests the gender effect may not be as general as they suggest.

The difference between the impact of Immoral Commission versus Immoral Omission is not statistically significant (p >.10 for the Wald test of coefficient equality)

Model (3) also indicates a marginally significant impact of higher self-reported life happiness predicting a lower probability that one burns money.

We thank Elven Priour for programming the experiment. The financial support of ANR (National Research Agency) for the FELIS program (Liberté et protection gratefully acknowledged.

surveying over 50,000 ://www.eurofound.europa.eu/observatories/eurwork/articles/


Éric Ricard Fractional

L P For

Éric Ricard
email: eric.ricard@unicaen.fr

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

This note deals with the perturbation theory of functional calculus of selfadjoint operators on Hilbert spaces. More precisely, given a function f : R → R, the problem is to get a control of f (x) -f (y) S for some symmetric norm on selfadjoint operators in terms of possibly another norm x -y S . This topic was developed from the 50's by the Russian school. Birman and Solomjak had a strong impact on it by the introduction of operator integrals in the 60's. Since then, this subject has been very active. Many mathematicians tried to enlarge the classes of functions f or norms involved. The list would be too long, but we can quote Arazy [START_REF] Arazy | Certain Schur-Hadamard multipliers in the spaces Cp[END_REF][START_REF] Arazy | Contractive projections in Cp[END_REF], Ando [START_REF] Ando | Comparison of norms |||f (A) -f (B)||| and[END_REF], and more recently the breakthroughs by Alexandrov-Peller [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF][START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF],... and Potapov-Sukochev [START_REF] Potapov | Operator-Lipschitz functions in Schatten-von Neumann classes[END_REF][START_REF] Potapov | Double operator integrals and submajorization[END_REF] and their coauthors [START_REF] Potapov | On the Arazy conjecture concerning Schur multipliers on Schatten ideals[END_REF],... Usually the results are stated for symmetric (quasi-)norms on compact operators, for instance the Schatten p-classes S p for 0 < p ∞. Nevertheless the noncommutative integration theory in von Neumann algebras also gives a natural framework to study these questions.

Our starting point is an inequality in [START_REF] Birman | Estimates of the spectrum of a difference of fractional powers of selfadjoint operators[END_REF], for any fully symmetric norm . S and any 0 < θ < 1, and x, y positive operators on some Hilbert space, i.e x, y ∈ B(H) +

x θ -y θ S |x -y| θ S . It was extended by Ando [START_REF] Ando | Comparison of norms |||f (A) -f (B)||| and[END_REF] to any operator monotone function f : R + → R + instead of x → x θ . Dodds and Dodds [START_REF] Peter | On a submajorization inequality of T. Ando. In Operator theory in function spaces and Banach lattices[END_REF] adapted the proof to semi-finite von Neumann algebras for all fully symmetric norms.

In the case of Schatten classes, Birman Koplienko and Solomjak's result or Ando's proof actually give that for p θ and x, y ∈ B(H) +

x θ -y θ p/θ

x -y θ p . This also holds for semi-finite Neumann algebras by [START_REF] Peter | On a submajorization inequality of T. Ando. In Operator theory in function spaces and Banach lattices[END_REF] or [START_REF] Potapov | Double operator integrals and submajorization[END_REF]. For general von Neumann algebras, Kosaki got the case p = θ in [START_REF] Kosaki | Applications of uniform convexity of noncommutative L p -spaces[END_REF] with an extra factor, a full argument can be found in [START_REF] Caspers | Noncommutative de Leeuw theorems[END_REF] or [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF].

Another remarkable extension was obtained in [START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF] for Schatten p-classes when 1 < p < ∞; it is shown that for any θ-Hölder function f on R, with 0 < θ < 1 and any selfadjoint x, y ∈ B(H) sa , one has [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF] f (x) -f (x) p/θ C p,f x -y θ p . In particular this holds if f (x) = |x| θ or f (x) = sgn(x)|x| θ . For them, the arguments can be adapted to general von Neumann algebras [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF] and one can also reach p = 1 in [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF].

Surprisingly, when p < 1 even for Schatten classes, very little is known. One can find some asymptotic estimates in [START_REF] Birman | Estimates of the spectrum of a difference of fractional powers of selfadjoint operators[END_REF] or [START_REF] Ju | Rotfel'd. Asymptotic behavior of the spectrum of abstract integral operators[END_REF] but [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF] seems to be unknown. Weaker related inequalities were also recently obtained in [START_REF] Sobolev | Functions of self-adjoint operators in ideals of compact operators[END_REF].

Among general results Raynaud [START_REF] Raynaud | On ultrapowers of non commutative Lp spaces[END_REF] proved that x → f (x) from L p to L p/θ is uniformly continuous on balls for f as above. In [START_REF] Pisier | The non-commutative Khintchine inequalities for 0 < p < 1[END_REF], for type II von Neumann algebras a strange quantitative estimate was obtained for its modulus of continuity.

Our main result is that (1) holds for all 0 < p ∞ for both f (x) = |x| θ and f (x) = sgn(x)|x| θ and all von Neumann algebras. We hope that the techniques developed here may be useful for related topics.

We do not address similar questions when θ > 1. When p 1, this is done in [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF] and when p < 1, some local results can be found in [START_REF] Arazy | Certain Schur-Hadamard multipliers in the spaces Cp[END_REF][START_REF] Potapov | On the Arazy conjecture concerning Schur multipliers on Schatten ideals[END_REF] for Schatten classes.

As usual to deal with such questions, one has to find norm estimates for some Schur multipliers, this is done in the second section. Next, they are used to derive the main result for semi-finite von Neumann algebras. The argument heavily rests on homogeneity of f . To generalize to type III algebras, one usually relies on the Haagerup reduction principle, but it involves approximations using conditional expectations that are not bounded when p < 1 and it seems difficult to use it in our situation. The only available tool we have is to use weak-type inequalities in semi-finite algebras to go to type III. The situation is so particular here that this can be done quite easily in section 4. We end up with some general remarks and extensions.

In the whole paper, we freely use noncommutative L p -spaces. One may use [START_REF] Pisier | Non-commutative L p -spaces[END_REF][START_REF] Pisier | Non-commutative vector valued Lp-spaces and completely p-summing maps[END_REF][START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF] and [START_REF] Fack | Generalized s-numbers of τ -measurable operators[END_REF] as general references. When τ is a normal faithful trace on a von Neumann algebra M, we use the classical definition of noncommutative L p associated to M

L p (M, τ ) = {x ∈ L 0 (M, τ ) | x p p = τ (|x| p ) < ∞}
, where L 0 (M, τ ) is the space τ -measurable operators (see [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF]). When dealing with more general von Neumann algebras, we rely on Haagerup's construction. Given a normal faithful semi-finite weight ϕ 0 on a von Neumann algebra M, Haagerup defined the noncommutative L p -space L p (M, ϕ 0 ) for 0 < p ∞ (see [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF]). His definition is independent of ϕ 0 (Corollary 38 in [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF]). When ϕ 0 is a normal faithful trace, his definition is equivalent to the previous one (up to a complete isometry) but the identifications are not obvious. Nevertheless for most of our statements, we won't need the reference to ϕ 0 or τ so we may simply write L p (M). When 0 < p < 1, L p (M) is a p-normed space so that for all families (a k ) in L p (M), As usual, we denote constants in inequalities by C pi if they depend only on parameters (p i ). They may differ from line to line.

Schur multipliers

A Schur multiplier with symbol M = (m i,j ) i∈I,j∈J over M I,J , the set of matrices indexed by sets I and J, is formally given by S M ((a i,j ) i∈I,j∈J ) = M • A = (m i,j a i,j ) i∈I,j∈J . Definition 2.1. Given a matrix M = (m i,j ) i∈I,j∈J of complex numbers, we say that M defines a pcompletely bounded Schur multiplier for some 0 < p ∞ if the map S M ⊗Id Lp(M) on S p I,J [L p (M)] is bounded for all von Neumann algebra M and we put

M pcb = sup M S M ⊗ Id Lp(M) .
Remark 2.2. For 1 < p = 2 < ∞, this is not exactly the usual definition of complete boundedness but it is formally stronger. Indeed an unpublished result of Junge states that

S M ⊗ Id Lp(M) S M cb if M is a QWEP von Neumann algebra.
We start by easy examples that can be found in [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF].

Lemma 2.3. Let (α k ) ∈ p (Z) for 0 < p 1 and assume that (f k ) ∈ ∞ (I) Z and (g k ) ∈ ∞ (J) Z are bounded families. Then M given by m i,j = k α k f k (i)g k (j) is a p-completely bounded Schur multiplier with M pcb (α k ) p sup k f k ∞ . g k ∞ .
Proof. It is clear that a rank one symbol M k = (f k (i)g k (j)) i∈I,j∈J defines a p-completely bounded Schur multiplier with norm f k ∞ . g k ∞ for all p and k. The result then follows by the p-triangular inequality.

We will often use permanence properties of pcb-Schur multipliers.

Lemma 2.4. Assume M = (m i,j ) i∈I,j∈J is a p-completely bounded Schur multiplier, then • M = (m i,j ) i∈I ,j∈J with I ⊂ I, J ⊂ J, then M pcb M pcb . • M = (m i,j ) (i,k)∈I×K,(j,l)∈J×L for any non empty sets K, L, then M pcb = M pcb .

Proof. We view 2 (I ), 2 (J ) as subspaces of 2 (I), 2 (J). Let P = (1 I (i)1 J (j)) i∈I,j∈J , this is a rank one p-completely bounded Schur multiplier with norm 1, and S M coincides with the restriction of S P • S M to matrices indexed by I × J .

The second point is classical using tensorisation with B( 2 (L), 2 (K)) ⊗ M instead of M.

Since finitely supported matrices are dense in S p I,J [L p (M)], we also have Lemma 2.5. If (M n ) ∈ M N I,J is a bounded sequence of pcb-Schur multipliers converging pointwise to some M , then M is also a pcb-Schur multiplier with M pcb lim M n pcb .

Remark 2.6. A p-completely bounded Schur multiplier M for p 1 is automatically q-completely bounded for p < q ∞. Indeed, the extreme points of the unit ball of S 1 are rank one matrices, but for those matrices the S 1 and S p norms coincide. Thus S M must be bounded on S 1 . But bounded Schur multipliers on S 1 are automatically 1-completely bounded (see [START_REF] Pisier | Non-commutative vector valued Lp-spaces and completely p-summing maps[END_REF]). Thus we get the result on S q for all 1 q ∞ by complex interpolation and duality. The case p < q 1 also follows by interpolation.

The following is a suitable adaptation of classical arguments (see [START_REF] Birman | Estimates of the spectrum of a difference of fractional powers of selfadjoint operators[END_REF][START_REF] Ju | Rotfel'd. The singular values of the sum of completely continuous operators[END_REF][START_REF] Ju | Rotfel'd. Asymptotic behavior of the spectrum of abstract integral operators[END_REF]). We use the measured space L 2 ([0, 2π] 2 , 1 (2π) 2 dm 2 ) where m 2 is the Lebesgue measure. Lemma 2.7.

Let K : [0, 2π] × [0, 2π] → C be a 2π-periodic continuous function such that for any d 0 ∂ d+1 (∂y) d ∂x K is continuous. Then M = (K(x, y)) x,y∈[0,2π] is a p-completely bounded Schur multiplier for all 0 < p 1 with for d > 1/p M pcb C 2 dp -1 + 2 1/p ∂ d+1 (∂y) d ∂x K 2 + ∂ d (∂y) d K 2 + ∂ ∂x K 2 + K 2 ,
where C is a universal constant. Moreover if M i = (K i (x, y)) x,y∈[0,2π] is a family of matrices as above, indexed by i ∈ I, then

M = (K i (x, y)) (x,i)∈[0,2π]×I,y∈[0,2π]
or its transpose satisfies

M pcb C 2 dp -1 + 2 1/p sup i ∂ d+1 (∂y) d ∂x K i 2 + ∂ d (∂y) d K i 2 + ∂ ∂x K i 2 + K i 2 .
Of course, this is relevant only if the above sup is finite.

Proof. We rely on Fourier expansions, put e k (x) = e ikx and h k,l (x, y) = e k (x)e l (y) . As (

h k,l ) k,l∈Z is an orthonormal basis in L 2 ([0, 2π] 2 , 1 (2π) 2 dm 2 ), we have the equality in L 2 , K = l,k∈Z α k,l h k,l where α k,l = K, h k,l = 1 (2π) 2 2π 0 2π 0 K(x, y)e -ikx e -ily dydx.
Assume for the moment that l = 0. Integrating by part in y, we get for d 0,

α k,l = 1 (il) d ∂ d (∂y) d K, h k,l . Let β k,l = (il) d α kl . When k = 0, another integration by part with respect to x gives β k,l = 1 ik 1 (2π) 2 2π 0 2π 0 ∂ d+1 (∂y) d ∂x K(x, y)e -ikx e -ily dydx. The Cauchy-Schwarz inequality gives that k =0 |β k,l | k =0 1 k 2 1/2 k =0 ∂ d+1 (∂y) d ∂x K, h k,l 2 1/2 C ∂ d+1 (∂y) d ∂x K 2 . Thus k∈Z |β k,l | C ∂ d+1 (∂y) d ∂x K 2 + ∂ d (∂y) d K 2 = C d independent from l and one can define a continuous function f l (x) = k∈Z 1 i d β k,l e k (x) bounded by C d .
In the same way to deal with l = 0, f 0 (x) = k∈Z α k,0 e k (x) is a continuous function. Indeed as above

k∈Z |α k,0 | k =0 1 k 2 1/2 k =0 ∂ ∂x K, h k,0 2 1/2 + |α 0,0 |, f 0 is bounded by C 0 = C ∂ ∂x K 2 + K 2 .
Choosing d > 1 p , we can conclude to the pointwise equality

(2) K(x, y) = f 0 (x)e 0 (y) + l =0 1 l d f l (x)e l (y).
The result follows directly from Lemma 2.3 by choosing

I = Z, α = (1 k =0 1 k d +1 k=0 ) k ∈ p (Z), f k as above and g k = e k . Obviously sup k g k ∞ = 1, sup k f k ∞ C d + C 0 and α p 2 dp-1 + 2 1/p .
The second statement also follows from Lemma 2.3 since in (2), the factorization in y and the sequence α is independent from K i . We have

K i (x, y) = f i 0 (x)e 0 (y) + l =0 1 l d f i l (x)e l (y), hence we can again take α = (1 k =0 1 k d + 1 k=0 ) k ∈ p (Z), f k (x, i) = f i k (x) and g k (y) = e k (y). We also have sup k f k ∞ C d + C 0 .
The same holds for the transpose of M as the condition in Lemma 2.3 is invariant by transposition.

The Sobolev constant (of order d) for K will mean the quantity

∂ d+1 (∂y) d ∂x K 2 + ∂ d (∂y) d K 2 + ∂ ∂x K 2 + K 2 . Let θ ∈]0, 1[, for x, y 0 recall that (3) x θ -y θ x -y = 1 0 θ (tx + (1 -t)y) 1-θ dt,
where the left hand side has to be understood as

θx θ-1 if x = y. Corollary 2.8. The matrix N = x θ -y θ x-y x 0,y∈[1,2]
defines a p-completely bounded Schur multiplier for 0 < p 1 with N pcb C p for some constant depending only on p.

Proof. First we start by showing that x θ -y θ

x-y 0 x 1/2,y∈ [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF][START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF] is a pcb-Schur multiplier.

We fix a

C ∞ function ϕ : [-π, π] → [0, 1] with support in [-1/4, 3/4] that is identically 1 on [0, 1/2] and another C ∞ function ψ : [0, 2π] → [0, 1] with support in [7/8, 3] that is identically 1 on [1, 2]. We define K(x, y) = ϕ(x)ψ(y) 1 x-y on [-π, π] × [0, 2π].
It is C ∞ and can be extended to a 2π-periodic C ∞ function. Thus Lemma 2.7 and a restriction yield that

1 x-y 0 x 1/2,y∈[1,2]
is a pcb-Schur multiplier. Then one just need to compose it with x θ -y θ 0 x 1/2,y∈ [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF][START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF] which is also clearly a pcb-Schur multiplier by Lemma 2.3.

Next we show that x θ -y θ

x-y x 1/2,y∈ [START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF][START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF] is also a pcb-Schur multiplier.

This time we fix a

C ∞ function ϕ : [0, 2π] → [0, 1] with support in [1/4, 3] that is identically 1 on [1/2, 2].
For i 0, one uses

K i (x, y) = ϕ(x)ϕ(y) (x+i) θ -y θ (x+i)-y on [0, 2π] 2 .
It is clear that K i can be extended to a C ∞ 2π-periodic function. By construction, for x and y in the support of ϕ and t ∈ [0, 1], the smallest value of t(x + i) + (1 -t)y is bigger than 1/4. Thus, the formula (3) shows that any derivative of order l of x θ -y θ

x-y on the support of K i is bounded by 4 l+1-θ θ(1 -θ)...(l -θ). Thus using the chain rule, one sees that K i and its derivatives up to order d + 1 are uniformly bounded independently of i and θ. Thus the same holds for the Sobolev constant in Lemma 2.7 for K i .

Lemma 2.7 gives that

K i (x, y) (x,i)∈[0,2π]×N,y∈[0,2π] is pcb. By Lemma 2.4, we can conclude by restricting x to [1/2, 3/2[×N [1/2, ∞[ (via (x, i) → x + i) and y to [1, 2].
The Corollary follows by gluing the two pieces together.

Corollary 2.9. For k ∈ Z, the matrix 1) for some constant depending only on p.

M k = x θ -y θ x-y x 0,y∈[2 -k-1 ,2 -k ] is a p-completely bounded Schur multiplier for 0 < p 1 with M k pcb C p 2 -k(θ-
Proof. This is obvious by homogeneity from Corollary 2.8 with a change of variables x ↔ 2 -k-1 x, y ↔ 2 -k-1 y.

Remark 2.10. One can exchange the roles of x and y.

It will be convenient to redefine M -1 , gathering all k -1:

Corollary 2.11. The matrix M -1 = x θ -y θ

x-y x 0,y 1 is a p-completely bounded Schur multiplier for 0 < p 1 with M -1 pcb C p 1 1-θ 1/p for some constant depending only on p.

Proof. Writing [1, ∞[= ∪ k 0 [2 k , 2 k+1
[ and using the previous Corollary for each piece, this follows from the p-triangular inequality as (2 k(θ-1) ) k 0 ∈ p (N). Since (2 k(θ-1) ) k 0 p c p (1 -θ) -1/p for some constant c p . We get that M -1 pcb c p C p (1 -θ) -1/p where C p comes from 2.9.

Remark 2.12. The kernel in formula ( 3) is positive definite because for x, y > 0

x θ -y θ x-y = c θ R+ t θ 1 x+t 1
y+t dt for some c θ > 0. Using this fact and similar arguments, one can check that there is some C so that M -1 pcb C for all 0 < θ < 1 and all p 1.

We now turn to another family of multipliers.

Corollary 2.13. For a 1, the matrix

H a = 1 a+x+y x,y∈[0,1]
is a p-completely bounded Schur multiplier for 0 < p 1 with H a pcb C p /a for some constant C p depending only on p.

Proof. As for Corollary 2.8 take a smooth function ϕ :

[-π, π] → [0, 1] that is supported on [-1/4, 5/4] such that ϕ(t) = 1 for t ∈ [0, 1]. Define K(x, y) = 1
a+x+y ϕ(x)ϕ(y) on [-π, π] and make it 2π-periodic so that it is C ∞ . Using the chain rule, one easily sees that the Sobolev norms from Lemma 2.7 are dominated by C p /a. Corollary 2.14. Given a, b 0 with a + b > 0, one has

x θ ± y θ x + y x a,y b pcb C p 1 1 -θ 1/p max{a, b} θ-1 ,
for some constant C p depending on p.

k (x + 1), y ↔ 2 k y 1 1 + t + x + y x∈I k ,y∈J k pcb = 2 -k 1 1 + 2 -k (1 + t) + x + y x∈[0,1[,y∈[0,1[ pcb .
Setting a = 1 + 2 -k (1 + t) ∈ [1, 3] and using Corollary 2.13 the latter multiplier is bounded by -θ) . A similar estimate holds for the same symbol

C p 2 -k . The multiplier ((1 + x) θ ± (t + y) θ )1 x∈I k ,y∈J k is bounded by a fixed multiple of 2 kθ . Thus (1+x) θ ±(t+y) θ 1+t+x+y x∈I k ,y∈J k pcb C p 2 -k(1
if x ∈ J k+1 and y ∈ I k or x, y ∈ [0, 1[ (with k = 0). The sets [0, 1[ 2 , J k+1 × I k , I k × J k for k 0 form a partition of [0, ∞[ 2 into
product sets. Thus, the p-triangular inequality gives

(1 + x) θ ± (t + y) θ 1 + t + x + y x 0,y 0 p pcb C p p + 2 k 0 C p p 2 -p(1-θ)k C p p 1 1 -θ .

Ando's inequality in semi-finite algebras

Schur multipliers are intimately related to perturbations of the functional calculus of selfadjoint operators. Illustrations can be found in [START_REF] Sh | Double operator integrals in a Hilbert space[END_REF][START_REF] Arazy | Certain Schur-Hadamard multipliers in the spaces Cp[END_REF][START_REF] Aleksandrov | Functions of operators under perturbations of class Sp[END_REF][START_REF] Aleksandrov | Hankel and Toeplitz-Schur multipliers[END_REF][START_REF] Potapov | Operator-Lipschitz functions in Schatten-von Neumann classes[END_REF][START_REF] Potapov | Double operator integrals and submajorization[END_REF][START_REF] Potapov | On the Arazy conjecture concerning Schur multipliers on Schatten ideals[END_REF] and many other references.

Indeed let f : R → R be a (continuous) function. Assume that x, y are selfadjoint elements in some semi-finite von Neumann algebra (M, τ ) with finite discrete spectra (x i ) i∈I and (y j ) j∈J and associated spectral projections p i ∈ M and q j ∈ M. As x = i x i p i and y = j y i q j , with

i p i = j q j = 1, one has (4) f (x) -f (y) = i,j p i (f (x i ) -f (y j ))q j = i,j p i f (x i ) -f (y j ) x i -y j (x -y)q j , where f (u)-f (v) u-v can take any value if u = v ∈ R. The map T : z → i,j p i f (xi)-f (yj ) xi-yj
zq j is very close to be a Schur multiplier with symbol the divided differences of f . Lemma 3.1. For any 0 < p ∞, let M = (m i,j ) be a p-completely bounded Schur multiplier then the following map

T M : M → M T M (z) = i,j m i,j p i zq j
extends to a bounded map on L p (M) with T M Lp(M)→Lp(M) M pcb .

Proof. Recall that we assume that I and J are finite, (p i ) i∈I and (q j ) j∈J are orthogonal projections in M summing up to 1. The maps π : M → B( 2 (J), 2 (I))⊗M and ρ : B( 2 (J), 2 (I))⊗M → M defined by

π(z) = (p i zq j ) i∈I,j∈J =    p 1 . . . p |I|    z q 1 . . . q |J| ρ((X i,j )) = i,j p i X i,j q j = p 1 . . . p |I| (X i,j )    q 1 . . . q |J|   
clearly extend to contractions at L p -levels respectively on M and B( 2 (J), 2 (I)) ⊗ M since the column and row matrices in the above products are contractions. Since T M = ρ•(S M ⊗Id Lp(M) )•π, we get the lemma.

We want to deal with homogeneous functions of selfadjoint operators, namely

f (x) = |x| θ or f (x) = sgn(x)|x| θ = x θ + -x θ -.
We aim to prove the following theorem that we call Ando's inequality. C p,θ x -y θ p . The reduction from type III to type II will be explained in the next section so we only deal with semi-finite algebras here.

Proof. The result for p = ∞ follows from Theorem 4.1 in [START_REF] Aleksandrov | Operator Hölder-Zygmund functions[END_REF].

The result for f (x) = sgn(x)|x| θ when 1 p < ∞ can be found in [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF]. The absolute value map x → |x| is bounded on L q (M) provided that 1 < q < ∞. This is a result by Davies [START_REF] Davies | Lipschitz continuity of functions of operators in the Schatten classes[END_REF] for Schatten classes, that can be extended to all semi-finite von Neumann algebras (see Remark 6.2 in [START_REF] Caspers | Weak type estimates for the absolute value mapping[END_REF] for instance). Thus for 1 p < ∞, with q = p/θ, we have an estimate |x| θ -|y| θ p/θ C p/θ sgn(x)|x| θ -sgn(y)|y| θ p/θ . Therefore the Theorem holds for f (x) = |x| θ using it for f (x) = sgn(x)|x| θ when 1 p < ∞.

We just need to prove the Theorem for p < 1.

If the result holds for the couples (p, θ 1 ) and (p/θ 1 , θ 2 ), it also holds for (p, θ 1 θ 2 ). Indeed for instance if Hence C p,θ1θ2 C p/θ1,θ2 C θ2 p,θ1 . By this transitivity, we reduce the proof to p θ; indeed if p > θ, then (p, θ) = (p.1, p.θ/p) and the estimate follows from that for (p, p) and (1, θ/p).

We do it only for f (t) = |t| θ as the other case is similar. We prove the inequality from case to case regarding M and the values of x and y.

Case 1: We assume that x, y ∈ M sa with finite discrete spectra and x-y ∞ 2 and x-y p/2 2. We prove that |x| θ -|y| θ p/θ C p,θ for some C p,θ > 0.

We denote by x = i∈I x i p i and y = i∈J y j q j the spectral decompositions of x and y, so that p i = 1 {xi} (x) and q i = 1 {yi} (y).

We will rely on the formula (4), decompose |x| θ -|y| θ = 1 i,j=-1 a i (|x| θ -|y| θ )b i where a -1 = 1 ]∞,0[ (x), a 0 = 1 {0} (x) and a 1 = 1 ]0,∞[ (x) and similarly for b i .

We use dyadic decompositions. Set

I k = [2 -k-1 , 2 -k [, J k =]0, 2 -k-1 [ for k 0 and I -1 = [1, ∞[, J -1 =]0, 1[, J -2 =]0, ∞[. The definition is made so that the sets I k × J k-1 and J k × I k are disjoint with union {(x, y) ∈]0, ∞[ 2 | max{x, y} ∈ I k }. Hence we have a partition ]0, +∞[ 2 = ∪ k -1 (I k × J k-1 ∪ J k × I k ). Define accordingly the maps T 1 k (z) = xi∈I k ,yj ∈J k-1 p i x θ i -y θ j xi-yj zq j , T 2 k (z) = xi∈J k ,yj ∈I k p i x θ i -y θ j
xi-yj zq j . For any 0 < a 1, Lemma 3.1 says that T 1 k La(M)→La(M) is dominated by

x θ i -y θ j xi-yj xi∈I k ,yj ∈J k-1 acb
. By Lemma 2.4, this norm is smaller than

x θ -y θ x-y x∈I k ,y 0 acb .
Using Remark 2.10 and Corollary 2.9 for k 0, this norm is bounded by C a 2 -k(θ-1) . When k = -1, using Corollary 2.11 instead, we also get a bound by C a,θ . We have similar estimates for

T 2 k La(M)→La(M) . Put r k = 1 I k (x), s k = 1 I k (y) as well as u k = 1 J k (x), v k = 1 J k-1 (
y), we can write (note that the sums are actually finite)

a 1 (|x| θ -|y| θ )b 1 = xi>0,yj >0 p i x θ i -y θ j x i -y j (x -y)q j = k -1 T 1 k (x -y) + T 2 k (x -y) = k -1 T 1 k (r k (x -y)v k ) + T 2 k (u k (x -y)s k ).
We use the above norm estimates for T j k with a = p/θ to get -θ) . By the p/θ-triangular inequality

T j k L p/θ (M)→L p/θ (M) C p,θ 2 k(1
a 1 (|x| θ -|y| θ )b 1 p/θ p/θ C p/θ p,θ k -1 2 k(1-θ)p/θ ( r k (x -y)v k p/θ p/θ + u k (x -y)s k p/θ p/θ ).
But by definition 0 r k x, u k x, yv k , ys k 2 -k for k 0 and x -y ∞ 2, so that we have

r k (x -y)v k ∞ r k x -yv k ∞ 2 -k , and also u k x -ys k ∞ 2 -k for all k -1 . But also r k (x-y)v k p/2 , u k (x-y)s k p/2
x-y p/2 2. Thus as θ/p = (θ/2).2/p+(1-θ/2)/∞, by the Hölder inequality

r k (x -y)v k p/θ , u k (x -y)s k p/θ 2.2 -k(1-θ/2) . This is enough to conclude that a 1 (|x| θ -|y| θ )b 1 p/θ p/θ 4C p/θ p,θ k -1 2 -kp/2 .
To deal with a 1 (|x| θ -|y| θ )b -1 , one can do exactly the same using Corollary 2.14 as the Schur multipliers involved have shape

a θ -b θ a+b a∈I k ,b∈J k-1 , a θ -b θ a+b a∈J k ,b∈I k . The terms a -1 (|x| θ -|y| θ )b 1 and a -1 (|x| θ -|y| θ )b -1 can be treated similarly.
It is a well known fact that complex interpolation remains valid for L p (M, τ ) in the range 0 < p 1, see Lemma 2.5 in [START_REF] Pisier | The non-commutative Khintchine inequalities for 0 < p < 1[END_REF] for what we need. Using it, one easily deals with the remaining terms as for instance

a 0 (|x| θ -|y| θ ) p/θ = a 0 |y| θ p/θ a 0 |y| θ p = a 0 y θ p = a 0 (x -y) θ p 2.
Gluing the pieces together, we find a constant C p,θ so that (5) holds in Case 1.

Case 2: We assume M finite, x, y ∈ M sa , y = x + q for some projection q with q p = 1, that is τ (q) = 1. Consider (x n ) a sequence in the von Neumann algebra generated by x that |x n | |x|, x nx ∞ → 0 and each x n has a finite spectrum, and similarly for (y n ) and y.

Obviously |x n | θ -|x| θ p/θ → 0 by the dominated convergence theorem as M is finite (similarly for y). As x n -y n t → q t = 1 for t = ∞, p/2, x n and y n satisfy the assumptions of Case 1 for n big enough. Going to the limit in (5) for x n , y n , we get (5) for x, y with the same C p,θ .

Case 3: We assume M finite, x, y ∈ M sa and y = x + tq for some projection q with τ (q) = 1 and t ∈ R.

If t > 0, this follows by homogeneity applying the result for y/t, x/t and q. If t < 0, one just needs to exchange x and y. The constant C p,θ is the same as in the previous cases.

Case 4: We assume M finite, x, y ∈ M sa and y = x + n i=1 t i q i where t i ∈ R and q i are orthogonal projections with τ (q i ) = 1.

Simply put x 0 = x and x k = x + k i=1 t i q i and use the p/θ-triangular inequality and Case 3 for x k-1 and x k .

f (x) -f (y) p/θ p/θ n i=1 f (x i-1 ) -f (x i ) p/θ p/θ C p/θ p,θ n i=1 t i q i θ p p/θ . But x -y p = n i=1 |t i | p 1/p
and we get this case as q i p = 1.

Case 5: We assume M finite, x, y ∈ M sa and y = x + n i=1 t i q i where t i ∈ R and q i are orthogonal projections with τ (q i ) ∈ Q.

Consider the von Neumann algebra ( M, τ ) where M = M⊗L ∞ ([0, 1]) with the trace τ = N τ ⊗ where N is such that for all i, N τ (q i ) ∈ N . Put z = z ⊗ 1 for z ∈ M. We have ỹ -x = n i=1 t i qi . For each i, and k N τ

(q i ), let q i,k = q i ⊗ 1 [(k-1)/(N τ (qi)),k/(N τ (qi))]
. The (q i,k ) k are orthogonal projections with τ (q i,k ) = 1 and N τ (qi) k=1 q i,k = qi . By Case 4:

N θ/p f (x)-f (y) L p/θ (M) = f (x)-f (ỹ) L p/θ ( M) C p,θ x-ỹ θ Lp( M) = C p,θ N 1/p x-y Lp(M) θ .
We still obtain the same constant C p,θ . Case 6: We assume M finite and x, y ∈ M sa . By considering again ( M, τ ) where M = M ⊗ L ∞ ([0, 1]) with the trace τ = τ ⊗ , in a masa containing it, one can approximate x-y for the L 1 -norm by elements of the form δ k = n k i=1 t i,k q i,k where τ (q i,k ) ∈ Q. With y k = x + δ k , y k converges to y in L 1 it also does in L p as M is finite. The Ando inequality for the index 1/θ also gives that f (y k ) -f (y) 1/θ C θ y k -y θ 1 , hence f (y k ) converges to f (y) in L p/θ . Since x, y k satisfy the assumptions of Case 5, the conclusion follows by going to the limit in [START_REF] Arazy | Certain Schur-Hadamard multipliers in the spaces Cp[END_REF].

Case 7:

We assume M semi-finite and x, y ∈ L p (M) sa . This is again a matter of approximation. By the semi-finiteness of M and functional calculus in a masa containing x, we may approximate x in L p by elements of the form x n = N i=1 t i q n commuting with x where q n are finite projections. We can as well assume that f (x n ) converges to f (x) in L p/θ , similarly for y n and y (these are purely commutative results). Then y n and x n are in some finite subalgebra of M and we have [START_REF] Arazy | Certain Schur-Hadamard multipliers in the spaces Cp[END_REF] for them by Case 6:

f (x n ) -f (y n ) p/θ C p,θ x n -y n θ p .
Taking again limits as n → ∞ gives the result. Remark 3.3. One can get a proof of the case p = ∞ following case 1 (since p/2 = ∞) and then case 6 directly. A careful analysis of the constant shows that lim sup θ→1 (1 -θ)C ∞,θ < ∞ as in [START_REF] Aleksandrov | Operator Hölder-Zygmund functions[END_REF].

We slightly extend the result to τ -measurable operators L 0 (M, τ ). With its measure topology [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF][START_REF] Fack | Generalized s-numbers of τ -measurable operators[END_REF], it becomes a complete Hausdorff topological * -algebra (Theorem 28 in [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF]) in which M is dense. We recall the Fatou Lemma in L 0 (Lemma 3.4 in [START_REF] Fack | Generalized s-numbers of τ -measurable operators[END_REF]) if v n → v in L 0 and then v q lim inf v n q for all q. Moreover, the functional calculus associated to our f (t) = |t| θ or f (t) = sgn(t)|t| θ is continuous on L sa 0 (Lemma 3.2 in [START_REF] Raynaud | On ultrapowers of non commutative Lp spaces[END_REF]). Theorem 3.4. Let 0 < θ < 1 and 0 < p ∞ then there exists C p,θ so that for any semifinite von Neumann algebra (M, τ ), and x, y

∈ L 0 (M, τ ) sa such that x -y ∈ L p (M, τ ), then |x| θ -|y| θ , sgn(x)|x| θ -sgn(y)|y| θ ∈ L p/θ (M, τ ) and |x| θ -|y| θ p/θ C p,θ x -y θ p , sgn(x)|x| θ -sgn(y)|y| θ p/θ C p,θ x -y θ p .
Proof. This is a matter of approximations.

We start by giving arguments to get the result when x, y ∈ M sa and x -y ∈ L p (M). First note that if a bounded sequence (x n ) n ∈ M sa goes to x for the strong-topology then (f (x n )) n also goes to f (x) for the strong-topology (Lemma 4.6 in [START_REF] Takesaki | Theory of operator algebras. I[END_REF]). Take any finite projection γ ∈ M and (q n ) n a sequence of finite projections that goes to 1 strongly, it follows that (γf (q n xq n )γ) n goes to γf (x)γ in L 2 . since L 2 ⊂ L 0 is continuous, we get that in the topology of L 0 , lim n γf (q n xq n )γ = γf (x)γ and similarly for y.

As q n xq n and q n yq n are in L p , we get γ(f (q n xq n ) -f (q n yq n ))γ p/θ C p,θ q n (x -y)q n θ p C p,θ x -y θ p . Using the Fatou lemma, we obtain γ(f (x) -f (y))γ p/θ C p,θ x -y θ p . It is a simple exercise to check that if z ∈ L sa 0 is so that sup γzγ p/θ C where the sup runs over all finite projections, then z ∈ L p/θ with norm less than C. This allows to conclude.

Next take any x, y ∈ L 0 such that x -y ∈ L p . Let q n = 1 |x| n ∧ 1 |y| n , then q n is an increasing sequence of projections to 1 with τ (1 -q n ) → 0 as x, y ∈ L 0 . Moreover q n xq n ∈ M goes to x in L 0 (similarly for y) (see [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF] page 20). By the continuity of the functional calculus in L 0 , (f (q n xq n )) and (f (q n yq n )) go to f (x) and f (y). Once again with the help of the Fatou lemma and the previous case

f (x) -f (y) p/θ lim inf f (q n xq n ) -f (q n yq n ) p/θ C p,θ lim inf q n (x -y)q n θ p C p,θ x -y θ p .

Ando's inequality in type III algebras

Before going to type III algebras, we need to extend Ando's inequality to weak-L p spaces. We will use the K-interpolation method see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF].

As usual, given two compatible quasi-Banach spaces X 0 and X 1 , we let for t > 0 and x ∈ X 0 +X 1

K t (x, X 0 , X 1 ) = inf{ x 0 X0 + t x 1 X1 ; x = x 0 + x 1 }.
For 0 < η < 1 and 0 < q ∞ set x η,q = t -η K t (x, X 0 , X 1 ) Lq(R + ,dt/t) = ∞ 0 (t -η K t (x, X 0 , X 1 )) q dt t 1/q , with the obvious modification when q = ∞.

The interpolated space (X 0 , X 1 ) η,q is {x ∈ X 0 + X 1 | x η,q < ∞} with (quasi)-norm . η,q . If (M, τ ) is a semi-finite von Neumann algebra, and x ∈ L 0 (M, τ ) we denote as usual its decreasing rearrangement by µ t (x) (see [START_REF] Fack | Generalized s-numbers of τ -measurable operators[END_REF]).

The noncommutative Lorentz spaces L p,q (M, τ ) for 0 < p < ∞ and 0 < q ∞ are defined as in the commutative case. The space L p,q (M, τ ) consists of all measurable operators x ∈ L 0 (M, τ ) so that x Lp,q = t 1/p µ t (f ) Lq(R + ,dt/t) < ∞. With the (quasi)-norm . Lp,q , it becomes a (quasi)-Banach space.

The results about real interpolation of commutative L p -spaces ([7] Theorem 5.3.1) remain available for semi-finite von Neumann algebras. Indeed for any 0 < p 0 < p 1 ∞, for all t > 0, x ∈ L p0 (M, τ )+L p1 (M, τ ), the quantities K t (x, L p0 (M, τ ), L p1 (M, τ )) and K t (µ(x), L p0 (R + ), L p1 (R + )) are equivalent with constants depending only on p 0 and p 1 , see [START_REF] Xu | Noncommutative Lp-spaces[END_REF] for details. As a consequence, we get that for p 0 = p 1 ∞ and 0 < q ∞, 0 < η < 1 with 1 p = 1-η p0 + η p1 , (L p0 (M, τ ), L p1 (M, τ )) η,q = L p,q (M, τ ) with equivalent norms (depending only on the parameters but not on (M, τ )).

We drop the reference to (M, τ ) to lighten notation. When x = x * ∈ L p0 + L p1 , we can also consider

K sa t (x, L p0 , L p1 ) = inf{ x 0 p0 + t x 1 p1 ; x = x 0 + x 1 with x i = x * i }. Lemma 4.1. Let 0 < p 0 < p 1 ∞, t > 0 and x ∈ (L p0 + L p1 ) sa , then K sa t (x, L p0 , L p1 ) K t (x, L p0 , L p1 ) K sa t (x, L p0 , L p1 )/2 max{1/p0,1}-1 . Proof. This is a standard fact. Let x = a 0 + a 1 with a i ∈ L pi such that a 0 p0 + t a 1 p1 K t (x, L p0 , L p1 ) + . Then x = b 0 + b 1 with b i = (a i + a * i )/2. We have b i pi 2 1/pi-1 a i pi if p i < 1 or b i pi
a i pi otherwise, hence we get the result letting ε → 0.

Remark 4.2. Similarly one can easily show that for x 0, K t (x, L p0 , L p1 ) is equivalent to inf{ x 0 p0 + t x 1 p1 ; x = x 0 + x 1 with x i 0}. Lemma 4.3. Let 0 < p 0 < p 1 ∞ and θ ∈]0, 1[ and x, y ∈ (L p0 + L p1 ) sa . Then for all t > 0 and f (s) = |s| θ or f (s) = sgn(s)|s| θ :

K t θ (f (y) -f (x), L p0/θ , L p1/θ ) C p0,p1,θ K t (y -x, L p0 , L p1 ) θ .
Proof. Thanks to the previous Lemma, it suffices to do it with K sa t instead of K t . Choose selfadjoint operators δ 0 ∈ L p0 and δ 1 ∈ L p1 so that y -x = δ 0 +δ 1 and δ 0 p0 +t δ 1 p1 2K sa t (x -y, L p0 , L p1 ).

Set

a 0 = f (y) -f (x + δ 1 ) = f (y) -f (y -δ 0 ) and a 1 = f (x + δ 1 ) -f (x), then f (y) -f (x) = a 0 + a 1 . By Theorem 3.4 for p i , we obtain a i pi/θ C pi,θ δ i θ pi . Since, a 0 p0/θ + t θ a 1 p1/θ C p0,p1,θ ( δ 0 1 + t δ 1 p1
) θ , we have found a suitable decomposition to conclude. Proposition 4.4. For all 0 < p < ∞, 0 < q ∞ and 0 < θ < 1, there exists C p,q,θ > 0 such that for x, y ∈ L sa p , with f (s) = |s| θ or f (s) = sgn(s)|s| θ : f (y) -f (x) L p/θ,q C p,q,θ y -x θ L p,qθ .

Proof. Put p 0 = p/2, p 1 = 2p and η = 2/3 so that 1 p = 1-η p0 + η p1 . We have

f (y) -f (x) L p/θ,q p/θ,q t -η K t (f (y) -f (x), L p0/θ , L p1/θ ) Lq(R + ,dt/t) , y -x L p,qθ p,qθ u -η K u (y -x, L p0 , L p1 ) L qθ (R + ,du/u) .
But by Lemma 4.3

t -η K t (f (y) -f (x), L p0/θ , L p1/θ ) Lq(R + ,dt/t) C p,θ t -η K t 1/θ (y -x, L p0 , L p1 ) θ Lq(R + ,dt/t) . But by a change of variable u = t 1/θ : t -η K t 1/θ (y -x, L p0 , L p1 ) θ Lq(R + ,dt/t) = θ 1/q u -η K u (y -x, L p0 , L p1 ) θ L qθ (R + ,du/u)
and we get the estimate.

We can conclude with the proof of Theorem 3.2 for type III algebras.

Proof. Assume that M is given with a n.s.f. weight ϕ with modular group σ. Let R = M σ R be its core (with the dual action σ), this is a semi-finite von Neumann algebra with a trace τ such that τ • σt = e -t τ . Then by definition L p (M, ϕ) is isometrically a subspace of L p,∞ (R, τ ), more precisely

L p (M, ϕ) = {x ∈ L 0 (R, τ ) | σt (x) = e -t/p x, ∀t ∈ R} ⊂ L p,∞ (R, τ
), with by definition x p = x Lp,∞(R) (see [START_REF] Terp | L p -spaces associated with von Neumann algebras[END_REF] Definition 13 and Lemma 5 or Lemma B in [START_REF] Kosaki | On the continuity of the map ϕ → |ϕ| from the predual of a W * -algebra[END_REF]).

Thus Theorem 3.2 for type III algebras follows from Proposition 4.4 for the weak-L p spaces noticing that f (L p (M, ϕ)) ⊂ L p/θ (M, ϕ) as σt is a representation and f is θ-homogeneous.

Further comments

By very classical arguments using the Cayley transform see [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF], estimates for the functional calculus are equivalent to some for commutators or anticommutators. Let M p,q denote the Mazur map for p < q given by M p,q (f ) = f |f | (p-q)/q . The 2 × 2-tricks from [START_REF] Éric | Hölder estimates for the noncommutative Mazur maps[END_REF] also give Proposition 5.2. Let 0 < p ∞ and 0 < θ < 1, then there exists a constant C p,θ so that for x, y ∈ L p (M):

M p,p/θ (x) -M p,p/θ (y)] p/θ C p,θ x -y θ p .

This improves the estimates for M p,q when p < q of Theorem 4.1 in [START_REF] Pisier | The non-commutative Khintchine inequalities for 0 < p < 1[END_REF] and gives the expected Hölder continuity.

Remark 5.3. The above two propositions are valid for all von Neumann algebras. For semi-finite ones, similar estimates are true for the Lorentz norms as in Proposition 4.4.

Contrary to the case p 1, at least for p 1/2, there is no constant C p so that for all 0 < θ < 1,

x θ -y θ x-y

x 1,y 1 pcb

C p .
Indeed if this is so, then taking θ = 1 -ε, x = e t/ε , y = e s/ε by Lemma 2.5, letting ε → 0, we would get that e -max{t,s} t,s 0 pcb C p . Using that 2 max{x, y} = |x -y| + x + y, we would deduce that e -|t-s| 0 t,s 1 pcb

C p .
By easy arguments going from a discrete situation to a continuous one, one can deduce that the same matrix has to be a Schur multiplier on S p (L 2 [0, 1]). The constant kernel 1 on [0, 1] 2 is in S p (L 2 [0, 1]) with norm one, thus we would get that e -|x-y| 0 x,y 1 S p (L2[0,1])

C p .
The operator T on L 2 [0, 1] with kernel K(x, y) = e -|x-y| is obviously positive and Hilbert-Schmidt with HS norm less than 1. One easily checks that the eigenvectors f λ associated to λ must satisfy λf = λf -2f . Letting λ = 2 1+α 2 with α > 0, the only possibilities are f λ (x) = ae iαx +be -iαx . But T (e iαx ) = 2 1+α 2 e iαxe -x 1+iα -e x e -1+iα 1-iα . Thus, λ is a eigenvalue iff e 2iα = 1-iα 1+iα 2

.

Set tan θ = α with θ ∈]0, π/2[, so that e 2iα = e -4iθ . The equation tan t = -2t+kπ admits a unique solution θ k in ]0, π/2[ for k > 0 so that tan θ k ≈ kπ. We deduce that the set of eigenvalues of T is ( 2 1+tan(θ k ) 2 ) k 1 with associated eigenvector f λ (x) = e iαx -1+iα 1-iα e -iαx . Thus T / ∈ S p (L 2 [0, 1]) when p 1/2.

As f λ / f λ 2 is uniformly bounded in C[0, 1], we can also deduce using Lemma 2.3 that for p > 1/2 e -max{t,s} 0 t,s 1 pcb < ∞ and e -max{t,s} t,s 0 pcb < ∞.

About Corollary 2.11, it is easier to see that the norm cannot be independent of 0 < θ < 1 for p 1. Indeed the multiplier x-y x+y x 1,y 1

, or equivalently x-y x+y x 0,y 0 using homogeneity, is not bounded for p = 1 (hence also for p < 1) as shown in [START_REF] Davies | Lipschitz continuity of functions of operators in the Schatten classes[END_REF].

  will use the notation S p I,J for the Schatten p-class on B( 2 (J), 2 (I)), this is naturally a subspace of L p (B( 2 (I ∪ J)), tr), where tr is the usual trace. Thus by S p I,J [L p (M)], we will mean the corresponding subspace of L p (B( 2 (I ∪J))⊗M, tr⊗ϕ 0 ), one can think of it as matrices indexed by I × J with coefficients in L p (M). We will often use non countable sets like I =]0, 1].

Proof.

  Without loss of generality we may assume a b. By a change of variable x ↔ a(1 + x) and y ↔ a(t + y), with t = b/a, it boils down to show that (1+x) θ ±(t+y) θ 1+t+x+y x 0,y 0 pcb is bounded independently of t ∈]0, 1]. We use a dyadic decomposition related to max{x, y}. Assume x ∈ I k = [2 k , 2 k+1 [ and y ∈ J k = [0, 2 k [. Then by homogeneity and a change of variables x ↔ 2

Theorem 3 . 2 .

 32 Let 0 < θ < 1 and 0 < p ∞ then there exists C p,θ so that for any von Neumann algebra M, and x, y ∈ L p (M) sa one has (5) |x| θ -|y| θ p/θ C p,θ x -y θ p , sgn(x)|x| θ -sgn(y)|y| θ p/θ

  |x| θ1 -|y| θ1 p/θ1 C p,θ1 x -y θ1 p and |z| θ2 -|t| θ2 p/(θ1θ2) C p/θ1,θ2 z -t θ2 p/θ1 , then one gets with z = |x| θ1 , t = |y| θ1 : |x| θ1θ2 -|y| θ1θ2 p/(θ1θ2) C p/θ1,θ2 C θ2 p,θ1 x -y θ1θ2 p .

Proposition 5 . 1 .

 51 Let 0 < p ∞ and 0 < θ < 1, then there exists a constant C p,θ so that forx ∈ L p (M) sa and b ∈ M with f (s) = |s| θ or f (s) = sgn(s)|s| θ : [f (x), b] p/θ C p,θ [x, b] θ p b 1-θ . If x, y ∈ L p (M) + and b ∈ M, then bx θ ± y θ b p/θ C p,θ bx ± yb θ p b 1-θ .