Jorge Armenta-Segura

Sergio Rajsbaum
email: rajsbaum@im.unam.mx

Jérémy Ledent
email: jeremy.ledent@strath.ac.uk

LANMR 2020

Keywords: Distributed algorithm, approximate consensus, fault-tolerance, epistemic logic

We investigate the two agents approximate agreement problem in a dynamic network in which topology may change unpredictably, and where consensus is not solvable. It is known that the number of rounds necessary and sufficient to guarantee that the two agents output values 1/k 3 away from each other is k. We distil ideas from previous papers to provide a self-contained, elementary introduction, that explains this result from the epistemic logic perspective.

Introduction

Problems of reaching agreement are central to distributed computing. Often consensus is needed to agree on the same value, for example, when agents need to agree on whether to commit or abort the results of a distributed database transaction. However, in many situations consensus is impossible to achieve, such as networks where agents may crash and delays are unpredictable [START_REF] Fischer | Impossibility Of Distributed Commit With One Faulty Process[END_REF], or read/write shared memory models [START_REF] Loui | Memory requirements for agreement among unreliable asynchronous processes[END_REF], or certain dynamic networks [START_REF] Coulouma | A characterization of oblivious message adversaries for which consensus is solvable[END_REF]. All these impossibility results are due to the same reason: the indistinguishability graph [START_REF] Attiya | Indistinguishability[END_REF] of the global states of a protocol trying to solve consensus is connected, while that of the consensus task is not [START_REF] Biran | A Combinatorial Characterization of the Distributed 1-Solvable Tasks[END_REF][START_REF] Herlihy | Distributed Computing Through Combinatorial Topology[END_REF][START_REF] Moses | A layered analysis of consensus[END_REF].

In this paper we are interested in many other situations, where approximate agreement is sufficient; for example, when sensors estimate a certain measurement or clock synchronization where agents maintain similar time estimates. Many variants of approximate agreement, and in various message passing and shared memory models have been considered since early on, e.g. [START_REF] Dolev | Reaching approximate agreement in the presence of faults[END_REF]. Approximate agreement is an interesting weakening of consensus that can be solved in many more situations. The task is parametrised by a real number ε > 0, and the agents must decide values that are at most ε away from each other. The time to reach a decision however, will be a function of how small ε is. Many algorithms have been proposed to try to minimize the time until decision is reached, e.g. in shared memory models [START_REF] Hoest | Towards a topological characterization of asynchronous complexity[END_REF], in networks [START_REF] Charron-Bost | Fast, Robust, Quantizable Approximate Consensus[END_REF], and others.

Consensus has been thoroughly studied from the epistemic logic perspective, and the close connection with common knowledge is well-known [START_REF] Fagin | Reasoning About Knowledge[END_REF]. Approximate agreement has been less studied from the epistemic logic perspective. It was recently shown that it is closely related to k-iterated knowledge in a certain shared memory model [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF], roughly, meaning that the agents must know, that they know, that they know that they know, and so forth (k times), each other's inputs, in order to reach certain degree of approximation of their decisions. This was shown within a general simplicial complex model for multi-agent epistemic logic for task solvability. In this paper we are interested in studying in more detail this result, as well as providing a self-contained, more elementary introduction to the topic.

We focus on one specific distributed computing model, that has received much attention recently, dynamic networks (see surveys [START_REF] Braud-Santoni | The next 700 impossibility results in time-varying graphs[END_REF][START_REF] Casteigts | Time-varying graphs and dynamic networks[END_REF][START_REF] Kuhn | Dynamic networks: models and algorithms[END_REF][START_REF] Winkler | An overview of recent results for consensus in directed dynamic networks[END_REF]). There is a fixed set of agents that operate in rounds communicating by sending messages to each other. Rounds are synchronous in the sense that the messages received at some round have been sent at that round. At each round, the communication graph is chosen arbitrarily among a set of directed graphs, G. This set G determines the network model. Hence the communication graph can change unpredictably from one round to the next. Dynamic networks are interesting for various reasons, in particular because they include as special cases other classic models, such as shared memory models [START_REF] Herlihy | From wait-free to arbitrary concurrent solo executions in colorless distributed computing[END_REF], so results in dynamic networks can often be extrapolated to other models.

Approximate agreement is solvable in a dynamic network model G if and only if each communication graph in G has a rooted spanning tree [START_REF] Charron-Bost | Approximate consensus in highly dynamic networks: The role of averaging algorithms[END_REF] (e.g. [START_REF] Biran | A Combinatorial Characterization of the Distributed 1-Solvable Tasks[END_REF][START_REF] Castañeda | A topological perspective on distributed network algorithms[END_REF]). In this work we consider such a G, where approximate agreement is solvable, but not consensus. Two agents, starting with binary input values, are to decide values in the interval of their inputs, which are at most 1/3 k apart from each other 5 . We rephrase this problem epistemically, as requiring that the agents reach k-iterated knowledge about their respective inputs. This knowledge can be achieved, if and only if, the protocol runs for k rounds. We show these results in the dynamic epistemic logic (DEL) [START_REF] Baltag | Dynamic epistemic logic[END_REF] framework of [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF], instantiated to a dynamic network. We follow recent work, in using the dual of a Kripke graph, a simplicial complex, as a model for multi-agent epistemic logic, that exposes the topological features that determine if a task is solvable [START_REF] Goubault | A dynamic epistemic logic analysis of the equality negation task[END_REF][START_REF] Van Ditmarsch | Knowledge and simplicial complexes[END_REF]. While the results we present here have been shown in these papers, we instantiate the results using graphs instead of simplicial complexes, making the development more accesible.

Organization. Before getting into epistemic logic, in Section 2 we present two algorithms that solve approximate agreement. In Section 3 we overview the DEL framework. In Section 4 we give the formal epistemic logic semantics to our dynamic network model, and we do the same for tasks, in Section 5. The solvability results are in Section 6. The conclusions are in Section 7. Additional details and proofs are in the Appendix.

Approximate agreement algorithms

We start by presenting algorithms, to guide the reader's intuition before diving into the formal semantics considerations of the subsequent sections.

Two agents, g, w (for gray, white), communicate with each other exchanging messages in a sequence of k rounds. In a round, each agent sends a message to the other agent, receives the message sent by the other agent, and then updates its local state. In each round, at most one of the two messages sent may be lost.

Let N = 3 k , k ≥ 0. In the N -approximate agreement task, after k rounds, agents decide values d g , d w , resp., of the form i/N , for an integer i, 0 ≤ i ≤ N . Each agent starts with an input value from the set {0, 1}. The decisions must be such that if the two input values are equal, then both output values are equal to the input value. Otherwise, the output values d g , d w satisfy |d g -d w | ≤ 1/N . The algorithm in Figure 1 solves N -approximate agreement in k rounds (e.g [START_REF] Charron-Bost | Fast, Robust, Quantizable Approximate Consensus[END_REF][START_REF] Függer | Tight bounds for asymptotic and approximate consensus[END_REF], [START_REF] Herlihy | Distributed Computing Through Combinatorial Topology[END_REF][Chapter 2]). A simple induction argument on the number of rounds proves its correctness. The examples on the right show two 4-round executions where agents agree on values 1/3 4 = 1/81 away from each other. The first example (represented in the top two rows) represent an execution where all messages from w are lost. The first column, with labels 0 and 1, corresponds to the initial input values. Then each subsequent column shows the values of the processes after each round. An edge indicates that a message was received. In the second execution (bottom two rows of the picture), both messages arrive in the first two rounds, but in the 3rd round the message from g to w is lost, and in the 4th round the message from w to g.

The algorithm of Figure 1 can be used by the agents to decide values arbitrarily close to each other, by running enough rounds, k. However, the size of messages sent grows with k. Remarkably, there is an algorithm that solves the problem sending 1-bit messages, see Figure 2, which is a reformulation and small adaptation of the algorithm given in [START_REF] Delporte-Gallet | Communication complexity of wait-free computability in dynamic networks[END_REF]. In a 1-bit protocol, each message received from the other process can be either 0, 1 or ⊥. Thus, in a k-round computation, we call the view of a process the sequence v ∈ {0, 1, ⊥} k of messages that were received. For instance with four rounds, v = (0, 0, ⊥, 1) indicates that the process received message "0" in the first two rounds, no message in the third round, and message "1" in the fourth round. To reduce notation, we often write such a view as a word, e.g. v = 00⊥1. Given a view v and a message m ∈ {0, 1, ⊥}, we write v • m for the new view where we append m at the end of v.

The algorithm in Figure 2 below is generic, in the sense that both processes simply collect their view during the computation, and at the very end a decision function δ computes the output depending on the view. To choose which bit to send, at each round, the process counts how many messages were received until now, nbMsg = #{m ∈ view | m ̸ = ⊥}, adds its input ℓ ∈ {0, 1}, and sends the parity of the result. The computation of δ is done locally at the end of the protocol, and does not involve communication between the processes. It is adapted from [START_REF] Delporte-Gallet | Communication complexity of wait-free computability in dynamic networks[END_REF], except that we deal with all four combinations of inputs in {0, 1} instead of fixing in advance one input edge. We use the following facts:

• The first time a process p receives a message, it can immediately guess the input of the other process p ′ .

Indeed, since all previous messages were lost by p, we know for sure that all messages were received by p ′ . • Process p can then use this information to "decode" the subsequent messages, and behave like in the algorithm of [START_REF] Delporte-Gallet | Communication complexity of wait-free computability in dynamic networks[END_REF].

One-bit messages N -approximate agreement (ℓ) /* input ℓ ∈ {0, 1} */ The rest of the paper is devoted to answering the question: what do the agents learn about their inputs after running protocols like these ones? For these two specific cases, do they learn something different? We will give a formal semantics based on dynamic epistemic logic to answer such questions. And as an application, we will show that these two algorithms are optimal in the number of rounds: in less than k rounds, it is impossible for the agents to produce outputs closer than 1/3 k . This result has been proved using combinatorial 2, for 1 and 2 rounds of computation. Process names are represented by colors; inputs ℓ ∈ {0, 1} are written inside the vertices; and views are written next to the vertices. For the two-round protocol, the value of δ(ℓ, view) at each node is given by projecting vertically downwards. These pictures represent protocol complexes in the usual sense of distributed computing, but also in the sense of the product-update model construction defined later. arguments e.g. [START_REF] Függer | Tight bounds for asymptotic and approximate consensus[END_REF][START_REF] Herlihy | Distributed Computing Through Combinatorial Topology[END_REF], here we will see a different explanation: in less than k rounds, they cannot acquire sufficient knowledge about their inputs. Also, while the two algorithms seem rather different, the agents learn exactly the same information about their inputs in both.

3 for i = 1 to k do 4 nbMsg = #{x ∈ view[1...(i -1)] | x ̸ = ⊥} 5 m = view[i] 6 if m ̸ = ⊥ then /* if m = ⊥,

Input complex

I 0 0 1 1 One round I[OB 1 ] 0 ⊥ 0 ⊥ 1 ⊥ 1 ⊥ 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 Protocol complex for two rounds I[OB 2 ] 0 ⊥⊥ 0 ⊥⊥ 1 ⊥⊥ 1 ⊥⊥ 0 0⊥ 0 1⊥ 1 0⊥ 1 1⊥ 0 0⊥ 1 0⊥ 0 1⊥ 1 1⊥
3 Fundamentals about dynamic epistemic logic (DEL)

A simplicial model for epistemic logic

We describe here the model for epistemic logic, based on chromatic simplicial complexes [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF][START_REF] Van Ditmarsch | Knowledge and simplicial complexes[END_REF]. This reformulates the usual semantics of formulas in Kripke models, in terms of simplicial models. See Appendix B and Theorem B.4.

Syntax

Let AP be a countable set of propositional variables and A a finite set of agents. The language of epistemic logic formulas L K (A, AP ), or just L K if A and AP are implicit, is generated by the following BNF grammar:

φ ::= p | ¬φ | (φ ∧ φ) | K a φ p ∈ AP , a ∈ A
The proof theory of epistemic logics can be found in [START_REF] Van Ditmarsch | Dynamic Epistemic Logic[END_REF]; in the rest of the paper, we focus on studying models. In the following, the number of agents A is two and we let A = {g, w}. The agents will be represented as colors, gray and white. We can then use terminology of graphs, a specialization of the case of larger number of agents that requires using simplicial complexes, see Appendix B.

Given a set V , an (undirected) graph6 C over a set of vertices V is defined by a non-empty finite set of edges. Each edge is a set of two vertices. The set of vertices of C is noted V(C), and the set of edges F(C). A chromatic graph ⟨C, χ⟩ consists of a graph C and a coloring map χ : V(C) → A, such that for all X ∈ F(C), the two vertices of X have distinct colors.

Simplicial models.

Let V be some countable set of values, and AP = {p a,x | a ∈ A, x ∈ V} be the set of atomic propositions. Intuitively, p a,x is true if agent a holds the value x. We write AP a for the atomic propositions of agent a.

A simplicial model M = ⟨C, χ, ℓ⟩ consists of a chromatic graph ⟨C, χ⟩, and a labeling ℓ : V(C) → P(AP ) that associates with each vertex v ∈ V(C) a set of atomic propositions concerning agent χ(v), i.e., such that

ℓ(v) ⊆ AP χ(v) . Given an edge X = {v 0 , v 1 } ∈ C, we write ℓ(X) = ℓ(v 0 ) ∪ ℓ(v 1 ).

Binary input model.

Each agent gets an input value from the set {0, 1}. Each agent knows its own input value, but doesn't know which value has been received by the other agents. The figure below from [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF] shows the binary input simplicial model and its associated Kripke model for two agents, and for comparison also for three agents (although we will not use it in this paper). Notice that every possible combination of 0's and 1's is a possible world.

In the Kripke model, the agents are called b, g, w, and the labeling L of the possible worlds is represented as a sequence of values, e.g., 101, representing the values chosen by the agents b, g, w (in that order). In the 3-agents case, the labels of the dotted edges have been omitted to avoid overloading the picture, as well as other edges that can be deduced by transitivity.

In the simplicial model, agents are represented as colors (black, grey, and white). The labeling ℓ is represented as a single value in a vertex, e.g., "1" in a grey vertex means that agent g has chosen value 1. The possible worlds correspond to edges in the 2-agents case, and triangles in the 3-agents case. Note that the simplicial model depicted on the right, with three agents, is out of the scope of this paper. Indeed, here we only work with two agents, so that the simplicial complex is nothing more than a graph.

∈ L K . M, X |= p iff p ∈ ℓ(X) M, X |= ¬φ iff M, X ̸ |= φ M, X |= φ ∧ ψ iff M, X |= φ and M, X |= ψ M, X |= K a φ iff for all Y ∈ F(C), a ∈ χ(X ∩ Y ) implies M, Y |= φ Group knowledge.
For a formula φ, we write Eφ = K g φ ∧ K w φ for the group knowledge of φ among the agents {g, w}. Let E k denote k nested E operators. An edge path is a sequence of (not necessarily distinct) edges, such that each two consecutive edges in the sequence intersect in a vertex. For an edge X let N k (X) be the set of edges reachable from X by an edge path of at most k edges. Thus, N 1 (X) = {X}, denoted by N (X). Also, N 2 (X) is equal to X together with the edges that intersect X, and in general N k (X) = N (N k-1 (X)). The figure below is an illustration of the cases k = 1, 2. It shows that M, X |= Eφ states that φ should hold in X and in its two neighboring edges, namely, in the edges that belong to N 2 (X), and analogously for k = 2.

Morphisms between models

Let C and D be two graphs. A simplicial map7 f : C → D maps the vertices of C to vertices of D, such that if X is an edge of C, f (X) is an edge of D. A chromatic simplicial map between two chromatic graphs is a simplicial map that preserves colors, i.e. χ(f (v)) = χ(v) for all v. A morphism of simplicial models f : M → M ′ is a chromatic simplicial map that preserves the labeling: ℓ ′ (f (v)) = ℓ(v). The next theorem from [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF] states that morphisms of simplicial models cannot "gain knowledge". Lemma 3.4 (knowledge gain) Consider simplicial models M = ⟨C, χ, ℓ⟩ and M ′ = ⟨C ′ , χ ′ , ℓ ′ ⟩, and a morphism f : M → M ′ . Let X ∈ F(C) be an edge of M , a an agent, and φ ∈ L CK a positive formula, i.e. which does not contain negations except, possibly, in front of atomic propositions. Then, M ′ , f (X) |= φ implies M, X |= φ.

Dynamic epistemic logic basic notions

Dynamic epistemic logic (DEL) is the study of modal logics of model change [START_REF] Baltag | Dynamic epistemic logic[END_REF][START_REF] Van Ditmarsch | Dynamic Epistemic Logic[END_REF]. A modal logic studied in DEL is obtained by using action models [START_REF] Baltag | The logic of common knowledge, public announcements, and private suspicions[END_REF]. An action can be thought of as an announcement made by the environment (not necessarily public). An action model describes all the possible actions that might happen, as well as how they affect the different agents. The product-update operation takes an epistemic model M and an action model A, and creates a new model M [A] that describes all the new possible worlds after an action from A has occurred in M . This classic version is described in Appendix D, here we present the simplicial model version [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF][START_REF] Van Ditmarsch | Knowledge and simplicial complexes[END_REF].

Action models

A simplicial action model ⟨T, χ, pre⟩ consists of a chromatic graph ⟨T, χ⟩, where the edges F(T ) represent communicative actions, and pre assigns to each edge X ∈ F(T ) a precondition formula pre(X) in L K .

Given two chromatic graphs C and T of dimension n, the Cartesian product C ×T is the following chromatic graph. Its vertices are of the form (u, v) with u ∈ V(C) and v ∈ V(T ) such that χ(u) = χ(v); the color of (u, v) is χ((u, v)) = χ(u) = χ(v). Its edges are of the form X ×Y = {(u 0 , v 0 ), . . . , (u k , v k )} where X = {u 0 , . . . , u k } ∈ C, Y = {v 0 , . . . , v k } ∈ T and χ(u i ) = χ(v i ).

Let M = ⟨C, χ, ℓ⟩ be a simplicial model, and A = ⟨T, χ, pre⟩ be a simplicial action model. The product update simplicial model

M [A] = ⟨C[A], χ[A], ℓ[A]
⟩ is a simplicial model whose underlying graph is a sub-graph of the Cartesian product C × T , induced by all the edges of the form X × Y such that pre(Y ) holds in X, i.e., M, X |= pre(Y ). The valuation ℓ : V(C[A]) → P(AP ) at a pair (u, v) is just as it was at u: ℓ[A]((u, v)) = ℓ(u).

An action model for dynamic networks

Distributed computing model

For two processes, the model G where approximate agreement is solvable, but not consensus, is unique.

G w G g G gw
It consists of the three digraphs G g , G w , G gw on two vertices, g, w, one with the antiparallel arrows, and the other two with an arrow from one vertex to the other, meaning that either both messages arrive, or only one. It is easy to see that in any submodel of G consensus can be solved (if it has at least one arrow). The model G is equivalent to several message passing and shared memory models that have been previously considered, see e.g. [START_REF] Attiya | The combinatorial structure of wait-free solvable tasks[END_REF][START_REF] Herlihy | Distributed Computing Through Combinatorial Topology[END_REF]. Each agent has some input value, and they communicate r rounds. In each round an agent sends a message to the other agent, and the message arrives by the end of the round, or not at all. Which messages arrive depends on which graph from G is selected (it is convenient to assume that each vertex of a graph of G has a loop; an agent learns its own messages. But we do not draw the loops in the figures). In every round, any of the three graphs in G can be selected.

Input model

An input model, I, represents all the possible input values. To illustrate the ideas, we use the input simplicial model of Section 3.1 where two agents A = {g, w} each has an input value from the set {0, 1}.

Actions

An action8 t ∈ T is given by b 0 , b 1 , c, where c is a sequence of r digraphs from G and b 0 , b 1 ∈ {0, 1} are binary values, representing the inputs of both processes. Such an action is written c b0b1 . The precondition pre(c b0b1 ) of this action is a formula expressing the fact that the inputs of the agents g, w are respectively b 0 , b 1 . Formally, if input x a denotes the atomic proposition expressing that agent a has input value x, then pre(c b0b1 ) = input b0 g ∧ input b1 w .

Distributed algorithm

An action t = c b0b1 includes all the information about the inputs and message deliveries of an execution, but not about the actual algorithm being executed by the agents. Namely, an algorithm specifies the contents of messages sent in each round, and the state transition of each agent. Each agent starts in an initial state, determined by its input value. At the end of a round, each agent changes to a new state following a deterministic transition function, based on its current state, and on the message received. The new state of the agent determines the message sent in the new round. The local state of agent a at the end of execution t is denoted view a (t).

For example, consider the execution c = G w , G gw , G g , meaning that in the first round only w receives a message; in the second round both processes receive a message; and in the last round only g receives a message. Picking b 0 = 0 and b 1 = 1 gives us the action c 01 , where g starts with value 0 and w starts with value 1. This is independent of the algorithm. In the algorithm of Figure 1, local states are simply the value of variable ℓ, so this action leads to the local states view g (c 01 ) = 4/27 and view w (c 01 ) = 3/27. In the one-bit algorithm of Figure 2, the local states are the views, so this action leads to the local states view g (c 01 ) = ⊥01 and view w (c 01 ) = 00⊥.

Indistinguishability

The indistinguishability relation t ∼ a t ′ is a function of both the actions t, t ′ and the algorithm. It is defined as t ∼ a t ′ iff view a (t) = view a (t ′ ).

Action model

For a given algorithm, we can reformulate the indistinguishability relation as a simplicial action model ⟨T, χ, pre⟩ where ⟨T, χ⟩ is a chromatic graph whose vertices are V(T ) = {⟨a, view a (c b0b1 )⟩ | a ∈ A, c b0b1 ∈ T } and whose edges are of the form, for each action c b0b1 ∈ T , X = {⟨b, view w (c b0b1 )⟩, ⟨g, view g (c b0b1 )⟩}. The precondition of such a facet is pre(X) = input b0 w ∧ input b1 g . We write A r for the simplicial action model of the r-round averaging algorithm of Figure 1, and OB r for the action model of the r-round one-bit algorithm of Figure 2. We also write DN r for a generic r-round dynamic network action model over an unspecified algorithm. The action models OB 1 and OB 2 are depicted in Figure 4 (as we mention below, DN r and I[DN r ] are always isomorphic). For r = 1, each of the four input edges has been subdivided into 3 "smaller" edges: one for each possible graph of G. The colors white, grey, of the vertices correspond respectively to agents w, g. The view of each vertex is written next to it. The precondition of the three top edges is true exactly in the top edge of the input model I on the left, and similarly for the other sides of the square. For r = 2, the situation is similar except that each edge of I is subdivided into 3 2 = 9 edges.

Perhaps surprisingly, the product update models I[A r ] and I[OB r ] are isomorphic, meaning that both algorithms from Figures 1 and2 acquire the same knowledge. They both subdivide each edge of the input complex into exactly 3 r edges, thus they are both also isomorphic to the well-known full-information protocol complex.

Theorem 4.1 Let I be the binary input model for two processes. For any number of rounds r, the product update models I[A r ] and I[OB r ] are isomorphic.

Properties about the action model

One last thing to notice about this construction is that, when we compute the product update model I[DN r ], we obtain a simplicial model whose underlying graph is the same as the one of DN r . So, starting from the input model I, the effect of applying DN r is to subdivide each edge of the input (the same thing happens for any other input model). Remarkably, the topology of the input graph is preserved. And in fact, there is a rate of subdivision speed, determined by the constant 1/3. These are the two basic properties that we will need in the analysis of approximate agreement (known in several specific contexts e.g. [START_REF] Delporte-Gallet | Communication complexity of wait-free computability in dynamic networks[END_REF][START_REF] Herlihy | Distributed Computing Through Combinatorial Topology[END_REF]). Remarkably, they hold for any algorithm, not only full information algorithms. Theorem 4.2 For any algorithm for two agents, if the input model I is connected, then the product update model I[DN r ] is connected. Furthermore, each edge is subdivided into at most 3 r edges.

We have seen that the two properties mentioned in this theorem hold for a full information algorithm. It is not hard to see that they hold for any algorithm, since the indistinguishability relation of a non-full information algorithm is a coarsening of the full information indistinguishability algorithm.

Tasks

The action models of the previous section, A r and OB r , describe how the knowledge of the processes evolve when we execute a specific algorithm. Both of those algorithms are solving the same distributed task, namely, approximate agreement. In this section, we introduce the notion of task, which is an abstract specification of the goal that we are trying to solve, independently of the particular algorithm used to solve it. Informally, a task specifies for each possible input configuration, what are the possible output values that the agents may decide. This is once again formalized using the notion of action model; however, in the case of tasks, the actions do not correspond to communicative events, but to decisions taken by the processes. Tasks have been studied since early on in distributed computability [START_REF] Biran | A Combinatorial Characterization of the Distributed 1-Solvable Tasks[END_REF]. The DEL semantics that we use here was first introduced in [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF].

Consider a simplicial model I = ⟨I, χ, ℓ⟩ called the initial simplicial model. Each edge of I, with its labeling ℓ, represents a possible initial configuration. We fix I, the binary inputs model of Section 3.1.

A task for I is a simplicial action model T = ⟨T, χ, pre⟩ for agents A, where each edge is of the form X = {⟨w, d w ⟩, ⟨g, d g ⟩}. The values d w , d g are taken from an arbitrary domain of output values. Each such X has a precondition that is true in one or more facets of I, interpreted as "if the input configuration is a facet in which pre(X) holds, and every agent a ∈ A decides on the value d a , then this is a valid execution".

Semantics of task solvability

Given the simplicial input model I and a communication model A such as DN r , we get the simplicial protocol model I[A], that represents the knowledge gained by the agents after executing A. To solve a task T , each agent, based on its own knowledge, should produce an output value, such that the edge with the output values corresponds to an edge of T , respecting the preconditions of the task.

The following gives a formal epistemic logic semantics to task solvability. Recall that a morphism δ of simplicial models is a chromatic simplicial map that preserves the labeling: ℓ ′ (δ(v)) = ℓ(v). Also recall that the product update model I[A] is a sub-graph of the Cartesian product I × A, whose vertices are of the form (i, ac) with i a vertex of I and ac a vertex of A. We write π I for the first projection on I, which is a morphism of simplicial models. The justification for this definition is the following. An edge X in I[A] corresponds to a pair (i, ac), where i is an edge of I representing input value assignments to the agents, and ac is an edge of A codifying the communication exchanges that took place. The morphism δ takes X to an edge δ(X) = (i, dec) of I[T ], where dec is the edge of T defining the set of decision values that the agents will choose in X. Moreover, pre(dec) holds in i, meaning that dec corresponds to valid decision values for input i. The commutativity of the diagram expresses the fact that both X and δ(X) correspond to the same input assignment i. Now consider a single vertex v ∈ X with χ(v) = a ∈ A. Then, agent a decides its value solely according to its knowledge in Here again topology appears: a simplicial map is the discrete equivalent of a continuous map, thus connectivity is preserved by simplicial maps (Lemma 3.3); hence the topological nature of task solvability.

I[A]: if another edge X ′ contains v, then δ(v) ∈ δ(X) ∩ δ(X ′ ),

Approximate agreement

Recall that in the approximate agreement problem agents are required to decide on values which are close to each other. We have seen in Section 2 that no matter how close to each other one requires the agents to decide, this task is solvable in the DN r model, taking a sufficiently large r. Many versions of this task have been considered e.g. [START_REF] Charron-Bost | Fast, Robust, Quantizable Approximate Consensus[END_REF]. We present here the version of [START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF], for two agents g and w, which is at the core of previously considered situations.

The input complex is the binary input complex for two agents of Section 3.1: so, every possible combination of 0 and 1 can be assigned to the two agents. Their goal will be to output real values in the interval [0, 1], such that: (i) if their input is the same, they both decide the same output, and (ii) if their input is different, they both decide on values d g and d w such that |d g -d w | ≤ ε, for some fixed parameter ε ∈ [0, 1]. A discrete version of this task is N -approximate agreement, where the output values are only allowed to be of the form k/N for 0 ≤ k ≤ N . The two decision values should be within 1/N of each other: |d g -d w | ≤ 1/N . And to simplify the presentation, let us assume N is odd, without loss of generality, in the sense that to solve ε-agreement, one has to select N large enough, so that 1/N ≤ ε.

Let I = ⟨I, χ, ℓ⟩ be the input simplicial model for two agents with binary inputs, and T = ⟨T, χ, pre⟩ be the following action model. The set of vertices of T is V

(T ) = {(a, k/N ) | a ∈ A and 0 ≤ k ≤ N }. The facets of T are edges X k,k ′ = {(g, k/N ), (w, k ′ /N )} with |k -k ′ | ≤ 1.
The color of a vertex is χ(a, k/N ) = a. The precondition pre(X 0,0 ) is true in the worlds 00, 01 and 10 of I; the precondition pre(X N,N ) is true in the worlds 11, 01 and 10; and all the other preconditions pre(X k,k ′ ) are true in the worlds 01 and 10. In the figure below are the input model I (left) and the action model T (right), for N = 5. Proof We choose the worlds in the "middle" of the model I[T ], as shown in the picture above. More formally, recall that the vertices of I[T ] are defined as tuples (a, i, d) where a is an agent, i its input value and d its decision value. The world X is defined as the edge {(g, 1, ⌊N/2⌋/N ), (w, 0, (⌊N/2⌋ + 1)/N )}, and Y = {(w, 1, ⌊N/2⌋/N ), (g, 0, (⌊N/2⌋ + 1)/N )}. Checking that the formula is satisfied in these worlds simply consists in computing the length of the shortest path to one of the 00 or 11 edges on the sides (Lemma 3.2).

2

Solvability of approximate agreement

We now study the solvability of approximate agreement in the r-round dynamic graph model DN r . Recall that each round subdivides each edge into three edges (Theorem 4. Proof After r rounds each of the four edges of the input model I has been subdivided into 3 r edges. Thus, every world is at a distance at most k -1 from the nearest world with inputs 00 or 11. 2

Putting the two lemmas together, we get the following result: Theorem 6.3 The N -approximate agreement task is not solvable in the r-round model I[DN r ], when N ≥ 3 r + 1.

Proof Assume for contradiction that the task is solvable. Then, we would have a map δ :

I[DN r ] → I[T ].
Our goal is to find a contradiction using Lemma 3.4. To achieve this, we should find a formula φ and a world Z of I[DN r ], such that φ is false in Z but true in δ(Z). We choose the formula φ := E k φ 01 for k = ⌈N/2⌉. Since N ≥ 3 r + 1 implies ⌈N/2⌉ ≥ ⌈3 r /2⌉, we know by Lemma 6.2 that this formula is false in every world Z of I[DN r ]. All that remains to do is prove that there exists a world of I[T ], which is in the image of δ, and where the formula φ is true.

Since I[DN r ] is connected, its image δ(I[DN r ]) is connected (by Lemma 3.3). Moreover, the world 00 and the world 11 of I[T ] must both be in the image of δ, because of the commutative diagram of Definition 5.1. By connectedness, at least one of the middle worlds X, Y of Lemma 6.1 must belong to the image of δ. By Lemma 6.1, this world satisfies φ, which concludes the proof.

2

Conversely, we have seen in Section 2 that N -approximate agreement is solvable in r rounds whenever N = 3 r . The proof of the above theorem sheds light on the required knowledge to solve approximate agreement: while consensus is about reaching common knowledge, approximate agreement is about reaching some finite level of nested knowledge.

Conclusions

We have considered a basic notion of approximate agreement, and a computational model for two agents, where the task is solvable for any desired level of precision. We first presented two algorithms, which solve the task for a given precision level, using the same number of communication rounds. With this concrete setting in mind, we have proceeded to give a formal semantics based on dynamic epistemic logic, both to our computational model, and to the approximate agreement task. We derived a lower bound result showing that these algorithms are optimal in the number of rounds. The lower bound is due to the impossibility of the agents gaining global knowledge about their inputs faster, and a consequence of the connectivity of the computational model's epistemic states. Although much of these results were previously known, we have made an effort to distil the essential ingredients of several previous papers, and combined them into a unified, self-contained way, providing thus a more elementary introduction to the area, based only of graph theory, instead of higher dimensional simplicial complexes.

Averaging N -approximate agreement (ℓ) 1 round r from 1 to k do 2 send(ℓ) 3 m 4 if m ̸ = ⊥ then 5 ℓFigure 1 .

 123451 Figure 1. Each agent p ∈ {g, w} runs this code. The input is ℓ ∈ {0, 1}. A message m = ⊥ indicates that no message was received from the other process.

3 round r from 1 to k do 4 send 5 mFigure 2 .

 3452 Figure 2. Approximate agreement with 1-bit messages. Each agent runs this code with input ℓ ∈ {0, 1}; the output function δ(ℓ, view) is detailed in Figure 3.

  Local computation of δ(ℓ, view) 1 other = ⊥ /* input value of the other process */ 2 d = 0 /* distance to move towards the other process */

keep the same d */ 7 if other = ⊥ then 8 other 9 if other = ℓ then return ℓ 10 mFigure 3 .

 789103 Figure 3. Decision function δ(ℓ, view) used at the end of the one-bit algorithm of Figure 2.

Figure 4 .

 4 Figure 4. Protocol complex of the 1-bit protocol from Figure2, for 1 and 2 rounds of computation. Process names are represented by colors; inputs ℓ ∈ {0, 1} are written inside the vertices; and views are written next to the vertices. For the two-round protocol, the value of δ(ℓ, view) at each node is given by projecting vertically downwards. These pictures represent protocol complexes in the usual sense of distributed computing, but also in the sense of the product-update model construction defined later.

  The definition below mimics the usual semantics of formulas in Kripke models, reformulated here in terms of simplicial models: Definition 3.1 We define the satisfaction relation M, X |= φ determining when a formula φ is true in some epistemic state (M, X). Let M = ⟨C, χ, ℓ⟩ be a simplicial model, X ∈ F(C) an edge of C and φ

Lemma 3 . 2

 32 For a simplicial model M , edge X, and any φ ∈ L K , we have that M, X |= E k φ, iff M, Y |= Eφ for every Y ∈ N k (X).

Lemma 3 . 3

 33 Let f : C → D be a simplicial map. If C ′ is a connected subgraph of C then f (C ′ ) is a connected subgraph of D.

Definition 5 . 1

 51 A task T is solvable using the algorithm A if there exists a morphism δ :I[A] → I[T ] such that π I • δ = π I , i.e., the diagram of simplicial complexes below commutes.

  meaning that a has to decide on the same value in both edges. Two observations about the diagram. First, by Lemma 3.4, we know that the knowledge of each agent can only decrease (or stay constant) along the δ arrow. So, any (positive) formula which is known in I[T ] should already be known in I[A]. In other words, the goal of the agents is to improve knowledge through communication, by going from I to I[A], in order to match the knowledge required by I[T ]. Second, the possibility of solving a task depends on the existence of a certain simplicial map from the graph of I[A] to the graph of I[T ]. We have already seen the appearance of a topological property in Theorem 4.2.

6

  simplicial model I[T ] is depicted in the next figure, for N = 5. The numbers depicted in the nodes are the atomic propositions describing the input values from I. The decision values (of the form k/5) are implicit, the first column of nodes corresponds to the decision value 0, the second column is decision value 1/5, and so on. Approximate agreement solvabilityThe world X on the figure is the one with the most knowledge in the following sense. We write φ 01 for the formula expressing that the two inputs are different. Recall (Section 3.1) that Eφ = K g φ ∧ K w φ for the group knowledge of φ among the agents {g, w}. Then, we have I[T ], X |= E 3 φ 01 . On the other hand, the world Y has less knowledge: we haveI[T ], Y |= E 2 φ 01 , but I[T ], Y ̸ |= E 3 φ 01 .Lemma 6.1 In the simplicial model I[T ] for the N -approximate agreement task, there are two worlds X, Y which satisfy the formula E k φ 01 , for k = ⌈N/2⌉.

  2). The picture below shows the input model I, the model I[DN ] after one round, and the model I[DN 2 ] after two rounds.

Lemma 6 . 2

 62 In the r-round model I[DN r ], there is no world X such that I[DN r ], X |= E k φ 01 , for k = ⌈3 r /2⌉.

Fixing the constant in ε to 1/3 k facilitates the presentation, avoiding collateral details, and can be done without loss of generality.

Our non-traditional notations, C for a graph and F (C) for the set of edges of C, come from the general setting for n agents where graphs are replaced by simplicial complexes, and edges are replaced by facets. This is consistent with the notations of previous papers[START_REF] Goubault | A simplicial complex model for dynamic epistemic logic to study distributed task computability[END_REF][START_REF] Van Ditmarsch | Knowledge and simplicial complexes[END_REF].

A simplicial map is usually called a homomorphism in graph theory.

For the moment, an action is simply an element of the set T ; we will later define a simplicial action model ⟨T, χ, pre⟩ whose set of edges is in bijection with T .

Partially supported by UNAM-PAPIIT grant IN106520.

A Approximate agreement algorithms

Here we provide additional details about the correctness of the algorithms described in Section 2. Recall that N = 3 k , agents start with values in {0, 1}, and they have to decide values at most 1/N apart, unless their inputs are equal, in which case their outputs should be equal to their inputs. First we show that the algorithm in Figure 1 solves N -approximate agreement in k rounds.

Theorem A. [START_REF] Attiya | The combinatorial structure of wait-free solvable tasks[END_REF] The algorithm Averaging Approximate Agreement is correct.

Proof Let me p be the initial value of agent p. For k = 1, suppose w.l.o.g. that the message from g arrives, then the final value of agent w is d w = (me w + 2me g )/3. If both messages arrives then d w = (me g +2me w )/3 so |d w -d g | = |(2me g -me g +me w -2me w )/3| = |(me g -me w )/3| ≤ 1/3. If just one message arrives then d g = me g , so |d w -d g | = |(me w + 2me g -3me g )/3| = |(me w -me g )/3| ≤ 1/3. Let d k p be the final value of agent p after k rounds, and suppose that |d k w -d k g | ≤ 1/3 k for some k, then, if message from g arrives in round k + 1 (w.l.o.g.), d k+1 w = (d k w + 2d k g )/3. The rest of the proof is analogous to induction base, analysing both cases whether message from w gets lost or not.

2

To prove the correctness of the One-Bit Messages algorithm of Figure 2, we rely on Theorem 4.1. So let us prove it first: Theorem 4.1. Let I be the binary input model for two processes. For any number of rounds r, the product update models I[A r ] and I[OB r ] are isomorphic.

Proof Recall that I[A r ] is the product update model for r-rounds of the Averaging algorithm in Figure 1, and I[OB r ] is the product update model for r-rounds of the One-Bit algorithm of Figure 2. It is well known that I[A r ] is isomorphic to the full-information protocol complex, which subdivides each edge of the input into 3 k edges, preserving the topology. Thus, we show that I[OB r ] does the same thing.

We proceed by induction on r. For the one-round algorithm, I[OB 1 ] is depicted in Figure 4, and indeed each edge of the input complex has been subdivided into three.

Let us now assume that I[OB r ] is indeed isomorphic to I[A r ], i.e., that each edge of I has been subdivided in 3 k small edges. Consider an arbitrary vertex of I[OB r ] (a grey vertex w.l.o.g.), of the form ⟨g, view g (t)⟩ for some action t = c b0,b1 . Let us write v = view g (t) for the view of this vertex. Moreover, let us assume that the two neighbors of this vertex, with views u and w, are about to send two distinct messages, 0 and 1 respectively. (One can check that this property is true everywhere in I[OB 1 ]: every vertex has neighbors that will send different messages in the next round.)

After one more round of the one-bit algorithm, we obtain six edges in I[OB r+1 ] as depicted below (we assume w.l.o.g. that the grey vertex sends the message "0"; the same picture can be drawn with a 1 instead). Moreover, we can check that the five vertices in the middle at round r + 1 still have the inductive property that both neighbors will send a different message at the next round. This local reasoning can be done around each node of I[OB r ]. Thus, every edge is subdivided into 3,which concludes our proof. 2

Now we can show that the algorithm in Figure 2 solves N -approximate agreement in k rounds.

Theorem A.2 The algorithm One-Bit Messages Approximate Agreement is correct.

and we have shown in Theorem A.1 that I[A r ] solves 3 rapproximate agreement, we know that the One-Bit algorithm can solve approximate agreement as long as we choose the right decision function δ. This is precisely what the algorithm from Figure 3 does: δ(ℓ, view) simulates the Average Approximate Agreement algorithm via the isomorphism exhibited in Theorem 4.1. A detailed proof of correctness can be found in [START_REF] Delporte-Gallet | Communication complexity of wait-free computability in dynamic networks[END_REF]. 2

B Simplicial models and Epistemic Logic

Here we include generalized notions for any number of agents from section 3.

Generalizing graphs to complexes

Given a set V , a simplicial complex C is a family of non-empty finite subsets of V such that for all X ∈ C, Y ⊆ X implies Y ∈ C. We say Y is a face of X. Elements of V (identified with singletons) are called vertices. Elements of C are simplexes, and those which are maximal w.r.t. inclusion are facets. The set of vertices of C is noted V(C), and the set of facets F(C). The dimension of a simplex X ∈ C is |X| -1, and a simplex of dimension n is called an n-simplex. A simplicial complex C is pure if all its facets are of the same dimension, n. In this case, we say C is of dimension n. A graph without isolated vertices is a pure sumplicial complex of dimension 1. Given the set A of agents (that we will represent as colors), a chromatic simplicial complex ⟨C, χ⟩ consists of a simplicial complex C and a coloring map χ : V(C) → A, such that for all X ∈ C, all the vertices of X have distinct colors.

Simplicial maps

Let C and D be two simplicial complexes. A simplicial map f : C → D maps the vertices of C to vertices of D, such that if X is a simplex of C, f (X) is a simplex of D. A chromatic simplicial map between two chromatic simplicial complexes is a simplicial map that preserves colors. Let S A be the category of pure chromatic simplicial complexes on A, with chromatic simplicial maps for morphisms.

Equivalence with Epistemic Logic

A Kripke frame M = ⟨S, ∼⟩ over a set A of agents consists of a set of states S and a family of equivalence relations on S, written ∼ a for every a ∈ A. Two states u, v ∈ S such that u ∼ a v are said to be indistinguishable by a. A Kripke frame is proper if any two states can be distinguished by at least one agent. Notice that being proper means that the intersection of all equivalence relations ∼ a is the identity; this may reveal interesting parallels with distributed knowledge (a formula that is true in all states in the intersection relation), see e.g. [START_REF] Fagin | Reasoning About Knowledge[END_REF]. Let M = ⟨S, ∼⟩ and N = ⟨T, ∼ ′ ⟩ be two Kripke frames. A morphism from M to N is a function f from S to T such that for all u, v ∈ S, for all a ∈ A, u ∼ a v implies f (u) ∼ ′ a f (v). We write K A for the category of proper Kripke frames, with morphisms of Kripke frames as arrows.

The following theorem states that we can canonically associate a proper Kripke frame with a pure chromatic simplicial complex, and vice versa. In fact, this correspondence extends to morphisms, and thus we have an equivalence of categories, meaning that the two structures contain the same information.

Theorem B.1 ([20]

) S A and K A are equivalent categories.

Example B.2 [[20]

] The picture below shows a Kripke frame (left) and its associated chromatic simplicial complex (right). The three agents, named b, g, w, are represented as colors black, grey and white on the vertices of the simplicial complex. The three worlds of the Kripke frame correspond to the three triangles (i.e., 2-dimensional simplexes) of the simplicial complex. The two worlds indistinguishable by agent b, are glued along their black vertex; the two worlds indistinguishable by agents g and w are glued along the grey-and-white edge.

g, w b

F G

Simplicial models

Let V be some countable set of values, A be a finite set of agents and AP = {p a,x |a ∈ A, x ∈ V} be the set of atomic propositions. Intuitively, p a,x is true if agent a holds the value x. We write AP a for the atomic propositions concerning agent a. A simplicial model M = ⟨C, χ, ℓ⟩ consists of a chromatic simplicial complex ⟨C, χ⟩, and a labeling ℓ : V(C) → P(AP ) that associates with each vertex v ∈ V(C) a set of atomic propositions concerning agent χ(v), i.e., such that ℓ(v) ⊆ AP χ(v) . Given a simplex X ∈ C, we write ℓ(X) = n i=0 ℓ(v i ). A morphism of simplicial models f : M → M ′ is a chromatic simplicial map that preserves the labeling: ℓ ′ (f (v)) = ℓ(v). We denote by SM A,AP the category of simplicial models over the set of agents A and atomic propositions AP .

Equivalence with epistemic logic

A Kripke model M = ⟨S, ∼, L⟩ consists of a Kripke frame ⟨S, ∼⟩ and a function L : S → P(AP ). Intuitively, L(s) is the set of atomic propositions that are true in the state s. A Kripke model is proper if the underlying Kripke frame is proper. A Kripke model is local if for every agent a ∈ A, s ∼ a s ′ implies L(s) ∩ AP a = L(s ′ ) ∩ AP a , i.e., an agent always knows its own values. Let M = ⟨S, ∼, L⟩ and M ′ = ⟨S ′ , ∼ ′ , L ′ ⟩ be two Kripke models on the same set AP . A morphism of Kripke models f : M → M ′ is a morphism of the underlying Kripke frames such that L ′ (f (s)) = L(s) for every state s in S. We write KM A,AP for the category of local proper Kripke models. We can now extend Theorem B.1 to an equivalence between simplicial models and Kripke models.

Theorem B.4 ([20]

) SM A,AP and KM A,AP are equivalent categories.

C Knowledge in simplicial models

Here we include additional details about group knowledge and knowledge gain from subsection 3.1.

Lemma 3.2 For a simplicial model M and edge X, we have that

Lemma 3.4 Consider simplicial models M = ⟨C, χ, ℓ⟩ and M ′ = ⟨C ′ , χ ′ , ℓ ′ ⟩, and a morphism f : M → M ′ . Let X ∈ F(C) be an edge of M , a an agent, and φ ∈ L CK a positive formula, i.e. which does not contain negations except, possibly, in front of atomic propositions. Then,

If φ = ψ ∧ θ for some formulas ψ and θ as depicted above, then M ′ , f (X) |= ψ and M ′ , f (X) |= θ. Therefore M, X |= ψ ∧ θ.

If φ = K a (ψ) with ψ being a formula like depicted above, let

Finally, every possitive formula can be seen as combinations of formulas like depicted above but linked with ∧. Hence, if φ = i∈N ψ i then for all i ∈ N, M ′ , f (X) |= ψ i so that M, X |= ψ i . Finally, M, X |= i∈N ψ i . 2

D Dynamic epistemic logic

Here we present the classic version of DEL.

An action model is a structure ⟨T, ∼, pre⟩,where T is a domain of action points, such that for each a ∈ A, ∼ a is an equivalence relation on T , and pre : T → L K is a function that assigns a precondition formula pre(t) to each t ∈ T . Let M = ⟨S, ∼, L⟩ be a Kripke model and A = ⟨T, ∼, pre⟩ be an action model. The product update model is M [A] = ⟨S[A], ∼ [A] , L[A]⟩, where each world of S[A] is a pair (s, t) with s ∈ S, t ∈ T such that pre(t) holds in s. Then, (s, t) ∼

Equivalence with simplicial action models

Recall from Theorem B.4 the two functors F and G that define an equivalence of categories between simplicial models and Kripke models. We have a similar correspondence between action models and simplicial action models. On the underlying Kripke frame and simplicial complex they are the same as before; and the precondition of an action point is just copied to the corresponding facet. The following proposition says that the "classic" product update agrees with the "fully simplicial" one.

E Dynamic networks and Tasks

Here we include additional details about Section 4 and 5.

Theorem 4.2 For any algorithm for two agents, the product update model M [DN r ] is a graph which is connected (assuming the input model is connected). Furthermore, each edge is subdivided into at most 3 r edges.

Proof Since underlying simplicial complex of M [DN r ] is the same as the one of G(DN r ), we just need to prove that G(DN r ) is connected.


Viviana Grasselli

On the definition of zero resonances for the Schr ödinger operator with optimal scaling potentials

We consider the Schrödinger operator -∆ + V on R n with potential in the Lorentz space L n/2,1 and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L 2 and that verify the equation (-∆ + V )ψ = 0, namely ker Ḣ1 (-∆ + V ). We prove that a function in this set is either in a weak Lebesgue space, L n n-2 ,∞ , or in L 2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V , furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to L 1,∞ or L 1 . We study dimensions n ≥ 3 and in dimension three we generalize a result proved by Beceanu.

Introduction

We consider the Schrödinger operator -∆ + V on R n with n ≥ 3. The evolution of solutions of the time-dependent Schrödinger equation is influenced by the the spectrum of the operator -∆ + V , in particular by the bottom of it and the nature of the zero frequency: whether it is in the spectrum and if it is an eigenvalue or a resonance. Roughly speaking, zero is a resonance if there is a solution to the equation (-∆ + V )ψ = 0 which does not decay fast enough to be in L 2 and is usually assumed to belong in some kind of weighted space. This solution is a resonant state.

Local energy decay for solutions of dispersive equations, for example, is tightly linked to the spectral properties of the Schrödinger operator. First results in this direction were proved in three dimensional weighted L 2 spaces in [START_REF] Rauch | Local decay of scattering solutions to Schrödinger's equation[END_REF], [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], in higher dimension in [START_REF]Spectral properties of schrödinger operators and time-decay of the wave functions. results in l2(r4)[END_REF], [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions results in L 2 (R m ), m ≥ 5[END_REF] and for more general elliptic operators in [START_REF] Murata | Asymptotic expansions in time for solutions of schrödinger-type equations[END_REF]. For an extensive overview on the subject the reader may refer to the survey article [START_REF] Schlag | Dispersive estimates for Schrödinger operators[END_REF]. A way to approach these estimates is through expansions of the resolvent around zero energy from which one can obtain an expansion of the evolution operator via Fourier transform. This is the strategy followed in [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], where the authors compute resolvent expansions in dimension three distinguishing the cases whether there are eigenvalues or resonance obstructions at zero. A unified approach for resolvent expansions applicable to all dimensions was found later in [START_REF] Jensen | A unified approach to resolvent expansions at thresholds[END_REF].

In this brief note we recover properties of zero resonances and eigenfunctions described in the seminal paper [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF] under weaker assumptions on V and with optimal behavior of such states as |x| tends to infinity. On top of this, we obtain results of the same flavor in dimension four, covering all cases where zero resonances exist and showing that for n ≥ 5 our solutions always belong to L 2 . We do not exhibit any exotic or new behavior but rather give what we consider to be a fairly simple proof of classical results for a more general class of potentials (essentially the optimal one). Our result in dimension three also has the advantage of generalizing a characterization found by Beceanu in [START_REF] Beceanu | Dispersive estimates in R 3 with threshold eigenstates and resonances[END_REF] as we will see more in detail later on.

In much of the literature, resonances are usually found to belong to weighted Sobolev or Lebesgue spaces where there is no scaling invariance. For example, in [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF] the authors define a resonance as a solution to (-∆ + V )ψ = 0 where the operator -∆ + V is meant to be extended to the weighted Sobolev space H 1,-s with weight ⟨x⟩ -s and a suitable s > 1/2. A similar definition can also be found in the leading paper on dispersive L 1 to L ∞ estimates by Journé, Soffer and Sogge ( [START_REF] Journé | Decay estimates for schrödinger operators[END_REF]). Therein a zero resonant state is taken in a weighted L 2 space, which is the same definition used in [START_REF] Yajima | Dispersive estimates for schrödinger equations with threshold resonance and eigenvalue[END_REF] in which the author studies L p to L q dispersive estimates. Similarly, in [START_REF] Rodnianski | Time decay for solutions of schrödinger equations with rough and timedependent potentials[END_REF], [START_REF] Goldberg | Dispersive bounds for the three-dimensional schrödinger equation with almost critical potentials[END_REF], [START_REF]Dispersive estimates for the three-dimensional schrödinger equation with rough potentials[END_REF] and [START_REF] Goldberg | Dispersive estimates for schrödinger operators in dimensions one and three[END_REF] the authors define resonances as functions that belong to the intersection of weighted L 2 spaces given by ∩ s>1/2 L 2,-s with weight ⟨x⟩ -s . Even in more recent papers ( [START_REF] Aafarani | Large time behavior of solutions to schrödinger equation with complex-valued potential[END_REF] , [START_REF] Soffer | l p boundedness of the scattering wave operators of schrödinger dynamics -part 2[END_REF]) with much stricter assumptions on the potentials than the one considered herein, the framework used to define resonances is the one of weighted L 2 spaces. Along the same line, one can consider the definition in [START_REF] Marzuola | Strichartz estimates and local smoothing estimates for asymptotically flat schrödinger equations[END_REF]. For a generic time dependent second order elliptic operator a zero resonant state (Definition 1.8) is taken in a rather complicated local smoothing space that for the case of dimension three and higher is contained in the weighted L 2 space with weight ⟨x⟩ -1 . As an example for the non euclidean case, in [START_REF] Wang | Asymptotic expansion in time of the schrödinger group on conical manifolds[END_REF] on a conical manifold we have a perturbation of a model operator both by a potential and a metric and resonances are defined as functions in weighted Sobolev spaces. In such work the author, as in [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], is mainly interested in the influence of zero resonant states on the singularities of the resolvent.

The fact that we can consider a zero resonant state as a function in a weak Lebesgue space is a very useful feature and it comes at a fairly small cost. Indeed, in various applications (like [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], [START_REF] Goldberg | Dispersive estimates for schrödinger operators in dimensions one and three[END_REF], [START_REF] Erdogan | Dispersive estimates for schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I[END_REF], [START_REF] Aafarani | Large time behavior of solutions to schrödinger equation with complex-valued potential[END_REF] or [START_REF] Soffer | l p boundedness of the scattering wave operators of schrödinger dynamics -part 2[END_REF]) the potentials must have a prescribed decay at infinity, that is |V (x)| ≲ ⟨x⟩ -β for some β > 2. Our assumption will be much less strict than requiring a specific pointwise decay and the results presented here only consider scaling invariant classes for the potentials. This can be a useful feature when working with nonlinear problems and we aim to further investigate properties on the bottom of the spectrum for Schrödinger operators with potentials that satisfy only scaling invariant assumptions.

More precisely, let L p,q (R n ) the Lorentz space defined on R n . It is defined as the space of functions such that the quasinorm denoted by ∥ • ∥ p,q is finite. We define the quasinorm via the distribution function

d f (t) = |{|f (x)| > t}| as ∥f ∥ p,q := p 1/q ˆ∞ 0 t q-1 (d f (t)) q/p dt 1/q for q < ∞ or ∥f ∥ p,∞ := sup t≥0 td f (t) 1/p
otherwise. As for the Lebesgue spaces, we identify two functions that are equal almost everywhere. (For some properties that we will use throughout the reader can refer to Appendix A). As mentioned before, for a prescribed decay the most general assumption is V (x) decaying like |x| -2-ε at infinity (notice that ⟨x⟩ -2-ε ∈ L n/2 (R n )). From now on, we will take V in a slightly smaller space than L n/2 (R n ), that is we will assume

V ∈ L n/2,1 (R n ) ⊂ L n/2 (R n ).
(1.1)

Assuming (1.1) is a little more restrictive than taking the potential in a Lebesgue space or in a Kato class, like in [START_REF] Bui | Dispersive and strichartz estimates for the three-dimensional wave equation with a scaling-critical class of potentials[END_REF] and [START_REF] Beceanu | Schrödinger dispersive estimates for a scaling-critical class of potentials[END_REF]. However, it is still less strict than what is assumed in [START_REF] Goldberg | Dispersive bounds for the three-dimensional schrödinger equation with almost critical potentials[END_REF], [START_REF]Dispersive estimates for the three-dimensional schrödinger equation with rough potentials[END_REF], or [START_REF]A dispersive bound for three-dimensional schrödinger operators with zero energy eigenvalues[END_REF], where L 3/2-ε ∩ L 3/2+ε ⊂ L 3/2,1 from Proposition A.1, and much more general than imposing a power-like decay, which we have seen is the custom in several cases. Moreover, our assumption is satisfied from the case V (x) = O(⟨x⟩ -2-ε ), which is a common assumption in the low frequencies asymptotic. Nonetheless, it does not seem to have been clearly observed that a larger class, such as L n/2,1 (R n ), gives the right framework to describe zero resonances and eigenfunctions more simply and optimally than in the usual approach [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF]. Indeed, the results presented in Theorems 1.1 and 1.2 may already be known to some experts, but we couldn't find any reference in the literature.

We will be interested in Ḣ1 (R n ) solutions of the equation

(-∆ + V )ψ = 0 (1.2)
with V ∈ L n/2,1 (R n ) and Ḣ1 (R n ) the homogeneous Sobolev space (see (2.1) for a definition). We will state our result for a function in Ḣ1 (R n ) which gives a simple and natural class of non necessarily L 2 functions where to look for solutions of (1.2). Moreover, -∆ is an isometry from Ḣ1 (R n ) to Ḣ-1 (R n ), a feature that we shall extensively use in Section 2. We also comment that by inverting -∆, equation (1.2) is equivalent to solving

ψ + (-∆) -1 V ψ = 0.
In his paper [START_REF] Beceanu | Dispersive estimates in R 3 with threshold eigenstates and resonances[END_REF] Beceanu solves this equation in L ∞ (R 3 ). As we just said, here we will rather solve it in Ḣ1 (R n ), or actually in L 2n n-2 ,∞ (R n ), in particular without seeking a priori bounded solutions.

The main results we will prove are the following.

Theorem 1.1. Let n ≥ 3, V ∈ L n/2,1 (R n ) and ψ ∈ Ḣ1 (R n ) a solution of the equation (-∆ + V )ψ = 0. i) If n ≥ 5 then ψ ∈ L 2
and hence all solutions are eigenfunctions.

ii)

If n = 3, 4 then lim |x|→∞ |x| n-2 ψ(x) = -c n ˆV (y)ψ(y)dy < ∞, with c n = n(n -2)|B(0, 1)|. Hence for n = 3 ψ ∈ L 3,∞ (R 3 )
and for n = 4 ψ ∈ L 2,∞ (R 4 ).

Remark 1. The conclusion in item i) can also be seen as a consequence of the decay given by item ii). However, we can give a direct proof of the fact ψ ∈ L 2 for n ≥ 5, without the need of computing the explicit limit of |x| n-2 ψ(x). Moreover, item i) is true under the weaker assumption

V ∈ L n/2 . Theorem 1.2. Let n ≥ 3, V ∈ L n/2,1 (R n ) and ψ ∈ Ḣ1 (R n ) a solution of the equation (-∆ + V )ψ = 0.
The following properties hold:

i) If ´V ψ = 0 then ψ = O( 1 |x| n-1
) near infinity and in particular ψ is in L 2 (R n ) and is a zero eigenfunction. Moreover, for n = 3, 4, ψ ∈ L 2 (R n ) if and only if ´V ψ = 0.

ii) If ´V ψ = ´yk V ψ = 0 for all k = 1, . . . , n then ψ = O( 1 |x| n ) near infinity. In particular, ψ is a zero eigenfunction and ψ ∈ L 1,∞ (R n ).

iii) If ´V ψ = ´yk V ψ = ´yk y l V ψ = 0 for all k, l = 1, . . . , n then ψ = O( 1 |x| n+1 ) near infinity. In particular, ψ is a zero eigenfunction and ψ ∈ L 1 (R n ).

Let us give a few comments about these results:

• as we mentioned earlier, Ḣ1 (R n ) is a pretty natural class of non L 2 functions to consider for this problem and in higher dimension n ≥ 5 we see from Theorem 1.1 that there are no solutions of (1.2) which do not belong to L 2 . This is due to the fact that Sobolev embeddings imply that the operator I + (-∆) -1 V maps L 2 into itself. Therefore, with respect to the existence of resonances Ḣ1 represents an optimal class of functions where to solve (1.2). However, from the point of view of regularity it is actually enough to require ψ ∈ L 2n n-2 ,∞ (R n ) and indeed we will perform all proofs for ψ in this larger class.

• In dimension three the simple assumption V ∈ L 3/2,1 (R 3 ) generalizes the result stated by Beceanu in [3] (Lemma 2.3) where the necessary and sufficient condition for ψ to be an eigenfunction is recovered only for potentials in

L 3/2,1 (R 3 ) ∩ L 1 (R 3 ).
Regarding the behavior at infinity, we give a more precise statement than just ψ ∈ ⟨x⟩ -1 L ∞ (R 3 ) since in Theorem 1.1 we give an explicit expression for the limit of |x|ψ for |x| → +∞.

• The conditions of orthogonality between V ψ and various other polynomial functions are not new ones. Indeed, they can also be found in [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], [START_REF]Spectral properties of schrödinger operators and time-decay of the wave functions. results in l2(r4)[END_REF] or [START_REF] Beceanu | Dispersive estimates in R 3 with threshold eigenstates and resonances[END_REF]. In particular in [START_REF] Jensen | Spectral properties of Schrödinger operators and time-decay of the wave functions[END_REF], [START_REF]Spectral properties of schrödinger operators and time-decay of the wave functions. results in l2(r4)[END_REF] it was already observed that the condition ´V ψ = 0 is the right one to discriminate between zero resonance or eigenvalue. However, in these works the authors consider conditions of pointwise decay on the potential that are more strict than our assumption. We also remark that the orthogonality conditions in item i) and ii) are actually necessary and sufficient for the decay of ψ, see Remark 6.

Notation. We will drop from the notations the dependence on the underlying space R n unless the situation requires it to make it explicit. We define the function a a

(x) = 1 |x| n-2 which belongs to L n n-2 ,∞ .
The main steps will be the following:

1. We will use the density of simple functions in L n/2,1 to decompose the potential V in

V = W + K
where W ∈ L n/2,1 is such that ∥W ∥ n/2,1 is arbitrarily small and K is a simple function, hence compactly supported and in any Lorentz space.

2. Thanks to the smallness of W in L n/2,1 we will be able to define and estimate G, the Green function of (-∆ + W ).

3. We will then solve in Ḣ1 (or more in general

L 2n n-2 ,∞ is sufficient) the equation (-∆ + W )ψ = -Kψ
where (-∆+W ) : Ḣ1 → Ḣ-1 is invertible and we will use the Green function computed in the previous step to write ψ(x) = -ˆG(x, y)K(y)ψ(y)dy.

In the rest of the paper ψ will be a

L 2n n-2 ,∞ solution of (-∆ + V )ψ = 0.
4. We will show that |x| n-2 ψ has finite limit at infinity and that such limit is given by -c n ´V ψ. Therefore, the value of such integral determines whether ψ is in L 2 or not. This is done in Section 3.

5.

With additional orthogonality conditions on V ψ we can repeat the argument of the previous step to prove that ψ has faster, but limited, decay until we can reach the L 1 space. This is done in Section 3.

2 Green function for a small potential Let Ḣs for s ∈ R the homogeneous Sobolev space of order s, defined as

Ḣs := {u ∈ S ′ : û ∈ L 1 loc , ∥u∥ Ḣs := ∥| • | s û∥ L 2 < ∞} (2.1)
where S ′ is the space of tempered distributions on R n . The following Sobolev embedding will be useful

Ḣ1 → L 2n n-2 , (2.2)
see Theorem 1.38 in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] for a proof.

Remark 2. We remark that the more general embedding

Ḣ1 → L 2n n-2 ,2 , (2.3)
is also true. It can be obtained with the same reasoning as in the proof of Theorem 1.38 in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] and using Young's inequality for Lorentz spaces in dimension n ≥ 3 ([5] Theorem 2.12):

∥| • | -α * f ∥ p,q ≤ C∥f ∥ r,q 1 + 1 p = 1 r + α n , where | • | -α ∈ L n/α,∞ .
We consider the operator

-∆ + V : Ḣ1 → Ḣ-1 ,
its selfadjoint realization on Ḣ1 is obtained via the quadratic form

q(u, u) = ⟨∇u, ∇u⟩ + ⟨V u, u⟩ = ˆ|∇u| 2 + V |u| 2 dx,
which is well defined on Ḣ1 . Indeed, given

V ∈ L n/2,1 ⊂ L n/2 and u ∈ Ḣ1 → L 2n n-2 we have V u ∈ L n/2 • L 2n n-2 ⊂ L 2n n+2 ⊂ Ḣ-1 , (2.4)
thanks to the dual of inclusion (2.2). We have just found that V maps Ḣ1 to Ḣ-1 , hence the scalar product ⟨V u, u⟩ is well defined when u ∈ Ḣ1 . From the continuous embeddings we also have

∥V ∥ Ḣ1 → Ḣ-1 ≤ ∥V ∥ n/2,1 .
We recall the following property.

Lemma 2.1. Let V ∈ L n/2,1 and δ ≪ 1 then there exist K, W ∈ L n/2,1 such that K is a simple function and ∥W ∥ n/2,1 ≤ δ.

We recall that simple functions are finite linear combinations of characteristic functions of sets of finite measure.

Proof. The statement follows directly from the density of simple functions in L n/2,1 , see Theorem 1.4.13 in [START_REF] Grafakos | Classical Fourier Analysis[END_REF].

From the previous lemma, we can decompose the potential V as

V = W + K ∥W ∥ n/2,1 ≪ 1.
Instead of looking for the solution of (-∆ + V )ψ = 0, in this section we will study ψ ∈ Ḣ1 as solution of the equation (-∆ + W )ψ = -Kψ.

We now consider the operator

(-∆ + W ) : Ḣ1 → Ḣ-1
where -∆ : Ḣ1 → Ḣ-1 is an isometry and

∥W ∥ Ḣ1 → Ḣ-1 ≤ ∥W ∥ n/2,1 ≪ 1.
Remark 3. The operator -∆ + W maps Ḣ1 into Ḣ-1 even under the weaker assumption W ∈ L n/2,2 , thanks to the embedding (2.3). However, the space L n/2,1 has the interest of being the dual of L n n-2 ,∞ , this will allow us to integrate W against the kernel of (-∆) -1 and to state that ´V ψ, limit of |x| n-2 ψ at infinity, is finite.

The smallness of W allows us to invert the operator via a Neumann series. Indeed, writing

(-∆ + W ) -1 = (-∆) -1 (I + W (-∆) -1 ) -1
we obtain a small perturbation of the identity and hence we can write, at least formally,

(-∆ + W ) -1 = j≥0 (-1) j (-∆) -1 (W (-∆) -1 ) j (2.5)
where the series is convergent thanks to the smallness of W .

We then apply an idea from [START_REF] Pinchover | On positive solutions of second-order elliptic equations, stability results, and classification[END_REF]: to construct the Green function we define the integral kernels corresponding to the operators in the series (2.5). We do so by recurrence, setting

G 0 (x, y) = c n 1 |x -y| n-2 , G j (x, y) = c n ˆ1 |x -z| n-2 W (z)G j-1 (z, y)dz
where c n = n(n -2)|B(0, 1)| is a constant depending on the dimension. Here G 0 is the kernel of (-∆) -1 , G 1 that is given by

G 1 (x, y) = c n ˆ1 |x -z| n-2 W (z)c n 1 |z -y| n-2 dz
is the kernel of (-∆) -1 W (-∆) -1 and so on, G j will be the kernel of (-∆) -1 (W (-∆) -1 ) j .

To bound the integrals defining G j we first remark a useful inequality.

Lemma 2.2. Let a(x) = 1 |x| n-2 . For x ̸ = y it holds ˆ1 |x -z| n-2 |W (z)| 1 |z -y| n-2 dz ≤ ∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 2 n-1 |x -y| n-2 .
Proof. We split the integral into the regions {|z -y|

≤ |x-y| 2 } and {|z -y| > |x-y| 2 } so that z ∈ B(y, |x-y| 2 ) implies |x -z| ≥ |x-y| 2 and we have ˆ1 |x -z| n-2 |W (z)| 1 |z -y| n-2 dz ≤ 2 n-2 |x -y| n-2 ˆ|W (z)| 1 |z -y| n-2 dz + 2 n-2 |x -y| n-2 ˆ1 |x -z| n-2 |W (z)|dz ≤ 2 n-2 |x -y| n-2 ∥W ∥ n/2,1 (∥| • -y| -(n-2) ∥ n n-2 ,∞ + | • -x| -(n-2) ∥ n n-2 ,∞ ) ≤∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 2 n-1 |x -y| n-2 .
The last inequality follows from the fact that L p,q quasinorm are invariant by translation.

Remark 4. In the previous lemma we stated some useful properties for the kernel of (-∆) -1 in R n , n ≥ 3 when integrated against a function in L n/2,1 . The two dimensional case, with its logarithmic behaviour in the kernel of (-∆) -1 needs a dedicated approach.

Theorem 2.3. Let W ∈ L n/2,1 with ∥W ∥ n/2,1 ≪ 1 sufficiently small, then G(x, y) := j≥0 (-1) j G j (x, y) (2.6)
is the Green function of -∆ + W and is such that

|G(x, y)| ≲ 1 |x-y| n-2 .
Remark 5. The theorem gives us a pointwise bound on the integral kernel of (-∆ + W ) -1 by the integral kernel of (-∆) -1 . We deduce that (-∆ + W ) -1 inherits any L p → L q or L p,q → L p ′ ,q ′ bound that (-∆) -1 enjoys.

Proof of Theorem 2.3. By Lemma (2.2) it is straightforward to bound G 1 by

|G 1 (x, y)| ≤ c 2 n ∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 2 n-1 |x -y| n-2 = 2 n-1 c n ∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 G 0 (x, y)
and setting C = 2 n-1 c n ∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 we obtain by induction

|G j (x, y)| ≤ C j G 0 (x, y).
Indeed, assuming |G j-1 (x, y)| ≤ C j-1 G 0 (x, y) and applying again Lemma 2.2 we directly obtain

|G j (x, y)| ≤c 2 n ˆ1 |x -z| n-2 |W (z)| C j-1 |z -y| n-2 dz ≤C j-1 c 2 n ∥a∥ n n-2 ,∞ ∥W ∥ n/2,1 2 n-1 |x -y| n-2 = C j G 0 (x, y).
The constant C is less than one thanks to the smallness of W and hence the series (2.6) is convergent. The bound on G follows directly from the one on G j .

Finally, we check that G is indeed the kernel of (- where to obtain (2.7) we used the fact that the series (2.5) is convergent with respect to the topology of B( Ḣ-1 , Ḣ1 ) and for (2.8) we used the absolute convergence of the series (2.6).

∆ + W ) -1 . Let φ, ψ ∈ C ∞ 0 two test functions and ⟨ , ⟩ the scalar product of L 2 ⟨ψ, (-∆ + W ) -1 φ⟩ =⟨ψ, j≥0 (-1) j (-∆) -1 (W (-∆) -1 ) j φ⟩ = j≥0 (-1) j ⟨ψ, (-∆) -1 (W (-∆) -1 ) j φ⟩ (2.7)

Properties of a zero resonant state

The aim of this section is to prove Theorems 1.1 and 1.2. Using the decomposition V = W + K we can write the resonance ψ as solution of (-∆ + W )ψ = -Kψ and by the Green function defined in Theorem 2.3 this solution is given by

ψ(x) = -ˆG(x, y)K(y)ψ(y)dy. (3.1)
First of all we prove Theorem 1.1, therefore obtaining that a resonant state ψ is actually an eigenfunction, when n ≥ 5, or is in a weak Lebesgue space, when n = 3, 4.

Proof of Theorem 1.1. • Case n ≥ 5. We consider ψ ∈ Ḣ1 solution of the eigenvalue equation (-∆ + V )ψ = 0, or equivalently (-∆ + W )ψ = -Kψ. We want to prove that ψ is in L 2 . We know that ψ is a solution of the equation

(I + (-∆) -1 W )ψ = -(-∆) -1 Kψ. (3.2)
Thanks to Hardy-Littlewood-Sobolev and Hölder inequalities we have

(-∆) -1 : L 2n n+4 ,2 → L 2 , W : L 2 → L 2n n+4 ,2
where ∥W ∥

L 2 →L 2n n+4 ,2 ≲ ∥W ∥ n/2,1 ≪ 1,
hence the operator (I + (-∆) -1 W ) can be inverted via a Neumann series and its inverse defines an operator from L 2 to L 2 . Applying this operator to (3.2) we find

ψ = -(I + (-∆) -1 W ) -1 (-∆) -1 Kψ with (I + (-∆) -1 W ) -1 : L 2 → L 2 . So ψ is in L 2 if (-∆) -1 Kψ is. Since n ≥ 5 we have the Sobolev embedding Ḣ2 → L 2n n-4 ,2
and its dual

L 2n n+4 ,2 → Ḣ-2 .
Now, K is a simple function, therefore it belongs to the space L n/3,∞ and by Hölder inequality where the last integral is finite. Indeed, K is a simple function, so K ∈ L p,q for any p and q.

∥(-∆) -1 Kψ∥ 2 = ∥Kψ∥ Ḣ-2 ≲ ∥Kψ∥ 2n n+4 ,2 ≲ ∥K∥ n/3,∞ ∥ψ∥ 2n n-2 ,
In particular K ∈ L

∥ψ∥ n n-2 ,∞ ≲ ∥χψ∥ n n-2 ,∞ + ∥(1 -χ)ψ∥ n n-2 ,∞ < ∞.
Now to determine the value of the limit we need to study the behavior of |x| n-2 G(x, y) for large |x| and y that ranges in a compact set (the support of K). Using the second resolvent identity

(-∆ + W ) -1 = (-∆) -1 -(-∆) -1 W (-∆ + W ) -1 (3.3)
we can write

|x| n-2 G(x, y) = c n |x| n-2 |x -y| n-2 -c n ˆ|x| n-2 4π|x -z| n-2 W (z)G(z, y)dz. (3.4)
The first term in |x| n-2 G(x, y) converges to c n , for the second term we split the integral in the regions B(0, |x|/2) and its complementary. First, taking |x| large enough if z ∈ B(0, |x|/2) c then we will have

|z -y| ≥ ||z| -|y|| = |z| -|y| ≥ |x| 2 -|y| > 0.
Using this bound we obtain

ˆB(0,|x|/2) c |x| n-2 4π|x -z| n-2 |W (z)G(z, y)|dz ≲ ˆB(0,|x|/2) c 1 |x -z| n-2 |W (z)| |x| n-2 ( |x| 2 -|y|) n-2 dz ≲ ˆB(0,|x|/2) c 1 |x -z| n-2 |W (z)|dz ≲∥a∥ n n-2 ,∞ ∥W 1 B(0,|x|/2) c ∥ n/2,1 |x|→+∞ -----→ 0.
(3.5)

The norm ∥W 1 B(0,|x|/2) c ∥ n/2,1 is defined by an integral of the distribution function. Then the convergence to zero is due to the fact that the superlevel of W 1 B(0,|x|/2) c tends to the empty set as |x| → +∞ and we can pass the limit in the integral thanks to the domination

d W 1 B(0,|x|/2) c (t) 2/n ≤ d W (t) 2/n ∈ L 1 (R + ).
On the other hand, for |x| → +∞ we have the pointwise convergence of

1 B(0,|x|/2) (z)c n |x| n-2 |x -z| n-2 W (z)G(z, y) → c n W (z)G(z, y)
and since the points z ∈ B(0, |x| 2 ) satisfy |x -z| > |x| 2 we have the domination Thanks to the previous proof we can derive the behavior of ψ at infinity. We can prove further decay for ψ, and therefore better integrability, under suitable orthogonality assumptions on ψ. To obtain decay for ψ we will prove that |x| α ψ is bounded at infinity for a suitable α.

1 B(0,|x|/2) (z) |x| n-2 4π|x -z| n-2 |W (z)G(z, y)| ≲ |W (z)| 1 |z -y| n-2 ∈ L 1 .

So again by dominated convergence

c n ˆB(0,|x|/2) |x| n-2 4π|x -z| n-2 W (z)G(z, y) |x|→+∞ -----→ c n ˆW (z)G(z, y)dz. ( 3
We take R > 0 sufficiently large such that

supp K ⊂ B(0, R) and |x| n-2 |ψ(x)| ∈ L ∞ (B(0, R) c ).
We recall that such R exists thanks to the fact that K is a simple function together with the fact that |x| n-2 |ψ(x)| has a finite limit. We study the behavior of ψ in B(0, R) c . To do so we define the spaces

B α = |x| -α L ∞ (B(0, R) c ) = {f : |x| α f ∈ L ∞ (B(0, R) c ) } (3.7)
with the natural norm

∥f ∥ Bα = ∥|x| α f 1 B(0,R) c ∥ ∞ .
In Proposition 3.2 we prove that ψ belongs to such spaces B α for suitable α.

We will use the following inequality. A proof can be found in Appendix B.

Lemma 3.1. Let N ∈ N and k = 0, . . . , N l = 0, . . . , k. There exist c kl ∈ R such that

1 |x -y| n-2 - 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l ≲ 1 |x -y| n-2 |y| N +1 |x| N +1 + |y| N +n-2 |x| N +n-2 for all x, y ∈ R n \ {0} and x ̸ = y. Notation. For y ∈ R n and α = (α 1 , . . . , α n ) ∈ N n we denote by y α = n k=1 y α k k . Proposition 3.2. Let ψ ∈ Ḣ1 a solution of the equation (-∆ + V )ψ = 0 and N = 0, 1, 2. Then ψ ∈ B N +n-1 , that is ψ decays like |x| -(N +n-1) at infinity, if ´yα V ψ = 0 for all multi- indices α ∈ N n , |α| ≤ N .
Proof. Using the integral kernel of ∆ -1 on R n we can write

ψ(x) = -c n ˆ1 |x -y| n-2 V (y)ψ(y)dy. (3.8)
By assumption for N ≤ 2 and k = 0, . . . N we have

ˆ k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l V (y)ψ(y)dy = 0.
Indeed, the terms in the sum are given by

|y| k+l (x • y) k-l |y| k-l = |y| 2l (x • y) k-l
and are therefore polynomials in y of degree k + l, since k + l ≤ N they vanish when we integrate them against V ψ.

Thanks to these vanishing quantities, we then rewrite (3.8) as

ψ(x) =c n ˆ 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l - 1 |x -y| n-2 V (y)ψ(y)dy = c n ˆ|y|≤R 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l - 1 |x -y| n-2 V (y)ψ(y)dy =:f (x) (3.9) + c n ˆ|y|≥R 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l - 1 |x -y| n-2 W (y)ψ(y)dy
where we recall that R is a sufficiently large radius such that

supp K ⊂ B(0, R), |x| n-2 |ψ(x)| ∈ L ∞ (B(0, R) c ).
Defining the operator

S : φ → c n ˆ|y|≥R 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l - 1 |x -y| n-2 W (y)φ(y)dy (3.10)
we can rewrite the previous identity as

ψ = f + Sψ. (3.11)
We will prove that the operator S is a contraction in the spaces B N +n-1 , B N +n-2 and that f ∈ B N +n-1 ⊂ B N +n-2 , hence the equation

φ = f + Sφ (3.12)
has a unique solution in B N +n-1 as well as a unique solution in B N +n-2 . This proves the statement. Indeed, let N = 0, then identity (3.11) together with item ii) in Theorem 1.1 tell us that ψ is the unique solution in B n-2 of the fixed point problem (3.12). The problem (3.12) has a unique solution in B N +n-1 as well, since

B N +n-1 ⊂ B n-2 N = 0, 1, 2 then ψ must be in B N +n-1 .
To conclude we must prove f ∈ B N +n-1 ⊂ B N +n-2 and that S is a contraction on B N +n-1 and B N +n-2 .

We use Lemma (3.1) to prove f ∈ B N +n-1 ⊂ B N +n-2 . Let |x| ≥ 2R, from the definition of f we first have

|f (x)| ≲ ˆ|y|≤R 1 |x -y| n-2 |y| N +1 |x| N +1 + |y| N +n-2 |x| N +n-2 |V (y)ψ(y)|dy ≲ 1 (|x| -R) n-2 1 |x| N +1 + 1 |x| N +n-2 ˆ|y|≤R |V ||ψ|dy hence |x| N +n-1 |f (x)| is bounded for |x| ≥ 2R. For R ≤ |x| ≤ 2R we rewrite f as f (x) =c n ˆ|y|≤R 1 |x| n-2 N k=0 k l=0 k+l≤N c kl |y| k+l |x| k+l • (x • y) k-l (|x||y|) k-l V (y)ψ(y)dy + ψ(x) + ˆ|y|≥R 1 |x -y| n-2 V (y)ψ(y)dy then since |x| n-2 ψ(x) ∈ L ∞ (B(0, R) c ), the bound |f (x)| ≲ ˆ|V (y)ψ(y)|dy + |ψ(x)| + ˆ|y|≥R |V (y)| |x -y| n-2 |y| n-2 |ψ(y)|dy implies that |x| N +n-1 |f (x)| is bounded for R ≤ |x| ≤ 2R .
We now prove that S is a contraction on B N +n-2 . Thanks to Lemma 3.1

|Sφ(x)| ≲ ˆ|y|≥R 1 |x -y| n-2 |y| N +1 |x| N +1 + |y| N +n-2 |x| N +n-2 |W (y)φ(y)|dy ≲ ˆ|y|≥R 1 |x -y| n-2 |y| N +1 |x| N +1 |W (y)φ(y)|dy + 1 |x| N +n-2 ∥φ∥ B N +n-2 ˆ|W (y)| |x -y| n-2 dy ≲ ˆ|y|≥R 1 |x -y| n-2 |y| N +1 |x| N +1 |W (y)φ(y)|dy (3.13) + 1 |x| N +n-2 ∥φ∥ B N +n-2 ∥W ∥ n/2,1
We split the domain of integration in the regions

{|y| ≥ R} = |y| ≥ R and |y| < |x| 2 ⊔ |y| ≥ R and |x| 2 ≤ |y| ≤ 2|x| ⊔ {|y| ≥ R and |y| > 2|x|} =: E < ⊔ E ≈ ⊔ E > (3.14)
and we remark the following properties

|y| -1 ≲ |x| -1 on E > ∪ E ≈ |x| -1 ≲ |y| -1 , |x -y| -1 ≲ |x| -1 on E < .
We use the inequalities

|y| N +1 |x -y| n-2 |x| N +1 |φ(y)| ≲        |y| N +n-2 |φ(y)| |x| N +n-2 |x -y| n-2 on E > ∪ E ≈ |y| N +n-2 |φ(y)| |x| N |y| n-2 |x -y| n-2 on E < ≲      1 |x| N +n-2 ∥φ∥ B N +n-2 1 |x -y| n-2 on E > ∪ E ≈ 1 |x| N ∥φ∥ B N +n-2 1 |y| n-2 |x -y| n-2 on E < obtaining ˆ|y|≥R 1 |x -y| n-2 |y| N +1 |x| N +1 |W (y)φ(y)|dy ≲∥φ∥ B N +n-2 1 |x| N +n-2 ˆE>∪E≈ |W (y)| |x -y| n-2 dy + 1 |x| N ˆE< |W (y)| |y| n-2 |x -y| n-2 dy ≲ 1 |x| N +n-2 ∥φ∥ B N +n-2 ∥W ∥ n/2,1
where we used Lemma 2.2 in the last inequality.

Having estimated the integral in (3.13) we can go back to the inequality on |Sφ(x)| from which we finally have

|Sφ(x)| ≲ 1 |x| N +n-2 ∥φ∥ B N +n-2 ∥W ∥ n/2,1 hence ∥Sφ∥ B N +n-2 ≲ ∥W ∥ n/2,1 ∥φ∥ B N +n-2
which implies that S is a contraction on B N +n-2 thanks to the smallness of ∥W ∥ n/2,1 .

As a last step, we need to prove that S is a contraction on B N +n-1 as well. We still use the subdivision (3.14). As we found before, thanks to Lemma 3.1

|Sφ(x)| ≲ ˆ|y|≥R 1 |x -y| n-2 |y| N +1 |x| N +1 + |y| N +n-2 |x| N +n-2 |W (y)φ(y)|dy.
We use the inequalities

|y| N +1 |x -y| n-2 |x| N +1 |φ(y)| = |y| N +n-1 |x -y| n-2 |x| N +1 |y| n-2 |φ(y)| ≲        ∥φ∥ B N +n-1 |x| N +n-1 • 1 |x -y| n-2 on E > ∪ E ≈ ∥φ∥ B N +n-1 |x| N +n-1 • 1 |y| n-2 on E < and |y| N +n-2 |x -y| n-2 |x| N +n-2 |φ(y)| = 1 |x -y| n-2 • |y| N +n-1 |x| N +n-2 |y| |φ(y)| ≲        ∥φ∥ B N +n-1 |x| N +n-1 • 1 |x -y| n-2 on E > ∪ E ≈ ∥φ∥ B N +n-1 |x| N +1 • 1 |y| n-2 |x -y| n-2 on E < , thanks to which we can bound |Sφ(x)| by |Sφ(x)| ≲ ∥φ∥ B N +n-1 |x| N +n-1 ˆE>∪E≈ |W (y)| |x -y| n-2 dy + ˆE< |W (y)| |y| n-2 dy + ∥φ∥ B N +n-1 |x| N +1 ˆE< |W (y)| |y| n-2 |x -y| n-2 ≲ ∥φ∥ B N +n-1 |x| N +n-1 ∥W ∥ n/2,1
where we used again Lemma 2.2 to obtain the last inequality. The previous bound is equivalent to

∥Sφ∥ B N +n-1 ≲ ∥W ∥ n/2,1 ∥φ∥ B N +n-1
which proves that S is a contraction on B N +n-1 thanks to the smallness of ∥W ∥ n/2,1 .

Remark 6. In the previous proof we showed that ´V ψ = 0 implies ψ ∈ B n-1 . We remark that also the opposite implication is true: if ψ ∈ B n-1 then the integral of V ψ must be zero. The same holds for the integral of V ψ with monomials of order one, indeed we show below that ´yk V ψ = 0 for all k = 1, . . . , n if and only if ψ ∈ B n . Assume ψ ∈ B n-1 , since ψ is also a solution of (-∆ + V )ψ = 0 we have ˆV (y)ψ(y) = ˆ∆ψ(y) = lim We also remark that we can not use the same reasoning to prove that under the assumption ψ ∈ B n+1 we have ´yα V ψ = 0 for all |α| = 2. Indeed, for k ̸ = l ˆyk y l V (y)ψ(y) = lim where the domination in the dominated convergence theorem is given by ψ itself. On the contrary, integrating against y 2 k we can not conclude that the integral vanishes since ˆy2 k V (y)ψ(y) = lim However, we remark the following fact. If ´yα V ψ = 0 for all α with |α| ≤ 2, the previous computations shows us that ´ψ = 0. We also know that under this orthogonality assumption ψ ∈ B n+1 , as stated in Theorem 1.2, hence a sufficiently fast decaying eigenfunction must be orthogonal to constants.

We are now able to prove Theorem 1.2.

Proof of Theorem 1.2. The proof is a direct consequence of the previous propositions.

We start by proving item i). If ´V ψ = 0, we can apply Proposition 3.2 for N = 0, obtaining ψ ∈ B n-1 and consequently ψ ∈ L 2 (B(0, R) c ). Moreover, ψ ∈ Ḣ1 ⊂ L |x| n-2 ≤ ψ ≤ c ′ |x| n-2 at infinity and therefore ψ is not in L 2 . We conclude that for dimensions three and four ´V ψ = 0 is also a necessary condition to have ψ ∈ L 2 . Now assume ´yk V (y)ψ(y)dy = 0 for all k = 1, . . . , n. We can apply Proposition 3.2 with N = 1 which implies ψ ∈ B n and hence ψ ∈ L 1,∞ , since 1 |x| n ∈ L 1,∞ . Item ii) is then proved. We conclude by proving item iii). Thanks to the assumptions on ´yk V (y)ψ(y)dy and ´yk y l V (y)ψ(y)dy for k, l = 1, . . . , n we can apply Proposition 3.2 with N = 2. Then

ψ ∈ B n+1 implies ψ ∈ L 1 (B(0, R) c ) while ψ ∈ L 2 , that we already have from item i) since ´V ψ = 0, implies ψ ∈ L 1 loc .
Alternatively, this last inclusion can be proved by interpolation. Since ψ is in B n+1 we have that |x|ψ is bounded by |x| -n at infinity and hence |x|ψ ∈ L 1,∞ . We can then write ψ

(x) = 1 |x| b(x) with 1 |x| ∈ L n,∞ , b ∈ L 1,∞ , which by Hölder inequality (A.3) implies ψ ∈ L n n+1 ,∞ . Having ψ ∈ L n n+1 ,∞ ∩ L 2,∞
we can conclude ψ ∈ L 1 by interpolation (Proposition A.1).

≤( ˆ(t

1 p 1 f * ) q 1 dt t ) q 2 q 1 +q 2 ( ˆ(t 1 p 2 f * ) q 2 dt t )
q 1 q 1 +q 2 from which(A.3) follows taking the power 1 q = q 1 +q 2 q 1 q 2

The following property is a sort of interpolation over the first index of the space.

Proposition A.1. Let f ∈ L p 0 ,∞ ∩ L p 1 ,∞ , then f ∈ L p,q for any p ∈ (p 0 , p 1 ) and any q ∈ (0, ∞].

Proof. We first consider q = ∞ then ∥f ∥ p,∞ = max{ sup

d f (t)≥1 td f (t) 1/p , sup 0≤d f (t)≤1 td f (t) 1/p }. (A.4)
For the case d f (t) ≤ 1 since p < p 1 we have

d f (t) 1/p-1/p 1 ≤ 1, conversely when d f (t) ≥ 1 d f (t) 1/p-1/p 0 ≤ 1.
We can therefore bound both suprema in (A.4) as

sup 0≤d f (t)≤1 td f (t) 1/p-1/p 1 +1/p 1 ≤ sup 0≤d f (t)≤1 td f (t) 1/p 1 = ∥f ∥ p 1 ,∞ < ∞ and sup d f (t)≥1 td f (t) 1/p-1/p 0 +1/p 0 ≤ sup t≥0 td f (t) 1/p 0 = ∥f ∥ p 0 ,∞ < ∞.
Now let q < ∞. We have the quantity td f (t) 1/p 1 which is bounded at infinity, then there exists a t > 0 such that d f (t) ≲ 1 t p 1 for any t ≥ t so for large enough t we have

td f (t) 1/p 0 ≲ t 1-p 1 /p 0 with 1 - p 1 p 0 < 0.
On the other hand td f (t) 1/p 0 is bounded around 0 so there exists t > 0 such that d f (t) ≲ 1 t p 0 for any 0 < t ≤ t from which for small t it holds

td f (t) 1/p 1 ≲ t 1-p 0 /p 1 with 1 - p 0 p 1 > 0.
Now we can proceed to estimate the L p,q norm, for λ ∈ (0, 1) we write (td f (t) 1/p 1 ) (1-λ)q ˆ∞ t (td f (t) 1/p 0 ) λq dt t ≲∥f ∥ λq p 0 ,∞ ˆt 0 t (1-p 0 /p 1 )(1-λ)q-1 dt + ˆt t . . . dt

1 p = (1 -λ) 1 p 1 + λ 1 p 0 ∥f ∥ p,q = ˆt 0 (t 1-λ+λ d f (t) (1-λ) 1 p 1 +λ 1 p 0 ) q dt t = ˆt 0 . . .
+ ∥f ∥ (1-λ)q p 1 ,∞
ˆ∞ t t (1-p 1 /p 0 )λq-1 dt ≃∥f ∥ λq p 0 ,∞ t (1-p 0 /p 1 )(1-λ)q | t 0 + ˆt t . . . dt + ∥f ∥ (1-λ)q p 1 ,∞ t (1-p 1 /p 0 )λq | ∞ t where all the terms are finite since (1 -p 0 /p 1 )(1 -λ)q > 0 and (1 -p 1 /p 0 )λq < 0.

B Proof of Lemma 3.1.

In this appendix we give a proof of Lemma 3.1.

(x • y) |x||y| k ≲ |s| N +1 |x| n-2 ≲ 1 |x| n-2 |y| 2 |x| 2 + |y| |x| N +1 ≲ 1 |x| n-2 • |y| N +1 |x| N +1 (B.2)
where in the last inequality we used the fact that |y| 2 |x| 2 ≪ |y| |x| . Finally, we remark that we can reorganize the sum in the previous inequality as

= j≥0 (- 1 )

 j≥01 j ˆψ(x)G j (x, y)φ(y) dxdy= ˆψ(x) j≥0 (-1) j G j (x, y)φ(y) dxdy(2.8)= ˆψ(x)G(x, y)φ(y) dxdy

n- 2 , 2 .

 22 2 < ∞ thanks to the inclusion Ḣ1 → L 2n We have obtained (-∆) -1 Kψ ∈ L 2 and hence ψ ∈ L 2 . • Case n = 3, 4. We recall from Theorem 2.3 that |G(x, y)| ≲ 1 |x-y| n-2 , then using (3.1) for large enough |x| |x| n-2 |ψ(x)| ≲ |x| n-2 ˆ1 |x -y| n-2 |K(y)ψ(y)|dy ≲ ˆ|K(y)ψ(y)|dy < ∞.

  ,∞ ) * and ψ ∈ L 2n n-2 ,∞ . We can then obtain ψ ∈ L n n-2 ,∞ . Let χ be a smooth cutoff which is equal to 1 on a large enough compact set, for the compact part it holds ψχ ∈ L n n-2 ,∞ since χ, ψ ∈ L 2n n-2 ,∞ . For the part at infinity we can bound ψ(1 -χ) by 1 |x| n-2 ∈ L n n-2 ,∞ . So applying (A.1) we conclude

. 6 )

 6 Summing together (3.5) and (3.6) in (3.4) we obtain lim |x|→+∞ |x| n-2 G(x, y) = c n -c n ˆW (z)G(z, y)dz and since |x| n-2 G(x, y) ≲ 1 and Kψ ∈ L 1 we can pass the limit in the integral in (3.1). This yields lim |x|→+∞ |x| n-2 ψ(x) = -c n ˆK(y)ψ(y)dy + c n ˆ ˆW (z)G(z, y)dz K(y)ψ(y)dy. Now exchanging the order of integration in the second term and using relation (3.1) we have ˆ ˆW (z)G(z, y)dz K(y)ψ(y)dy = ˆW (z) ˆG(z, y)K(y)ψ(y)dydz = -ˆW (z)ψ(z)dz from which the statement follows since V = W + K.

  ε→0 ˆ∆ψ(y)χ(εy)with χ a smooth cutoff on B(0, 1). After integration by parts we obtain ˆV (y)ψ(y) = lim ε→0 ˆε2 ψ(y)∆χ(εy) = 0 where the limit is zero by dominated convergence, thanks to the domination|ε 2 ψ(y)∆χ(εy)| = |εy| 2 |∆χ(εy)| 1 |y| 2 |ψ(y)| ≲ |ψ(y)| |y| 2 with |ψ(y)| |y| 2 ≲ 1 |y| n+1integrable on the support of ∆χ (it would actually be sufficient that ψ belongs to B n-2+δ for some positive δ).Similarly, if ψ ∈ B n we have ˆyk V (y)ψ(y) = lim ε→0 ˆyk ∆ψ(y)χ(εy) = lim ε→0 ˆ(y k ε 2 ∆χ(εy) + 2ε∂ y k χ(εy))ψ(y) = 0 again by dominated convergence since |(y k ε 2 ∆χ(εy) + 2ε∂ y k χ(εy))ψ(y)| =|ε 2 y k |y|∆χ(εy) + 2|εy|∂ y k χ(εy)| 1 |y| |ψ(y)| ≲ |ψ(y)| |y| with |ψ(y)| |y| ≲ 1 |y| n+1 integrable on the support of ∂ y k χ (as before, ψ ∈ B n-1+δ for δ > 0 would be enough).

ε→0

  ˆyk y l ∆ψ(y)χ(εy) = lim ε→0 ˆ(y k y l ε 2 ∆χ(εy) + 2εy k ∂ y l χ(εy) + 2εy l ∂ y k χ(εy))ψ(y) = 0

2 k ε 2

 22 ∆χ(εy) + 4y k ε∂ y k χ(εy) + 2χ(εy))ψ(y) =2 ˆψ(y)dy.

loc ⊂ L 2 loc since 2n n- 2 ≥

 2 2 for n ≥ 3. For n = 3, 4, if ´V ψ ̸ = 0 the limit of |x| n-2 ψ is finite and non zero thanks to Theorem 1.1. Hence, c

Lemma B. 1 . 1 ≪ 1 .d

 111 Let N ∈ N and k = 0, . . . , N l = 0, . . . , k. There exist c kl ∈ R such that1 |x -y| n-2 -y) k-l (|x||y|) k-l ≲ 1 |x -y| n-2 |y| N +1 |x| N +1 + |y| N +n-2 |x| N +n-2for all x, y ∈ R n \ {0} and x ̸ = y.Proof. We divide the proof according to the size of |y| with respect to |x|.We first consider the case |y| ≫ |x| and use the trivial bound(x•y) |x||y| ≤ n-2 1 + |y| N +n-2 |x| N +n-2 ≲ 1 |x -y| n-2 • |y| N +n-2 |x| N +n-2where we used the bounds |x -y| ≲ |y| and 1 ≲ |y| |x| .Now we consider the region |y| ≃ |x|. Again, (x•y) |x||y| ≤ 1, moreover |y| |x| ≃ 1 and |x-y| ≲ |x|. Therefore, we bound trivially by1 |x -y| n-2n-2 ≃ 1 |x -y| n-2 • |y| N +1 |x| N +1 .To conclude, we look at the case |y| ≪ |x|. We have|x -y| n-2 =(|x| 2 + |y| 2 -2x • y)We use the Taylor expansion around zero for the function k s k + O(|s| N +1 ) d k :We apply this expansion with s = |y| 2 |x| 2 -2 |y| |x| (x•y) |x||y| which satisfies the bound |s| ≪ 1, given that we are in the region |y| ≪ |x|. From (B.1) we have 1 |x -y| n-2 = 1 |x| n-2 • N k=0 d k s k + O(|s| N +1 ) and given the choice of s 1 |x -y| n-2 -

  |y| l |x| l • (x • y) k-l (|x||y|) k-l

Acknowledgments: This work received support from the University Research School EUR-MINT (State support by the National Research Agency' Future Investments program, reference number ANR-18-EURE-0023). The author would like to thank Professor H. Mizutani for useful discussions on this topic and the anonymous referee for his/her comments that allowed to improve the paper, in particular Lemma 3.1.

A Facts about Lorentz spaces

We collect here a few properties of Lorentz spaces that we have used. The following statements hold on R n for any n.

First of all we recall the definition of the quasinorm

otherwise. Observe that L p,p = L p and that L p,∞ is the weak L p space. The quantity we just defined is only a quasinorm, since it holds ∥f + g∥ p,q ≲ p,q ∥f ∥ p,q + ∥g∥ p,q (A.1) (see inequality (1.4.9) in [START_REF] Grafakos | Classical Fourier Analysis[END_REF]). Only for p > 1 and any q ∈ [1, ∞] the space L p,q is normable ( [START_REF] Hunt | On l(p,q) spaces[END_REF]).

Remark 7. We recall that the quasinorm of the Lorentz space can also be defined via the decreasing rearrangement f

Lorentz spaces are growing spaces with respect to the second index, in particular we have the chain of inclusions L p,q 1 ⊂ L p,q 2 for any q 1 < q 2 (A.2) (see Proposition 1.4.10 in [START_REF] Grafakos | Classical Fourier Analysis[END_REF]).

We also have a two indexed Hölder inequality ∥f g∥ L p,q ≤ ∥f ∥ L p 1 ,q 1 ∥g∥ L p 2 ,q 2 (A.3) for any p 1 , q 1 , p 2 , q 2 such that 1 p 1 + 1 p 2 = 1 p and 1 q 1 + 1 q 2 = 1 q . Inequality (A.3) can be easily proved using the definitions given in Remark 7. For q = ∞ it follows directly from the definition of ∥ • ∥ p,∞ , while for q ∈ (0, ∞) it is obtained using inequality ´f α g β dt t ≤ ( ´f dt t ) α ( ´g dt t ) β for α and β which sum to 1. In particular taking α = q 2 q 1 +q 2 and β = q 1 q 1 +q 2 ∥f g∥ q p,q = ˆ(t

We obtain the term in (B.3) thanks to the following remark: