Isabelle Servin
email: isabelle.servin@cea.fr

Alexandre Teolis

Arnaud Bazin

Paule Durin

Olha Sysova

Corinne Gablin

Benoît Saudet

Didier Leonard

Olivier Soppera

Jean-Louis Leclercq

Yann Chevolot

Raluca Tiron

Thierry Delair

Stéphane Trombotto

Water-soluble bio-sourced resists for DUV lithography in a 200/300 mm pilot line environment

Keywords: Lithography, photoresist, chitosan, water-based, semiconductor, industrial pre-transfer, sustainability 1

Water-based bio-sourced resists are promising candidates as alternatives for deep ultraviolet (DUV) lithography by replacing current photoresists issued from petro-chemistry for microelectronics application. Chitosan films produced from seafood industry wastes enable patterning processes free of organic solvent and alkali-based developers, by substitution with water. After demonstrating high-resolution patterning at lab-scale after transfer into silica 10 mm wafer, we investigate here the industrial pre-transfer chitosan-based photoresist on the 300 mm pilot line scale at CEA-Leti for 193 nm DUV lithography.

Introduction

Environmental sustainability and user-toxicity have become a critical concern in the semiconductor industry [1,2]. Photolithography, as the major process of transferring patterns for the nanofabrication, requires the use of high volume of organic solvents and alkali-based solutions as developers, hazardous for health and environment. Environmentally friendly water-based bio-sourced resists may constitute improved alternatives to current synthetic petroleum-derived photoresists for deep ultraviolet (DUV) lithography [3][4][5].

Among bio-sourced materials, chitosan is a promising candidate as photoresist. Chitosan is a natural polysaccharide derived from chitin, which is the second most abundant natural biopolymer on earth behind cellulose and mostly provided by seafood industry wastes [6]. Its high versatility thanks to its non-toxicity, biodegradability, film-forming properties and UV sensitivity permits to address a wide range of applications. In our previous works, it was shown that chitosan films allow patterning processes free of organic solvent and alkali-based developers, by substitution with a green solvent (deionized (DI) water). High-resolution patterning with 100-nm thick chitosan films was obtained at lab-scale down to 50 nm lines by writing with a 30 kV electron beam system after transfer into silicon wafer [START_REF] Caillau | Proc. of SPIE Advanced Lithography[END_REF] and 5 µm by exposing on DUV lithography tool [START_REF] Sysova | [END_REF].

Here, the lithographic process with chitosan films was firstly investigated at production scale on the 300 mm pilot line scale at CEA-Leti. Finally, we started to quantify the environmental impacts of the photolithography process based on water-soluble bio-sourced resists using the life cycle analysis (LCA) methodology for assessing its environmental relevance compared to current processes using synthetic petroleum-derived resists.

Experimental

Materials

The conventional resist, soluble in organic solvent was spin-coated on Si wafer followed by resist rinsing with PGMEA to clean both edge and back-side of the wafer and developed after exposure with alkaline solution TMAH 2.38% and finally rinsed with DI water. The bio-sourced resist differs from conventional one by the compositions, by substituting the solvent by DI water for dissolution and development ( Chitosan resist is a positive-tone water developable biosourced resist directly coated on the substrate (Si) for photolithography by modifying its intrinsic properties. The photosensitive thin layer has a 100 nm-thickness.

Chitosan obtained from squid pens was provided by Mahtani Chitosan from batch type 114 with a degree of acetylation (DA) of 2 % determined by 1 H NMR, a massaverage molar mass (Mw) of 619 kg/mol and a dispersity (Ð) of 1.6 determined by size-exclusion chromatography. Chitosan was acetylated at 0.5 % (w/v) in a hydroalcoholic mixture of 1, 2-propanediol and AcOH (50/50 v/v). An amount of acetic anhydride was added stoichiometrically to D-glucosamine (GlcN) unit of chitosan and mixed for 18 h under vigorous mechanical stirring to reach the targeted DA. Chitosan solution was then filtered through cellulose membranes with decreasing pore sizes from 3 µm to 0.45 µm. Acetylated chitosan was finally precipitated with NH4OH, washed with deionized water and freeze-dried. Acetylated chitosan with a DA of 35 %, a Mw of 693 kDa and a dispersity of 1.8 was obtained.

Chitosan films (DA 35 %, Mw 693 kDa) were obtained by spin-coating of a 0.5 % (w/w) solution of acetylated chitosan. Chitosan was solubilized with AcOH aqueous solution (stoichiometry to GlcN unit) for 18 h. Solution was then filtered through 0.45 µm cellulose membranes prior using.

Lithography process & equipment

The lithographic process consists of three main steps: (a) chitosan film coating, (b) DUV exposure on scanner, (c) development step to reveal binary patterns (Figure 1).

Following spin-coating of high Mw chitosan film, the DUV exposure induces depolymerization of chitosan by scission of glycosidic linkages, decreasing Mw then revealing patterns by resist removal into DI water during the development step. Resist was spin-coated on a SOKUDO DUO coat and develop track from SCREEN in a dedicated aqueous spin-coat unit. The resist was then exposed on a 193 nm dry scanner NIKON NSR 307E with an NA 0.85 and using a binary mask with multiple patterns (lines, trenches, contact, plots) with varying Critical Dimension (CD) and pitch. Finally, the wafer was developed in a standard develop unit

Characterization methods

The degree of acetylation (DA) of chitosan was determined by 1 H NMR spectroscopy (AVANCE III 400 MHz, Bruker) according to the Hirai et al. method [9].

The averages molar masses (Mw, Mn) and the dispersity Ð were determined by size exclusion chromatography (SEC) using a differential refractometer (Optilab T-rex, Wyatt) coupled online with a MALLS detector (Dawn Heleos II, Wyatt). The refractive index increment (dn/dc) was adjusted according to Schatz results [10].

Chitosan film thicknesses on Silicon bare wafers were controlled with a KLA-Tencor ellipsometer using a Cauchy model.

Post-lithography imaging and CD of the patterns were measured with a HITACHI CG4000 top-view CD-SEM, with an acceleration voltage of 300 V.

The profiles were carried out with a Bruker InSight 3D AFM (Atomic Force Microscopy), a non-destructive technique.

Lab-scale results

At lab-scale, we have previously demonstrated high-resolution sub-micron patterning with chitosan resists (DA 2%) down to 50 nm lines by writing with a 30 kV electron beam system or 300 nm with 248 nm DUV tool [START_REF] Caillau | Proc. of SPIE Advanced Lithography[END_REF]. Features of 5 µm lines exposed on 193 nm DUV lithography tool were transferred by plasma etching into Silica substrate preserving pattern fidelity [START_REF] Sysova | [END_REF] as shown in Figure 2. 4 Pilot line results

Scale-up challenges

Chitosan solutions at the state-of-the-art have been designed and tuned to meet the microelectronics specifications (clean room rules, thickness target, adhesion...).

Nevertheless, the scalability from small coupons to large scale on 300 mm Si wafer size remains challenging. The industrial pre-transfer of chitosan resist requires to control and fit several key performance criteria listed in Table 2.

Key parameters

Clean room controls

The introduction of new material as chitosan requires a strict control of sources of contamination (metallic and particles) to fit the cleanroom specifications defined by the manufacturing of integrated circuits (IC). Metallic contamination analyzed with Total reflection X-Ray Fluorescence (TXRF) technique reveals that all metal contaminants measured on back-side wafer are below the defined limits authorized in clean room (Table 3).

Metal

(At/cm 2 ) Al < 5 12 Co, Cr, Cu, Fe, Mn, Ni, V < 1 11 Ca, K, Mg, Na < 1 12 Ti, Zn < 5 11 Particles control measured with Unpatterned Surface Inspection System from KLA-Tencor SP3 tool shows equivalent particles number on back-side wafer before and after film spin-coating of bio-sourced chitosan film compared to a process of reference (wafer without chemistry).

Thus, contamination controls show the compatibility of chitosan with the clean room rules for both metallic and particles contaminants. Moreover, the Edge Bead Removal (EBR) and back-side (BS) chemical cleaning with DI water reveals to be efficient with a complete edge bead elimination and a back-side substrate without resist residues.

Spin-coating optimization

Chitosan films of 100 nm thickness were prepared onto 300 mm silicon wafers by resist spin-coating in manual dispense mode in a dedicated aqueous-based unit. Commonly, adhesion promoters are used prior process steps to promote wetting and adhesion of the photoresist onto the surface. This step is unnecessary with chitosan resist due to its high affinity with silanol groups on the surface wafer [11]. Amine, hydroxyl and carbonyl groups of chitosan form hydrogen bonds with the surface providing a strong adhesion onto the substrate.

Despite its great affinity with Si surface, chitosan resist difficultly spreads onto the wafer. The high viscosity (~0.5 Pa.s) of the resist caused by the high molecular weight chitosan requires optimizing the spin-coating process. A slightly acidic solution was firstly spread on the surface to promote the full coverage of the resist.

Here, a two-step dispense of the solution was performed according to the method proposed by Young et al [12]. The method applied for chitosan resists consisted of a first dispense of the solution (200 RPM, 10 s). The spinning of the wafer was stopped and a second volume was dispensed. The wafer was then spun up at the rotational speed of 1000 RPM during 120 s for final thickness adjustment. The 300 mm wafer was fully covered without defects (Figure 3a). A soft-bake at 150 °C for 5 min was finally applied to remove residual solvent and densify the film. The resulting coating had a thickness of 94 nm with a uniformity of ± 9 nm (Figure 3b). The thickness uniformity should be improved to aim the same target as conventional resists (3 < 1nm). Here, no surfactants were added to the resist for the smoothing of the film. Formulation with additives for modifying the surface tension could be considered to get better uniformity. On other hand, the automatic dispense of the resist could greatly improve the uniformity thickness and should be envisioned in future works.

DUV exposures: patterning

DUV exposures have been carried out using the 193 nm dry scanner to pattern ~100-nm thick chitosan films.

Irradiation doses from 600 to 900 mJ/cm² through a binary mask with multiples features were applied on chitosan films, acting as a positive resist under DUV irradiation by chain depolymerisation. Investigations by SEC of 100-nm thick chitosan films showed a strong decrease of the Mw down to 14 kg/mol with a Ð of 2.0 after an irradiation at 600 mJ/cm². Low molecular weight chitosan chains lead to an increase of the film solubility and are removed during the development step in deionized water (pH ~ 6,5 -7) while unexposed areas are not diluted and remain [13,14].

Thus, the binary patterns revealed after development show a good quality imaging down to 1 µm line/space (L/S) patterns (Figure 4). The optimal dose is between 600 and 700 mJ/cm², which is greater than conventional resists (< 100 mJ/cm²). Photosensitivity enhancement will be required prior to its industrial implementation. Addition of PhotoAcid Generators (PAG) may improve the lithographic performances of chitosan-based resists. Chitosan solubility for higher molecular weight chains increases in acidic environments [15]. The generation of acid species from PAG under UV exposure should promote the dissolution of longermolecular-weight chitosan chains, and consequently permits a full-pattern opening at lower doses. The 2 µm L/S pattern profile analyzed by 3D-AFM shows fully opened features with no remaining chitosan between lines (Figure 5). A dose of 700 mJ/cm² gives a height pattern of 70 nm. The thickness loss in the non-irradiated area is about 25 % corresponding to the dark erosion effect. Below 1 µm resolution, the patterns were not fully opened or are deformed. This lack of resolution of the resist may be due to surface tension during development step inducing resist deformation. Further improvements should be performed to obtain the maximum photoresist contrast.

Environmental impacts

This Life Cycle Analysis (LCA) aims to compare the environmental impacts of the lithographic process using a chitosan-based bio-resist compared to a conventional process using a petro-sourced resist. The functional unit of our system is referred to a batch of 25 Silicon 300 mm wafers size with equivalent DUV lithographic performances as standard one.

The LCA is based on ISO 14040 standard [16] and the impact assessment method recommended by the European Commission (PEF): EF 3.0 Method (adapted). The software used is Simapro version 9.3.0.3 coupled with the Ecoinvent 3.8 database. The LCA carried out is attributional and only uses "cut-off" data.

The LCA boundaries extends from the extraction of the raw materials till the end of lithographic process step for the two systems, by excluding some common steps, namely Silicon wafer production or energy consumptions and materials required to operate cleanroom infrastructure. Data collection listed for all the inputs and outputs included the energy, the resources and the materials, required for Life Cycle Inventory (LCI) phase come from on-site process, tools parameters, from publications [17], patents, or data provided by manufacturers for raw materials synthesis.

The LCA unique score of the two processes has revealed the same six main impact factors among 16, whose the sum is at least equal to 80%, namely depletion of fossil fuels, climate change, ecotoxicity in fresh water, ionizing radiation, depletion of mineral and metallic resources and water use.

The LCA comparison shows that the environmental impact of the chitosan process is on average 10% less polluting than the classic one on trilayer stack (Figure 6). The impact is lower for all the indicators except that of ionizing radiation due to high electricity consumption during the synthesis of the chitosan, resulting from the use of nuclear energy in France. However, this indicator is certainly overestimated as chitosan preparation is performed in R&D laboratory and not at an industrial scale.

For processes using chitosan resist, the need of underlayers (SOC-Si-ARC) [START_REF] Wei | Proc. of SPIE Advanced Lithography[END_REF] are not yet been proved. In this case, their removal would contribute to a drastic reduction down to 81% in the overall environmental impact, due to the preponderant role of the organic solvent (PGMEA) which serves to dilute the underlayers/resist, and wafer cleaning operations.

Conclusion

The lithographic process based on water-soluble chitosan resist was developed on a 300 mm pilot line. The industrial pre-transfer of chitosan films on 300 mm Si wafer for DUV lithography has been successfully proved. The essential key points were highlighted:

-Clean room controls within IC industry -Solution ready-to-use for spin coating on 300 mm track using a dedicated aqueous-based unit -Spin-coating of a film with thickness of ~ 100 nm ± 9 nm without defects -Photopatterning capability with 193 nm DUV lithography down to 1 µm with preserved patterns profile.

-Reduce the environmental footprint by using a biosourced resist and replacing PGMEA and TMAH organic products by DI water.

The latest achievements are very encouraging to consider eco-friendly chitosan-based resists for lithography process. Prior to its implementation, it requires improving sensitivity and resolution with formulation with added PAGs. Etch plasma transfer in silicon and resist stripping are under investigation on 300 mm industrial tool.

Figure 1 .

 1 Figure 1. Lithographic process steps with chitosan films. (a) Spin-coating of chitosan film (b) Exposure at 193 nm (c) Development step with DI water.

Figure 2 .

 2 Figure 2. Patterning of chitosan at lab-scale. (a) Film deposition on Silica "coupons" 10*10 mm -AFM images of 5 µm lines patterns (b) after 193 nm DUV exposure (c) after plasma etch transfer into silica. Adapted from[START_REF] Sysova | [END_REF]

Figure 3 .

 3 Figure 3. (a) Chitosan film deposited by spin-coating on a 300 mm Si wafer. (b) Associated film thickness profile along the wafer.

Figure 3 .

 3 Figure 3. Top-view CD-SEM images with chitosan dense trenches patterns Half-pitch (HP) (a) HP 5 µm (b) HP 2 µm (c) HP 1 µm (d) HP 0.7 µm after DUV exposures at 700 mJ/cm².

Figure 4 .

 4 Figure 4. Patterning with chitosan resist. (a) Top-view CD-SEM image of 2 µm dense trenches. (b) 3D-AFM profile of 2 µm patterns.

Figure 5 .

 5 Figure 5. LCA comparison based on a batch of 25 wafers  Conventional resist on trilayer stack  Chitosan on trilayer stack  Chitosan on Si substrate.

Table

Table 1 .

 1 Photoresist

	Composition Conventional resist	Bio-sourced resist
	Photoresist	Petrochemical industry	Polysaccharides
		Polyacrylates,	extracted	from
		polystyrenic…	natural sources
	Solvent	Organic solvents	De-ionized water
		PGMEA, Ethyl Lactate,	Acetic acid
		PGME…
	Developer	Alkaline solution	De-ionized water
		TMAH 2.38%

composition comparison: conventional organic resist vs bio-sourced resist

Table 2 .

 2 Key parameters of new resist qualification in semiconductor.

	Clean room rules	Contamination	control
		(metallic, particles)
	Compatibility on substrate	Adhesion, film coating on Si
	Lithography step	Thickness uniformity, profiles
		patterns after development
	Performances	Resolution, sensitivity, LWR
	Etch transfer capability	Profiles preservation, resist
		budget, resist stripping
	Material in solution	Shelf life, purity, viscosity

Table 3 .

 3 Metallic contaminants control.

Acknowledgment

The authors gratefully acknowledge the financial support from the French National Research Agency (ANR) within the ANR-19-CE43-0009 'LITHOGREEN' project.

The authors will acknowledge Ahmed Gharbi, Rémi Cocquand (CEA-Leti) and Agnès Crépet (CNRS-IMP)

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Aurélien Deya

ON THE 1D STOCHASTIC SCHRÖDINGER PRODUCT AURÉLIEN

Keywords: Stochastic Schrödinger equation, fractional noise, renormalization. 1

We exhibit various restrictions about the wellposedness of the Schrödinger product

where Ψ refers to the so-called linear solution of the stochastic Schrödinger problem. We focus more specifically on the case where Ψ satisfies

where 9

B is a white noise in space with fractional time covariance of index H ¡ 1 2 . As an important consequence of our analysis, we obtain that if H is close to 1 2 (that is 9 B is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem pıft ¡ ∆qu λ u p u q 9 B, u 0 0, λ t¡1, 1u, p, q ¥ 1, t R, x T, using only a first-order expansion of the solution ("u Ψ z").

Presentation of the problem

In this paper, we propose to point out some limitations in the analysis of the general class of non-linear stochastic Schrödinger models pıf t ¡ ∆qu λ u p u q 9 B, up0, .q 0, t R, x T,

for fixed integers p, q ¥ 1, λ t¡1, 1u, and where 9 B is a stochastic noise. The study of noise influence on NLS models is a recurring topic in the SPDE literature. The most widely covered situation -by far -is that of a white noise in time with suitably colored spatial covariance. Provided such a noise is regular enough, the solution of (1.1) is expected to take values in a space of functions (almost surely); the powerful Itô integration tools then become available, which even opens the possibility to treat multiplicative perturbations (see e.g. [START_REF] De Bouard | The stochastic nonlinear Schrödinger equation in H 1[END_REF][START_REF] Cheung | Stochastic nonlinear Schrödinger equations on tori[END_REF][START_REF] Oh | On the stochastic nonlinear Schrödinger equations at critical regularities[END_REF][START_REF] Oh | On the stochastic nonlinear Schrödinger equations with non-smooth additive noise[END_REF] for additive-noise models, and [START_REF] Barbu | Stochastic nonlinear Schrödinger equations[END_REF][START_REF] De Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF][START_REF] Brzézniak | On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact Riemannian manifold[END_REF][START_REF] Hornung | The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates[END_REF] for a multiplicative noise).

In contrast with this "functional" case, we are here interested in rougher "distributional" situationsfor which literature turns out to be much more scarce. Namely, in the continuation of [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF], we will focus on examples for which the equation can only be handled in a space of negative-order distributions, and for which renormalization procedures are also required.

In order to implement these ideas, let us consider, throughout the analysis, the case of a white-inspace fractional-in-time noise. Thus, for some (fixed) index H p 1 2 , 1q, 9 B is here defined as the centered Gaussian noise with covariance given by the formula

E 9
Bps, xq 9 Bpt, yq

% |t ¡ s| 2H¡2 δ txyu (1.2)
or otherwise stated: for all test-functions ϕ, ψ on R ¢ T, E

x 9 B, ϕyx 9 B, ψy

% » T dx » R 2 dsdt |t ¡ s| 2¡2H ϕps, xqψpt, xq.
(1.3) Such a noise can be equivalently described through the expansion 9

Bpt, xq ķZ 9 β pkq t e ¡ıkx , (1.4) where the 9 β pkq 's are independent (real-valued) fractional noises on R, with common Hurst index H ¡ 1 2 .

Now, it is a classical fact that the expected regularity of the solution u in (1.1) is prescribed by that of the associated linear problem pıf t ¡ ∆qΨ 9 B, Ψp0, .q 0, t R, x T.

(1.5)

We will thus rely on the following preliminary lemma to identify the distributional cases of interest.

Lemma 1.1. In the above setting, the following assertions hold true.

piq If H ¡ 3 4 , then for every T ¡ 0, one has Ψ L 2 pr0, T s ¢ Tq almost surely. piiq If H ¤ 3 4 , then for every T ¡ 0, one has E

Ψ 2 L 2 pr0,Ts¢Tq % V.
The statement of item piiq corresponds to a slightly extended version of [11, Proposition 2.1, item (ii)] (take H 0 H and H 1 1 2 therein), and it can be proved with the same arguments. As for item piq, we have included a sketch of its proof in Section 6.1, for the sake of clarity.

Based on the result of Lemma 1.1, we henceforth focus on the case where H p 1 2 , 3 4 s, that is the situation where the solution Ψ of (1.5), and accordingly the solution u of (1.1), cannot be defined as functions.

Before we describe the so-called first-order strategy at the core of our investigations, let us introduce a few notations and spaces that will be used throughout the paper.

Notation 1.2. From now on and for the rest of the paper:

We use the classical convention 1 r0,as : ¡1 ra,0s if a 0. We fix a smooth symmetric function χ : R Ñ r0, 1s such that χ 1 on r¡1, 1s and supppχq r¡ 3 2 , 3 2 s. For every function f : R Ñ R and every λ R, we denote the Fourier transform (in time) as Fpf qpλq : » R dx e ¡ıλx f pxq. We denote by I χ the local integration operator, that is for all f : R Ñ R and t R, I χ f ptq : ¡ıχptq » t 0 ds χpsqf psq. (1.6) We also denote by I χ p., .q the Fourier kernel associated with this operator, and defined through the identity

F I χ f ¨pλq » R dλ 1 I χ pλ, λ 1 qFpfqpλ 1 q. (1.7)
For every function ϕ : T Ñ R and every k Z, we set where the notation x.y refers to xλy : p1 |λ| 2 q 1 2 . Also, we set X b : X b 0 and X c : X 0 c .

We denote by pΩ, F, Pq the complete probability space which accommodates the fractional noise 9 B under consideration (or equivalently the one-parameter fractional noises p 9 β pkq q in (1.4)). For any Banach space E, we define L V,¡ pΩ; Eq : £ p¥1 L p pΩ; Eq. Last but not least, we denote by 9 B pnq the (spatial) regularization of 9 B derived from (1.4) through the formula (1.10)

1.1. The first-order strategy and related questions.

Recall that we concentrate here on situations where the solution u of (1.1) is not expected to be a function, so that there is no a priori interpretation of the product term u p u q .

To overcome this problem, a natural idea -which has been extensively implemented in parabolic frameworks, as well as in [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF][START_REF] Forlano | Stochastic nonlinear Schrödinger equation with almost space-time white noise[END_REF] -is to consider some first-order expansion of the solution. We propose to explain the details behind this approach at the level of the approximated equation first.

For 9 B pnq defined as in (1.10), let u pnq be the (well-defined) solution of the approximated equation pıf t ¡ ∆qu pnq λ pu pnq q p pu pnq q q 9 B pnq , u pnq p0, .q 0, t R, x T.

(1.11)

With the notation in (1.6), and setting pnq : I χ e ı∆. 9

B pnq ¨, v pnq ptq : e ıt∆ u pnq ptq, (1.12) it is easy to check that we can (locally) recast (1.11) under the mild formulation v pnq ptq pnq ptq I χ M pp,qq v pnq , . . . , v pnq ¨ptq,

(1. [START_REF] Hornung | The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates[END_REF] where the product operator M pp,qq is defined in Fourier coordinates by the formula M pp,qq pv p1q , . . . , v ppq , w p1q , . . . , w pqq q k ptq m ķ1,...,kp

k1 ... kpm ¸ 1,..., p 1 ... pm¡k e ıtΩ k,k, v p1q k1 ptq ¤ ¤ ¤ v ppq kp ptqw p1q 1 ptq ¤ ¤ ¤ w pqq q ptq, (1.14)
with Ω k,k, : ¡k 2 ¡ pk 2 1 . . . k 2 p q p 2 1 . . . (1.15)

In light of (1.13), a natural first-order transformation of the problem simply consists in the consideration of the difference process z pnq : v pnq ¡ pnq , which now satisfies the equation z pnq ptq I χ M pp,qq z pnq pnq , . . . , z pnq pnq ¨ptq, (1.16) where the right-hand side can be readily expanded by multilinearity.

The whole point of this change of perspective can be roughly summed up as follows: taking the action of the operator I χ M pp,qq into account, we hope the difference-process z pnq to be sufficiently regular (and at least more regular than pnq or v pnq ) so that the product terms in (1.16) can now make sense as n Ñ V, with a suitable control in the scale X b c (see (1.9)).

As a first step of this analysis, we are naturally led to examine the "deterministic" component arising from the multilinear expansion of (1.16), that is the operation

z Þ ÝÑ I χ M pp,qq z, . . . , z ¨.
The wellposedness and stability of such a product in X b c is classically guaranteed by the condition b ¡ 1 2 (see e.g. [START_REF] Kenig | Quadratic forms for the 1-D semilinear Schrödinger equation[END_REF]). For this reason, we shall henceforth assume that b ¡ 1 2 and thus concentrate on the restricted scale

2 X b c , b p 1 2 , 1q, c r0, 1q @ .
Then, when it comes to the analysis of the "stochastic" components in (1.16) (i.e., the terms involving at least one pnq in the expansion of the right-hand side), the following important observation immediately comes to mind.

Observation.

In order to guarantee the convergence of equation (1.16), one should at least be able to address the following two problems:

pP1q Does the sequence of -explicit -stochastic processes I χ M pp,qq pnq , . . . , pnq ¨(1.17) converge in L V,¡ pΩ; X b c q as n Ñ V (for b, c to be determined) ? pP2q Does the stochastic Schrödinger product operation L pnq : z Þ ÝÑ I χ M pp,qq z, pnq , . . . , pnq ¨(1.18) converge as a random operator from X b c to X b c as n Ñ V (for b, c to be determined) ? The two above problems pP1q and pP2q will be our guidelines in the subsequent study. However, for a proper examination of these questions in the rough stochastic framework, the above formulations both need to be refined, which is the purpose of the two next sections. Nevertheless, in order not to deviate from the existing SPDE literature (especially the known results in the heat or wave settings), we shall impose this renormalization procedure to be explicit. In other words, the transformation should only give rise to explicit renormalizing constants at the level of the approximated equation.

In order to achieve this objective, our strategy will obey the following two rules:

pC1q We do allow the use of (natural) renormalization procedures for the explicit process

I χ M pp,qq pnq , . . . , pnq ¨.
pC2q We do not allow any renormalization (and so any deformation) for the general product operation L pnq : z Þ ÝÑ I χ M pp,qq z, pnq , . . . , pnq ¨.

Remark 1.3. Observe that the condition pC2q immediately rules out any "a priori deformation of the product", such as a Skorohod-type interpretation of the problem (see e.g. [START_REF] Chen | Moment estimates for some renormalized parabolic Anderson models[END_REF]), or the renormalization trick implemented in [START_REF] Forlano | Stochastic nonlinear Schrödinger equation with almost space-time white noise[END_REF] for the cubic model, namely:

pıf t ¡ ∆qu ¢ |u| 2 ¡ » T |u| 2 u 9 B, up0, .q Φ, t R, x T.
It is clear indeed that the correction term ³ T |u| 2 derived from the latter transformation is not explicit, i.e. it is not explicitly defined in terms of 9 B.

With these considerations in mind, and following the condition pC1q, we can refine the formulation of the problem pP1q as follows: pP1 I q Can the sequence of stochastic processes I χ M pp,qq pnq , . . . , pnq be suitably renormalized so as to converge in L V,¡ pΩ; X b c q (for b, c to be determined) ?

1.1.2. Stochastic Schrödinger product as a random operator.

Let us now go back to the formulation of the problem pP2q, related to the control of the (Schrödinger) product operation L pnq in (1.18). Following the above condition pC2q, we intend to tackle this product operation directly, that is without any renormalization trick.

Recall that by (1.14), one has with Ω k,k, given by (1.15).

L pnq pzq k I χ ¢ t Þ Ñ m ķ1,.
Based on this expression, one can morally expect the desired regularization effect (in space) to stem from the integration (in time) of the exponential factor e ıtΩ k,k, in (1.19). We are thus naturally led to decompose the expression of L pnq as a sum of a "degenerate" (Schrödinger) product operator

L ¥,pnq pzq k : m ķ1,...,kp k1 ... kpm ¸ 1 ,..., p 1 ... p m¡k 1 tΩ k,k, 0u I χ ¡ t Þ Ñ z k1 ptq pnq k2 ptq ¤ ¤ ¤ pnq kp ptq pnq 1 ptq ¤ ¤ ¤ pnq q ptq © , (1.20)
and a "non-degenerate" (Schrödinger) product L U,pnq pzq k :

I χ ¢ t Þ Ñ m ķ1,...,kp k1 ... kpm ¸ 1 ,..., p 1 ... p m¡k 1 tΩ k,k, $0u e ıtΩ k,k, z k1 ptq pnq k2 ptq ¤ ¤ ¤ pnq kp ptq pnq 1 ptq ¤ ¤ ¤ pnq q ptq . (1.21)
Let us now particularize the formulation of the problem pP2q to each of these two components.

As far as L ¥,pnq is concerned, observe that the degeneracy condition 1 tΩ k,k, 0u actually offers drastic simplifications in the expression of the product, by introducing a strong correlation between the indexes k, k, (see for instance (1.26) in the case p q 1). Thanks to these simplifications, one can legitimately hope for a direct analysis of the operator and a direct treatment of the following question:

pP2 I q Does the operator norm L ¥,pnq

X b c ÑX b c converge (in L V,¡ pΩq) as n Ñ V (for suitable b, c) ?
Unfortunately, due to the much higher sophistication of the non-degenerate component L U,pnq (see Remark 1.6 for a brief comparison between L U,pnq and L ¥,pnq when p q 1), the evaluation of the random -implicitly defined -norm L U,pnq

X b c ÑX b c
turns out to be a much more difficult task. In fact, capturing the value of an operator norm in LpX b c , X b c q (or in any other distributions scale) is known to be a rarely attainable objective in general, and this observation is all the more true in our random setting.

For this reason, we shall instead focus on the analysis of a more tractable estimate of L U,pnq

X b c ÑX b c .
In order to introduce the latter quantity, observe that L U,pnq can be written in Fourier coordinates as

F L U,pnq pzq k ¨pλq ķ1 » dλ 1 K pnq χ ¨kk1 pλ, λ 1 qFpz k1 qpλ 1 q, (1.22)
for some specific kernel K pnq χ (see the expression (2.6) below for a description of K pnq χ in the case p q 1). Now, based on this kernel formulation, it is easily checked (see Section 6.2.1 for details) that for every

z X b c , L U,pnq pzq X b c ¤ z X b c ¤ P pnq b,c , (1.23)
where P pnq b,c is given by

P pnq b,c : ķ 1,k I 1 » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk I 1 y 2c xλ I 1 y 2b § § § § ķ » dλ txky 2c xλy 2b u K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk I 1 pλ, λ I 1 q § § § § 2 .
(1.24)

With bound (1.23) in mind, the -explicit -estimate P pnq b,c is thus the quantity that will serve us as a landmark in the analysis of L U,pnq , and more exactly as an approximation of the operator norm L U,pnq X b c ÑXc . Along this idea, let us particularize the previous control issue pP2q to L U,pnq through the following simplified version of the problem.

pP2 P q Does the quantity P pnq b,c converge in L V,¡ pΩq (or even in L 1 pΩq) as n Ñ V (for suitable b, c) ?

Remark 1.4. In accordance with the developments in [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF], we consider the approximation of L U,pnq

X b
c ÑXc by P pnq b,c as a full part of the first-order strategy here described. Note that similar kernel-based estimates of operator norms are also extensively used in the so-called theory of random tensors developed by Deng, Nahmod and Yue (see e.g. item (5) of Proposition 5.1 or the proof of Proposition 6.1 in [START_REF] Deng | Random tensors, propagation of randomness, and nonlinear dispersive equations[END_REF]).

We have included a short discussion about the sharpness of this approximation in Sections 6.2.1 and 6.2.2 below. In particular, it is therein shown that in the Young integration setting, the consideration of the corresponding quantity P pnq allows us to recover the well-known threshold value H 1 2 for the Hurst index.

Objective of the study.

In the case of the simplest possible nonlinearity in (1.1) (i.e., p q 1), the above-described firstorder strategy is precisely the method that was implemented -with success -in [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF], for a fractional noise of index H ¡ 5 8 . In particular, the three questions pP1 I q, pP2 I q and pP2 P q all received positive answers in the latter situation.

Our aim in the present study is to show that this first-order approach is however not sufficient to cover the whole range 1 2 H 1, thus advocating for more sophisticated developments in rougher situations than those treated in [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF]. This conclusion will be derived from a close examination of the challenging convergence issues pP1 I q, pP2 I q and pP2 P q. Just as in [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF], we will restrict to the most elementary nonlinearity in (1.1), i.e. p q 1, keeping in mind that even stronger restrictions should apply for larger p, q. Notation 1.5. We set from now on M : M p1,1q (see (1.14)), and so Mpv, wq k ptq :

ķ1 e ıtΩ k,k 1 v k k1 w k1 , with Ω k,k1 : |k k 1 | 2 ¡ |k 1 | 2 ¡ |k| 2 2kk 1 .
(1.25)

In particular, 1 tΩ k,k 1 0u 1 tk0utk1ku , and so L ¥,pnq pzq k ķ1

1 tk0utk1ku I χ ¡ t Þ Ñ z k1 ptq pnq k1¡k ptq © , (1.26) while L U,pnq pzq k 1 tk$0u I χ ¢ t Þ Ñ ķ 1$k e ıtΩ k,k 1 ¡k z k1 ptq pnq k1¡k ptq . (1.27)
Remark 1.6. By writing L ¥,pnq pzq k °k1 L ¥,pnq pzq kk1 and similarly L U,pnq pzq k °k1 L U,pnq pzq kk1 , one immediately sees from (1.26)-(1.27) that pL ¥,pnq pzq kk1 q morally reduces to two (time-dependent) vectors, while pL ¥,pnq pzq kk1 q consists of an almost full (time-dependent) matrix. This of course gives us an idea of how much more sophisticated the non-degenerate operator L U,pnq with respect to L ¥,pnq .

Main results

With the above presentation of the problem in mind, we are in a position to state our main results related to the three central questions pP1 I q, pP2 I q and pP2 P q. Recall that we have fixed p q 1 for the rest of the paper. Let us start with the examination of the tree process pnq : I χ M pnq , pnq ¨, which we shall refer to as the (Schrödinger) product tree in the sequel.

As we evoked it earlier, this quantity needs to be renormalized before we can study its convergence. To this end, we will successively rely on three classical rescaling steps: pRq A first partial space averaging Mp pnq , pnq q ÝÑ Mp pnq , pnq q ¡ pnq » pnq , in the spirit of Bourgain's renormalization method for the cubic NLS model (see e.g. [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF][START_REF] Colliander | Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 pTq[END_REF]).

pR I q Then a more standard space averaging:

Mp pnq , pnq q¡ pnq » pnq ÝÑ Mp pnq , pnq q :

Mp pnq , pnq q¡ pR P q Finally, a stochastic Wick renormalization trick:

Mp pnq , pnq q ÝÑ Mp pnq , pnq q ¡ E Mp pnq , pnq q $ . Remark 2.1. Observe that the rescaling terms in Steps pR I q and pR P q only involve reduced quantities, i.e. quantities depending on at most two of the three parameters pt, x, ωq, which can indeed be expected from any reasonable renormalization trick.

On the other hand, the rescaling term in pRq still appeals to the "fully-dependent" quantity pnq .

However, the latter can easily be turned into a linear correction drift at the level of the approximated equation, making it acceptable in the procedure (see [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF]Section 1.2] for details).

As a result of the three steps above, we derive the following renormalized version of the product tree:

pnq k ptq : I χ ¡ Mp pnq , pnq q k p.q ¡ E Mp pnq , pnq q k p.q $ © ptq, (2.2)
where the renormalized product Mp pnq , pnq q is defined by (2.1). Based on the expression (1.25) of M, this renormalized product can also be written in Fourier coordinates as Mp pnq , pnq q k ptq :

1 tk$0u ķ 1 $0 e ıtΩ k,k 1 pnq k k1 pnq k1 . (2.3)
In other words, we are here dealing with the "non-degenerate" part of pnq , i.e. the summands for which the quantity Ω k,k1 does not vanish.

Our main result about this (renormalized) product tree can now be stated as follows.

Proposition 2.2. Assume that H p 1 2 , 3 4 s. Then the following picture holds true. piq For every 0 ¤ c 1 2 , one has

sup n¥1 E pnq 2 Xc % V. piiq For every 1 2 b 2H ¡ 1 2 , one has sup n¥1 E pnq 2 X b % V. piiiq If b 2H ¡ 1 2 , then E pnq 2 X b % nÑV ÝÑ V.
The above properties thus provide us with the exact time regularity of (namely p2H ¡ 1 2 q¡), which,

given the central role of the process in the dynamics of (1.1), can be seen as an important result of independent interest. In view of our present objective, the optimal condition 1 2 b 2H ¡ 1 2 will be our first restriction in the application of the first-order strategy described in Section 1.1.

Problem (P2'): about the Schrödinger product operator L ¥,pnq .

Let us now turn to the issues related to the Schrödinger product operator L pnq , starting from its degenerate -and relatively simple -component L ¥,pnq (see (1.26)).

Our main result about L ¥,pnq will actually be derived from a careful examination of the action of the operator on suitable (Gaussian) processes. The property can be summed as follows.

Proposition 2.3. Assume that H p 1 2 , 3 4 q and fix b p 1 2 , 1q.

If 0 ¤ c 3 2 ¡ 2H, then there exists a sequence pZ pnq q n¥1 of random functions such that

sup n¥1 E Z pnq q X b c % V for every q ¥ 2, and E L ¥,pnq Z pnq 2 X b c % nÑV ÝÑ V.
(2.4)

In particular, if 0 ¤ c 3 2 ¡ 2H, then for every p ¡ 2, one has E L ¥,pnq p X b c ÑX b c % nÑV ÝÑ V.
Proposition 2.4 therefore offers a partial answer to the guideline question pP2 I q. In particular here, the result gives birth to our second restriction on the spaces X b c involved in the first-order analysis: namely, one must have c ¥ 3 2 ¡ 2H.

Problem (P2"

): about the Schrödinger product operator L U,pnq .

As explained in the above Section 1.1.2, we only focus here on the convergence issue for the approxi-

mation P pnq b,c of L U,pnq X b c ÑX b c
given by the expression

P pnq b,c : ķ 1 ,k I 1 » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk I 1 y 2c xλ I 1 y 2b § § § § ķ » dλ txky 2c xλy 2b u K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk I 1 pλ, λ I 1 q § § § § 2 , ( 2.5)
where the Fourier kernel K pnq χ is defined through the identity

F L U,pnq pzq k ¨pλq ķ1 » dλ 1 K pnq χ ¨kk1 pλ, λ 1 qFpz k1 qpλ 1 q.
In the setting p q 1 under consideration, it is readily checked that K pnq χ is in fact given by

K pnq χ ¨kk1 pλ, λ 1 q : 1 tk$0u 1 tk1$ku » R dλ 2 F pnq k1¡k ¨pλ 2 qI χ pλ, Ω k,k1¡k λ 1 ¡ λ 2 q, (2.6)
where the integration kernel I χ p., .q has been introduced in (1.7).

Our main result about P pnq b,c , partially responding to the problem pP2 P q, reads as follows. Proposition 2.4.

Let H p 1 2 , 3 4 q, b p 1 2 , 1q and c p0, 1q. If c ¥ b ¡ 1 4 , then one has E § § P pnq b,c § § % nÑV ÝÑ V.
Proposition 2.4 therefore points out our third (and last) restriction on the class of spaces X b c suitable for a first-order strategy: one must impose that c b ¡ 1 4 .

Conclusion.

By gathering the constraints exhibited in Propositions 2.2, 2.3 and 2.4, we can conclude our investigations about the limits of the first-order strategy for the stochastic Schrödinger problem (1.1).

Corollary 2.5. If 1 2 H ¤ 9
16 , then for any pair pb, cq p 1 2 , 1q ¢ r0, 1q, one has either pnq

X b nÑV ÝÑ V, L ¥,pnq X b c ÑX b c nÑV ÝÑ V, or P pnq b,c nÑV ÝÑ V in L V,¡ pΩq.
(2.7)

In particular, for

1 2 H ¤ 9
16 , the stochastic Schrödinger problem (1.1) cannot be treated with the first-order strategy described in Section 1.1.

Proof. For none of the two explosions in (2.7) to happen, it is necessary that b

2H ¡ 1 2 (Proposition 2.2), c ¥ 3 2 ¡ 2H (Proposition 2.3) and c b ¡ 1 4 (Proposition 2.4), which can be summed up as 3 2 ¡ 2H ¤ c b ¡ 1 4 2H ¡ 3 4 ,
and hence one must have H ¡ 9

16 .

These results thus call for the development of more sophisticated methods, such as paracontrolled or random-tensor-type strategies, in order to cover the whole range H ¡ 1 2 for 9 B. To be more specific, we do not expect any possible improvement regarding the constraints on the Schrödinger product tree (Proposition 2.2), and we only advocate for a more sophisticated treatment of the product operation z Þ Ñ I χ M pp,qq pz, pnq q. These further developments could for instance be derived from a suitable "ansatz" formulation of the problem (see [START_REF] Deng | Random tensors, propagation of randomness, and nonlinear dispersive equations[END_REF]Section 5.2]), which we plan to investigate in a future study.

Finally, it is worth noting that there is no hope to reach the case of a space-time white noise 9 B through the present renormalization method, as it can be seen from the following explosion result. Proposition 2.6. Assume that H 1 2 , that is 9 B is a space-time white noise on R ¢ T. Then, with the notation of Section 2.1 and for every b ¥ 1 2 , it holds that

E pnq 2 X b % nÑV ÝÑ V.
The latter property can be established with similar arguments as those in the proof of Proposition 2.2, item piiiq. For the sake of conciseness, we prefer to leave the details as an exercise to the reader.

The rest of the paper is organized as follows. In Section 3, we focus on the analysis of the Schrödinger product tree and on the proof of Proposition 2.2. Sections 4 and 5 are then devoted to the study of the (stochastic) Schrödinger product operator L pnq : Section 4 contains the proof of Proposition 2.3, while Section 5 contains the proof of Proposition 2.4. Finally, the Appendix section contains a partial proof of Lemma 1.1 (Section 6.1), as well as a discussion about the relevance of P pnq b,c as an approximation of L pnq X b c ÑXc (Section 6.2).

Convergence of the Schrödinger product tree

We start with the study of the convergence issue for the renormalized Schrödinger product tree pnq k ptq : I χ ¡ Mp pnq , pnq q k p.q ¡ E Mp pnq , pnq q k p.q $ © ptq,

introduced in Section 2.1. We recall in particular that the (rescaled) linear solution pnq has been intro- duced in (1.12), while the renormalized product operator M is defined in Fourier coordinates by Mpv, wq k ptq :

1 tk$0u ķ 1$0 e ıtΩ k,k 1 v k k1 w k1 , with Ω k,k1 : |k k 1 | 2 ¡ |k 1 | 2 ¡ |k| 2 2kk 1 . (3.2)
As a preliminary step, let us point out some useful expressions and estimates for the covariance of the process pnq , at the core of expression (3.1).

Covariance of the linear solution.

One has by definition

Fp pnq k qpλq 1 txky¤2 n u F I χ e ¡ı.k 2 9
β pkq ¨¨pλq and we can use the integration kernel I χ p., .q (see (1.7)) to express this quantity as

Fp pnq k qpλq 1 txky¤2 n u » dλ 1 I χ pλ, λ 1 qF e ¡ı.k 2 9 β pkq ¨pλ 1 q 1 txky¤2 n u » dλ 1 I χ pλ, λ 1 q » dt e ¡ıpλ1 k 2 qt 9 β pkq t .
Based on this expression, and since

E 9 β pkq t 9 β pk I q t I % 1 tkk I u |t ¡ t I | 2H¡2 , we get that E Fp pnq k qpλqFp pnq k I qpλ I q % 1 txky¤2 n u 1 txk I y¤2 n u » dλ 1 dλ I 1 I χ pλ, λ 1 qI χ pλ I , λ I 1 q » dtdt I e ¡ıpλ1 |k| 2 qt e ıpλ I 1 |k I | 2 qt I E 9 β pkq t 9 β pk I q t I % 1 tkk I u 1 txky¤2 n u » dλ 1 dλ I 1 I χ pλ, λ 1 qI χ pλ I , λ I 1 q » dtdt I e ¡ıpλ1 |k| 2 qt e ıpλ I 1 |k| 2 qt I |t ¡ t I | 2H¡2 c 1 tkk I u 1 txky¤2 n u » R dξ |ξ| 2H¡1 » dλ 1 dλ I 1 I χ pλ, λ 1 qI χ pλ I , λ I 1 q » dtdt I e ¡ıpλ1 |k| 2 ξqt e ıpλ I 1 |k| 2 ξqt I c 1 tkk I u 1 txky¤2 n u » R dξ |ξ| 2H¡1 I χ pλ, ¡ξ ¡ k 2 qI χ pλ I , ¡ξ ¡ k 2 q. (3.3)
For a more detailed expression of this quantity, let us introduce the following notation.

and so we can rephrase (3.3) as

E Fp pnq k qpλqFp pnq k I qpλ I q % c 1 tkk I u 1 txky¤2 n u » dtdt I χptqχpt I qe ¡ıλt e ıλ I t I » R dξ |ξ| 2H¡1 Ξ t p¡ξ ¡ k 2 qΞ t Ip¡ξ ¡ k 2 q. (3.6)
Besides, based on (3.5), it is readily checked that § § Ipλ, λ 1 q § § À

1 xλyxλ ¡ λ 1 y , (3.7)
and therefore the expression in (3.3) leads us to the following uniform estimate (with respect to n):

for all k, k I Z and λ, λ I R, § § § § E Fp pnq k qpλqFp pnq k I qpλ I q % § § § § À 1 tkk I u 1 xλyxλ I y » R dξ |ξ| 2H¡1 1 xξ λ k 2 y 1 xξ λ I k 2 y À 1 tkk I u 1 xλyxλ I y 1 xλ ¡ λ I y 1¡ε 1 xλ k 2 y 2H¡1 1 xλ I k 2 y 2H¡1 & , ( 3.8)
where the second inequality follows from [START_REF] Deya | Renormalization of a 1d quadratic Schrödinger model with additive noise[END_REF]Lemma 3.8].

With similar arguments, we obtain that

E Fp pnq k qpλqFp pnq k I qpλ I q % c 1 tkk I u 1 txky¤2 n u » R dξ |ξ| 2H¡1 I χ pλ, ¡ξ ¡ k 2 qI χ pλ I , ξ ¡ k 2 q, ( 3.9)
and then, for all k, k I Z and λ, λ

I R, § § § § E Fp pnq k qpλqFp pnq k I qpλ I q % § § § § À 1 tkk I u 1 xλyxλ I y » R dξ |ξ| 2H¡1 1 xξ λ k 2 y 1 xξ ¡ λ I ¡ k 2 y À 1 tkk I u 1 xλyxλ I y 1 xλ λ I 2k 2 y 1¡ε 1 xλ k 2 y 2H¡1 1 xλ I k 2 y 2H¡1 & . (3.10)
We are now in a position to tackle the proof of our main asymptotic result.

Proof of Proposition 2.2.

First, using the integration kernel I χ p., .q (see (1.7)) and the Fourier expression of M (see (3.2)), we can write the Fourier transform of the process under consideration as

F I χ Mp pnq , pnq q k ¨pλq » R dλ I I χ pλ, λ I qF Mp pnq , pnq q k ¨pλ I q 1 tk$0u ķ 1$0 » R dλ I I χ pλ, λ I q » dt e ¡ıλ I t e ıtΩ k,k 1 pnq k k1 ptq pnq k1 ptq 1 tk$0u ķ 1$0 » dλ 1 Fp pnq k1 qpλ 1 q » dλ 2 Fp pnq k k1 qpλ 2 q » R dλ I I χ pλ, λ I q
» dt e ¡ıλ I t e ıtΩ k,k 1 e ıtλ2 e ¡ıtλ1 , and thus

F I χ Mp pnq , pnq q k ¨¨pλq 1 tk$0u ķ 1 $0 » dλ 1 dλ 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qFp pnq k1 qpλ 1 qFp pnq k k1 qpλ 2 q.
Based on this expression, and by applying Wick's formula, we can compute

E § § F I χ Mp pnq , pnq q k ¨pλq § § 2 % 1 tk$0u ķ 1 $0 » dλ 1 dλ 2 ķ I 1 $0 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k I 1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k k1 qpλ 2 qFp pnq k I 1 qpλ I 1 qFp pnq k k I 1 qpλ I 2 q % § § §E F I χ Mp pnq , pnq q k ¨pλq $ § § § 2 1 tk$0u ķ 1$0 » dλ 1 dλ 2 ķ I 1 $0 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k I 1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k I 1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k I 1 qpλ I 2 q % 1 tk$0u ķ 1$0 » dλ 1 dλ 2 ķ I 1 $0 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k I 1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k k I 1 qpλ I 2 q % E Fp pnq k k1 qpλ 2 qFp pnq k I 1 qpλ I 1 q % . (3.11)
As far as the last term of this expansion is concerned, observe that according to (3.9), one has

1 tk$0u E Fp pnq k1 qpλ 1 qFp pnq k k I 1 qpλ I 2 q % E Fp pnq k k1 qpλ 2 qFp pnq k I 1 qpλ I 1 q % 1 tk$0u 1 tk1k k I 1 u 1 tk k1k I 1 u E Fp pnq k1 qpλ 1 qFp pnq k k I 1 qpλ I 2 q % E Fp pnq k k1 qpλ 2 qFp pnq k I 1 qpλ I 1 q % 1 tk$0u 1 tk1k k I 1 u 1 t2k k I 1 k I 1 u E Fp pnq k1 qpλ 1 qFp pnq k k I 1 qpλ I 2 q % E Fp pnq k k1 qpλ 2 qFp pnq k I 1 qpλ I 1 q % 0.
For the same reason,

E Fp pnq k1 qpλ 1 qFp pnq k I 1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k I 1 qpλ I 2 q % 1 tk1k I 1 u E Fp pnq k1 qpλ 1 qFp pnq k1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k1 qpλ I 2 q % .
Thus, going back to (3.11) and recalling the definition (3.1) of pnq , we deduce that

E § § F pnq k ¨pλq § § 2 % E § § F I χ Mp pnq , pnq q k ¨pλq § § 2 % ¡ § § §E F I χ Mp pnq , pnq q k ¨pλq $ § § § 2 1 tk$0u ķ 1$0 » dλ 1 dλ 2 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k1 qpλ I 2 q % . (3.12)
Using the estimates (3.7) and (3.8), we get that for all 0 ¤ c 1 2 and ε ¡ 0 small enough,

E pnq 2 Xc % À ķ$0 ķ 1$0 xky 2c » dλ xλy 2 » dλ 1 xλ 1 y dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y » dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y À ķ$0 ķ 1$0 xky 2c » dλ xλy 2 § § § § » dλ 1 xλ 1 y » dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y § § § § 2 À ķ$0 ķ 1$0 xky 2c » dλ xλy 2 § § § § » dλ 1 xλ 1 y 1 xλ ¡ Ω k,k1 λ 1 y 1¡ ε 2 § § § § 2 À ķ$0 ķ 1$0 xky 2c » dλ xλy 2 1 xλ ¡ Ω k,k1 y 2¡2ε À ķ$0 xky 2c ķ 1$0 1 xΩ k,k1 y 2¡2ε À ķ 1 xky 2¡2c¡2ε ķ1 1 xk 1 y 2¡2ε V,
which already proves item piq of the statement.

Let us now focus on the norm in X b , which, thanks to (3.12), can be expanded as

E pnq 2 X b % ķ$0 » dλ xλy 2b ķ 1$0 » dλ 1 dλ 2 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k1 qpλ I 2 q % . 3.2.1. Proof of item piiq. Assume here that 1 2 b 2H ¡ 1
2 . Using the estimates (3.7) and (3.8), we get that

E pnq 2 X b % À ķ$0 » dλ xλy 2¡2b ķ 1$0 » dλ 1 xλ 1 y dλ 2 xλ 2 y » dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y 1 xλ 1 k 2 1 y 2H¡1 1 xλ I 1 k 2 1 y 2H¡1 & 1 xλ 2 pk k 1 q 2 y 2H¡1 1 xλ I 2 pk k 1 q 2 y 2H¡1 & À A b B b , with A b : ķ$0 ķ 1 $0 » dλ xλy 2¡2b » dλ 1 xλ 1 y dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y 1 xλ 1 k 2 1 y 2H¡1 1 xλ 2 pk k 1 q 2 y 2H¡1 » dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y
and B b :

ķ$0 ķ 1 $0 » dλ xλy 2¡2b » dλ 1 xλ 1 y dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y 1 xλ 1 k 2 1 y 2H¡1 » dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y 1 xλ I 2 pk k 1 q 2 y 2H¡1 .
Since A b and B b no longer depend on n, we only need to prove that these two quantities are finite.

For A b , note first that

» dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y À » dλ I 1 xλ I 1 y 1 xλ ¡ Ω k,k1 λ I 1 y 1¡ ε 2 À 1 xλ ¡ Ω k,k1 y 1¡ε .
On the other hand,

» dλ 1 xλ 1 y dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y 1 xλ 1 k 2 1 y 2H¡1 1 xλ 2 pk k 1 q 2 y 2H¡1 À » dλ 1 xλ 1 y 1 xλ 1 k 2 1 y 2H¡1 » dλ 2 xλ 2 y 1 xλ 2 pk k 1 q 2 y 2H¡1 À 1 xk 1 y 4H¡2¡ε 1 xk k 1 y 4H¡2¡ε .
Thus, going back to the definition of A b , we obtain that for ε ¡ 0 small enough,

A b À ķ 1 $0 1 xk 1 y 4H¡2¡ε ķ$0 1 xk k 1 y 4H¡2¡ε » dλ xλy 2¡2b 1 xλ ¡ Ω k,k1 y 1¡ε (3.13) À ķ 1 $0 1 xk 1 y 4H¡2b¡2ε ķ$0 1 xk k 1 y 4H¡2¡ε 1 xky 2¡2b¡ε .
Since b 2H ¡ 1 2 , one has 4H ¡ 2b ¡ 1, and so we can deduce that

A b À ķ 1 $0 1 xk 1 y 8H¡4b¡1¡4ε V,
for ε ¡ 0 small enough.

As far as B b is concerned, one has

» dλ 1 xλ 1 y dλ 2 xλ 2 y 1 xλ ¡ Ω k,k1 ¡ λ 2 λ 1 y 1 xλ 1 k 2 1 y 2H¡1 À » dλ 1 xλ 1 y 1 xλ 1 k 2 1 y 2H¡1 1 xλ ¡ Ω k,k1 λ 1 y 1¡ε À ¢ » dλ 1 xλ 1 y 1 xλ 1 k 2 1 y 4H¡2 1 2 ¢ » dλ 1 xλ 1 y 1 xλ ¡ Ω k,k1 λ 1 y 2¡2ε 1 2 À 1 xk 1 y 4H¡2¡ε 1 xλ ¡ Ω k,k1 y 1 2
and in the same way

» dλ I 1 xλ I 1 y dλ I 2 xλ I 2 y 1 xλ ¡ Ω k,k1 ¡ λ I 2 λ I 1 y 1 xλ I 2 pk k 1 q 2 y 2H¡1 À 1 xk k 1 y 4H¡2¡ε 1 xλ ¡ Ω k,k1 y 1 2 , which gives
B b À ķ 1$0 1 xk 1 y 4H¡2¡ε ķ$0 1 xk k 1 y 4H¡2¡ε » dλ xλy 2¡2b 1 xλ ¡ Ω k,k1 y .
Thus, we are in the same position as in (3.13), and we can use the same arguments to assert that B b V.

This achieves to prove that for every b such that

q % E Fp pnq k k1 qpλ 2 qFp pnq k k1 qpλ I 2 q % 1 txk1y¤2 n u » R dξ |ξ| 2H¡1 » dtdt I χptqχpt I qe ıλ1t e ¡ıλ I 1 t I Ξ t p¡ξ ¡ k 2 1 qΞ t Ip¡ξ ¡ k 2 1 q 1 txk k1y¤2 n u » R dη |η| 2H¡1 » dsds I χpsqχps I qe ¡ıλ2s e ıλ I 2 s I Ξ s p¡η ¡ pk k 1 q 2 qΞ s Ip¡η ¡ pk k 1 q 2 q. (3.14)
Then observe that by (3.5),

» dλ 1 dλ 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qe ıλ1t e ¡ıλ2s » R
and in a similar way

» dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ I 2 ¡ λ I 1 qe ¡ıλ I 1 t I e ıλ I 2 s I δ ts I t I u χpt I qe ¡ıt I Ω k,k 1 » R
dw e ıλw χpwq1 r0,ws pt I q.

» dλ 1 dλ 2 » dλ I 1 dλ I 2 I χ pλ, Ω k,k1 λ 2 ¡ λ 1 qI χ pλ, Ω k,k1 λ I 2 ¡ λ I 1 q E Fp pnq k1 qpλ 1 qFp pnq k1 qpλ I 1 q % E Fp pnq k k1 qpλ 2 qFp pnq k k1 qpλ I 2 q % » R dξ |ξ| 2H¡1 » R dη |η| 2H¡1 ¢ » R dv e ¡ıλv χpvq » dt 1 r0,vs ptqe ıtΩ k,k 1 χptq 3 Ξ t p¡ξ ¡ k 2 1 qΞ t p¡η ¡ pk k 1 q 2 q ¢ » R dw e ıλw χpwq » dt I 1 r0,ws pt I qe ¡ıt I Ω k,k 1 χpt I q 3 Ξ t Ip¡ξ ¡ k 2 1 qΞ t Ip¡η ¡ pk k 1 q 2 q
and so, going back to (3.12), we get the expression

E pnq 2 X b % ķ$0 ķ 1 $0 1 txk1y¤2 n u 1 txk k1y¤2 n u » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ xλy 2b § § Q Ω k,k 1 ξ k 2 1 ,η pk k1q 2 pλq § § 2 ,
where we have set

Q L β,β Ipλq :
» dt e ¡ıλt χptq » t 0 ds e ısL χpsq 3 Ξ s p¡βqΞ s p¡β I q.

In particular,

E pnq 2 X b % ¥ ķ¥1 ķ 1¥1 1 txk1y¤2 n u 1 txk k1y¤2 n u » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § Q Ω k,k 1 ξ k 2 1 ,η pk k1q 2 pλq § § 2 ¥ ķ¥1 ķ 1 ¥1 1 txk1y¤2 n u 1 txk k1y¤2 n u Ω 1 2b k,k1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § Q Ω k,k 1 ξ k 2 1 ,η pk k1q 2 pλΩ k,k1 q § § 2 .
For all L ¥ 1, we can use an integration-by-parts argument to decompose Q L β,β IpλLq as Using this decomposition, we can write

Q L β,
E pnq 2 X b % ¥ ķ¥1 ķ 1 ¥1 1 txk 1 y¤2 n u 1 txk k 1 y¤2 n u Ω 1 2b k,k 1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § 2 ķ¥1 ķ 1 ¥1 1 txk 1 y¤2 n u 1 txk k 1 y¤2 n u Ω 1 2b k,k 1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § R Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § 2 ¡ 2 ķ¥1 ķ 1 ¥1 Ω 1 2b k,k 1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § § § R Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § ¥ ķ¥1 ķ 1 ¥1 1 txk 1 y¤2 n u 1 txk k 1 y¤2 n u Ω 1 2b k,k 1 » dξ 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § 2 ¡ 2 ķ¥1 ķ 1 ¥1 Ω 1 2b k,k 1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § § § R Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § .
By applying Lemma 3.2 below, we get that for all Ω ¥ 1, k, ¥ 1 and ε ¡ 0 small enough,

» dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω ξ k,η pλq § § § § R Ω ξ k,η pλq § § À 1 Ω 4¡ε » |λ|¥2 dλ |λ| 4¡2b » dξ |ξ| 2H¡1 » dη |η| 2H¡1 1 xξ ky 1 xη y » |λ|¤2 dλ |λ| 2¡2b |λ ¡ 1| 1¡ε » dξ |ξ| 2H¡1 » dη |η| 2H¡1 ¢ 1 xξ kyxη y 1 xξ pk Ωqyxη y 1 xξ pk Ωqyxη p ¡ Ω ¡ ξ ¡ kqy 1 xη p ¡ Ωqyxξ ky 1 xη p ¡ Ωqyxξ pk Ω ¡ η ¡ qy & À 1 Ω 4¡ε ,
where we have used the fact that for all 0 ¤ α β ¤ 1 such that α β ¡ 1, one has

sup aR » R dξ |ξ| α 1 xξ ay β V. Therefore, for ε ¡ 0 small enough, ķ¥1 ķ 1 ¥1 Ω 1 2b k,k 1 » dξdη |ξ| 2H¡1 |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § § § R Ω k,k 1 ξ k 2 1 ,η pk k 1 q 2 pλq § § À ķ¥1 ķ 1 ¥1 1 Ω 3¡2b¡ε k,k 1 À ¢ ķ¥1 1 k 3¡2b¡ε 2 V, due to b 1.
We are thus left with the analysis of M pnq :

ķ¥1 ķ 1¥1 1 txk1y¤2 n u 1 txk k1y¤2 n u Ω 1 2b k,k1 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω k,k 1 ξ k 2 1 ,η pk k1q 2 pλq § § 2 . (3.19)
To this end, write for all Ω ¥ 1,

» dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω ξ k 2 1 ,η pk k1q 2 pλq § § 2 1 |Ω| 2 » dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2¡2b § § § § » dt e ¡ıΩpλ¡1qt χptq 4 Ξ t p¡ξ ¡ k 2 1 qΞ t p¡η ¡ pk k 1 q 2 q § § § § 2 1 |Ω| 2 » dtdt I χptq 4 χpt I q 4
¢ » dλ |λ| 2¡2b e ¡ıλΩpt¡t I q e ıΩpt¡t I q J k1 pt, t I qJ k k1 pt, t I q, where we have set

J pt, t I q : » dξ |ξ| 2H¡1 Ξ t p¡ξ ¡ 2 qΞ t Ip¡ξ ¡ 2 q.
Then, by elementary transformations,

1 |Ω| 2 » dtdt I χptq 4 χpt I q 4 ¢ » dλ |λ| 2¡2b e ¡ıλΩpt¡t I q e ıΩpt¡t I q J k1 pt, t I qJ k k1 pt, t I q 1 |Ω| 1 2b » dtdt I |t ¡ t I | 2b¡1 χptq χpt I q 4 e ıΩpt¡t I q J k1 pt, t I qJ k k1 pt, t I q 1 |Ω| 1 2b » dtds |s| 2b¡1 χptq 4 χpt ¡ sq 4 e ıΩs J k1 pt, t ¡ sqJ k k1 pt, t ¡ sq 1 |Ω| 3 » dtds |s| 2b¡1 χptq 4 χ ¡ t ¡ s Ω © 4 e ıs J k1 ¡ t, t ¡ s Ω © J k k1 ¡ t, t ¡ s Ω © . ( 3.20)
Let us now expand the two quantities J k1 t, t ¡ s Ω ¨and J k k1 t, t ¡ s Ω ¨. On the one hand,

J k1 ¡ t, t ¡ s Ω © » dξ |ξ| 2H¡1 Ξ t p¡ξ ¡ k 2 1 qΞ t¡ s Ω p¡ξ ¡ k 2 1 q » dξ |ξ| 2H¡1 ¢ » t 0 dr χprqe ıpξ k
J k k1 ¡ t, t ¡ s Ω © 1 pk k 1 q 4H¡2 » du |u| 2¡2H e ¡ıu » du I 1 r0,t¡ s Ω s pu I q1 r0,ts ¡ u I u pk k 1 q 2 © χpu I qχ ¡ u I u pk k 1 q 2 © .
By injecting these two expansions into (3.20), we deduce the expression

» dξ |ξ| 2H¡1 » dη |η| 2H¡1 » dλ |λ| 2b § § M Ω ξ k 2 1 ,η pk k1q 2 pλq § § 2 1 |Ω| 3 1 k 4H¡2 1 1 pk k 1 q 4H¡2 » dt χptq 4 » ds |s| 2b¡1 e ıs χ ¡ t ¡ s Ω © 4 ¢ » dr |r| 2¡2H e ır » dr I 1 r0,t¡ s Ω s pr I q1 r0,ts ¡ r I r k 2 1 © χpr I qχ ¡ r I r k 2 1 © ¢ » du |u| 2¡2H e ¡ıu » du I 1 r0,t¡ s Ω s pu I q1 r0,ts ¡ u I u pk k 1 q 2 © χpu I qχ ¡ u I u pk k 1 q 2 © 1 |Ω| 3 1 k 4H¡2 1 1 pk k 1 q Q b,H pΩ, k, k 1 q, (3.21)
where we have set

Q b,H pΩ, k, k 1 q : » dt » dr I » du I χptq 4 χpr I qχpu I q ¢ » ds |s| 2b¡1 e ıs 1 r0,t¡ s Ω s pr I q1 r0,t¡ s Ω s pu I qχ ¡ t ¡ s Ω © 4 ¢ » dr |r| 2¡2H e ır 1 r0,ts ¡ r I r k 2 1 © χ ¡ r I r k 2 1 © ¢ » du |u| 2¡2H e ¡ıu 1 r0,ts ¡ u I u pk k 1 q 2 © χ ¡ u I u pk k 1 q 2 © .
Using the subsequent Lemma 3.3, we can easily check that

Q b,H pΩ k,k1 , k, k 1 q Ñ q b,H as k, k 1 Ñ V,
where q b,H :

¢ » ds |s| 2b¡1 e ıs § § § § » dr |r| 2¡2H e ır § § § § 2 ¢ » dt χptq 8 » t 0 dr I χpr I q 2 » t 0 du I χpu I q 2 .
Since q b,H ¡ 0, there exists K ¥ 1 large enough such that for all

k 1 ¥ k ¥ K, one has Q b,H pΩ k,k1 , k, k 1 q ¥ q b,H 2 .
Therefore, going back to (3.19) and (3.21), we get that

M pnq Á ķ¥K 1 |k| 2¡2b ķ 1 ¥k 1 txk1y¤2 n u 1 txk k1y¤2 n u 1 |k 1 | 4H¡2b 1 |k k 1 | 4H¡2 .
and since K does not depend on n, we deduce that

lim inf nÑV M pnq Á ķ¥K 1 |k| 2¡2b ķ 1¥k 1 |k 1 | 4H¡2b 1 |k k 1 | 4H¡2 Á ķ¥K 1 |k| 8H¡4b¡1 ¢ 1 k ķ 1¥k 1 | k1 k | 4H¡2b 1 |1 k1 k | 4H¡2 Á ķ¥K 1 |k| 8H¡4b¡1 » V 1 dx |x| 4H¡2b |1 x| 4H¡2 Á ķ¥K 1 |k| 8H¡4b¡1 » V 1 dx |x| 8H¡2b¡2 .
At this point, recall that b 2H ¡ 1 2 , and so the latter bound reduces in fact to

ķ¥K 1 |k| 8H¡4b¡1 » V 1 dx |x| 8H¡2b¡2 ¢ ķ¥K 1 |k| ¢ » V 1 dx |x| 4H¡1
V, which leads us to the desired conclusion lim nÑV M pnq V.

Auxiliary lemmas.

The few technical results below have been used in the proof of Proposition 2.2.

Lemma 3.2. For all L ¥ 1 and β, β I R, let M L β,β I and R L β,β I be the functions on R defined by (3.17) and (3.18), respectively. Then the following estimates hold true:

piq For all |λ| ¥ 2, one has § § M L β,β Ipλq § § À 1 L 2 |λ| 2 and § § R L β,β Ipλq § § À 1 L 2 |λ| 2 1 xβyxβ I y . piiq For all |λ| ¤ 2, one has § § M L β,β Ipλq § § À 1 L 2¡ε 1 |λ||λ ¡ 1| 1¡ε and § § R L β,β Ipλq § § À 1 L 2 |λ| 1 xβyxβ I y 1 xL βyxβ I y 1 xL βyxL β ¡ β I y 1 xL ¡ β I yxβy 1 xL ¡ β I yxL β ¡ β I y & . (3.22)
Proof. piq If |λ| ¥ 2, then by an elementary integration-by-parts argument, we get that

§ § M L β,β Ipλq § § À 1 L 2 |λ| 2 » dt § § f t pχp.q 4 qptq § § § § Ξ t p¡βq § § § § Ξ t p¡β I q § § § § χptq § § 4 § § f t pΞ . p¡βqqptq § § § § Ξ t p¡β I q § § § § χptq § § 4 § § Ξ t p¡βq § § § § f t pΞ . p¡β I qqptq § § % À 1 L 2 |λ| 2 ,
where we have used the uniform bound

|Ξ t p¡βq| |f t Ξ t p¡βqq| À 1.
(3.23)

In the same way, § § R L β,β Ipλq § § À piiq For M L β,β Ipλq, we can use again (3.23) and an integration-by-parts argument to write, for |λ| ¤

2, § § M L β,β Ipλq § § À 1 L|λ| § § § § » dt e ¡ıLtpλ¡1q χptq 4 Ξ t p¡βqΞ t p¡β I q § § § § 1¡ε À 1 L 2¡ε |λ||λ ¡ 1| 1¡ε .
As for R L β,β Ipλq, observe first that |s| α e ıs 2 1 r0,t¡ s L s pr I q ¡ 1 r0,ts pr I q @ 1 r0,t¡ s L s pu I qχ

¡ t ¡ s L © § § § § 1 r0,ts pr I q § § § § » R ds |s| α e ıs 2 1 r0,t¡ s L s pu I q ¡ 1 r0,ts pu I q @ χ ¡ t ¡ s L © § § § § 1 r0,ts pr I q1 r0,ts pu I q § § § § » R ds |s| α e ıs 4 χ ¡ t ¡ s L © ¡ χ t ¨B § § § § (3.29)
For the first integral in (3.29), observe for instance that for almost every s ¥ 0, 1 r0,t¡ s L s pr I q ¡ 1 r0,ts pr I q 1 tr I ¥0u 2 which, combined with the previous decomposition 1 r0,t¡ s L s pu I q 1 tu I ¥0u 1 ts¤Lpt¡u I qu ¡ 1 tu I 0u 1 ts¥Lpt¡u I qu , yields 2 1 r0,t¡ s L s pr I q ¡ 1 r0,ts pr I q @ 1 r0,t¡ s L s pu I q ¡ 1 r0,t¡ s L s pr I q ¡ 1 r0,ts pr I q @ 1 r0,t¡ s L s pu I qχ

1 tt¥r I s L u ¡ 1 tt¥r I u @ ¡ 1 tr I 0u 2
1 tt¤r I s L u ¡ 1 tt¤r I u @ ¡ 2
1 tr I ¥0u
¡ t ¡ s L © § § § § À 1 L α max ¢ 1 |t ¡ r I | α , 1 |t ¡ u I | α
, which corresponds to the desired bound.

The bound for the second integral in (3.29) immediately follows from the same arguments, that is from the combination of (3.30) with (3.32): we get here that §

§ § § » R ds |s| α e ıs 2 1 r0,t¡ s L s pu I q ¡ 1 r0,ts pu I q @ χ ¡ t ¡ s L © § § § § À 1 L α |t ¡ u I | α .
As for the third integral in (3.29), we can write

§ § § § » V 0 ds s α e ıs 3 χ t ¡ s L ¨¡ χptq A § § § § § § § § α ı » V 0 ds s α 1 e ıs 3 χ ¡ t ¡ s L © ¡ χptq A 1 ıL » V 0 ds s α e ıs χ I ¡ t ¡ s L © § § § § À 1 L α » V 0 ds s α 1 § § χpt ¡ sq ¡ χptq § § 1 L » 4L 0 ds s α À 1
L α , which completes the proof of (3.27).

The proof of (3.28) stems from a similar strategy: observe for instance the decomposition, valid for almost all t, r ¡ 0,

1 r0,ts ¡ r I r L © ¡ 1 r0,ts pr I q 1 t¡ r L ¤r I ¤t¡ r L u ¡ 1 t0¤r I ¤tu 1 t0¤t¤ r L u 2 1 t¡ r L ¤r I ¤t¡ r L u ¡ 1 t0¤r I ¤tu @ 1 tt¡ r L u 2 1 t¡ r L ¤r I ¤0u ¡
§ § § § » A2 A1 dr |r| α e ıλr ϕ ¡ r L © § § § § À }ϕ} V }ϕ I } V K 1¡α , (3.31)
and for all 0

B 1 ¤ B 2 ¤ V, L ¥ 1, § § § § » B2 B1 dr r α e ıλr ϕ ¡ r L © § § § § À 1 B α 1 ϕ V K 1¡α L α ϕ I V .
(3.32)

Proof. For (3.32), let us write, with an elementary integrations by parts,

» B2 B1 dr r α e ıλr ϕ ¡ r L © 1 ıλ 1 B α 2 e ıλB2 ϕ ¡ B 2 L © ¡ 1 ıλ 1 B α 1 e ıλB1 ϕ ¡ B 1 L © α ıλ » B2 B1 dr r α 1 e ıλr ϕ ¡ r L © ¡ 1 ıλL » B2 B1 dr r α e ıλr ϕ I ¡ r L © , which, since supp ϕ I r¡K, Ks, gives § § § § » B2 B1 dr r α e ıλr ϕ ¡ r L © § § § § À 1 B α 1 ϕ V ϕ V » V B1 dr r α 1 1 L » KL 0 dr r α § § §ϕ I ¡ r L © § § § À 1 B α 1 ϕ V K 1¡α L α ϕ I V .
As for (3.31), it suffices to observe that for every A r0, Vs, one has

§ § § § » A 0 dr r α e ıλr ϕ ¡ r L © § § § § ¤ 1 t0¤A¤1u }ϕ} V » 1 0 dr r α 1 tA¥1u }ϕ} V » 1 0 dr r α § § § § » A 1 dr r α e ıλr ϕ ¡ r L © § § § § & À }ϕ} V 1 tA¥1u § § § § » A 1 dr r α e ıλr ϕ ¡ r L © § § § § .
We can then use (3.32) to (uniformly) bound the latter integral and derive (3.31).

4. Proof of Proposition 2.3

Proof of Proposition 2.3.

Fix b p 1 2 , 1q and c r0, 3 2 ¡ 2Hq. For every n ¥ 1, define Z pnq through the formula

F Z pnq k ¨pλq : 1 tk$0u 1 xky 3¡4H 1 xλy 2 F pnq k ¨pλq.
For every q ¥ 1, one has

E Z pnq 2q X b c % À E ¢ ķ xky 2c » dλ xλy 2b § § F Z pnq k ¨pλq § § 2 q & À ķ1,...,kq 1 xk 1 y 6¡8H¡2c ¤ ¤ ¤ 1 xk q y 6¡8H¡2c » dλ 1 xλ 1 y 4¡2b ¤ ¤ ¤ dλ q xλ q y 4¡2b E § § F pnq k1 ¨pλ 1 q § § 2 ¤ ¤ ¤ § § F pnq kq ¨pλ q q § § 2 % À ¢ ķ 1 xky 6¡8H¡2c » dλ xλy 4¡2b E § § F pnq k ¨pλq § § 2q % 1 q q À ¢ ķ 1 xky 6¡8H¡2c » dλ xλy 4¡2b E § § F pnq k ¨pλq § § 2 % q ,
where we have used the fact that F pnq k ¨pλq is a Gaussian variable to derive the last inequality. Now we can use the basic covariance estimate (3.8) to assert that

ķ 1 xky 6¡8H¡2c » dλ xλy 4¡2b E § § F pnq k ¨pλq § § 2 % À ķ 1 xky 6¡8H¡2c » dλ xλy 6¡2b 1 xλ k 2 y 2H¡1 À ķ 1 xky 4¡4H¡2c V, due to c 3 2 ¡ 2H.
We have thus checked the first part of (2.4): namely, for every q ¥ 2,

sup n¥0 E Z pnq q X b c % V.
We now intend to show that

E L ¥,pnq Z pnq 2 X 0 % nÑV ÝÑ V, (4.1)
which, since b, c ¥ 0, will immediately prove the second part of (2.4).

Let us start the analysis by writing, along (1.22),

F L ¥,pnq pZ pnq q k ¨pλq ķ 1 $0 1 tk0utk1ku 1 xk 1 y 3¡4H » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 F pnq k1¡k ¨pλ 2 ¡ λ 1 qF pnq k1 ¨pλ 2 q,
and so

E L ¥,pnq Z pnq 2 X 0 % ķ » dλ E § § F L ¥,pnq pZ pnq q k ¨pλq § § 2 % ¥ » dλ E § § F L ¥,pnq pZ pnq q 0 ¨pλq § § 2 % ¥ » dλ E § § § § ķ 1 $0 1 xk 1 y 3¡4H » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 F pnq k1 ¨pλ 2 ¡ λ 1 qF pnq k1 ¨pλ 2 q § § § § 2 & ¥ » dλ § § § § ķ 1$0 1 xk 1 y 3¡4H » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 E F pnq k1 ¨pλ 2 ¡ λ 1 qF pnq k1 ¨pλ 2 q % § § § § 2 : I pnq ,
where we have used Jensen's inequality to derive the last inequality. It remains us to prove that I pnq nÑV ÝÑ V.

(4.2) To this end, we can use the notation of the subsequent Lemma 4.1 to write

I pnq » dλ § § § § ķ 1 $0 1 txk 1 y¤2 n u 1 xk1y 3¡4H » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 1 |k1| 4H¡2 M pλ2 ¡ λ1, λ2q R k 1 pλ2 ¡ λ1, λ2q & § § § § 2 » dλ § § § § ¢ ķ1 1 txk 1 y¤2 n u 1 xk1y 3¡4H 1 |k1| 4H¡2 ¢ » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 M pλ2 ¡ λ1, λ2q ķ 1 $0 1 txk 1 y¤2 n u 1 xk1y 3¡4H » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 R k 1 pλ2 ¡ λ1, λ2q § § § § 2 ¥ |S pnq | 2 » dλ § § § § » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 M pλ2 ¡ λ1, λ2q § § § § 2 » dλ § § § § ķ 1 $0 1 txk 1 y¤2 n u 1 xk1y 3¡4H » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 R k 1 pλ2 ¡ λ1, λ2q § § § § 2 ¡ 2 |S pnq | ¤ » dλ § § § § » dλ1 Iχpλ, λ1q » dλ2 xλ2y 2 M pλ2 ¡ λ1, λ2q § § § § § § § § ķ 1 $0 1 txk 1 y¤2 n u 1 xk1y 3¡4H » dλ I 1 Iχpλ, λ I 1 q » dλ I 2 xλ I 2 y 2 R k 1 pλ I 2 ¡ λ I 1 , λ I 2 q § § § § & ,
where we have set

S pnq : ķ1 1 txk1y¤2 n u 1 xk 1 y 3¡4H 1 |k 1 | 4H¡2 .
Note that the previous inequality entails in particular

I pnq ¥ |S pnq | 2 » dλ § § § § » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 M pλ 2 ¡ λ 1 , λ 2 q § § § § 2 ¡ 2 |S pnq | ¤ » dλ » dλ 1 § § I χ pλ, λ 1 q § § » dλ 2 xλ 2 y 2 § § M pλ 2 ¡ λ 1 , λ 2 q § § & ķ 1$0 » dλ I 1 § § I χ pλ, λ I 1 q § § » dλ I 2 xλ I 2 y 2 § § R k1 pλ I 2 ¡ λ I 1 , λ I 2 q § § & . ( 4.3)
By combining the definition (4.6) and the estimate (4.7) below, it is easy to check that the integral

» dλ » dλ 1 § § I χ pλ, λ 1 q § § » dλ 2 xλ 2 y 2 § § M pλ 2 ¡ λ 1 , λ 2 q § § & ķ 1$0 » dλ I 1 § § I χ pλ, λ I 1 q § § » dλ I 2 xλ I 2 y 2 § § R k1 pλ I 2 ¡ λ I 1 , λ I 2 q § § &
is finite. Indeed, on the one hand, it holds that

» dλ 1 § § I χ pλ, λ 1 q § § » dλ 2 xλ 2 y 2 § § M pλ 2 ¡ λ 1 , λ 2 q § § À 1 xλy » dλ 1 xλ ¡ λ 1 y » dλ 2 xλ 2 y 3 1 xλ 2 ¡ λ 1 y À 1 xλy » dλ 1 xλ ¡ λ 1 yxλ 1 y À 1 xλy 2¡ε ,
and on the other hand

ķ 1$0 » dλ I 1 § § I χ pλ, λ I 1 q § § » dλ I 2 xλ I 2 y 2 § § R k1 pλ I 2 ¡ λ I 1 , λ I 2 q § § À ķ 1$0 » dλ I 1 » dλ I 2 xλ I 2 y 2 § § R k1 pλ I 2 ¡ λ I 1 , λ I 2 q § § À ķ 1$0 » dλ I 1 » dλ I 2 xλ I 2 y 2 § § R k1 pλ I 1 , λ I 2 q § § À ķ 1 $0 1 |k 1 | 4H¡2ε À 1,
where we have used (4.7). We can thus rephrase (4.3) as

I pnq ¥ c 0 |S pnq | 2 ¡ c 1 |S pnq | |S pnq | 2 c 0 ¡ c 1 |S pnq | & , ( 4.4)
where c 1 ¥ 0 is a finite constant and c 0 :

» dλ § § § § » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 M pλ 2 ¡ λ 1 , λ 2 q § § § § 2 .
Observe that S pnq nÑV ÝÑ V, and therefore, based on (4.4), it only remains us to guarantee that c 0 ¡ 0.

To this end, write Finally, since χ ¥ 0 and χ 1 on r¡1, 1s, we obtain that The following decomposition result has been used in the proof of Proposition 2.3.

c 0 » dλ § § § § » dλ 1 I χ pλ, λ 1 q » dλ 2 xλ 2 y 2 M pλ 2 ¡ λ 1 , λ 2 q § § § § 2 c » dλ § § § § » dt e ¡ıλt χptq » dλ 1 Ξ t pλ 1 q » dλ 2 xλ 2 y 2 » ds χpsq 2 Ξ s pλ 2 ¡ λ 1 qΞ s pλ 2 q § § § § 2 c » dt χptq 2 § § § § » ds χpsq 2 » dλ 2 xλ 2 y 2 ¢ » dλ 1 Ξ t pλ 1 qΞ s pλ 2 ¡ λ 1 q Ξ s pλ 2 q § § § § 2 . Then » dλ 1 Ξ t pλ 1 qΞ s pλ 2 ¡ λ 1 q » dr χprq1 r0,
c 0 ¥ c » 1 0 dt χptq 2 § § § § » t
Lemma 4.1. In the setting of Proposition 2.3, one has for every pk 1 , k I 1 q $ p0, 0q,

E F pnq k1 ¨pλqF pnq k I 1 ¨pλ I q % 1 tk1k I 1 u 1 txk1y¤2 n u 1 |k 1 | 4H¡2 M pλ, λ I q R k1 pλ, λ I q & , ( 4.5)
where, using the notation introduced in (3.4), we have set

M pλ, λ I q : c » R ds χpsq 2 Ξ s pλqΞ s pλ I q (4.6)
for some constant c ¡ 0, and the function R is such that for all k 1 $ 0 and ε ¡ 0,

» dλ 1 » dλ 2 xλ 2 y 2 § § R k1 pλ 1 , λ 2 q § § À 1 |k 1 | 4H¡2ε .
1 u 1 txk1y¤2 n u » R dξ |ξ| 2H¡1 F A λ,λ I ¨pξ k 2 1 q c 1 tk1k I 1 u 1 txk1y¤2 n u » R dξ |ξ ¡ k 2 1 | 2H¡1 F A λ,λ I ¨pξq, (4.8)
where we have set With this notation, let us define

M pλ, λ I q : c » R dξ F A λ,λ I ¨pξq c A λ,λ Ip0q and R k1 pλ, λ I q : c » R dξ 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 & F A λ,λ I ¨pξq,
which, going back to (4.8), yields the decomposition (4.5).

In order to check (4.7), let us write § § R k1 pλ, λ I q § § À

» 1 2 k 2 1 ¡V dξ § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § § § F A λ,λ I ¨pξq § § » V 1 2 k 2 1 dξ § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § § § F A λ,λ I ¨pξq § § . (4.9)
On the one hand, for every ξ p¡V,

1 2 k 2 1 q, one has |ξ ¡ k 2 1 | ¥ 1 2 k 2 1 , and so § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § À § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § ε § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § 1¡ε À 1 |k 2 1 | p2H¡1qε 1 |k 2 1 | 2Hp1¡εq |ξ| 1¡ε À |ξ| 1¡ε |k 1 | 4H¡2ε , which entails » 1 2 k 2 1 ¡V dξ § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § § § F A λ,λ I ¨pξq § § À 1 |k 1 | 4H¡2ε » R dξ |ξ| 1¡ε § § F A λ,λ I ¨pξq § § . (4.10)
On the other hand,

» V 1 2 k 2 1 dξ § § § § 1 |ξ ¡ k 2 1 | 2H¡1 ¡ 1 |k 2 1 | 2H¡1 § § § § § § F A λ,λ I ¨pξ ¨ § § |k 1 | 4¡4H » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § § § F A λ,λ I ¨pk 2 1 ξq § § . (4.11)
By injecting (4.10) and (4.11) into (4.9), we obtain that § § R k1 pλ, λ I q § § À 1

|k 1 | 4H¡2ε » R dξ |ξ| 1¡ε § § F A λ,λ I ¨pξq § § |k 1 | 4¡4H » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § § § F A λ,λ I ¨pk
As a result, § § F A λ,λ I ¨pξq § § À § § F f λ I ¨pξq § § § § F g λ ¨pξq § § . Then it is readily checked that § § F f λ I ¨pξq § § § § § § » R ds e ¡ıξs χpsq » R dt I e ¡ıλ I t I χpt I q1 r0,t I s psq § § § § § § § § » R dt I e ¡ıλ I t I χpt I q » t I 0 ds e ¡ıξs χpsq § § § § À 1 xλ I y 1 xξ λ I y .
In the same way, § § F g λ ¨pξq § § À 1 xλy

1 xξ λy , which yields § § F A λ,λ I ¨pξq § § À 1 xλy 1 xλ I y 1 xξ λy 1 xξ λ I y . Therefore » R dξ |ξ| 1¡ε § § F A λ,λ I ¨pξq § § À 1 xλyxλ I y » R dξ |ξ| 1¡ε xξ λ I yxξ λy , while » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § § § F A λ,λ I ¨pk 2 1 ξq § § À 1 xλy 1 xλ I y » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § 1 xλ k 2 1 ξy 1 xλ I k 2 1 ξy .
Going back to (4.12), we obtain the estimate § § R k1 pλ, λ I q § § À 1 xλy

1 xλ I y 1 |k 1 | 4H¡2ε » R dξ |ξ| 1¡ε xξ λ I yxξ λy |k 1 | 4¡4H » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § 1 xλ k 2 1 ξy 1 xλ I k 2 1 ξy & .
Once endowed with this bound, the derivation of (4.7) easily follows. Namely, for k 1 $ 0,

» dλ 1 » dλ 2 xλ 2 y 2 § § R k1 pλ 1 , λ 2 q § § À 1 |k 1 | 4H¡2ε » dλ 1 xλ 1 y » dλ 2 xλ 2 y 3 » R dξ |ξ| 1¡ε xξ λ 1 yxξ λ 2 y 1 |k 1 | 4H¡4 » dλ 1 xλ 1 y » dλ 2 xλ 2 y 3 » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § 1 xλ 1 k 2 1 ξy 1 xλ 2 k 2 1 ξy À 1 |k 1 | 4H¡2ε » R dξ |ξ| 1¡ε » dλ 1 xλ 1 y 1 xλ 1 ξy » dλ 2 xλ 2 y 3 1 xλ 2 ξy 1 |k 1 | 4H¡4 » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § » dλ 1 xλ 1 y 1 xλ 1 k 2 1 ξy » dλ 2 xλ 2 y 3 1 xλ 2 k 2 1 ξy À 1 |k 1 | 4H¡2ε » R dξ |ξ| 1¡ε xξy 2¡ ε 2 1 |k 1 | 4H¡4 » V 1 2 dξ § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § 1 xk 2 1 ξy 2¡ε À 1 |k 1 | 4H¡2ε » R dξ |ξ| 1¡ε xξy 2¡ ε 2 » V 1 2 dξ |ξ| 2¡ε § § § § 1 |ξ ¡ 1| 2H¡1 ¡ 1 § § § § & À 1 |k 1 | 4H¡2ε .
5. Proof of Proposition 2.4

Notation.

As a preliminary observation, note that due to (3.3), the covariance of the kernel K pnq satisfies

E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kI k I 1 pλ I , λ I 1 q % 1 tk,k I $0u 1 tk1$ku 1 tk I 1 $k I u » dλ 2 dλ I 2 I χ pλ, Ω k,k1¡k λ 1 ¡ λ 2 qI χ pλ I , Ω k I ,k I 1 ¡k I λ I 1 ¡ λ I 2 qE Fp pnq k1¡k qpλ 2 qFp pnq k I 1 ¡k I qpλ I 2 q % 1 tk1¡kk I 1 ¡k I u E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kI k I 1 pλ I , λ I 1 q % . ( 5.1)
For future reference, let us also rephrase the above expression (when k k I , k 1 k I 1 ) using (3.3), which gives

E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk1 pλ I , λ I 1 q % 1 tk$0u 1 tk1$ku » dλ 2 dλ I 2 I χ pλ, Ω k,k1¡k λ 1 ¡ λ 2 qI χ pλ I , Ω k,k1¡k λ I 1 ¡ λ I 2 qE Fp pnq k1¡k qpλ 2 qFp pnq k1¡k qpλ I 2 q % 1 tk$0u 1 tk1$ku 1 txk1¡ky¤2 n u » R dξ |ξ| 2H¡1 A ξ pk,λq pk 1 , λ 1 qA ξ pk,λq pk 1 , λ I 1 q, (5.2)
where we have set

A ξ pk,λq pk 1 , λ 1 q : » dλ 2 I χ pλ, Ω k,k1¡k λ 1 ¡ λ 2 qI χ pλ 2 , ¡ξ ¡ pk 1 ¡ kq 2 q.
5.2. Proof of Proposition 2.4.

We fix b, c p0, 1q such that b ¡ c ¤ 1 4 and consider the related quantity P pnq b,c in (2.5). Using Jensen's inequality, we immediately obtain that

E P pnq b,c $ ķ1,k I 1 Z » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk I 1 y 2c xλ I 1 y 2b E § § § § ķZ » R dλ txky 2c xλy 2b u K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk I 1 pλ, λ I 1 q § § § § 2 & ¥ ķ1,k I 1 Z » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk I 1 y 2c xλ I 1 y 2b § § § § ķZ » R dλ txky 2c xλy 2b u E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk I 1 pλ, λ I 1 q & § § § § 2 : P pnq b,c .
Then, thanks to (5.1), we can assert that

P pnq b,c ķ1Z » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk 1 y 2c xλ I 1 y 2b § § § § ķZ » R dλ txky 2c xλy 2b u E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk1 pλ, λ I 1 q & § § § § 2 ķ1Z » R 2 dλ 1 xk 1 y 2c xλ 1 y 2b dλ I 1 xk 1 y 2c xλ I 1 y 2b ķ,k I Z » R 2 dλdλ I txky 2c xλy 2b utxk I y 2c xλ I y 2b u E K pnq χ ¨kk1 pλ, λ 1 q K pnq χ ¨kk1 pλ, λ I 1 q & E K pnq χ ¨kI k1 pλ I , λ 1 q K pnq χ ¨kI k1 pλ I , λ I 1 q & ,
which, combined with (5.2), yields that

P pnq b,c ķ,k I $0 » R 2 dλdλ I txky 2c xλy 2b utxk I y 2c xλ I y 2b u ķ1tk,k I u 1 txk1¡ky¤2 n u 1 txk1¡k I y¤2 n u » R dλ 1 xk 1 y 2c xλ 1 y 2b » R dλ I 1 xk 1 y 2c xλ I 1 y 2b » R dξ |ξ| 2H¡1 A ξ pk,λq pk 1 , λ 1 qA ξ pk,λq pk 1 , λ I 1 q » R dξ I |ξ I | 2H¡1 A ξ I pk I ,λ I q pk 1 , λ 1 qA ξ I pk I ,λ I q pk 1 , λ I 1 q ķ,k I $0 » R 2 dλdλ I txky 2c xλy 2b utxk I y 2c xλ I y 2b u ķ1tk,k I u 1 txk1¡ky¤2 n u 1 txk1¡k I y¤2 n u » R dξ |ξ| 2H¡1 » R dξ I |ξ I | 2H¡1 § § § § » R dλ 1 xk 1 y 2c xλ 1 y 2b A ξ pk,λq pk 1 , λ 1 qA ξ I pk I ,λ I q pk 1 , λ 1 q § § § § 2 .
As a result,

P pnq b,c ¥ ķ$0 xky 4c ķ 1 $k 1 txk1¡ky¤2 n u » R 2 dλdλ I » R dξ |ξ| 2H¡1 » R dξ I |ξ I | 2H¡1 § § § § » R dλ 1 xk 1 y 2c xλ 1 y 2b A ξ pk,λq pk 1 , λ 1 qA ξ I pk,λ I q pk 1 , λ 1 q § § § § 2 ¥ ķ$0 xky 4c ķ 1 $k 1 txk1¡ky¤2 n u » R dλ 1 xk 1 y 2c xλ 1 y 2b » R dλ I 1 xk 1 y 2c xλ I 1 y 2b » R dξ |ξ| 2H¡1 » R dλ A ξ pk,λq pk 1 , λ 1 qA ξ pk,λq pk 1 , λ I 1 q & » R dξ I |ξ I | 2H¡1 » R dλ I A ξ I pk,λ I q pk 1 , λ 1 qA ξ I pk,λ I q pk 1 , λ I 1 q & ¥ ķ$0 xky 4c ķ 1 $k 1 txk1¡ky¤2 n u » R dλ 1 xk 1 y 2c xλ 1 y 2b » R dλ I 1 xk 1 y 2c xλ I 1 y 2b § § § § » R dξ |ξ| 2H¡1 » R dλ A ξ pk,λq pk 1 , λ 1 qA ξ pk,λq pk 1 , λ I 1 q § § § § 2 .
In turn, this entails that

P pnq b,c ¥ 2¤k¤2 n xky 4c » ¡2k 1 2 ¡2k¡ 1 2 dλ 1 xk ¡ 1y 2c xλ 1 y 2b » ¡2k 1 2 ¡2k¡ 1 2 dλ I 1 xk ¡ 1y 2c xλ I 1 y 2b § § § § » R dξ |ξ| 2H¡1 » R dλ A ξ pk,λq pk ¡ 1, λ 1 qA ξ pk,λq pk ¡ 1, λ I 1 q § § § § 2 Á 2¤k¤2 n xky 4c xky 4c xky 4b » ¡2k 1 2 ¡2k¡ 1 2 dλ 1 » ¡2k 1 2 ¡2k¡ 1 2 dλ I 1 § § § § » R dξ |ξ| 2H¡1 » R dλ A ξ pk,λq pk ¡ 1, λ 1 qA ξ pk,λq pk ¡ 1, λ I 1 q § § § § 2 Á 2¤k¤2 n xky 4c xky 4c xky 4b » 1 2 ¡ 1 2 dλ 1 » 1 2 ¡ 1 2 dλ I 1 § § § § » R dξ |ξ| 2H¡1 » R dλ A ξ pk,λq pk ¡ 1, λ 1 ¡ 2kqA ξ pk,λq pk ¡ 1, λ I 1 ¡ 2kq § § § § 2 .
(5.3)

At this point, observe that due to Ω k,¡1 2k, one has in fact

A ξ pk,λq pk ¡ 1, λ 1 ¡ 2kq » dλ 2 I χ pλ, Ω k,¡1 λ 1 ¡ 2k ¡ λ 2 qI χ pλ 2 , ¡ξ ¡ 1q » dλ 2 I χ pλ, λ 1 ¡ λ 2 qI χ pλ 2 , ¡ξ ¡ 1q : Bpξ, λ, λ 1 q,
and therefore, going back to (5.3), we deduce that

P pnq b,c Á ¢ 2¤k¤2 n xky 4c xky 4c xky 4b ¢ » 1 2 ¡ 1 2 dλ 1 » 1 2 ¡ 1 2 dλ I 1 § § § § » R dξ |ξ| 2H¡1 » R dλ Bpξ, λ, λ 1 qBpξ, λ, λ I 1 q § § § § 2 .
Recall now that b ¡ c ¤ 1 4 . Thus, in order to ensure that P pnq b,c nÑV ÝÑ V, we only need to guarantee that

» 1 2 ¡ 1 2 dλ 1 » 1 2 ¡ 1 2 dλ I 1 § § § § » R dξ |ξ| 2H¡1 » R dλ Bpξ, λ, λ 1 qBpξ, λ, λ I 1 q § § § § 2 ¡ 0.
(5.4)

To this end, observe for instance that for pλ 1 , λ I 1 q p0, 0q,

» R dξ |ξ| 2H¡1 » R dλ Bpξ, λ, 0qBpξ, λ, 0q » R dξ |ξ| 2H¡1 » R dλ § § Bpξ, λ, 0q § § 2 ,
and then, for pξ, λq p¡1, 0q,

Bp¡1, 0, 0q » dλ 2 I χ p0, ¡λ 2 qI χ pλ 2 , 0q » dλ 2 ¢ » R dt χptq » t 0 ds e ıλ2s χpsq ¢ » R dt I e ıλ2t I χpt I q » t I 0 ds I χps I q » R dt χptq » t 0 ds χpsqχp¡sq » ¡s 0 ds I χps I q » R dt χptq » t 0 ds χpsq 2 » s 0 ds I χps I q ¡ 0,
where we have used the fact that χ is symmetric and χ 1 on r¡1, 1s.

For obvious continuity reasons, we deduce that (5.4) is satisfied, and accordingly one has We know that the linear solution Ψ is explicitly given by the convolution formula Ψptq » t 0 e ¡ı∆pt¡sq 9

Bpsq, and hence, using the notation introduced in (1.8),

E Ψ 2 L 2 pr0,Ts¢Tq % ķ » T 0 E § § Ψ k ptq § § 2 % dt ķ » T 0 E § § § § » t 0 e ¡ık 2 s 9 β pkq t¡s ds § § § § 2 & dt ķ » T 0 » t 0 ds » t 0
ds I e ¡ık 2 ps¡s I q E , as it can be seen from a straightforward application of Cauchy-Schwarz inequality in (6.6).

A classical example where the three quantities 0,0 1 c 2 0.707. 6.2.2. Example: Young integration as random operator. As an additional justification of our focus on P pnq , let us consider the (well-known) case of the fractional integration operator.

Thus, consider a fractional noise 9

B on R with index H p0, 1q, as well as a smooth approximation Then define the (local) fractional integration operator in a standard way: for any regular z : R Ñ R and t R, L pnq z ¨ptq : χptq » t 0 ds χpsqzpsq 9 B pnq psq, (6.9) where χ : R Ñ R only stands for a smooth localizing cut-off function.

In Fourier mode, the operator L pnq can easily be written along the pattern of (6.5), that is as F L pnq z ¨pλq » dλ 1 K pnq pλ, λ 1 q pFzqpλ 1 q, for some unique kernel K pnq . Setting For the sake of conciseness, we prefer to leave the proof of this assertion as an exercise to the reader.

8 )

 8 We consider the general scale of anisotropic Besov spaces tX b c , b, c r0, 1qu defined through the norm xky 2c @ § § Fpz k qpλq § § 2 , (1.9)

9 B

 9 pnq

2. 1 .

 1 Problem (P1'): control of the renormalized Schrödinger product tree.

dv e ¡ıλv χpvq » dr 1

 1 r0,vs prqχprq » dλ 1 dλ 2 e ırpΩ k,k 1 λ2¡λ1q e ıλ1t e ¡ıλ2s » R dv e ¡ıλv χpvq » dr 1 r0,vs prqχprqe ırΩ k,k 1 ¢ » dλ 1 e ¡ırλ1 e ıλ1t ¢ » dλ 2 e ırλ2 e ¡ıλ2s δ tstu χptqe ıtΩ k,k 1 » R dv e ¡ıλv χpvq1 r0,vs ptq,

(3. 16 )

 16 Combining (3.14), (3.15) and (3.16), we obtain that

  χpsq 3 Ξ s p¡βqΞ s p¡β I q.

dr I 1 » dr e ırk 2 1

 12 r0,t¡ s Ω s pr I qχpr I q » dr 1 r0,ts pr I rqχpr I rqe ıpξ k pξ I 1q » dr I 1 r0,t¡ s Ω s pr I q1 r0,ts pr I rqχpr I qχpr I rq 1

» t 0 ds e ısL χpsq 3 Ξsp¡βqΞsp¡β I q 1 ıL e ıtL χptq 3 ds e ısL χpsq 3 Ξ00Lemma 3 . 3 .|s| α e ıs 1 piiq|s| α e ıs 4 1piq© 1 |s| α e ıs 4 1

 03133331414 Ξtp¡βqΞtp¡β βq χpsq4 Ξsp¡β I q ¡ 1 ıL » t 0 ds e ıspL¡β I q χpsq 4 Ξsp¡βq, and so, by(3.24), we get that s p¡βqΞ s p¡β I q ds e ıspL βq χpsq 4 Ξ s p¡β I q § ds e ıspL¡β I q χpsq4 Ξ s p¡βq § § § § & .Then, using the same elementary arguments as for the derivation of (3.7), we obtain that § Fix α p0, 1q. piq It holds that sup L¥1 t,r I ,u I r¡2,2s r0,t¡ s L s pr I q1 r0,t¡ s L s pu I qχ ¡ For all L ¥ 1 and t, r I , u I r¡2, 2s, one has § r0,t¡ s L s pr I q1 r0,t¡ s L s pu I qχ ¡ t ¡ s L © ¡ 1 r0,ts pr I q1 r0,ts pu I qχ t Let L ¥ 1 and t, r I , u I r¡2, 2s. One has, for almost every s R, 1 r0,t¡ s L s pr I q 1 tr I ¥0u 1 ts¤Lpt¡r I qu ¡ 1 tr I 0u 1 ts¥Lpt¡r I qu , and so we can write 1 r0,t¡ s L s pr I q1 r0,t¡ s L s pu I q ¡ 1 tr I ¥0u 1 ts¤Lpt¡r I qu ¡ 1 tr I 0u 1 ts¥Lpt¡r I qu ©¡ 1 tu I ¥0u 1 ts¤Lpt¡u I qu ¡ 1 tu I 0u 1 ts¥Lpt¡u I qu © 1 tr I ,u I ¥0u 1 ts¤Lrpt¡r I qpt¡u I qsu 1 tr I ,u I 0u 1 ts¥Lrpt¡r I qpt¡u I qsu . Therefore » R ds |s| α e ıs 1 r0,t¡ s L s pr I q1 r0,t¡ s L s pu I qχ ¡ apply the estimate (3.31) in Lemma 3.4 below to deduce (3.25). We can then use the same arguments to prove (3.26), by noting that 1 r0,ts ¡ r I r L tt¥0u 1 t¡r I L¤r¤¡r I L Ltu ¡ 1 tt 0u 1 t¡r I L Lt¤r¤¡r I Lu . piiq Regarding (3.27), we naturally start with the bound § r0,t¡ s L s pr I q1 r0,t¡ s L s pu I qχ ¡ t ¡ s L © ¡ 1 r0,ts pr I q1 r0,ts pu I qχ t

2 1α e ıs 2 1α e ıs 2

 222 t0¤u I ¤r I ¤tu 1 tLpt¡r I q¤s¤Lpt¡u I qu ¡ 1 tr I ¤tu 1 tu I 0u 1 ts¥Lpt¡u I qu 1 ts¥Lpt¡r I qu @ . r0,t¡ s L s pr I q ¡ 1 r0,ts pr I use the estimate (3.32) in Lemma 3.4 to assert that §

dλ 1 e 2 »§ 2 ,

 122 ırλ1 e ıvpλ2¡λ1q » dv χpvq 2 1 r0,ts pvq1 r0,vs psqe ıvλ2 , dr χprq1 r0,rs psq » dv χpvq 2 1 r0,ts pvq1 r0,vs psqe ¡|r¡v| § § § where we have used the classical identity » dλ xλy 2 e ¡ıλt c e ¡|t| .

(4. 7 )ds e ıpξ k 2 1 1 RR

 721 Proof. Recall that by(3.3), one hasE F pnq k1 ¨pλqF pnq k I 1 ¨pλ I q % c 1 tk1k I 1 u 1 txk1y¤2 n u » R dξ |ξ| 2H¡1 I χ pλ, ¡ξ ¡ k 2 1 qI χ pλ I , ¡ξ ¡ k 2 1 q, qs χpsq1 r0,ts psq » R dt I e ¡ıλ I t I χpt I q » R ds I e ¡ıpξ k 2 1 qs I χps I q1 r0,t I s ps I q c dt e ıλt χptq1 r0,ts psq » R dt I e ¡ıλ I t I χpt I q1 r0,t I s ps I dt e ıλt χptq1 r0,ts psq » R dt I e ¡ıλ I t I χpt I q1 r0,t I s rq & c 1 tk1k I

A

  λ,λ Iprq : » R ds χpsqχps rq » R dt e ıλt χptq1 r0,ts psq » R dt I e ¡ıλ I t I χpt I q1 r0,t I s ps rq.

6 . 1 .

 61 Proof of Lemma 1.1, item piq.

for every s ¡ 1 k 2 , 2 . 1 .

 1221 we can use an integration-by-parts argument to derive that § Estimate of the product-operator norm. We gather here a few elements to justify our consideration of the quantity P pnq b,c in (2.5) as an estimate of the operator normL General operator estimates in Besov spaces. Given a general kernel K L 2 pR d ¢ R d q, define the operator L K : L 2 pR d q Ñ L 2 pR d q by FpL K f qpλq » R d dλ 1 Kpλ, λ 1 qpFfqpλ 1 q, λ R d . (6.5)For each b pb 1 , . . . , b d q, consider the anisotropic Besov space H b on R d related to the Fourier multiplier xλy b :¢ d i1 |λ i | 2bi

1 2 . 1 xλ 1 y b Kpλ, λ 1 q xλ 1 y b pFfqpλ 1 xλy 2 b 4 bRemark 6 . 1 .

 12112461 If K has no specific a priori structure, then we are essentially confined to very general transformations toward the operator norm ofL K (from H b to H b I ): namely, I Kpλ, λ 1 qKpλ, λ I 1 q & xλ 1 y b pFfqpλ 1 q ¨ xλ I1 y b pFfqpλ I which provides us with the general bound L K H b ÑH b I ¤ P 1 ,b I, where P b,b I : The operator norm L K H b ÑH b I can be more basically -but less sharply -bounded by the Hilbert-Schmidt norm of L K , that is the quantity Q

L K H b ÑH b I , P 1 4 b,b I and Q 1 2 bL 2

 422 ,b I can be compared is given by the elementary kernel Kpt, sq : 1 t0¤s¤t¤1u in L 2 pRq, or more exactly by the integration operator Lf ptq : 1 r0,1s ptq » t 0 f psq ds.In this case, it can be checked that L

9 B pnq given for instance by 9 B

 99 pnq ptq : stands for the Fourier transform of a Wiener process W on R.

z 2 X 1 4» 2 . ( 6 . 10 )Proposition 6 . 2 . 2 , then for every b p 1 2

 2126106221 b : » dλ xλy 2b § § Fpzq § § 2 and applying (6.7), we deduce that L pnq X b ÑX b À P pnq b q , with P pnq b : dλ xλy 2b K pnq pλ, λ 1 qK pnq pλ, λ IFocusing exclusively on this explicit quantity P pnq b allows us to recover the classical "Young" dichotomy of fractional integration theory. In the above setting, the following picture holds true:piq If H ¡ 1

  Proof of item piiiq. Assume now that b 2H ¡ 1

	3.2.2. 2 .
	Using the expression in (3.6), we can write
	E	Fp pnq k1 qpλ 1 qFp pnq k1 qpλ I 1
				1 2	b 2H ¡ 1 2 , one has
		sup n¥1	E	pnq 2 X b	% V,
	as desired.

  ısL χpsq 3 Ξ s p¡βqΞ s p¡β I q M L β,β Ipλq R L β,β Ipλq, ¡ıLtpλ¡1q χptq 4 Ξ t p¡βqΞ t p¡β I q

	β IpλLq ds e where » » t dt e ¡ıλLt χptq 0 M L β,β Ipλq : » 1 ıλL dt e (3.17

  Fix α p0, 1q. For all test-function ϕ : R Ñ R with support in r¡K, Ks (K ¥ 1) and all λ t¡1, 1u, it holds that

	sup
	A1,A2r¡V,Vs
	L¥1
	1 tt¡ r L ¤r I ¤tu

@ 1 tr¥L|t|u 1 tr¥¡Lr I u 1 tr¤Lpt¡r I qu ¡ 1 tr¥L|t|u 1 t0¤r I ¤tu 1 tr¥L|r I |u 1 tr Ltu 1 tr I ¤0u ¡ 1 tr¥L|t¡r I |u 1 tr Ltu 1 tr I ¤tu , which paves the way toward the application of (3.32), just as above. Lemma 3.4.


Weiping Zhou

Arnaud Voituriez
email: arnaud.voituriez@cnrs.fr

Total Synthesis of (-)-HM-3 and (-)-HM-4 using an Asymmetric Gold-Catalyzed [3,3]-Sigmatropic Rearrangement of Sulfonium

The asymmetric total synthesis of two sesquiterpenoids, (-)-HM-3 and (-)-HM-4, isolated from phytopathogenic fungus has been developed using as key step a gold-catalyzed [3,3]-sigmatropic rearrangement of sulfonium. During this transformation, an excellent chirality transfer was observed between the chiral sulfoxide substrate and the newly formed quaternary stereogenic center, with 96% enantiomeric excess. The implementation of this efficient methodology allowed to isolate the two natural products in only 7-8 chemical steps with overall yields reaching 12-16%.

 It appeared that the compound HM-4 was also isolated in several liverworts and that these sesquiterpenoids and other derivatives could possess interesting biological activities as anti-oxidant and antibiotics. [3] Scheme 1. Structure of natural products (-)-HM-3 and (-)-HM-4.

Despite their interesting chemical structures and potentially relevant biological properties, it was not until 2014 that the first asymmetric total synthesis of these compounds was accomplished. [4] Indeed, Yoshida's group described a methodology based on a palladiumcatalyzed hydroarylation of enantioenriched allenic alcohol, following by an Eschenmoser-Claisen rearrangement, for the isolation of enantioenriched natural products (-)-HM-3 and (-)-HM-4. Although using a very elegant synthetic approach in the synthesis of these densely functionalized molecules, we thought that they could be isolated in fewer chemical steps and increased the overall yields (15 steps, 3.7% overall yield for (-)-HM-4, and 16 steps, 3% overall yield for (-)-HM-3). [4] The proposed retrosynthetic analysis for the isolation of (-)-HM-4 and (-)-HM-3 was described in Scheme 2.

Scheme 2. Proposed retrosynthetic strategy to access (-)-HM-3 and (-)-HM-4.

Firstly, (-)-HM-3 (1) could be obtained by the selective acylation of (-)-HM-4 (2). In the retrosynthetic strategy, we planned to access (-)-HM-4 from the -dimethylated cyclopentenone (R)-7, coming from enantioenriched cyclopentenone (R)-6, possessing a C4quaternary stereocenter (Scheme 2). It is noteworthy that these intermediates are common with the strategy previously proposed by Yoshida. [4] This compound 6 could come from the intermediate 5 via an aldolization reaction. The desired chiral 4-oxopentanal derivative 5 will be easily accessible by a gold-catalyzed asymmetric [3,3]-sigmatropic rearrangement, [START_REF] Smith | [END_REF] starting from chiral vinyl sulfoxide (R,Z)-3 and propargyl silane 4. Indeed, we have planned to use our recently developed asymmetric methodology [6] to directly access this key building block. We also postulated the formation of the stereocenter with a high enantiomeric excess, with transfer of the chirality from the sulfur atom to the newly formed quaternary carbon center.

With this retrosynthetic approach in mind, we first studied the synthesis of vinyl sulfoxide substrate 3 using a Horner-Wadsworth-Emmons olefination reaction (Scheme 3a). Starting with diethyl ((p-tolylsulfinyl)methyl)phosphonate S1 [7] and commercially available 1- (2,3-dimethoxy-4-methylphenyl)ethan-1-one S2, in the presence of n-butyllithium in diethylether, the two olefinic compounds, (R,Z)-3 and (R,E)-3, were synthesized in quantitative yield in approximately 1/1 ratio. A simple column chromatography on silica gel allowed the isolation of pure isomers, and the measurement of enantiopurity was ascertained by the use of chiral HPLC analysis (see ESI for details). Then, these two substrates were engaged separately in the key step of the synthesis, namely the gold(I)-catalyzed [3,3]sigmatropic rearrangement/ aldol condensation sequence (Scheme 3b). Vinyl sulfoxide (R,Z)-3 reacted with triisopropyl(prop-2-yn-1yl)silane 4, in the presence of 10 mol% of commercially available JohnPhosAu(MeCN)SbF6 as gold(I) catalyst, in 1,2-dichloroethane. The use of 5 mol% gold catalyst [8] or a Brønsted acid such as triflimide (HNTf2) decreased the reaction yield. After completion of the [3,3]-sigmatropic rearrangement/hydrolysis step, the solvent was removed. At this stage, the crude mixture was directly used for the intramolecular aldol condensation/elimination reaction. To this end, an inorganic base such as potassium carbonate (K2CO3) was added in tert-butanol to the crude mixture and this two-step procedure afforded (R)-6 in 67% yield and 96% ee. The same one-pot transformation starting from vinyl sulfoxide (R,E)-3 furnished compound (S)-6 in 55% yield and 96% ee. Concerning the mechanism of this transformation (Scheme 4a), we first have the coordination of the propargyl substrate 4 [START_REF] Eberhart | Selected uses of substrate 4 in synthesis and sigmatropic rearrangements: a)[END_REF] to the cationic gold(I) catalyst. The generation of such carbocationic intermediates (I / II) was favored by the stabilization brought by the silicon atom, also called silyl effect. [10] Subsequently, nucleophilic O-addition of vinyl sulfoxide 3 to this intermediate could take place smoothly to form the sulfonium intermediate (III). The latter could then undergo a [3,3]-sigmatropic rearrangement, to form thionium intermediate (IV) and in the following, hydrolysis, protodemetallation and protodesilylation steps should provide the desired chiral 4-oxopentanal derivative 5. Furthermore, the stereospecific formation of chiral quaternary carbon centers will be based on the sulfur-directed chair-like transition state of sulfonium (IIIa) and (IIIb) (Scheme 4b). In these intermediates, the orientation of the methyl and aryl substituents were determined by the double bond geometry of vinyl sulfoxides (Z)-3 and (E)-3. Consequently, starting with the same R-sulfoxide 3, the absolute configuration of the new carbon stereogenic center will depend of the E/Z-geometry of the olefin. Due to the terminal hydrolysis of the thionium ion, the chiral sulfoxide auxiliary may be considered as a "traceless auxiliary", which makes this asymmetric transformation even more interesting. To complete the total syntheses, we started from enone (R)-6 and introduced two methyl substituents in -position of carbonyl function in 75% yield, using classical conditions (NaH, MeI, THF/DMF) (Scheme 5). Next, sulfuration of carbonyl function of (R)-7, using Lawesson's reagent and one-pot reduction (Raney nickel, H2, EtOH) gave the cyclopentane derivative. This strategy has replaced favorably the multi-step reduction strategy, as described in previous reports. [4] Meanwhile, using our protocol, the sulfur product was obtained after filtration over a pad of silica gel and was then immediately subjected to hydrogenation. Scheme 5. Total synthesis of (-)-HM-3 1, (-)-HM-4 2 and formal synthesis of aquaticenol 8.

After a complete reduction of both C=S bond and C=C olefinic bond, the product was then isolated after filtration without the need for further purification. Finally, demethylation of methoxy-aryl substituents was promoted by treatment with BBr3 at -40 °C. Consequently, (-)-HM-4 (2) was delivered in 60% yield over 3 steps. Overall, this natural product was isolated in 7 steps and 16% yield. 1 H and 13 C NMR data were in complete agreement with those reported for both the natural and the synthetic product. [1,4] However, the []D value corresponds to the average values reported in the literature: {[]D 20 -44.6 (c 1.0, CHCl3); lit.: []D 20 -24.6 (c 0.16, CHCl3) [1] and []D 20 -57.4 (CHCl3) [4] }.

Finally, regioselective acetylation of the most exposed hydroxyl group was conducted at room temperature with the use of acetic anhydride (Ac2O) and pyridine, providing (-)-HM-3 (1) in 73% yield. Overall, (-)-HM-3 was synthesized in 8 steps and 12% yield. The 1 H and 13 C NMR spectra and []D 20 were in accordance with those of natural product {[]D 20 -33.4 (c 0.60, CHCl3); lit. [4] : []D 20 -33.3 (c 0.44, CHCl3)}. Furthermore, the straightforward asymmetric synthesis of (-)-HM-4 (2) may provide a direct access to the dimeric natural product aquaticenol, isolated from Lejeunea aquatica. [3c] Indeed, starting from 2, and after the addition of AlCl3 in nitromethane and then DDQ, Asakawa et al. isolated the axially chiral biaryl natural product aquaticenol 8, as a mixture of diastereoisomers. [3c,11] In conclusion, the second asymmetric synthesis of two natural products, (-)-HM-3 and (-)-HM-4 was achieved by a shorter route, with higher overall yields and high optical purity. This synthetic pathway allowed a formal synthesis of the disesquiterpenoid aquaticenol. The synthetic protocol described in this study clearly indicates that the development of a new, efficient and concise methodology has been beneficial in significantly improving the speed and effectiveness of these total syntheses.

Scheme 3 .

 3 Scheme 3. Synthesis of chiral vinyl sulfoxide substrates 3 and enantioenriched cyclopentenones 6. [Au] = JohnPhosAu(MeCN)SbF6, TIPS = triisopropylsilyl, p-Tol = para-tolyl, DCE = 1,2-dichloroethane.

We thank the China Scholarship Council for a Ph.D. grant to W. Z. and the CNRS for financial support.