Manfred Schlosser

Florence Mongin

Nh

Pyridine Elaboration through Organometallic Intermediates: Regiochemical Control and Completeness

Keywords: Pyridine, Metalation, Heterosubstituents, Electrophiles, Lithium Compounds, Superbase Chemistry

Pyridines carrying heterosubstituents (such as carboxy, amido, amino, alkoxy or trifluoromethyl groups or solely individual halogen atoms) can be readily and site selectively metalated. Subsequent reaction with a suitable electrophile opens rational access to a wealth of new building blocks for the synthesis of biologically active compounds. This approach relies on organometallic methods, which are both efficacious and extremely flexible as far as the substitution site and the product structure are concerned.

Introduction

Pyridines belong to the most prominent and most important heterocycles. Derivatives such as nicotine, nicotinamide (niacin) and nicotinamide adenine dinucleotide diphosphate (NADP) or pyridoxine (vitamin B 6 ) occupy biological key positions. In addition, countless pyridine congeners are registered as pharmaceutically or agriculturally active principles. In general, a fair degree of structural complexity characterizes such compounds. This calls for highly selective, flexible and efficacious methods of synthesis. _________________________________________ [a] Institute of Chemical Sciences and Engineering (ISIC) Ecole Polytechnique Fédérale, BCh CH-1015 Lausanne, Switzerland e-mail : manfred.schlosser@epfl.ch fax: ++41-21-6939365 [b] Synthèse et Electrosynthèse organiques Université de Rennes 1, CNRS Bâtiment 10A, Case 1003, Campus scientifique de Beaulieu F-35042 Rennes, France e-mail : florence.mongin@univ-rennes1.fr fax : +33-2-23-23-69-55

Pyridine itself can be extracted from coal tar or made by condensation of crotonaldehyde and formaldehyde with ammonia in the presence of air. [START_REF] Joule | Heterocyclic Chemistry[END_REF] It undergoes electrophilic substitutions such as nitration, sulfonation and halogenation exclusively at the 3-position. [START_REF] Joule | Heterocyclic Chemistry[END_REF] 3-Bromopyridine can be converted into 3-acetonylpyridine by a single electron-transfer mediated reaction with the potassium enolate of acetone in liquid ammonia and into 3aminopyridine using ammonia in the presence of copper sulfate at elevated temperatures. [START_REF] Joule | Heterocyclic Chemistry[END_REF] Halogen atoms located at the 2-or 4-position are readily displaced by all kinds of nucleophiles under mild conditions and without requiring transition element assistance. [START_REF] Joule | Heterocyclic Chemistry[END_REF] Pyridine reacts with sodium or potassium amide to afford 2-aminopyridine (Tchitchibabin process) and with organometallics such as butyllithium or phenyllithium to give the corresponding 2-alkyl-or 2-arylpyridine after metal hydride elimination or direct oxidation. [START_REF] Joule | Heterocyclic Chemistry[END_REF][START_REF] Ziegler | [END_REF] In contrast, Grignard reagents add at the position of N-silylated pyridines. [3] There is a trick to reorient electrophilic substitution from the ordinary 3-position also to the 4-position. As long as unprotonated, pyridine N-oxides accommodate the reagent cleanly at the N-remote site. [4] Catalytic hydrogenation smoothly reduces the resulting 4-nitropyridine N-oxide to 4-aminopyridine. [5] The same conditions can be applied to convert 3-nitropyridine to 3-aminopyridine. Conversely, 2-nitropyridine is prepared from the directly accessible 2aminopyridine by treatment with peroxosulfuric acid. [6] All three aminopyridines (Scheme 1) can be further transformed in many ways, in particular through to corresponding diazonium salts.

N O N O NO 2 N NH 2 N N N NO 2 NH 2 N N NH 2 NO 2
Scheme 1. The three aminopyridines derived directly or indirectly from the parent compounds.

Problems arise as soon as a second substituent has to be attached to the pyridine nucleus. Regioisomeric mixtures will be obtained almost inevitably. Sometimes the selectivity is still amazingly high. For example, 3-methoxypyridine provides a 10 : 1 mixture of 2-and 6nitration products and no 4-or 5-isomers at all (Scheme 2). [7] More typically, however, product mixtures result that are worthless in practical terms. Thus, 2-aminopyridine gives rise Scheme 3.

Mono-, di-and tribrominated derivatives formed upon bromination of 2aminopyridine.

Site selectivity could be easily achieved of course if the electrophile would be allowed to react with a specific metal-bearing pyridine rather than with the unmodified heterocycle having several vacant and hence potentially eligible positions. Pyridylmagnesium halides (Scheme 4) and pyridyllithiums (Scheme 4) are readily generated by the reductive insertion of a metal (such as magnesium) [9] into the carbon-halogen bond of a given bromopyridine or by permutational halogen/metal interconversion of the latter with isopropylmagnesium chloride [10] or butyllithium. [11] Dibromopyridines could serve as particularly attractive starting materials as they offer the possibility to replace one halogen by a first electrophile and the other one by a second electrophile. Unfortunately, this approach suffers from serious drawbacks. First of all, halogens of the same kind can be alternatively and selectively exchanged against metal only in exceptional cases, so far just in 2,5-dibromopyridine and 2,3,5-tribromopyridine (Scheme 4). [12] Ordinarily, the two halogen atoms (as present, e.g., in 3,4-dibromopyridine) are attacked randomly. Moreover, the use of dibromopyridines as key starting materials would only defer the problem as their regioselective preparation is in general not trivial. Finally, unexpected complications may be encountered. Thus, the exchange by-product 1-bromobutane may react, even unnoticed by the authors, [13] with the newly generated organometallic intermediate. The lesson to be drawn from this dilemna is simple: maintain the organometallic route as a guiding principle for the electrophilic substitution of heterocycles but introduce the metal by metalation (i.e., hydrogen/metal permutation) rather than by halogen/metal exchange (be it insertion or permutation). The metalation and subsequent substitution, in particular functionalization of a variety of pyridines will be extensively reviewed in the following Sections. To resume already now, the metalation of pyridine itself or its alkyl-or aryl-bearing congeners is not very tempting under a synthesis point of view but is a highly appealing option if the metalation is assisted and oriented by electronegative substituents.

Metalation

Pyridine and Picolines

Brandsma et al. [14] have reported the metalation of pyridine using our superbasic LIC-KOR [15,16] mixture (LIC = butyllithium, plus KOR = potassium tert-butoxide) as the reagent.

The observed blue color, the moderate yields and notably the found product composition of 2-, 3-and 4-isomers raise doubts about the assumed deprotonation process and argue rather for a single electron-transfer (SET) mechanism. The latter is definitively operative when pyridine is treated by lithium diisopropylamide (LIDA) in diethyl ether (DEE) and in the presence of hexamethylphosphoric triamide (HMPT) as evidenced by the formation of 2,2'bipyridine in 50% yield. [17] On the other hand, deprotonation can be brought about when facilitated by coordinative neighboring group assistance. When treated, under in situ trapping conditions, simultaneously with lithium 2,2,6,6-tetramethylpiperidide (LITMP) and tributylchlorostannane in tetrahydrofuran (THF), 2,2'-bipyridyl gives via the intermediates 1 and 2 the mono-3-and di-3,3'-stannylated derivatives in 50% and 14% yield, respectively. In the same way, 2,4'-bipyridyl can be converted via the species 3 into the 2'-stannane in 64% yield (Scheme 5). [18] N Coordination of lithium to the ring nitrogen atom is without doubt also at the origin of the remarkable selectivity of Caubère's base, [19] the 1 : 1 mixture of butyllithium with lithium 2-(dimethylamino)ethoxide (LIDMAE), toward pyridines. To favor complexation, the reaction is generally conducted in hexanes, but also DEE and THF can be employed. Pyridine, [19c] 4methylpyridine (-picoline) and 3,5-dimethylpyridine (,'-lutidine) [19f] are metalated exclusively at the 2-position generating intermediates 4 (Scheme 6). Despite this impressive record, Caubère's base does nevertheless not always behave regioselectively. For example, 3,4-dimethylpyridine (,-lutidine) undergoes lithiation concomitantly at the 2-and 6-position and at the 4-methyl group as well (in a 78 : 5 : 17 ratio). [19h] Moreover, the reagent has to be used in large excess (3-to 20-fold) which inevitably consumes large amounts of the electrophile and limits the scale (to 1 -5 mmol of substrate in most cases). Under such circumstances alternative possibilities should be kept in mind. They are briefly outlined in the next Section.

N N N N N N N N N N N N N N N 3 3' 3 3' [a] [b] Li Sn(C 4 H 9 ) 3 1 [a] 2 Li Li [b] Sn(C 4 H 9 ) 3 [a] [b] 3
N N N [a] [b] 4 R R Li R El [R = H, CH 3 ]

Pyridine Oxides and Pyridine Borates

As a systematic investigation has revealed, [20] pyridine N-oxides are selectively metalated at the 2-position (intermediates 5, Scheme 7). Unfortunately the yields are poor (14 -44%).

Disubstituted derivatives rank among the most abundant by-products.

N N N [a] [b] 5 Li El [R = H, 2-, 3-and 4-H 3 C, 4-H 3 CO, 4-Cl] R O O O R R Scheme 7. Metalation of pyridine N-oxides. Reaction conditions: [a] LiC 4 H 9 (in DEE or THF); [b] El-X = CO 2 , (H 5 C 6 ) 2 C=O, (H 2 C) 5 C=O.
-Oxidopyridinium betaines also are amenable to lithiation at the 2-position (intermediates 6, Scheme 8). [21] The zwitterionic substrates form readily when the pyridine and hexafluoroacetone are combined.

N N N [a] [b] 6 Li El [R = H, C(CH 3 ) 3 ] R R R CF 3 F 3 C O CF 3 F 3 C O Scheme 8.
Metalation of pyridine/hexafluoroacetone adducts. Reaction conditions: [a] LITMP in THF at -107 °C; [b] El-X = H 5 C 6 -CH=O, (F 3 C) 2 C=O, I 2 , ClSi(CH 3 ) 3 .

Intermediates 5 and 6 suggest a coordinative interaction between the oxido anion and lithium. However, mere electrostatic effects suffice to activate the 2-position toward proton abstraction accomplished with strong bases as evidenced by the behavior of pyridine, [22] 4methylpyridine [22] and 4-(dimethylamino)pyridine [23] adducts with boron trifluoride (intermediates 7, Scheme 9).

N N N [a] [b] 7 Li El [R = H, CH 3, N(CH 3 ) 2 ] BF 3 BF 3 R R R Scheme 9.
Metalation of pyridine/boron trifluoride adducts. Reaction conditions: [a] LITMP in DEE at -75 °C; [b] El-X = H 5 C 6 -CH=O, (H 5 C 6 ) 2 C=O, (CH 2 ) 5 C=O.

The potential of the latter method is far from being explored and exploited. In the absence of boron trifluoride, -picoline is deprotonated at the methyl group. The scope of lithium ditert-butyl-2,2,6,6-tetramethylpiperid-1-ylzincate for the lithiation of heterocycles at Nadjacent sites, so far demonstrated with pyridine, quinoline and isoquinoline on a quasianalytical 1.0 mmol scale [24] deserves to be tested further.

Pyridinecarboxylic Acids and Congeners

Both ethyl pyridine-3-and -4-carboxylate were found to be very prone to proton abstraction by lithium diisopropylamine at the 4-and 2-position, respectively. However, the lithiated species proved unstable, being lost by instantaneous autocondensation. [25] Such problems can be circumvented when pyridinecarboxylic acids themselves or halogenated derivatives thereof are treated with two molar equivalents of LITMP in THF at -75 °C or -50 °C. [26] The thus generated intermediates 8 and 9 (Scheme 10) can be effectively trapped by a variety of electrophiles.

N N N 8a 8b 8c N N N 9a 9b 9c COOLi Li COOLi Li Li COOLi COOLi Li Br COOLi Li Cl Cl COOLi Li
Scheme 10. Organolithium species generated by treatment of pyridine-2-, -3-and -4-carboxylic acid, 2-and 6-chloropyridine-3-carboxylic acid and 5-bromopyridine-3-carboxylic acid with LITMP (2 equiv.).

Abstraction of a "mobile proton", this time from an NH entity, precedes also the neighboring group assisted metalation of secondary pyridinecarboxamides (e.g., intermediates 10 and 11, Scheme 11) [27] which occurs analogously as established in the benzene series. [28] N N N

11a 11b

Li

C Li OCH 3 C=N-CH 2 C 6 H 5 Li LiO OCH 3 C=N-C 6 H 5 LiO N-C 6 H 5 LiO Scheme 11. 3-Lithiated N-benzyl or N-phenyl pyridinecarboxamides.
Although in the benzene series sec-butyllithium is required, [29] tertiary pyridinecarboxamides undergo smooth deprotonation at a vacant 3-or 4-position even with LIDA as the base (intermediates 12 and 13, Scheme 12). [30] The N-substituents are generally isopropyl groups but also ethyl groups are appropriate.

N N N 12a 12b 12c: X = H 13: X = OCH 3 Li C C Li C=O Li X NR 2 O [R = CH(CH 3 ) 2 , C 2 H 5 ] NR 2 O R 2 N Scheme 12. 3-or 4-Lithiated N,N-diisopropyl (or N,N-diethyl) pyridine-2-, -3-or -4- carboxamides.
Oxazolines are imino-ethers incorporated into a ring structure and hence are masked carboxylic acids, which are immediately formed upon acidic hydrolysis. Following precedents encountered in the benzene [31a] and thiophene series, [31b] 3-and 4-(4,5-dihydro-4,4dimethyloxazol-2-yl)pyridines are readily metalated at the 4-and 3-position, respectively (intermediates 13 and 14, Scheme 13). [32] It is advisable to employ relatively weak bases such as methyllithium or LITMP as most organomagnesiums or organolithiums tend to add nucleophilically to the 4-position. and4-(4,5dihydro-4,4-dimethyloxazol-2-yl)pyridines.

Amino-and Amido-Substituted Pyridines

Guided once more by leads existing in the benzene series, [33] N-pivaloyl [34] or N-tertbutoxycarbonyl [35] protected 2-, 3-and 4-aminopyridines have been effectively metalated using butyllithium or tert-butyllithium (intermediates 15 -18, Scheme 14). The organometallic species produced in this way open an easy access to a variety of valuable, biologically active compounds. The metalation of (N,N-dialkylamino)alkyl substituted pyridines has been only scarcely examined up to now. Both 2-and 4-(dimethylamino)pyridine [19g] and 3-(N-methylpyrrolidin-2-yl)pyridine (nicotine) [36a] were found to be attacked by an excess of Caubère's base in hexanes exclusively at the 6-position, whereas lithiation of 6-chloronicotine with butyllithium occurred cleanly at the 5-position [36b] (intermediates 19 and 20, Scheme 15).

N N 19 20

Cl Li

N

Li N Scheme 15. Metalation of nicotine and 6chloronicotine at the 6-and 5-position, respectively.

Hydroxy-and Alkoxy-Substituted Pyridines

Methoxypyridines and other alkoxypyridines are readily metalated by lithium dialkylamides such as LIDA or LTMP and by aryllithiums such as mesityllithium or phenyllithium (intermediates 21 -24, Scheme 16). [37] Caubère's base promotes the metalation of 4-methoxypyridine at the 3-position. [19g] Neat alkyllithiums, in particular butyllithium, tend to add nucleophilically at the 2-or 6-position, a sometimes quantitative process. [19b,37b,38] N

N N 21a: X = H 22 : X = H 3 CO 21b 21c N N 23 24 OR Li OR Li OR OCH 2 OCH 3 Li X Li OCH 3 Li [OR = OCH 3 , OC 2 H 5 , OCH 2 C 6 H 5 ]
Scheme 16. Lithiated alkoxypyridines.

Two remarkable optional site selectivities deserve attention. As mentioned in the preceding paragraph, 2-methoxypyridine is attacked by lithium amides or aryllithiums exclusively at the 3-position, [37a,b,e] but by Caubère's base solely at the 6-position. [19c] 3-Ethoxy and 3-methoxypyridine get deprotonated, respectively, by LIDA and by Caubère's base [19g] at the 2-position [37a] whereas 3-(methoxymethoxy)pyridine reacts at the more acidic 4-position. [39] O-Lithiated hemiaminals can be easily formed by nucleophilic addition of either an organolithium to a dialkylformamide or of a lithium dialkylamide to a carbaldehyde. As first demonstrated by H.W. Gschwend et al., [40] the lithium -dialkylaminomethoxide unit provides neighboring group assistance to the metalation of adjacent aromatic positions. In the intramolecular competition with a methoxy group, the chelating N-(2-dimethylaminoethyl)-Nmethylamino chain proves to be a superior and the less flexible N'-methyl-N-piperidazyl ring an inferior ortho-directing substituent. They give rise to the intermediates 24a and 24b, respectively (Scheme 17). [41] N

N Li N OLi N OLi H 3 CO H 3 CO Li 24a 24b N N Scheme 17. C-Lithiated lithium -(6-methoxy- pyridyl)--(dialkylaminomethoxides.

Halo-and Trifluoromethyl-Substituted Pyridines

The metalation of halopyridines was pioneered by G. W. Gribble et al. [42] Using LIDA as the base, 2-bromopyrid-3-yllithium, [43b] 3-bromopyrid-4-yllithium, [42,43a] 4-bromopyrid-4-yllithium [43c] and 3,5-dibromopyrid-4-yllithium [44] have been made accessible. All these intermediates (25 -26, Scheme 18) are fairly stable in THF at -100 °C. Some of the reported regiohomogenities have to be met with caution, however. As a careful reexamination has revealed, [45] species 25a is contaminated by the isomeric 2bromopyrid-4-yllithium in an approximate 9 : 1 ratio. Traces of 3-bromopyrid-2-yllithium are still produced at -100 °C along with 3-bromopyrid-4-yllithium and the amount of undesired compounds increases at higher temperatures.

2-Chloropyrid-3-yllithium (27a), 3-chloropyrid-4-yllithium (27c) and 4-chloropyrid-3yllithium (27d) are readily generated when the corresponding chloropyridines are treated with LIDA [42] (Scheme 19). In contrast, when butyllithium in the presence of N,N,N',N'tetramethylethylenediamine (TMEDA) [43a] or Caubère's base [46] is employed, 3-chloropyridine gives, respectively, mainly or exclusively 3-chloropyrid-2-yllithium (27b). Optional site selectivity [12,47] of metalation is characteristic for several dichloropyridines.

Whatever the base, 2,3-, [48] 2,4- [49] and 3,5- [50] dichloropyridine are only deprotonated at the 3and 4-position (intermediates 28, Scheme 20).

N N N 28a 28b 28c Cl Cl Li Cl Li Cl Li Cl Cl
Scheme 20. Site-immutable metalation of three dichloropyridines.

However, 2,6-dichloropyridine affords 2,6-dichloropyrid-3-yllithium (29a) and 2,6dichloropyrid-4-yllithium (29b) in a 10 : 1 or 1 : 3 ratio depending on whether LIDA or butyllithium serves as the base [51] (Scheme 21). Low temperature protocols should equally enable the generation and subsequent transformation of tri-and tetrachloropyridyllithiums. The only example of that kind known so far is 2,3,6-trichloropyrid-4-yllithium. [52] 2-Fluoropyridine [42,53] and 4-fluoropyridine [54] are readily metalated at the 3-position (intermediates 32a and 32d, Scheme 22). Deprotonation of 3-fluoropyridine [55] occurs cleanly at the most acidic 4-position after long exposure times and in relative basic media whereas the 2-position is favored when neighboring group assistance by coordination to the ring nitrogen atom is effective. Thus, TMEDA-activated butyllithium metalates 3-fluoropyridine solely at the 4-position when applied in THF, but mainly at the 2-position when in DEE [55] (intermediates 32c and 32b, respectively, Scheme 22). When TMEDA is replaced by the nonchelating 1,8-diazabicyclo[2.2.2]octane (DABCO), the 2-position is exclusively attacked in DEE. [55c] 2-Fluoropyridine has four different vacant positions. To illustrate the concept of "regiochemically exhaustive substitution", [12,56] the carboxy group was introduced into each of these empty sites passing through the organometallic intermediates 32a and 33a -33c [57] (Scheme 23). The execution relies on our "toolbox methods" [12] and, more precisely, the clever use of two favorite protective groups. Bulky trialkylsilyl entities do not only impede the access of any reagent to the site they occupy themselves but also screen sterically the directly adjacent positions. Thus, 3-chloro-2-fluoro-4-(trimethylsilyl)pyridine can only be deprotonated at the 6-position . In contrast, the protective chlorine substituent eliminates the 3-position as an acidic site but, at the same time, activates the adjacent 4-position. In this way, a trimethylsilyl group or a second chlorine atom can be readily introduced there to deflect the metalation to the 6-and 5-position, respectively. [57] After quenching of the intermediates with the desired electrophile, the protective groups can be conveniently removed by protodesilylation or reduction (Scheme 23).

N N N 33a 32a 33b F Cl Li Cl Li N 33c Cl Cl F Li F F Li N N N F COOH COOH N F HOOC F F HOOC N F [a] [b,c, d,e] [b,c] [b,a] [f,g] [f] [f,h] [f,h] (H 3 C) 3 Si Scheme 23.
Regiochemically exhaustive functionalization of 2-fluoropyridine. Reaction conditions: [a] LIDA in THF at -75 °C; [b] Cl 2 FCCClF 2 ; [c] LiC 4 H 9 in THF at -75 °C; [d] ClSi(CH 3 ) 3 ;

[e] Excess LiC 4 H 9 + LiOCH 2 CH 2 N(CH 3 ) 2 in hexanes at -75 °C; [f] (1.) CO 2 , (2.) aq. HCl; [g] (H 9 C 4 ) 4 NF hydrate in THF at +25 °C; [h] HCOONH 4 + cat. Pd/C in ethanol at +25 °C (stirred slurry).

Analogously, 3-fluoropyridine was converted through the organometallic intermediates 32c and 34a -c into the 3-fluoropyridine-2-, -4-, -5-and -6-carboxylic acids [58] (Scheme 24).

Chlorine and trimethylsilyl groups play again a crucial role in directing the metal to the targeted spot. 4-Chloro-3-fluoropyridine features another impressive example of optionally site selective metalation. LIDA in THF deprotonates the 5-position, but LITMP in (poorly coordinating) DEE the 2-position. The latter outcome opens another entry to 3-fluoropyridine-2-carboxylic acid. [58] N N N

34c 34b 32c Cl F F F Li N 34a F Cl Li Li Li N N N F COOH N COOH HOOC HOOC N [a] [b,e] [c,d] [b,a] [f,h,g] [f,h] [f] [f,g] F F F F (H 3 C) 3 Si (H 3 C) 3 Si Scheme 24.
Regiochemically exhaustive functionalization of 3-fluoropyridine. Reaction conditions: [a] LIDA in THF at -75 °C; [b] Cl 2 FCCClF 2 ; [c] ClSi(CH 3 ) 3 ; [d] LITMP in THF at -75 °C; [e] Excess LiC 4 H 9 + LiOCH 2 CH 2 N(CH 3 ) 2 in hexanes at -75 °C; [f] (1.) CO 2 , (2.) aq. HCl; [g] HCOONH 4 + cat. Pd/C in methanol at +50 °C (stirred slurry); [h] (H 9 C 4 ) 4 NF hydrate in THF at +25 °C.

2,4-Difluoropyridine [59] and 3,5-difluoropyridine [60] undergo metalation of course at the location flanked by the two halogen atoms. 2,6-Difluoropyridine [61] and 3,4difluoropyridine [62] react with strong bases at, respectively, the 3-and 5-position, as expected.

The regioexhaustive functionalization of 2,3-difluoropyridine [63] is once more tributary to the smart deployment of chlorine and trialkylsilyl protective groups followed by site-specific metalation (intermediates 35ac, Scheme 24), carboxylation and ultimate "makeup take off".

An alternative route to 5,6-difluoropyridine-3-carboxylic acid leads through 2,3-difluoro-4iodopyridine which is subjected to deprotonation-triggered heavy-halogen migration [14,64] prior to neutralization, iodine/lithium permutation and reaction with carbon dioxide. [57] N N 35c 35a The same principles can be applied to 2,5-difluoropyridine. [57] The three possible carboxylic acids are obtained via the intermediates 36a -c, Scheme 26). The 3,6-difluoropyridinecarboxylic acid can be prepared either by exploiting the shielding effect of the trimethylsilyl group (Scheme 26) or also by taking advantage of the ambivalent reactivity of 4-chloro-2,5-difluoropyridine which undergoes metalation with the superbasic LIC-KOR mixture at the 3-position (as shown in Scheme 26), but with LITMP in diethyl ether (DEE) at the 6-position. [57] N N 36c 36a

F F Li N 35b F Cl F Li N N F COOH N F HOOC N [a] [c,d] [b,a] [e,
Li N 36b

K/Li Cl F F N N COOH N F F N [a] [d,e] [b,c] [f,h] [f] [f,g] (H 3 C) 3 Si COOH F Li F F HOOC F F F F F F F Scheme 26.
Regiochemically exhaustive functionalization of 2,5-difluoropyridine. Reaction conditions: [a] LIDA in THF at -75 °C; [b] Cl 2 FCCClF 2 ; [c] LIDA + N,N,N',N",N"pentamethyldiethylenetriamine (PMDTA) + KOC(CH 3 ) 3 in THF at -75 °C; [d] ClSi(CH 3 ) 3 ; [e] LITMP in DEE at -75 °C; [f] (1.) CO 2 (2.) aq. HCl; [g] Zn (powder) in 25% aq. NH 3 at +25 °C (stirred slurry); [h] (H 9 C 4 ) 4 NF hydrate in THF at +25 °C.

In the preceding four paragraphs several chlorofluoropyridylmetals are mentioned which contain the two lightest halogen elements simultaneously and wherein the metal resides at a chlorine-adjacent position (3-chloro-2-fluoropyrid-4-yllithium, 4,5-dichloro-6fluoropyrid-3-yllithium, 4-chloro-5-fluoropyrid-3-yllithium, 4-chloro-5,6-difluoropyrid-3yllithium and 4-chloro-2,6-difluoropyrid-3-ylpotassium/lithium). To this list 3,5-dichloro-2fluoropyrid-4-yllithium, [63] 5-chloro-2,3-difluoropyrid-4-yllithium [63] and 3-chloro-5fluoropyrid-4-yllithium [62] may be added.

The first three out of six possible trifluoropyridyllithiums have been described recently: 4,5,6-trifluoropyrid-3-yllithium [62] (37b) and 2,4,6-trifluoropyrid-3-yllithium [59] (37c). Two of the three possible tetrafluoropyridyllithiums are known: 2,3,4,6tetrafluoropyrid-3-yllithium [59,65] (38a) and 2,3,5,6-tetrafluoropyrid-4-yllithium [66] (38b) (Scheme 27).

N N N 37a 37b 37c N N 38a 38b Li F Li F F Li F F F F F Li F F F F F F F Li F F Scheme 27.
A few tri-and tetrafluoropyridyllithiums.

Trifluoromethyl-substituted aromatic or heterocyclic building blocks attract more and more attention. [67] The required starting material can be readily made either by treatment of a trichloromethyl precursor with hydrogen fluoride, [68] a suitable carboxylic acid with sulfur tetrafluoride [69] or by bromine or iodine displacement with in situ generated trifluoromethylcopper. [70] If a metal can be selectively introduced into such a structure, subsequent functionalization is a mere trifle.

Ten of twelve possible (trifluoromethyl)pyridyllithiums have been set free by bromine/lithium or iodine/lithium permutation using butyllithium in tetrahydrofuran or toluene. [71] They all proved chemically and thermally stable at temperatures around or below -50 °C. Such species can also be generated by hydrogen/metal rather than halogen /metal interconversion also the success of the metalation option depends closely on the exact reaction conditions. 2-(Trifluoromethyl)pyrid-3-yllithium (39a, Scheme 28) is obtained cleanly by treatment of 2-(trifluoromethyl)pyridine with LITMP in THF after 6 h. [72] After short exposure times or in DEE inevitably regioisomeric mixtures are produced. Caubère's base in DEE metalates 2-(trifluoromethyl)pyridine effectively at the 6-position (intermediate 39b). [72] 4-(Trifluoromethyl)pyridine can be smoothly lithiated with LITMP in THF at -75 °C at the 3-position and with Caubère's base at the 2-position (intermediates 39c and 39d, respectively, Scheme 28).

N N N CF 3 Li Li CF 3 N CF 3 CF 3 Li Li 39a 39b 39c 39d
Scheme 28. Several (trifluoromethyl)pyridyllithiums from 2-and 4-(trifluoromethyl)pyridine by metalation (hydrogen/metal permutation).

According to a literature report, [73] not only 2,5-, 3,4-and 3,5bis(trifluoromethyl)pyridine but also 3-(trifluoromethyl)pyridine can be selectively metalated ). [72] 3-(Trifluoromethyl)-2-(trimethylsilyl)pyridine can be isolated in 30% yield when generated under in situ trapping conditions. [74] This compound can be readily converted by halodesilylation [75] into 2-bromoor 2-iodo-3-(trifluoromethyl)pyridine (X = Br, I) and, although not yet attempted, presumably also by nitrodesilylation [76] into 2-nitro-3-(trifluoromethyl)pyridine (X = NO 2 ) and by Hiyama coupling [77] into 2-aryl-3-(trifluoromethyl)pyridines (X = aryl, Scheme 29). LIDA and ClSi(CH 3 ) 3 in THF at -75 °C. [c] with Br 2 or ICl. [d] (1.) CO 2 (2.) aq. HCl.

N N CF 3 CF 3 N 41 40 CF 3 Li [a or b] [a] H R Li N CF 2 H R N CHF 2 R N CF 3 N CF 3 Si(CH 3 ) 3 X [b] [c]
The metalation/functionalization sequence can also be applied to (trifluoromethyl)pyridines carrying additional heterosubstituents. For example, when 2amino-3-chloro-5-(trifluoromethyl)pyridine was N-pivaloyl protected and subsequently treated with two equivalents of LIDA, the resulting 4-lithiated intermediate (42, Scheme 30) reacts with iodine or benzaldehyde and, after deprotection, affords the final products in satisfactory over-all yield. [78] N

N NH Cl Cl Li N El 42 F 3 C F 3 C N=C NH Cl F 3 C [COR = COC(CH 3 ) 3 ; El = I, CH(OH)C 6 H 5 ] R OLi R O O R Scheme 30.
Metalation and subsequent electrophilic substitution of N-pivaloyl protected 2-amino-3-chloro-5-(trifluoromethyl)pyridine.

The readily available [70,[78][79][80] chloro(trifluoromethyl)pyridine are challenging model compounds to test the scope of the methods elaborated for performing regiochemically exhaustive functionalizations. In the entire series, one site is always kinetically more acidic than the others and thus can be directly deprotonated. This is, for instance, the 4-position of 5chloro-2-(trifluoromethyl)pyridine. [80] The corresponding derivatives can be hence obtained with minimal effort. The open question is whether or not the same set of organometallic methods will routinely enable the preparation of the two other isomers. As the following examples demonstrate, this is the case indeed.

Fast proton abstraction occurs again from the 4-position, when 2-chloro-5-(trifluoromethyl)pyridine is treated with LIDA in the presence of lithium N,Ndiisopropylcarbamate and lithium bromide in THF at -75 °C. [79] Reaction of the intermediate 43 with elemental iodine provides 2-chloro-4-iodo-5-(trifluoromethyl)pyridine from which the organometallic precursor species (43) can be regenerated by iodine/lithium permutation.

On the other hand, the iodo compound can be subjected to a LIDA-promoted heavy halogen migration [64] which displaces the iodine from the 4-to the 6-position. Quenching of the lithiated intermediate 44a produces a neutral 6-iodo compound which can be converted into the corresponding carboxylic acid by halogen/metal permutation (intermediate 45a) and subsequent carboxylation. Curiously, the iodine reappears at the 3-rather than at the 6position when its migration is triggered with the "slim" lithium piperidide (LIPIP) rather than with the bulky LIDA. In this way, the 2-chloro-5-(trifluoromethyl)pyridine-3-carboxylic acid can also be readily made (via intermediates 44b and 45b, Scheme 31). [79] 2-Chloro-5-(trifluoromethyl)pyridine represents an atypical case as it covers the sole example of a reagent-dependent directionally diverging heavy halogen migration known so far. In general, protective groups are required to steer the metal to the targeted location.

N N 44a 43 Li N 44b Li I Cl F 3 C N N I N Cl F 3 C N [a] [f] [b] [f] I I Cl Li Cl Cl I Cl Cl F 3 C F 3 C F 3 C F 3 C F 3 C N N Li N Cl F 3 C [g,h] [c] [g,h] M M Cl Cl F 3 C F 3 C 45a 43 45b N N COOH N Cl F 3 C [i] [i] [i] COOH HOOC Cl Cl F 3 C F 3 C
Li Li [d] [e] Carboxylation gives the second acid whereas iodination followed by deprotonation triggered heavy halogen migration, [64] neutralization and iodine/magnesium permutation gives another organometallic intermediate ( 48) [79] which opens the entry to the last missing carboxylic acid (Scheme 32). against lithium (intermediate 53) followed by neutralization and halogen/metal permutation using lithium tributylmagnesate [81] (Scheme 34). [79] N N 51 50 demonstrate the general feasibility of regioselective functionalizations by means of the "toolbox methods". [12] At the same time, one recognizes the remarkable thermal stability conferred upon pyridyllithiums by the trifluoromethyl group and an additional halogen substituent. Thus, neither 2-bromo-4-(trifluoromethyl)pyrid-3-yllithium (56a) nor 5-bromo-2-(trifluoromethyl)pyrid-4-yllithium (56b) and not even 2-iodo-4-(trifluoromethyl)pyrid-3yllithium (57a) nor 4-iodo-2-(trifluoromethyl)pyrid-3-yllithium (57b) are prone to the elimination of, respectively, lithium bromide or lithium iodide when kept at -75 °C for prolonged periods of time.

N 52 Li I Cl N N N Cl N [a] [b,c] [d,e] [f,g] [f] [h,i,f] Cl F 3 C Cl Cl F 3 C Cl Cl Li HOOC F 3 C Li F 3 C F 3 C COOH F 3 C COOH F 3 C Si(

Summary and Outlook

The methods outlined above enable the effortless and regiochemically exhaustive functionalization of heterosubstituted pyridines (Scheme 36) and hence open an entry to a great variety of attractive building blocks for research and development in the life sciences arena. This potential is particularly appealing when fluorine-bearing pyridine derivatives are targeted as the smallest halogen is a unique tool to engineer and fine-tune biological properties. [81] Regiochemically exhaustive substitution of, for example, a 3heterosubstituted pyridine. Despite all the impressive applications so far featured, we should not ignore inherent problems. The metalation of pyridines is a tightrope walk. Pyridine is more acidic than benzene by 10 kcal/mol in the gas phase [82] and, based on rate estimates regarding the basecatalyzed hydrogen isotope exchange, by some 7.6 kcal/mol in the condensed phase [83,[START_REF] Reutov | CH-Acids[END_REF] . The kinetics reflect such differences in thermodynamics. 2-Fluoropyridine is metalated by secbutyllithium or LITMP in tetrahydrofuran at -75 °C, after statistical correction, 720 or, respectively, 520 times faster than benzene [START_REF] Mongin | [END_REF] . At the same time the proneness of the substrate to undergo nucleophilic additions or substitutions increases steeply when a benzenic derivative is replaced by a pyridinic one. It requires skills and cleverly selected working conditions to maneuver successfully between the desired and undesired reaction mode.

If adequate precautions are taken, the organometallic approach to structure elaboration offers singular advantages. The possibility to attach substituents, in particular functionality, selectively to any vacant position ("regiochemical exhaustiveness" [12,56] ) has already been mentioned above. Product flexibility is another inestimable virtue. The metal present in the reaction intermediate may be replaced by any electrophilic component, carbon dioxide, the popular trapping agent, being just one out of hundreds if not thousands of candidates. Finally, organometallic reactions use to proceed rapidly and often quantitatively. This is an invitation to realize shortcuts in synthesis sequences and to contract several individual steps to a one-pot protocol. [86] Traditionally pyridines play a privileged role in the realm of heterocycles. Therefore, they were in the focus of the present article. However, the concepts and methods described can be of course applied to other six-or five-membered N-heterocycles or O-and S-heterocycles as well.

  Scheme 4. Site specifically activated pyridines by bromine/magnesium or bromine/lithium exchange (reductive insertion or permutational interconversion): 2-, 3-and 4-pyridylmetals and bromopyridylmetals.

  Scheme 13. 4-or 3-Lithiated 3-and 4-(4,5dihydro-4,4-dimethyloxazol-2-yl)pyridines.

  Scheme 14. 3-or 4-Lithiated N-pivaloyl or Ntert-butoxycarbonyl protected 2-, 3-and 4aminopyridines.

Scheme 18 .

 18 Scheme 18. Lithiated bromopyridines.

  Scheme 19. Lithiated monochloropyridines.

Scheme 21 .

 21 Scheme 21).

  Scheme 22. Selective metalation of 2-, 3-and 4fluoropyridine.

at the 2 -

 2 (or 6-)position and subsequently trapped as the carboxylic acids in satisfactory yields (55 -85%). The repetition of some parts of this work proved frustrating. The true yield of 3-(trifluoromethyl)pyridine-2-carboxylic acid (via intermediate 40, Scheme 29) reproducibly obtained by several collaborators was 1.0 -1.5% rather than the claimed 75%. The main component identified in the tar-like product mixture is 1-butyl-5-(difluoromethyl)pyridine, obviously formed as a consequence of nucleophilic addition of the organometallic reagent to the heterocyclic substrate (intermediate 41, Scheme 29

  Scheme29.Reaction of 3-(trifluoromethyl)pyridine with strong bases: nucleophilic adition vs. deprotonation. Reaction conditions: [a] LiC 4 H 9 in DEE at -75 °C.[b] LIDA and ClSi(CH 3 ) 3 in THF at -75 °C.[c] with Br 2 or ICl.[d] (1.) CO 2 (2.) aq. HCl.

[

  Scheme 31.Regiochemically exhaustive functionalization of 2-chloro-5-(trifluoromethyl)pyridine. Reaction conditions:[a] LIDA + LiOOC-N( i C 3 H 7 ) 2 + LiBr in THF at -75 °C; [b] I 2 ;[c] LiC 4 H 9 in THF at -75 °C;[d] LIDA in THF at -75 °C;[e] LiN(CH 2 ) 5 (LIPIP) in THF at -75 °C (for 20 h);[f] rapid spontaneous swap of Li and I places; [g] H 2 O or HOCH 3 ; [h] ClMgCH(CH 3 ) 2 (or LiC 4 H 9 ) in THF at -75 °C; [i] (1.) CO 2 (2.) aq. HCl.

  Scheme 32.Regiochemically exhaustive functionalization of 3-chloro-4-(trifluoromethyl)pyridine. Reaction conditions: [a] LITMP in DEE at -100 °C;[b] ClSi(CH 3 ) 3 ;[c] LITMP in THF at -75 °C;[d] I 2 ; [e] H 2 O or HOCH 3 or acid; [f] LiC 4 H 9 in toluene at -75 °C; [g] (1.) CO 2 (2.) aq. HCl; [h] (1.) aq. sodium hydroxide at 100 °C (2.) aq. HCl; [i] (H 9 C 4 ) 4 NF hydrate in refluxing THF.

The regioexhaustive functionalization of 2 -

 2 Scheme 35. Regioexhaustive functionalization of 2-bromo-6-(trifluoromethyl)pyridine. Reaction conditions: [a] LIDA in THF at -85(!) °C; [b] ClSi(CH 3 ) 3 ; [c] LITMP in THF at -75 °C; [d] I 2 ; [e] LIDA in THF at -75 °C; [f] (1.) CO 2 (2.) aq. HCl; [g] (H 9 C 4 ) 4 NF hydrate in refluxing THF; [h] H 2 O or HOCH 3 or acid; [i] ClMgCH(CH 3 ) 2 in THF at 0 °C.

  Scheme 36.Regiochemically exhaustive substitution of, for example, a 3heterosubstituted pyridine.

  Cl 2 FCCClF 2 ;[c] ClSi(CH 3 ) 3 ;[d] Excess LiC 4 H 9 + LiOCH 2 CH 2 N(CH 3 ) 2 in hexanes at -75 °C; [e] (1.) CO 2 , (2.) aq. HCl; [f] HCOONH 4 + cat. Pd/C in ethanol at +25 °C (stirred slurry); [g] (H 9 C 4 ) 4 NF hydrate in THF at +25 °C.

		F
		F
	(H 3 C) 3 Si
	Li	F	F
	g]	[e]	[e,f]
		F	F
	HOOC	F	F
	Scheme 25.	Regiochemically exhaustive
	functionalization Reaction conditions: [a] LIDA in THF at -75 °C; of 2,3-difluoropyridine.
	[b]

  CH 3 ) 3

	Scheme 34. Regioexhaustive functionalization
	of 2-chloro-6-(trifluoromethyl)pyridine. Reaction conditions: [a] LIDA in THF at -85(!) °C; [b] ClSi(CH 3 ) 3 ; [c] LITMP in THF at -75 °C; [d] I 2 ; [e] LIDA in THF at -75 °C; [f] (1.) CO 2 (2.) aq.
	HCl; [g] (H 9 C 4 ) 4 NF hydrate in THF at +25 °C;
	[h] H 2 O or HOCH 3 or acid; [i] LiMg(C 4 H 9 ) 3
	(0.33 molar equivalents).

Acknowledgment

Financial support was provided by the Schweizerische Nationalfonds zur Förderung der wissenschaftlichen Forschung (grant 20-63'584-00), the Bundesamt für Berufsbildung und Technologie (KTI-Projekt 5474.1 KTS) and the Bundesamt für Bildung und Wissenschaft (grant 97.0083 linked to the TMR project FMRXCT-970129; grant C02.0060 linked to a COST D24 project).


Vuk Milišić
email: milisic@math.univ-paris13.fr

Ulrich Razafison
email: ulrich.razafison@univ-fcomte.fr

Weighted Sobolev spaces for the Laplace equation in periodic infinite strips

Keywords: AMS Classification: Primary 47B37; Secondary 35C15, 42B20, 31A10

come

Weighted Sobolev spaces for the Laplace equation in periodic infinite strips

Vuk Milisic, Ulrich Razafison

Introduction

In this article, we solve the Laplace equation in a 1-periodic infinite strip in two space dimensions: ∆u = f , in Z :=]0, 1[×R.

(1.1)

As the domain is infinite in the vertical direction, one introduces weighted Sobolev spaces describing the behavior at infinity of the solution. This behavior is related to weighted Sobolev properties of f .

The usual weights, when adapted to our problem, are polynomial functions at infinity and regular bounded functions in the neighborhood of the origin: they are powers of ρ(y 2 ) := (1 + y 2 2 ) 1/2 and, in some critical cases, higher order derivatives are completed by logarithmic functions (ρ(y 2 ) α log β (1 + ρ(y 2 ) 2 )).

The literature on the weighted Sobolev spaces is wide [START_REF] Hanouzet | Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace[END_REF][START_REF] Amrouche | Dirichlet and neumann exterior problems for the n-dimensional Laplace operator an approach in weighted sobolev spaces[END_REF][START_REF] Girault | The Stokes problem and vector potential operator in three-dimensional exterior domains: an approach in weighted Sobolev spaces[END_REF][START_REF] Girault | The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of R 3[END_REF][START_REF] Boulmezaoud | On the Laplace operator and on the vector potential problems in the half-space: an approach using weighted spaces[END_REF][START_REF] Boulmezaoud | On the Stokes system and on the biharmonic equation in the half-space: an approach via weighted Sobolev spaces[END_REF][START_REF] Amrouche | Isotropically and anisotropically weighted Sobolev spaces for the Oseen equation[END_REF][START_REF] Razafison | The stationary Navier-Stokes equations in 3D exterior domains. An approach in anisotropically weighted L q spaces[END_REF][START_REF] Amrouche | The stationary Oseen equations in R 3 . An approach in weighted Sobolev spaces[END_REF][START_REF] Amrouche | Weighted Sobolev spaces for a scalar model of the stationary Oseen equations in R 3[END_REF] and deals with various types of domains. To our knowledge, this type of weights has not been applied to problem (1.1). The choice of the physical domain comes from periodic singular problems : in [START_REF] Jäger | On the roughness-induced effective boundary condition for an incompressible viscous flow[END_REF][START_REF] Neuss | Effective laws for the poisson equation on domains with curved oscillating boundaries[END_REF][START_REF] Milišić | Blood-flow modelling along and trough a braided multi-layer metallic stent[END_REF], a zoom around a domain's periodic ǫ-perturbation leads to set an obstacle of size 1 in Z and to consider a microscopic problem defined on a boundary layer. The behavior at infinity of this microscopic solution is of importance: it provides an averaged feed-back on the macroscopic scale (see [START_REF] Bresch | High order multi-scale wall-laws, Part I: the periodic case[END_REF] and references therein). This paper presents a systematic analysis of such microscopic problems. We intend to give a standard framework to skip tedious and particular proofs related to the unboundedness of Z .

We provide isomorphisms of the Laplace operator between our weighted Sobolev spaces. It is the first step among results in the spirit of [START_REF] Amrouche | Dirichlet and neumann exterior problems for the n-dimensional Laplace operator an approach in weighted sobolev spaces[END_REF][START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF][START_REF] Amrouche | Nečasová, Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition[END_REF]. Since error estimates for boundary layer problems [START_REF] Jäger | On the roughness-induced effective boundary condition for an incompressible viscous flow[END_REF][START_REF] Milišić | Blood-flow modelling along and trough a braided multi-layer metallic stent[END_REF][START_REF] Neuss | Effective laws for the poisson equation on domains with curved oscillating boundaries[END_REF] are mostly performed in the H s framework we focus here on weighted Sobolev spaces W m,p α,β (Z ) with p = 2. There are three types of tools used : arguments specifically related to weighted Sobolev spaces [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF][START_REF] Giroire | Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales[END_REF], variational techniques from the homogenization literature [START_REF] Milišić | Blood-flow modelling along and trough a braided multi-layer metallic stent[END_REF][START_REF] Babuška | Solution of interface problems by homogenization. parts I and II[END_REF][START_REF] Jäger | On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness[END_REF] and some potential theory methods [START_REF] Mcowen | The behavior of the Laplacian on weighted Sobolev spaces[END_REF]. A general scheme might illustrate how these ideas relate one to each other :

-a Green function G specific to the periodic infinite strip is exhibited for the Laplace operator. The convolution of f with G provides an explicit solution to (1.1). A particular attention is provided to give the weakest possible meaning to the latter convolution under minimal requirements on f .

-variational inf-sup techniques specially adapted to the weighted spaces provide existence and uniqueness theorems for a restricted range of weights. This leads to first series of isomorphisms results in the variational context.

-these arguments are then applied to weighted derivatives and give natural regularity shift results in H m α,# (Z ) for m ≥ 2 (see below).

-by duality and appropriate use of generalized Poincaré estimates (leading to interactions -orthogonality or quotient spaces -with various polynomial families), one ends with generic isomorphism results that read Theorem 1.1. For any m ∈ Z, for any α ∈ R such that α ∉ Z + 1 2 , the mapping

∆ : H m+2 α,# (Z )/P ′ ∆ [m+3/2-α] → H m α,# (Z )⊥P ′ ∆ [-m-1/2+α]
is an isomorphism.

The spaces P

′ ∆

[m-1/2-α] of harmonic polynomials included in H m α,# (Z ) are defined in section 2. The previous theorem states that if one looks for a solution u of (1.1) that decays fast enough at infinity, then uniqueness is insured but the counter part is that the datum f must satisfy a polar condition. On the contrary, if the previous condition on u is released, then uniqueness is obtained up to harmonical polynomials and the datum f do not have to satisfy any polar condition. These properties are wellknown when studying elliptic equations in weighted Sobolev spaces (see for instance [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF]).

-for all values of α ∈ R and m ∈ Z such that α ∉ Z + 1 2 , we identify explicit solutions obtained via convolution with solutions given in Theorem 1.1. This gives our second main result: Theorem 1.2. Let m ∈ Z and α ∈ R such that α ∉ Z + 1 2 , and f ∈ H m α,# (Z )⊥P

′ ∆ q[-m-1/2-α] . Then G * f ∈ H m+2 α,# (Z )
is the unique solution of the Laplace equation (5.20) up to a polynomial of

P ′ ∆ [m+3/2-α]
. Moreover, we have the estimate

G * f H m+2 α,# (Z )/P ′ ∆ [m+3/2-α] ≤ C f H m α,# (Z ) .
-For α > m+1/2, α ∉ Z+ 1 2 , when the datum f ∈ H m α,# (Z ) does not satisfy the polar condition f ⊥P ′ ∆

[-m-1/2+α] , we set up an appropriate decomposition of f , and construct the solution using previous results. A singular behavior of this solution at infinity is identified : it behaves like a linear combination of sgn(y 2 ) and |y 2 |. As for corner domains [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF], when the singular part is removed, one recovers the weighted regularity estimates. This provides Theorem 7.1. This result is of particular interest in the context of homogenization.

The paper is organized as follows. In section 2, we define the basic functional framework and some preliminary results. In section 3, we adapt weighted Poincaré estimates to our setting. Then (section 4), we introduce a mixed Fourier transform (MFT) : it is a discrete Fourier transform in the horizontal direction and a continuous transform in the vertical direction. At this stage, we prove a series of isomorphisms in the non-critical case (section 5) by variational techniques. The MFT operator allows an explicit computation of a Green function (section 6). We show, as well, weighted and standard estimates of the convolution with the latter fundamental solution. Then we identify any of the solutions above with the convolution between G and the data f , this proves Theorem 1.2. Finally, when the polar conditions is not satisfied by the data, we construct a solution in section 7. In the appendices, we provide either technical proofs of some of the claims of the paper (see Appendices A, B, C, D and F), or results that we did not find in the literature but that are of interest in this context (Appendix E).

Notation, preliminary results and functional framework 2.1 Notation and preliminaries

We denote by Z the two-dimensional infinite strip defined by Z =]0, 1[×R. We use bold characters for vector or matrix fields. A point in R 2 is denoted by y = (y 1 , y 2 ) and its distance to the origin by r

= |y | = y 2 1 + y 2 2 1/2 .
Let N denotes the set of non-negative integers, N * = N \ {0}, Z the set of all integers and Z * = Z \ {0}. We denote by [k] the integer part of k. For any j ∈ Z, P ′ j stands for the polynomial space of degree less than or equal to j that only depends on y 2 . If j is a negative integer, we set by convention P ′ j = {0}. We define P ′ ∆ j the subspace of harmonic polynomials of P ′ j . The support of a function ϕ is denoted by supp(ϕ). We recall that D(R) and D(R 2 ) are spaces of C ∞ functions with compact support in R and R 2 respectively, D ′ (R) and D ′ (R 2 ) their dual spaces, namely the spaces of distributions. We denote by S (R) the Schwartz space of functions in C ∞ (R) with rapid decrease at infinity, and by S ′ (R) its dual, i.e. the space of tempered distributions. We recall that, for m ∈ N, H m is the classical Sobolev space and we denote by H m # (Z ) the space of functions that belong to H m (Z ) and that are 1-periodic in the y 1 direction. Given a Banach space B with its dual B ′ and a closed subspace X of B , we denote by B ′ ⊥X the subspace of B ′ orthogonal to X , i.e.:

B ′ ⊥X = { f ∈ B ′ , ∀v ∈ X , 〈 f , v〉 = 0} = (B /X ) ′ .
We introduce τ λ the operator of translation of λ ∈ Z in the y 1 direction. If Φ : R 2 → R is a function, then we have

(τ λ Φ)(y ) := Φ(y 1 -λ, y 2 ).
For any Φ ∈ D(R 2 ), we set ωΦ := λ∈Z τ λ Φ, which is the y 1 -periodical transform of Φ. The mapping y 1 → ωΦ(y ) belongs to C ∞ (R) and is 1-periodic. Observe that there exists a function θ satisfying θ ∈ D(R) and ωθ = 1.

More precisely, consider a function ψ ∈ D(R) such that ψ > 0 on the interior of its support. Then we simply

set θ := ψ ωψ . The function θ is called a periodical D(R)-partition of unity. If T ∈ D ′ (R 2 ), then for all ϕ ∈ D(R 2 ), we set 〈τ λ T, ϕ〉 := 〈T, τ -λ ϕ〉, λ ∈ Z.
Similarly, if T ∈ D ′ (R 2 ) has a compact support in the y 1 direction, then the y 1 -periodical transform of T , denoted by ωT , is defined by 〈ωT, ϕ〉 := 〈T, ωϕ〉, ∀ϕ ∈ D(R 2 ). These definitions are well-known, we refer for instance to [START_REF] Vo-Khac | Distributions, Analyse de Fourier, opérateurs aux dérivées partielles[END_REF] and [START_REF] Schwartz | Théorie des distributions[END_REF].

Remark 2.1. Let Φ be in D(R 2 ). Then we have

τ λ ∂ i Φ = ∂ i (τ λ Φ), ∀λ ∈ Z, (2.2
)

ω(∂ i Φ) = ∂ i (ωΦ) (2.3)
and

τ λ Φ L 2 (Z ) ≤ Φ L 2 (R 2 ) . (2.4)
As a consequence of equality (2.3), if u ∈ D ′ (R 2 ), we also have

ω(∂ i u) = ∂ i (ωu). (2.5)
The next lemma is used to prove the density result of Proposition 2.3.

Lemma 2.1. Let K be a compact of R 2 . Let u be in H m (R 2 ) and have a compact support included in K . Then we have

ω u H m (Z ) ≤ N (K ) u H m (R 2 ) ,
where N (K ) is an integer only depending on K .

Proof. Let us first notice that, since K is compact, there is a finite number of λ ∈ Z such that supp(τ λ u) ∩ [0, 1] × R is not an empty set. This number is bounded by a finite integer N (K ) that only depends on K . It follows that ω u is a finite sum and

ω u L 2 (Z ) ≤ N (K ) u L 2 (R 2 ) .
The end of the proof then follows from (2.5).

We now define the following spaces

C ∞ # (Z ) = { ϕ : Z → R, y 1 → ϕ(y ) ∈ C ∞ ([0, 1]) 1-periodic, y 2 → ϕ(y ) ∈ C ∞ (R)}, D # (Z ) = { ϕ : Z → R, y 1 → ϕ(y ) ∈ C ∞ ([0, 1]) 1-periodic, y 2 → ϕ(y ) ∈ D(R)}, S # (Z ) = { ϕ : Z → R, y 1 → ϕ(y ) ∈ C ∞ ([0, 1]) 1-periodic, y 2 → ϕ(y ) ∈ S (R)}.
The dual spaces of D # (Z ) and S # (Z ) are denoted by D ′ # (Z ) and S ′ # (Z ) respectively.

Weighted Sobolev spaces in an infinite strip

We introduce the weight ρ = ρ(y 2 ) := 1 + y 2 2 1/2 . Observe that, for λ ∈ N and for γ ∈ R, as |y 2 | tends to infinity, we have

∂ λ ρ γ ∂y λ 2 ≤ C ρ γ-λ . ( 2.6)
For α ∈ R, we define the weighted space

L 2 α (Z ) := H 0 α,# (Z ) = {u ∈ D ′ (Z ), ρ α u ∈ L 2 (Z )},
which is a Banach space equipped with the norm

u L 2 α (Z ) = ρ α u L 2 (Z ) . Proposition 2.1. The space D(Z ) is dense in L 2 α (Z ).
Proof. Let u be in L 2 α (Z ). Then, by definition of the space L 2 α (Z ), it follows that ρ α u ∈ L 2 (Z ). Therefore there exists a sequence (v n ) n∈N ⊂ D(Z ) such that v n converges to ρ α u in L 2 (Z ) as n → ∞. Thus, setting

u n = ρ -α v n , we see that u n converges to u in L 2 α (Z ) as n → ∞.
Remark 2.2. Observe that we have the algebraic inclusion D(Z ) ⊂ D # (Z ). It follows that the space D # (Z ) is dense in L 2 α (Z ).

For a non-negative integer m and a real number α, we set

k = k(m, α) =    -1 if α ∉ {1/2, ..., m -1/2} m -1/2 -α if α ∈ {1/2, ..., m -1/2}
and we introduce the weighted Sobolev space

H m α,# (Z ) = {u ∈ D ′ # (Z ); ∀λ ∈ N 2 ,0 ≤ |λ| ≤ k, ρ -m+|λ| (ln(1 + ρ 2 )) -1 ∂ λ u ∈ L 2 α (Z ), k + 1 ≤ |λ| ≤ m, ρ -m+|λ| ∂ λ u ∈ L 2 α (Z )},
which is a Banach space when endowed with the norm

u H m α,# (Z ) = 0≤|λ|≤k ρ -m+|λ| (ln(1 + ρ 2 )) -1 ∂ λ u 2 L 2 α (Z )
(2.7)

+ k+1≤|λ|≤m ρ -m+|λ| ∂ λ u 2 L 2 α (Z ) 1/2 .
We define the semi-norm

|u| H m α,# (Z ) = |λ|=m ∂ λ u 2 L 2 α (Z ) 1/2
. Let us give an example of such weighted space. Consider the case m = 1, then if α = 1/2, we have

H 1 α,# (Z ) = {u ∈ D ′ # (Z ); u ∈ L 2 α-1 (Z ), ∇u ∈ L 2 α (Z )}
and

H 1 1/2,# (Z ) = {u ∈ D ′ # (Z ); ln(1 + ρ 2 ) -1 u ∈ L 2 α-1 (Z ), ∇u ∈ L 2 α (Z )}. (2.8)
Observe that the logarithmic weight function only appears for the so-called critical cases α ∈ {1/2, ..., m -1/2}. Local properties of the spaces H m α,# (Z ) coincide with those of the classical Sobolev space H m # (Z ). When α ∉ {1/2, ..., m -1/2} we have the following algebraic and topological inclusions :

H m α,# (Z ) ⊂ H m-1 α-1,# (Z ) ⊂ ... ⊂ L 2 α-m (Z ).
(2.9)

When α ∈ {1/2, ..., m -1/2}, the logarithmic weight functions appear, so that we only have the inclusions

H m α,# (Z ) ⊂ ... ⊂ H m-α+1/2 1/2,# (Z ). (2.10)
Observe that the mapping

u ∈ H m α,# (Z ) → ∂ λ u ∈ H m-|λ| α,# (Z ) (2.11)
is continuous for λ ∈ N 2 . Using (2.6), for α, γ ∈ R such that α ∉ {1/2, ..., m -1/2} and αγ ∉ {1/2, ..., m -1/2}, the mapping

u ∈ H m α,# (Z ) → ρ γ u ∈ H m α-γ,# (Z ) (2.12)
is an isomorphism.

Proposition 2.2. Let q be the greatest non-negative integer such that y q 2 ∈ H m α,# (Z ). Then q reads :

• if m ∈ N * q = q(m, α) :=    m -3/2 -α, if α + 1/2 ∈ {i ∈ Z; i ≤ 0}, [m -1/2 -α], otherwise.
(2.13)

• if m = 0 one has :

q = q(0, α) :=    -3/2 -α, if α + 1/2 ∈ Z, [-1/2 -α], otherwise. (2.14)
In the table below, we give some of polynomials belonging to P q(m,α) for different values of m and for α in the range [-5 2 , 5 2 ].

m \ α -5 2 ; -3 2 -3 2 ; -1 2 -1 2 ; 1 2 1 2 ; 3 2 3 2 ; 5 2 P ′ q(0,α) P ′ 1 P ′ 0 0 0 0 P ′ q(1,α) P ′ 2 P ′ 1 P ′ 0 0 0 P ′ q(2,α) P ′ 3 P ′ 2 P ′ 1 P ′ 0 0 Table 1: Polynomial spaces included in H m α,# (Z )

for various values of α and m

We state now a density result whose proof is given in Appendix A.

Proposition 2.3. The space D # (Z ) is dense in H m α,# (Z ).

The above proposition implies that the dual space of H m α,# (Z ) denoted by H -m -α,# (Z ) is a subspace of D ′ # (Z ).

Weighted Poincaré estimates

Let R be a positive real number. For β = -1, one uses the standard Hardy estimates:

∞ R | f (r )| 2 r β d r ≤ 2 β + 1 2 ∞ R | f ′ (r )| 2 r β+2 d r, ( 3.15)
while for the specific case when β = -1, one switches to

∞ R | f (r )| 2 (ln r ) 2 r d r ≤ 4 3 2 ∞ R | f ′ (r )| 2 r d r.
We now introduce the truncated domain

Z R :=]0, 1[×(] -∞, -R[∪]R, +∞[).
Using the above Hardy inequalities in the y 2 direction, we can easily prove the following lemma.

Lemma 3.1. Let α, R > 1 be real numbers and let m ≥ 1 be an integer. Then there exists a constant C m α such that

∀ϕ ∈ D # (Z R ), ϕ H m α,# (Z R ) ≤ C m α ∂ 2 ϕ H m-1 α,# (Z R ) . (3.16)
As a consequence, we also have

∀ϕ ∈ D # (Z R ), ϕ H m α,# (Z R ) ≤ C m α |ϕ| H m α,# (Z R ) .
(3.17)

For the particular case when α ∈ {1/2, ..., m-1/2}, (3.17) cannot hold without introducing logarithmic weights in the definition of the space H m α,# (Z ). Proceeding as in [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF] (Theorem 8.3), we have the Poincaré-type inequalities : Theorem 3.1. Let α be a real number and m ≥ 1 an integer. Let j := min(q(m, α), m -1) where q(m, α) is defined by Proposition 2.2. Then there exists a constant C > 0, such that for any u ∈ H m α,# (Z ), we have

inf λ∈P ′ j u + λ H m α,# (Z ) ≤ C |u| H m α,# (Z ) . (3.18)
In other words, the semi-norm |.| H m α,# (Z ) defines on H m α,# (Z )/P ′ j a norm which is equivalent to the quotient norm.

In order to present the last results of the section, we introduce the space:

V α (Z ) = {v ∈ L 2 α (Z ), div v = 0}. (3.19)
Then, as a consequence of Theorem 3.1, we have the following isomorphism results on the gradient and divergence operators, whose proof is similar to the proof of Proposition 4.1 in [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF].

Proposition 3.1. Assume that α ∈ R. 1. If α > 1/2, then
• the gradient operator is an isomorphism:

∇ : H 1 α,# (Z ) → L 2 α (Z )⊥V -α (Z ),
• the divergence operator is an isomorphism:

div : L 2 -α (Z )/V -α (Z ) → H -1 -α,# (Z ). 2. If α ≤ 1/2, then
• the gradient operator is an isomorphism:

∇ : H 1 α,# (Z )/R → L 2 α (Z )⊥V -α (Z ),
• the divergence operator is an isomorphism:

div : L 2 -α (Z )/V -α (Z ) → H -1 -α,# (Z )⊥R.

The mixed Fourier transform (MFT)

The purpose of this section is to define the MFT. We shall use this tool in the next section in order to state a general result of uniqueness of (1.1) and in section 6 in order to compute the fundamental solution.

Rapidly decreasing functions

Let set Γ := Z × R and let us write the locally convex linear topological spaces :

S (Γ) := φ : Z × R → R; ∀k ∈ Z φ(k, •) ∈ S (R)
and sup

k∈Z,l ∈R k α l β ∂ γ l γ φ(k, l ) < ∞, ∀(α, β, γ) ∈ N 3 .
The space S (Γ) is endowed with the semi-norms

φ α,β,γ := sup k∈Z,l ∈R k α ′ l β ′ ∂ γ ′ l γ ′ φ(k, l ) , ∀α ′ ≤ α, β ′ ≤ β, γ ′ ≤ γ.
We define also

l 2 (Z; L 2 (R)) := u ∈ S ′ (Γ) s.t. k∈Z |u(k, •)| 2 L 2 (R) < ∞ . Proposition 4.1. S (Γ) is dense in l 2 (Z; L 2 (R)). Proof. Since u ∈ l 2 (Z; L 2 (R)), then ∀ǫ > 0 ∃k 0 > 0 such that |k|>k 0 u(k, •) 2 L 2 (R) < ǫ 2 .
We set

u δ :=      u δ (k, •) if |k| < k 0 , 0 otherwise , where u δ (k, •) is a standard smooth approximation of u(k, •) in D(R). Then, we choose δ(k 0 ) such that ∀k such that |k| ≤ k 0 u(k, •) -u δ (k, •) 2 L 2 (R) < ǫ 2k 0 , which finally gives u -u δ 2 l 2 (Z;L 2 (R)) ≤ |k|≤k 0 u(k, •) -u δ (k, •) 2 L 2 (R) + |k|>k 0 u(k, •) 2 L 2 (R) < ǫ
which proves the claim, since u δ trivially belongs to S (Γ).

Definition 4.1. We define the MFT operator F : S # (Z ) → S (Γ) as

F (ϕ)(k, l ) = Z ϕ(y)e -i ( k y 1 + l y 2 ) d y, ∀(k, l ) ∈ Γ,
where k = 2πk, and the same holds for l .

We list below some basic tools concerning the MFT needed in the remaining of the paper. They can be proved following classical arguments ( [START_REF] Yosida | Functional analysis[END_REF], [START_REF] Rudin | Analyse réelle et complexe[END_REF], [START_REF] Reed | Methods of modern mathematical physics. II. Fourier analysis, self-adjointness[END_REF]).

Proposition 4.2. [(i)]

1. The operator F : S # (Z ) → S (Γ) is an isomorphism and the inverse operator is explicit:

F -1 ( φ)(y) = k∈Z R φ(k, l )e i ( k y 1 + l y 2 ) d l , ∀ φ ∈ S (Γ), ∀y ∈ Z .

One has in a classical way, for any

( f , g ) ∈ S # (Z ) × S # (Z ), that Z f • g d y = k∈Z R F ( f )(k, l ) • F (g )(k, l )d l , and F ( f * g ) = F ( f )F (g ), ∀(k, l ) ∈ Γ.

Tempered distributions and MFT

In a quite natural manner we extend the concepts introduced above to tempered distributions.

Definition 4.2. [(i)]

1. A linear form T acting on S (Γ) is such that ∃(α, β, γ) ∈ N 3 T ( φ) ≤ C φ α,β,γ , ∀ φ ∈ S (Γ).
2. The MFT applied to a tempered distribution T ∈ S ′ # (Z ) is defined as

< F (T ), φ > S ′ , S := F (T )( φ) := T (F T ( φ)) =< T, F T ( φ) > S ′ # ×S # , ∀ φ ∈ S (Γ),
where

F T ( φ) := k∈Z R φ(k, l )e -i ( k y 1 + l y 2 ) d l , ∀ φ ∈ S (Γ), ∀y ∈ Z .
In the same way, we define the reciprocal operator denoted F -1 which associates to each distribution

T ∈ S ′ (Γ) a distribution F -1 ( T ) s.t. < F -1 ( T ), ϕ > S ′ ×S =< T , F (ϕ) > S ′ × S where φ(k, l ) = φ(-k, -l ).
We sum up properties extending classical results of Fourier analysis to our MFT.

Proposition 4.3. [(i)] 1. If T ∈ S ′ # (Z ) then F (T ) ∈ S ′ (Γ). Similarly, if T ∈ S ′ (Γ) then F -1 ( T ) ∈ S ′ # (Z ).
2. The Dirac measure belongs to S ′ # (Z ) and :

F (δ 0 ) = 1, ∀k ∈ Z, a.e. ℓ ∈ R.

the MFT acts on the derivatives in a polynomial fashion:

∀T ∈ S ′ # (Z ), F (∂ α T ) = i |α| ( kα + l α )F (T ), for any multi-index α ∈ N 2 . 4. Plancherel's Theorem : if f ∈ L 2 (Z ) then the Fourier transform F (T f ) is defined by a function F ( f ) ∈ l 2 (Z, L 2 (R)) i.e. F (T f ) = T F ( f ) ∀ f ∈ L 2 (Z ) and F ( f ) l 2 (Z;L 2 (R)) = f L 2 (Z ) .

The Laplace equation in a periodic infinite strip

In this section we study the problem

   -∆u = f in Z , u is 1-periodic in the y 1 direction (5.20)
in the variational context. Firstly, we characterize of the kernel of the Laplace operator.

Proposition 5.1. Let m ≥ 0 be an integer, α be a real number and j = min{q(m, α), 1} where q(m, α) is defined

by Proposition 2.2. A function u ∈ H m α,# (Z ) satisfies ∆u = 0 if and only if u ∈ P ′ ∆ j .
Proof. Since j ≤ 1, it is clear that if u ∈ P ′ ∆ j , then ∆u = 0. Conversely, let u ∈ S ′ # (Z ) satisfy ∆u = 0. We apply the MFT : in the sense defined in Definition 4.2, one has ( k2 + l 2 )F (u) = 0, ∀k ∈ Z, a.e. l ∈ R, which implies that

F (u) =      p j =0 δ j y 2 =0 (l ) if k ≡ 0, 0 otherwise, p being a non negative integer. A simple computation shows that F -1 (F (u)) = p j =0 (i y 2 ) j , indeed 〈F -1 (F (u)) , ϕ〉 S ′ # ,S # = 〈F (u), F (ϕ)〉 S ′ (Γ), S (Γ) = 〈 p j =0 δ j 0 , F (ϕ)(0, •)〉 S ′ (R)×S (R) = p j =0 (-1) j 〈δ 0 , ∂ j φ〉 S ′ (R)×S (R) = p j =0 (-1) j 〈δ 0 , ((i y 2 ) j ϕ)〉 S ′ (R)×S (R) = p j =0 (-1) j R (i y 2 ) j ϕ(y )d y = 〈 p j =0 (-i y 2 ) j , ϕ〉 S ′ # ×S # ,
where φ denotes the usual 1-dimensional Fourier transform of ϕ on R.

∆ : H 1 α,# (Z )/R → H -1 α,# (Z )⊥R, ( 5.21
)

is an isomorphism. Proof. Let us first check that if u belongs to H 1 α,# (Z ) for -1/2 ≤ α ≤ 1/2, then ∆u belongs to H -1 α,# (Z ). Indeed, for any ϕ ∈ D # (Z ), we can write |〈∆u, ϕ〉 D ′ # (Z ),D # (Z ) | = |〈∇u, ∇ϕ〉 L 2 α (Z ),L 2 -α (Z ) | ≤ ∇u L 2 α (Z ) ∇ϕ L 2 -α (Z ) = ∇u L 2 α (Z ) |ϕ| H 1 -α,# (Z ) . Since -1/2 ≤ α ≤ 1/2, then thanks to Theorem 3.1, the semi-norm |.| H 1 -α,# (Z ) is a norm on H 1 -α,# (Z )/R.
Therefore, we deduce that ∆u ∈ H -1 α,# (Z )⊥R. Observe now that for any function u ∈ H 1 α,# (Z ), there exists a constant k s.t.

u -k L 2 α-1 (Z ) = inf c∈R u -c L 2 α-1 (Z ) .
Since the functional is convex wrt c one derives easily that k(u) necessarily satisfies :

k(u) = Z uρ (2α-2) d y Z ρ (2α-2) d y .
A simple use of Jensen inequality proves that

(k(u)) 2 ≤ C (α) u 2 H 0 α-1,# (Z ) .
Next, we first prove the statement for -1/2 ≤ α ≤ 0. To do so, we prove the inf-sup condition. Set ω := ρ α and define

v(u) := ω 2 (u -k(u)). Let us prove that v(u) ∈ H 1 -α,# (Z ) for any -1/2 ≤ α ≤ 0. It is clear that v(u) L 2 -α-1 (Z ) = u -k(u) L 2 α-1 (Z ) .
Consequently, for the particular case α = -1/2 we also have

ln(1 + ρ 2 ) -1 v(u) L 2 -1/2 (Z ) ≤ C u -k(u) L 2 -3/2 (Z ) . (5.22)
Note that inequality (5.22) is needed, since logarithmic weight appears in the space

H 1 1/2,# (Z ) (see (2.8)). Furthermore, direct computations show that ∇v(u) = 2ω∇ω(u -k(u)) + ω 2 ∇u and ∂ y 2 ω = αy 2 ρ α-2 which implies that |∇ω| ≤ C ρ α-1 . It follows that ∇v(u) L 2 -α (Z ) ≤ C u -k(u) H 1 α,# (Z ) .
This shows that for any

-1/2 ≤ α ≤ 0, v(u) ∈ H 1 -α,# (Z ) and v(u) H 1 -α,# (Z ) ≤ C u -k(u) H 1 α,# (Z ) .
(5.23)

Now, one has :

a(u, v) = Z ∇u • ∇vd y = Z |∇u| 2 ω 2 + ∇u • (∇ω 2 )(u -k(u)) d y .
One focuses on the last term of the rhs, uses the divergence formula and writes

Z ∇u • (∇ω 2 )(u -k(u))d y = - 1 2 Z ∆(ω 2 )(u -k(u)) 2 d y = - Z (∆ω)ω + |∇ω| 2 (u -k(u)) 2 d y Using again that ∂ y 2 ω = αy 2 ρ α-2 one has explicitly that (∆ω)ω + |∇ω| 2 = 2(α 2 -1) y 2 ρ 2 + α ρ (2α-2) .
Since -1/2 ≤ α ≤ 0, we clearly have

(∆ω)ω + |∇ω| 2 ≤ 0.
This allows to write:

a(u, v(u)) ≥ Z |∇u| 2 ω 2 d y .
Thanks to Theorem 3.1 and inequality (5.23), we can write

a(u, v(u)) ≥ Z |∇u| 2 ω 2 d y ≥ C u H 1 α,# (Z )/R v(u) H 1 -α,# (Z ) .
The above inequality allows to obtain the inf-sup condition which proves that the Laplace operator defined by

∆ : H 1 α,# /R → H -1 α,# (Z )⊥R (5.24)
is an isomorphism provided that -1/2 ≤ α ≤ 0. By duality and transposition, the Laplace operator defined by (5.24) is also an isomorphism for 0 ≤ α ≤ 1/2. This ends the proof.

It remains to extend the isomorphism result (5.21) to any α ∈ R and to any m ∈ Z. For this sake, we state some regularity results for the Laplace operator in Z .

Theorem 5.2. Let α be a real number such that -1/2 < α < 1/2 and let ℓ be an integer. Then the Laplace operator defined by

∆ : H 1+ℓ α+ℓ,# (Z )/R → H -1+ℓ α+ℓ,# (Z )⊥R (5.25)
is an isomorphism.

Proof. Owing to Theorem 5.1, the claim is true for ℓ = 0. Assume that it is true for ℓ = k and let us prove that it is still true for ℓ = k + 1. The Laplace operator defined by (5.25) is clearly linear and continuous. It is also injective : if u ∈ H k+2 α+k+1,# (Z ) and ∆u = 0 then u is constant. To prove that it is onto, let f be given in

H k α+k+1,# (Z )⊥R.
According to (2.9), f belongs to H -1+k α+k,# (Z )⊥R. Then the induction assumption implies that there exists u ∈ H 1+k α+k,# (Z ) such that ∆u = f . Next, we have

∆(ρ∂ i u) = ρ∂ i f + ∂ i u∆ρ + 2∇ρ • ∇(∂ i u).
(5.26)

Using (2.6), (2.9), (2.11) and (2.12), all the terms of the right-hand side belong to H -1+k α+k,# (Z ). This implies that ∆(ρ∂ i u) belongs to H -1+k α+k,# (Z ). Therefore, ∆(ρ∂ i u) also belongs to H -2+k α+k-1,# (Z ). Moreover, u belonging to H 1+k α+k,# (Z ) implies that ρ∂ i u belongs H k α+k-1,# (Z ) and for any ϕ ∈ H 2-k -α-k+1,# (Z ), we have

〈∆(ρ∂ i u), ϕ〉 H -2+k α+k-1,# (Z )×H 2-k -α-k+1,# (Z ) = 〈ρ∂ i u, ∆ϕ〉 H k α+k-1,# (Z )×H -k -α-k+1,# (Z ) .
Since R ⊂ H 2-k -α-k+1,# (Z ), we can take ϕ ∈ R which implies that ∆ϕ = 0. It follows that ∆(ρ∂ i u) belongs to H -1+k α+k,# (Z )⊥R. Thanks to the induction assumption, there exists v in H 1+k α+k,# (Z ) such that

∆v = ∆(ρ∂ i u).
Hence, the difference v -ρ∂ i u is a constant. Since the constants are in H 1+k α+k,# (Z ), we deduce that ρ∂ i u belongs to H 1+k α+k,# (Z ) which implies that u belongs to H 2+k α+k+1,# (Z ).

Remark 5.1. Let us point out that the above theorem excludes the values α ∈ {-1 2 , 1 2 } : due to logarithmic weights, the space H 1 1/2,# (Z ) is not included in L 2 -1/2 (Z ). By duality, this also implies that the space L 2 1/2 (Z ) is not included in H -1 -1/2,# (Z ). Therefore, in (5.26) for α = 1/2, the term ∂ i u∆ρ does not belong to H -1 1/2,# (Z ). Thus, in this paper, the extension of (5.21) to any α ∈ R will exclude some values of α that are called critical values and belong to the set Z + 1 2 . Furthermore, tough (5.21) is valid for the critical values α = ± 1 2 , we shall exclude these values here for sake of clarity. We deal with the critical cases in a forthcoming paper. Remark 5.2. Applying the theorem above to the particular case when ℓ = 1, we derive a weighted version of Calderón-Zygmund inequalities [START_REF] Calderón | A singular integral operators and differential equations[END_REF]. More precisely, let α be a real number such that 1/2 < α < 3/2, then there exists a constant C > 0, such that for any u ∈ D # (Z ) :

∂ i ∂ j u L 2 α (Z ) ≤ C ∆u L 2 α (Z ) .
(5.27)

Using (5.27) and thanks to the Closed Range Theorem of Banach, we prove that :

Theorem 5.3. Let α be a real number satisfying 1/2 < α < 3/2. For m ∈ N, m ≥ 3, the following operator

∆ : H m α,# (Z )/P ′ m-2 → H m-2 α,# (Z )/P ′ m-4
(5. 28)

is an isomorphism.
The next claim is then a straightforward consequence of the latter result.

Theorem 5.4. Let α be a real number satisfying 1/2 < α < 3/2. For m ∈ N, m ≥ 3, the following operator

∆ : H m α,# (Z )/P ′ ∆ m-2 → H m-2 α,# (Z ) (5.29)
is an isomorphism.

The next theorem extends Theorem 5.1.

Theorem 5.5. Let α be a real number satisfying 1/2 < α < 3/2 and let ℓ ≥ 1 be a given integer. Then the Laplace operators defined by

∆ : H 1 -α+ℓ,# (Z )/P ′ ∆ 1-ℓ → H -1 -α+ℓ,# (Z )⊥P ′ ∆ -1+ℓ
(5.30)

and

∆ : H 1 α-ℓ,# (Z )/P ′ ∆ -1+ℓ → H -1 α-ℓ,# (Z )⊥P ′ ∆ 1-ℓ (5.31) are isomorphisms.
Proof. Observe first that when ℓ = 1, the result is proved in Theorem 5.1. Observe next that if m ≥ 2 is an integer, the mapping

∆ : H m α,# (Z )/P ′ ∆ m-2 → H m-2 α,# (Z )⊥P ′ ∆ -m+2
is onto. Indeed, if m = 2, this isomorphism is exactly defined by (5.25) with ℓ = 1. If m ≥ 3, it is defined by (5.29). Now through duality and transposition, the mapping

∆ : H -m+2 -α,# (Z )/P ′ ∆ -m+2 → H -m -α,# (Z )⊥P ′ ∆ m-2
is onto. Next, using the same arguments as in the proof of Theorem 5.2, we are able to show that for any integer ℓ ≥ 1, the operator

∆ : H -m+2+ℓ -α+ℓ,# (Z )/P ′ ∆ -m+2 → H -m+ℓ -α+ℓ,# (Z )⊥P ′ ∆ m-2
is an isomorphism. Choosing m = ℓ + 1, the operator defined by (5.30) is an isomorphism. By duality and transposition, the mapping defined by (5.31) is onto as well.

Remark 5.3. Summarizing theorems 5.1 and 5.5, we deduce that, for any α ∈ R such that α ∉ Z + 1 2 , the mapping

∆ : H 1 α,# (Z )/P ′ ∆ [1/2-α] → H -1 α,# (Z )⊥P ′ ∆ [1/2+α]
(5.32) is an isomorphism . As a consequence, for any m ∈ Z, for any α ∈ R such that α ∉ Z + 1 2 , the mapping

∆ : H m+2 α,# (Z )/P ′ ∆ [m+3/2-α] → H m α,# (Z )⊥P ′ ∆ [-m-1/2+α]
(5.33) is an isomorphism. This result is stated in Theorem 1.1.

We end this section with a proposition that gives a property of all distribution u ∈ D ′ # (Z ) satisfying ∇u ∈ L 2 -1 2 (Z ). The proof of the proposition is given in E.

Proposition 5.2. Let u

∈ D ′ # (Z ) such that ∇u ∈ L 2 -1 2 (Z ). Then u ∈ H 1 -1 2 ,#
(Z ) and there exists a constant C > 0

such that u H 1 -1 2 ,# (Z )/R ≤ C ∇u L 2 - 1 2
(Z ) .

The Green function and convolution in weighted spaces

The purpose of this section is threefold:

(i) we look for the fundamental solution for the Laplace equation in Z

(ii) we estimate the convolution with this solution in weighted Sobolev spaces (iii) we identify the solution of (1.1) found in the previous section with the convolution between the fundamental solution ans the datum f . This amount to prove Theorem 1.2.

To achieve the first goal, we use the MFT defined in Section 4.

The Green function for periodic strips

We aim at solving the fundamental equation:

-∆G = δ 0 in Z , (6.34) stated here in the sense of tempered distributions. Then G should satisfy, in S (Γ) ′ :

( k2 + l 2 )F (G) = 1.
The proposition below gives the expression of the Green function. Moreover, one has G 1 ∈ L 1 (Z ) ∩ L 2 (Z ) and thus G ∈ L 2 loc (Z ). The Green function can be written in a more compact expression : G(y) = -1 4π ln(2 cosh(2πy 2 ) -cos(2πy 1 ) ).

Notice that the Green function is odd with respect to y.

The proof of this proposition is given in B. One can easily obtain the asymptotic behavior of the Green function. As |y 2 | tends to infinity, we have

|G(y )| ≤ C |y 2 |, |∇G(y )| ≤ C and |∂ 2 G(y )| ≤ C e -2π|y 2 | ≤ C ρ(y 2 ) -σ , ∀σ ∈ R. (6.35)
Note that the estimate on the derivatives of second order on G is proved in F (Lemma 1). Moreover, as |y 2 | tends to infinity, we also have

|G 1 (y )| ≤ C ρ(y 2 ) -σ , ∀σ > 0, ∀y 1 ∈ (0, 1). (6.36)
Before proving weighted estimates on the Green function, we shall define the convolution if f ∈ D # (Z ), i.e.

G * f = Z G(x -y ) f (y ) d y = Z G(y ) f (x -y ) d y .
Thus the convolution G * f belongs to C ∞ # (Z ). Moreover, thanks to (6.34), we have ∆(G * f ) = f in Z.

First properties of the convolution with the fundamental solution

Definition 6.1. For any function f in L 2 (Z ) the horizontal average f reads :

f (y 2 ) := 1 0 f (y 1 , y 2 )d y 1 .
We continue with a property of the convolution between the Green function and data that have zero horizontal average. Lemma 6.1. For any h ∈ D # (Z ) such that h = 0 one has :

∂ α G 2 * h = 0, a.e. in Z , α ∈ N 2 s.t. |α| ∈ {0, 1, 2}.
Proof. G 2 is a tempered distribution, the convolution with h ∈ D # (Z ) makes sense :

< G 2 * h,ϕ > D ′ # (Z )×D # (Z ) =< G 2 , (h * ϕ) > D ′ # (Z )×D # (Z ) = 1 2 Z |x 2 | Z h(x + y)ϕ(y)d yd x = 1 2 R |x 2 | Z h(x 2 + y 2 ) ϕ(y)d yd x 2 = 0.
The same proof holds for derivatives as well. As this is true for every ϕ ∈ S # (Z ), the result is proved.

By the same way one has the complementary counterpart : Lemma 6.2. For any h ∈ D # (Z ) such that h only depends on y 2 one has :

∂ α G 1 * h = 0, a.e. in Z , α ∈ N 2 s.t. α ∈ {0, 1, 2}.
Lemma 6.3. define the horizontal Fourier transform, given f ∈ L 2 (Z )

F k ( f )(k, y 2 ) := 1 0 f (y 1 , y 2 )e -i k y 1 d y 1 , ∀k ∈ Z, a.e y 2 ∈ R.

If we define the convolution operators : for

g ∈ L 1 (Z ) ∩ L 2 (Z ) and f ∈ L 2 (Z ) g * y f := Z g (x -y) f (y)d y, (g * y 2 f )(x 1 , x 2 , z 1 ) := R g (x 1 , x 2 -y 2 ) f (z 1 , y 2 )d y 2 ,
then one has

F k (g * y f )(k, x 2 ) = (F k g * y 2 F k f )(k, x 2 ), ∀k ∈ Z, a.e. x 2 ∈ R.
Proof. If f ∈ L 2 (Z ) and g ∈ (L 1 ∩ L 2 )(Z ) then the convolution with respect to both variables is well defined.

Setting

( f * y 1 g )(x 1 , x 2 , z 2 ) := R f (x 1 -y 1 , x 2 )g (y 1 , z 2 )d y 1 ,
almost everywhere in y 2 and for every k ∈ Z one has

F k f * y 1 g (x 1 , x 2 -y 2 , y 2 ) = (F k f )(k, x 2 -y 2 )(F k g )(k, y 2 ).
Integrals in the vertical direction commute with the horizontal Fourier transform due to Fubini's theorem.

So one shall write :

R

F k f * y 1 g (k, x 2 -y 2 , y 2 )d y 2 = F k R ( f * y 1 g )(k, x 2 -y 2 , y 2 )d y 2 = F k ( f * y g )(k, x 2 ),
which ends the proof.

Lemma 6.4. For f ∈ L 2 β (Z ), one has f 2 L 2 β (Z ) = k∈Z F k ( f ) 2 L 2 β (R) , ∀β ∈ R.
The proof the above lemma is given in C.

∂ α G * ( f -f ) L 2 β ′ (Z ) ≤ C f -f L 2 β (Z ) , ∀β ′ < β - 1 2 , |α| ∈ {0, 1},
where the constant C is independent of the data.

Proof. Set again h := ff . By Lemma 6.4 :

∂ α G * h 2 L 2 β ′ (Z ) = ∂ α G 1 * h 2 L 2 β ′ (Z ) = k∈Z * F k (∂ α y G 1 * y h) 2 L 2 β ′ (R) = k∈Z * (i k) α 1 F k (∂ α 2 y 2 G 1 ) * y 2 F k (h) 2 L 2 β ′ (R) .
For all k ∈ Z * , thanks to Lemma 6.5, a.e. x in Z ,

(i k) α 1 F k (∂ α 2 y 2 G 1 ) * y 2 F k (h) ≤ R e -| k||y 2 -x 2 | | k| 1-|α| F k (h)(k, y 2 )d y 2 ≤ C F k (h)(k, •) L 2 β (R) ρ(x 2 ) -β | k| 3 2 -|α| ≤ C F k (h)(k, •) L 2 β (R) ρ(x 2 ) -β , for α multi-index s.t. |α| ∈ {0, 1}. Then ∂ α G * h 2 L 2 β ′ (Z ) ≤ k∈Z * F k (h)(k, •) 2 L 2 β (R) R ρ(x 2 ) 2β ′ -2β d x 2 ≤ C f L 2 β (Z ) , if β ′ < β -1/2.
We first have the following result.

Theorem 6.1. Let α be a real number such that α > 3/2 and f ∈ D # (Z ), then for any ǫ > 0 one has

G 2 * f -H (•, f ) L 2 α-2-ǫ (Z ) ≤ C f L 2 α (Z )
where

H (x 2 , f ) = 1 2 |x 2 | R f (y 2 )d y 2 -sgn(x 2 ) R y 2 f (y 2 )d y 2 .
Furthermore, if f is orthogonal to polynomials of

P ′ 1 , namely f ∈ D # (Z )⊥P ′ 1 , then H (., f ) = 0 and the above estimate is reduced to G 2 * f L 2 α-2-ǫ (Z ) ≤ C f L 2 α (Z ) .
Proof. As the argumentation is purely one-dimensional, we denote by f the average of f in the y 1 direction omitting the overline symbol. Note that we can write the convolution |x 2 | * f explicitly:

(|x 2 | * f ) = x 2 -∞ (x 2 -y 2 ) f (y 2 )d y 2 - ∞ x 2 (x 2 -y 2 ) f (y 2 )d y 2 = x 2 x 2 -∞ f (y 2 )d y 2 - ∞ x 2 f (y 2 )d y 2 + - x 2 -∞ y 2 f (y 2 )d y + ∞ x 2 y 2 f (y 2 )d y 2 .
Pointwize considerations show that

1 2 |x 2 | * f ∼ H (x 2 , f ), when |x 2 | → ∞.
Indeed,

|x 2 | * f |x 2 | ∼ x 2 x 2 -∞ f (y 2 )d y 2 - ∞ x 2 f (y 2 )d y 2 |x 2 | → R f (y 2 )d y 2 Âȃwhen |x 2 | → ∞ and - x 2 -∞ y 2 f (y 2 )d y 2 + ∞ x 2 y 2 f (y 2 )d y 2 → -sgn(x 2 ) R y 2 f (y 2 )d y 2 ȃwhen |x 2 | → ∞.
A more specific calculation shows that actually

1 2 |x 2 | * f -H (x 2 , f ) =|x 2 | x 2 -∞ f (y 2 )d y 2 ✶ R -(x 2 ) - ∞ x 2 f (y 2 )d y 2 ✶ R + (x 2 ) + sgn(x 2 ) x 2 -∞ y 2 f (y 2 )d y 2 ✶ R -(x 2 ) + ∞ x 2 y 2 f (y 2 )d y 2 ✶ R + (x 2 ) .
Provided that α belongs to the correct interval, an easy use of Hölder estimates shows then that:

1 2 |x 2 | * f -H (x 2 , f ) ≤ C ρ 3 2 -α (x 2 ) f L 2 α (Z ) , ∀x 2 ∈ R
which then multiplying by the appropriate weight on both sides and integrating wrt x 2 gives the desired result.

The below lemma is used in the next theorem and also to get weighted estimates on the convolution with the derivative of the Green function.

Lemma 6.6. Let α ∈ R, f ∈ D # (Z ) and ǫ a positive real parameter.

1. If α > 1 2 and if f is orthogonal to R, namely f ∈ D # (Z )⊥R, then sgn * f L 2 α-1-ǫ (Z ) ≤ C f L 2 α (Z ) . 2. If α ≤ 1 2 , then sgn * f L 2 α-1-ǫ (Z )/R ≤ C f L 2 α (Z ) .
Proof.

[(i)]
Suppose in a first step that f does not satisfy the polar condition f ⊥R. For α > 1 2 , the constants belong to L 2 -α , this implies in an obvious way that R f (y 2 )d y 2 is well defined. Then one writes :

(sgn(x 2 ) * f )(x) = y 2 <x 2 f (y 2 ) d y 2 - y 2 >x 2 f (y 2 ) d y 2 .
For x 2 tending to infinity, f * sgn(x 2 ) behaves as sgn(x 2 ) R f . Indeed

(sgn(x 2 ) * f )(x) -sgn(x 2 ) f (s)d s = 2 x 2 -∞ f (s)d s ✶ R -(x 2 ) - ∞ x 2 f (s)d s ✶ R + (x 2 ) ≤ 2 f L 2 α (Z ) x 2 -∞ ρ -2α (y 2 )d y 2 1 2 ✶ R -(x 2 ) + ∞ x 2 ρ -2α (y 2 )d y 2 1 2 ✶ R + (x 2 ) ≤ C f L 2 α (Z ) ρ 1 2 -α (x 2 ).
Notice that since α > 1/2, the integrals

x 2 -∞ ρ -2α (y 2 )d y 2 and

∞

x 2 ρ -2α (y 2 )d y 2 are defined. Then taking the square, multiplying by ρ 2β and integrating wrt x

sgn(x 2 ) * f -sgn(x 2 ) f (s)d s 2 L 2 β (Z ) ≤ C f 2 L 2 α (Z ) R ρ 1-2α+2β (x 2 )d x 2 ,
which is bounded, provided that β < α -1. Taking into account the polar condition f ⊥R gives then the first statement.

1. 2. We use a duality argument. For any ϕ ∈ D # (Z ), one can write

|〈sgn * f , ϕ〉 D ′ # (Z ),D # (Z ) | = |〈 f , sgn * ϕ〉 L 2 α (Z ),L 2 -α (Z ) | ≤ f L 2 α (Z ) sgn * ϕ L 2 -α (Z ) ≤ f L 2 α (Z ) ϕ L 2 -α+1+ǫ (
Z )⊥R , the last inequality being true using the first statement of the theorem provided that -α + 1 + ǫ > 1/2, which is the case if α ≤ 1 2 and ǫ > 0. This proves the second statement.

Similarly to Theorem 6.1, thanks to the previous lemma, one gets Theorem 6.2. Assume α be a real number such that

1 2 < α ≤ 3 2 . If f ∈ D # (Z )⊥R then for any ǫ > 0 one has G 2 * f L 2 α-2-ǫ (Z )/R ≤ C f L 2 α (Z ) .
The proof of this above theorem is given in D.

We can now summarized Theorem 6.1, Lemma 6.6 and Theorem 6.2.

Theorem 6.3. Assume α ∈ R and f ∈ D # (Z )⊥P ′ ∆ q(0,-α)
, where q(0, -α) is defined by Proposition 2.2. Then for any ǫ > 0, we have

G 2 * f L 2 α-2-ǫ (Z )/P ′ ∆ [ 3 2 -α] ≤ C f L 2 α (Z ) .
Proof. Let us first observe that the space of the harmonical polynomials of L 2 α-2-ǫ (Z ) that only depend on y 2 is the space P ′ ∆ q(0,α-2-ǫ) . But since ǫ > 0 is arbitrary, then using (2.14), we have

P ′ ∆ q(0,α-2-ǫ) = P ′ ∆ [ 3 2 -α]
.

Furthermore, using again (2.14), we also have

P ′ ∆ q(0,-α) =      R if 1/2 < α ≤ 3/2, P ′ 1 if α > 3/2 and P ′ ∆ [ 3 2 -α] =      R if 1/2 < α ≤ 3/2, {0} if α > 3/2.
Therefore thanks to Theorem 6.1 and Theorem 6.2, the statement is proved for α > 1 2 . Now as for Lemma 6.6, for the case α ≤ 1 2 , we use a duality argument. For any ϕ ∈ D # (Z ), one can write

|〈G 2 * f , ϕ〉 D ′ # (Z ),D # (Z ) | = |〈 f ,G 2 * ϕ〉 L 2 α (Z )⊥P ′ ∆ q(0,-α) ,L 2 -α (Z )/P ′ ∆ q(0,-α) | ≤ f L 2 α (Z ) G 2 * ϕ L 2 -α (Z )/P ′ ∆ q(0,-α) ≤ f L 2 α (Z ) ϕ L 2 -α+2+ǫ (Z )⊥P ′ ∆ [ 3 2 -α]
,

the last inequality being true provided that -α + 2 + ǫ > 1 2 which is the case if α ≤ 1 2 .

We extend the above statement to data that belong to L 2 α (Z ).

Theorem 6.4. Let α ∈ R and recall that q(0, -α) is defined by (2.14). Then for any function f ∈ L 2 α (Z )⊥P ′ ∆ q(0,-α) , one defines by density the convolution G * f as a function in

H 1 α-1-ǫ,# (Z )/P ′ ∆ [ 3 2 -α]
, for any ǫ > 0, and one has the estimates

G * f H 1 α-1-ǫ,# (Z )/P ′ ∆ [ 3 2 -α] ≤ C f L 2 α (Z )⊥P ′ ∆ q(0,-α)
,

where the constant C does not depend on the data.

Proof. The result is obtained by density. Proceeding as in the proof of Theorem 6.2, one is able to construct a regular sequence f δ ∈ D # (Z ) orthogonal to P ′ ∆ q(0,-α) converging strongly to f in the L 2 α (Z ) norm. Then Lemma 6.1 and 6.2 provide the decomposition :

G * f δ = G * ( f δ -f δ ) +G * f δ = G 1 * ( f δ -f δ ) +G 2 * f δ . ( 6.37)
Then it suffices now to apply Proposition 6.2, Lemma 6.6 and Theorem 6.3 to obtain that

G * f δ H 1 α-1-ǫ (Z )/P ′ ∆ [ 3 2 -α] ≤ C G 1 * ( f δ -f δ ) H 1 α-1-ǫ,# (Z ) + G 2 * f δ H 1 α-1-ǫ (Z )/P ′ ∆ [ 3 2 -α] ≤ C f δ L 2 α (Z )⊥P ′ ∆ q(0,-α) ≤ C f L 2 α (Z )⊥P ′ ∆ q(0,-α)
.

Then passing to the limit wrt δ extends the convolution to any function

f ∈ L 2 α (Z )⊥P [α-1 2 ]
We generalize the latter convolutions to weak data, namely, to f ∈ H -1 α,# (Z ).

Proposition 6.3. Let α ∈ R and f ∈ H -1 α,# (Z )⊥P ′ ∆
q(1,-α) , where q(1, -α) is defined by (2.13). Then, for any ǫ > 0, we have

G * f L 2 α-1-ǫ (Z )/P ′ ∆ [ 1 2 -α] ≤ C f H -1 α,# (Z ) ,
where C is independent on f .

Proof. The proof is again based on a duality argument. Indeed, thanks to Theorem 6.4, for any ϕ ∈ D # (Z ), we can write

〈 f ,G * ϕ〉 H -1 α,# (Z )⊥P ′ ∆ q(1,-α) ×H 1 -α,# (Z )/P ′ ∆ q(1,-α) ≤ f H -1 α,# (Z ) G * ϕ H 1 -α,# (Z )/P ′ ∆ q(1,-α) ≤ C f H -1 α,# (Z ) ϕ L 2 -α+1+ǫ (Z )⊥P ′ ∆ [ 1 2 -α]
, which proves the claim.

We are now in a position to give the proof of Theorem 1.2.

Sketch of the proof of Theorem 1.2.

Observe first that since we assume that α ∉ Z + 1 2 , then the polynomial spaces P ′ ∆ q(1,-α) and P ′ ∆ [1/2+α] coincide. We shall now decompose the proof into two steps:

1. In the first step we prove the statement for m = -1 which amount to prove that if α ∈ R satisfies

α ∉ Z + 1 2 , and f ∈ H -1 α,# (Z )⊥P ′ ∆ [1/2+α] , then G * f ∈ H 1 α,# (Z )
G * f H 1 α,# (Z )/P ′ ∆ [1/2-α] ≤ C f H -1 α,# (Z ) . Since f ∈ H -1 α,# (Z )⊥P ′ ∆ [1/2+α]
, then on the one hand, Proposition 6.3 yields that G * f belongs to

L 2 α-1-ǫ (Z )/P ′ ∆ [ 1 2 -α]
,

for any ǫ > 0, and on the other hand, isomorphism (5.32) shows that there exists a unique u in

H 1 α,# (Z )/P ′ ∆ [ 1 2 -α] ⊂ L 2 α-1-ǫ (Z )/P ′ ∆ [ 1 2 -α]
. Therefore thanks to Proposition 5.

1, u -G * f is a polynomial of P ′ ∆ [ 1 2 -α] ⊂ H 1 α,# (Z ) which in turn shows that G * f belongs to H 1 α,# (Z )/P ′ ∆ [ 1 2 -α]
.

2. In the second step we first assume m ≥ 0,

α ∈ R satisfying α ∉ Z + 1 2 and let f ∈ H m α,# (Z )⊥P ′ ∆ [-m-1/2+α] . Then it is clear that f ∈ H -1 α-m-1,# (Z )⊥P ′ ∆ [-m-1/2+α] .
-G * f ∈ P ′ ∆ [m+3/2-α] .
Finally using a duality argument we prove the statement for m ≤ 0 and α ∈ R satisfying α ∉ Z + 1 2 .

Solutions for data not satisfying any polar condition

We recall here Lemma 4.9 p. 40 [START_REF] Alliot | Étude des équations stationnaires de Stokes et Navier-Stokes dans des domaines extérieurs[END_REF],

Lemma 7.1. Let E be a Banach space, F a dense subset in E , M a subspace of finite dimension k of E ′ and ( f 1 , . . . , f k ) a basis in M there exists a sequence (g 1 , . . . , g k ) in F s.t.

< f i , g j > E ′ ,E = δ i j ,
and every element ϕ ∈ E can be decomposed as ϕ = g 0 + h s.t.

g 0 = k i =1 < h, f i > g i , h := ϕ -g 0
where g 0 ∈ F and h ∈ E ⊥M .

We now give a result that solves (1.1) when the datum f does not satisfy any polar condition. where the constant does not depend on the data nor on u and χ is a cut-off function equal to 1 in the neighborhood of zero and vanishing out of a compact support.

2. Let θ be a periodical D(R)-partition of unity and let (α j ) j ∈N be a sequence such that α j ∈ D(R 2 ), α j ≥ 0, R 2 α j (x )d x = 1 and the support of α j is included in the closed ball of radius r j > 0 and centered at (0, 0) where r j → 0 as j → ∞. It is well known that as j → ∞, α j converges in the distributional sense to the Dirac measure. We set w(y ) = θ(y 1 )u(y ). Then w belongs to H m (R 2 ) and has a compact support. Moreover, since ωθ = 1, we have ωw = ω(θu) = (ωθ)u = u. We define ϕ j = w * α j . Then ϕ j belongs to D(R 2 ) and converges to w in H m (R 2 ) as j tends to ∞. Let ψ j = ωϕ j , then ψ j belongs to D # (Z ) and thanks to Lemma 2.1, ψ j converges to ω w = u in H m α,# (Z ) as j tends to ∞.

B Proof of Proposition 6.1

We define:

G = lim N →∞ G N = lim N →∞ |k|<N J k e i k y 1 , with J k = R e i l y 2 k2 + l 2 d l = lim R→∞ R -R e i l |y 2 | k2 + l 2 d l .
The absolute value in the last right hand side is added as follows:

R

e i l y 2 k2 + l 2 d l = ∞ 0 e i l y 2 + e -i l y 2 k2 + l 2 d l = 2 ∞ 0 cos( l y 2 ) k2 + l 2 d l = 2 ∞ 0 cos( l |y 2 |) k2 + l 2 d l .
Proceeding as in example 1 p. 58 [START_REF] Reed | Methods of modern mathematical physics. II. Fourier analysis, self-adjointness[END_REF], one has, extending the integral to the complex plane, that

J k ≡ lim R→∞ C R e i z|y 2 | k2 + z2 d z = lim R→∞ C R e i z|y 2 | ( z + i |k|)( z -i |k|) d z = lim R→∞ 1 2π C R f(z) z -i |k| d z = i f(i |k|) = e -|
Re(z) Im(z) C R (-R, 0)
Figure 1: Path of integration in the complex plane thus one concludes formally that

J 0 = R e i l y 2 l 2 d l = - 1 2 |y 2 |.
The L 2 bound is achieved thanks to the Parseval formula

G 1 2 L 2 (Z ) = F (G 1 ) 2 ℓ 2 (Z;L 2 (R)) = k∈Z * R 1 ( k2 + l2 ) 2 d ℓ ≤ C k∈Z * 1 | k| 3 ≤ C ′ .
One recalls the expansion in series of the logarithm :

ln(1 -z) = - ∞ k=1 z k k , ∀z ∈ C : |z| < 1.
Thus for y = 0

G 1 (y ) = - 1 2π
Reln 1e 2π(-|y 2 |+i y 1 ) = - C Proof of Lemma 6.5

We set |x| = R and let R 0 be a real such that R 0 > 1. We define three regions of the real line :

D 1 := B (0, R/2), D 2 := B (0, 2R) \ B (0, R/2), D 3 := R \ B (0, 2R)
Decomposing the convolution integral in these three parts, one gets :

I (x) = 3 i =1 I i (x), where I i (x) = D i e -|a||x-y| ρ(y) -b d y.
We have two cases to investigate : R > R 0 and R ≤ R 0 .

1

. If R > R 0 (a) On |y| ≤ R/2, |x -y| ∼ R so that I 1 (x) ∼ e -|a|R |y|≤ R 2 (1 + |y|) -b d y.
According to the value of b, one has

I 1 (x) ≤ C e -|a|R          1 if b > 1, R if 0 ≤ b ≤ 1, R 1-b if b < 0. As R > 1, there holds e -|a|R ≤ ([σ] + 1)! (|a|R) σ . Setting σ(b) := b ✶ ]1,+∞[ (b) + (1 + b) ✶ [0,1] (b) + ✶ ]-∞,0[ (b), one recovers the claim in that case. (b) On R/2 ≤ |y| ≤ 2R, |y| ∼ R I 2 (x) ∼ R -b
b! ( b -p)! (|a|R) -p ≤ C e -|a|R |a| R b b! b ≤ C (b)R -b |a| .
In the latter inequality we used decreasing properties of the exponential function in order to fix the correct behavior for large R.

• either b > 0 and one has directly that

D Proof of Theorem 6.2

As is the proofs of Theorem 6.1 and Lemma 6.6, the proof is essentially 1D. Let us choose f ∈ D # (R)⊥R and a It is easy to check that ϕ δ ⊥R and that for every ǫ 1 there exists δ 0 s.t. δ < δ 0 ,

test function ϕ ∈ L 2 2-α+ǫ (R)⊥R. We construct (ϕ δ ) δ ∈ D(R)⊥R s.t. ϕ δ → ϕ in L 2 2-α+ǫ (
ϕ δ -ϕ L 2 2-α+ǫ (R) ≤ ψ δ -ϕ L 2 2-α+ǫ (R) + -1 δ -∞ ϕd y + ∞ 1 δ ϕd y ψ L 2 2-α+ǫ (R) ≤ ǫ 1 /2 + 2δ 3 2 +ǫ-α ϕ L 2 2-α+ǫ ψ L 2 2-α+ǫ (R) ≤ ǫ 1
where we used that

R ψ δ d y = R ϕ✶ -1 δ , 1 δ * Θ δ d y = R ϕ✶ -1 δ , 1 δ d z R Θ δ d y
Θ δ being the standard mollifier whose mean value is one, presented in (ii) part of the proof of Proposition 2.3, we used also the fact that

-1 δ , 1 δ ϕd z = - -∞,-1 δ ϕd z + 1 δ ,∞ ϕd z .
due to the orthogonality condition R ϕd y = 0. Then |x| * ϕ δ is infinitely differentiable and one can apply the Taylor expansion with the integral rest :

R (|x| * ϕ δ )(x) f (x)d x = R (|x| * ϕ δ )(0) + x 0 (ϕ δ * sgn)(s)d s f (x)d x ≡ R f (x) x 0 (ϕ δ * sgn)(s)d sd x.
Note that the regularization of ϕ plays here a crucial role since |x| * ϕ(0 3 2 ]. Proceeding as in the proof of Lemma 6.6, one has

) = R |xÂȃ|ϕ(x)d x does not make sense if ϕ ∈ L 2 2-α+ǫ (R) for α ∈] 1 2 ,
x 0 ϕ δ * sgn(s) d s ≤ ϕ δ L 2 2-α+ǫ (Z ) ρ α-1 2 -ǫ (x), leading to x 0 (ϕ δ * sgn)(s)d s L 2 -α (R) ≤ C ϕ δ L 2 2-α+ǫ (R) , with α < 3 2 + ǫ. Thus one has R (|x| * ϕ δ )(x) f (x)d xd y ≤ C f L 2 α (R) ϕ δ L 2 2-α+ǫ (R) .
Moreover, one needs that R ∈ L 2 -α (R), which is true if α > 1 2 . It is not difficult to prove by similar arguments that -α (R) topology. As in the proof of Lemma 6.6, by Fubini,

x 0 (sgn * ϕ)(s)d s = 2 x 0 s -∞ ϕ(t )d t ✶ s<0 (s) - ∞ s ϕ(t )d t ✶ s>0 (s) d s =: 2 x 0 g(s)d s.
As g is a C 1 function on any compact set in R, one can integrate by parts on (0, x) :

x 0 g(s)d s = [g(s)s] x 0 - x 0 g ′ (s) s d s.
Using this in the right hand side of the previous limit, one writes By Hölder estimates the integrals above are well defined. Fubini's theorem allows then to switch integration order. Using the orthogonality condition on ϕ and on f , one may easily show that Proof. We take u ∈ V α and by Proposition 1 we construct ϕ ∈ H 1 1 2 ,# (Z )/R satisfying ∇ ⊥ ϕ = u. We approximate it by a sequence ϕ j ∈ D # (Z ), as ∇ ⊥ is a continuous operator one has ∇ ⊥ ϕ j → ∇ ⊥ ϕ, strongly in L 2 α (Z ).

J := R f (x) x 0 (ϕ * sgn)(s)d sd x = 2 R f (x) g(x)x - x 0 g ′ (s) s d s d x = 2 R - f (x) x x -∞ ϕ(t )d t d x - R + f (x)x ∞ x ϕ(t )d t d x - R f (x) x 0 ϕ(s) s d s = 2 R f (x) x x -∞ ϕ(t )d t d x -2 R f (x)
                                     A = 2 R x f (x) x x -∞ ϕ(t )d t d x = R f (x)x
this ends the proof.

We are now ready to prove the proposition. Let u ∈ D ′ # (Z ) such that ∇u ∈ L 2 where the constant C does not depend on Proof. As g is in D # (Z ) it belongs to H m α,# (Z ) for all m ∈ Z and α ∈ R. Applying Theorem 1.2, one obtains that G * g is H 2 loc,# (Z ) which implies that on every compact set K ′ one has

∂ γ G * g L 2 α (K ′ ) ≤ C g L 2 α (Z ) .
Using then Lemma 2, one has that

∂ γ G * g L 2 α (Z \K ′ ) ≤ C g L 1 (Z ) If K ′ is

Proposition 6 . 1 .

 61 The Green function G solving (6.34) is a tempered distribution and it readsG(y) = 1 2 k∈Z * e -| k||y 2 |+i k y 1 | k| -|y 2 | = G 1 (y) +G 2 (y), ∀y ∈ Z ,

Lemma 6 . 5 . 6 . 3 Non-homogeneous estimates Proposition 6 . 2 .

 656362 Let a ∈ R s.t. |a| ≥ 1 then for any real b, one has I (x) := R e -|a||x-y| ρ(y) -b d y ≤ C ρ(x) -b |a| , x ∈ R. Let us set β a real number. Then for every f ∈ D # (Z )

  is a solution of the Laplace equation (5.20) unique up to a polynomial of P ′ ∆ [1/2-α] . Moreover, we have the estimate

k||y 2 |

 2 2| k| , where C R := {z ∈ C; |z| = R and Im(z) > 0} ∪ ([-R; R] × { 0}) as depicted in fig. 1 and f(z) := e i z|y 2 | /( z + i |k|), being a holomorphic function inside C R . The 1D-Fourier transform of the tempered distribution |y| is -2/ l 2 , (R, 0)

e

  -|a||x-y| d y, because |x -y| ≤ 3R, one has I 2 (x) ∼ R -b /|a|. (c) On the rest of the line |y| > 2R, |x -y| ∼ |y|, so that I 3 (x) ∼ |y|>2R e -|a||y| |y| -b d y. Two situations occur : • either b ≤ 0 then we set b := [-b] + 1 and one has |y|>2R e -|a||y| |y| -b d y ≤ C e -|a|R |a| R b b p=0

|y|>2Re

  -|a||y| |y| -b d y ≤ R -b |y|>R e -|a||y| d y.2. Otherwise, if R < R 0 then ρ(xy) ∼ ρ(y), and thusI (x) ∼ R e -|a||y| ρ(y) -b d y < C |a| .

  R)⊥R : by truncation and smoothing we setψ δ ∈ D # (Z ) s.t. ψ δ → ϕ in L2 2-α+ǫ (R)⊥R strongly. Then we choose an arbitrary function ψ ∈ D # (Z ) s.t. R ψd y = 1 and set ϕ δ (y) := ψ δ (y) -R ψ δ d y ψ.

lim δ→0 x 0 (

 0 ϕ δ * sgn)(s)d s = x 0 (ϕ * sgn)(s)d s strongly in the L 2

x 0 ϕ

 0 (s) s d s d x =: A -B.

2 Proposition 1 . 1 1 1 1 1 1Proposition 2 .

 211112 s) f (x)d x -s -∞ (xs) f (x)d x d s = R ϕ(s) R |x -s| f (x)d xd s = R ϕ(s)(|x| * f )(s)d s. We define V := {u ∈ D # (Z ) s.t. div u = 0}, ∇ ⊥ := (∂ y 2 , -∂ y 1 ) T a subspace of H 1 α,# (Z ) denoted V α := {u ∈ L 2 α (Z ) s.t. div u = 0}. Let α = 12 given a function u ∈ V α then there exists a function ϕ ∈ H s.t.∇ ⊥ ϕ = u, a.e y ∈ Z .Proof. We use the isomorphisms of the Laplace operator of Theorem 5.1. A simple computation gives that rot u ∈ H -⊥R thus : looking for ϕ s.t.∆ϕ = rot u, y ∈ Z .there exists a unique ϕ ∈ H /R solving this problem. Then applying the operator ∇ ⊥ to this equation one gets:∆(∇ ⊥ ϕu) = 0which by Proposition 5.1 implies the claim. If α = 1 2 , the set V is dense in V α .

-1 2 ( 2 , 1 2.FLemma 1 . 1 1 - 1 2 2 ≤Lemma 2 .Theorem 1 .

 221111221 Z ). Then for any ϕ ∈ D # (Z ), we have 〈∇u, ϕ〉 = -〈u, div ϕ〉, which implies that ∇u is orthogonal to V . Due to the density of V in V1 we deduce that ∇u is also orthogonal to V Thanks to Proposition 3.1, there exists w ∈ H 1 -1 2 ,# (Z ), such that ∇w = ∇u. Therefore there exists K ∈ R such that w = u + K ∈ H 1 -1 2 ,# (Z ). Now since constants belong to H 1 -1 2 ,# (Z ), we deduce that u ∈ H 1 Weighted estimates for higher order derivatives of the convolution with G If |y 2 | > 4 then for all multi-index γ s.t. |γ| = 2 one has|∂ γ G(y)| ≤ C e -2π|y 2 |The constant C does not depend on y 2 .Proof. A simple computation of second order derivatives gives :|∂ γ G(y )| ≤ C cosh(2πy 2 ) 1 -cos(2πy1 ) cosh(2πy 2 ) 2 and the constant C does not depend on y 2 . If |y 2 | > 4 > ln(4)/2π then cosh(2πy 2 ) > 2 and thus -(y )| ≤ C cosh(2πy 2 ) -1 C ′ e -2π|y 2 | . If g ∈ D # (Z ) then if we denote K := supp(g ) and one denotes ℓ := sup y ∈K |y 2 | one has that(∂ γ G * g )(y) ≤ C g L 1 (Z ) e -π|y 2 | , ∀y s.t. |y 2 | > max(2ℓ, 2), ∀γ ∈ N 2 s.t. |γ| = 2 Proof. Using that if z s.t. |z 2 | > 2ℓ then |z 2y 2 | > 1 2 |z 2 | > ℓthus applying the previous lemma, one has then also that|(∂ γ G)(zy )| ≤ C e -2π|z2 -y 2 | ≤ C e -π|z 2 | , ∀z s.t. |z 2 | > 2ℓ. Now multiplying (∂ γ G)(zy ) by g (y ) and integrating then with respect to y on K , one has the claim. If g ∈ D # (Z ) then ∂ γ G * g ∈ L 2 α (Z ) for any α ∈ R, and one has ∂ γ G * g L 2 α (Z ) ≤ C g L 2 α (Z ) , ∀γ ∈ N 2 s.t. |γ| = 2.

Theorem 2 .

 2 chosen big enough. As g is compactly supported, all weighted norms are equivalent, which ends the proof.By similar arguments one can prove as well :If g ∈ D # (Z ) then ∂ γ G * g ∈ L 2α (Z ) for any α ∈ R and any multi-index γ s.t. |γ| ≥ 2, and one has∂ γ G * g L 2 α (Z ) ≤ C g L 2 α (Z ) .

  On the one hand, from the first step G * f belongs to the space H1

	α-m-1,# (Z )/P	′ ∆ [m+3/2-α] . On the other hand, from (5.33), there exists a unique
	u in H m+2 α,# (Z )/P ′ ∆ [m+3/2-α] , a subset of H 1 α-m-1,# (Z )/P	′ ∆ [m+3/2-α] . Then using Proposition 5.1 we have
	u

Proof. By Lemma 7.1, one has the decomposition f = h + g where h ∈ H m α,# (Z )⊥P ′∆ [α- 1 2 -m] and g ∈ D # (Z ).

By (5.33) there exists a unique u h ∈ H m+2 α,# (Z )/P ′ ∆ [m+ 3 2 -α] solving -∆u h = h. On the other hand, by Theorem 6.1 combined with the proof of Lemma 6.6, and Theorem 2, in F, one gets easily, because of the compact support of g , that u g = G * g satisfies

(Z ) using the isomorphisms already established one concludes that there exists a

(Z ) and using isomorphisms results in this latter space by uniqueness one identifies u ≡ u f modulo polynomials in P ′∆ 1 which ends the proof.

Appendices A Proof of Proposition 2.3

For the proof of Proposition 2.3, the ideas of the proof come from [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF] and [START_REF] Nguetseng | Espaces de distributions sur des ouverts périodiques et applications[END_REF]. Let u be in H m α,# (Z ).

1. We first approximate u by functions with compact support in the

For ℓ ∈ N, we introduce the function Φ ℓ , defined by

, e ℓ ], λ ∈ N, owing that t ≤ 1 + t 2 ≤ 2 t and ln t ≤ ln(2+ t 2 ) ≤ 3 ln t , we prove that (see [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF], Lemma 7.1):

where C is a constant independent of ℓ. For a.e. y ∈ Z , we set u ℓ (y ) = u(y )Φ ℓ (y 2 ). Then, proceeding as in [START_REF] Amrouche | Weighted sobolev spaces and laplace's equation in R n[END_REF] (Theorem 7.2), one checks easily that u ℓ belongs to H m α,# (Z ), has a compact support in the y 2 direction, and that u ℓ converges to u in H m α,# (Z ) as ℓ tends to ∞. Thus, the functions of H m α,# (Z ) with compact support in the y 2 direction are dense in H m α,# (Z ) and, we may assume that u has a compact support in the y 2 direction.


Claire Berger

Jean-Yves Veuillen

Laurence Magaud

Pierre Mallet

Valerio Olevano

Xavier Blase

Milan Orlita

Paulina Plochocka

Clement Faugeras

Gines Martinez

Electronic properties of epitaxial graphene

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Electronic properties of epitaxial graphene

Claire Berger, Jean--Yves Veuillen, Laurence Magaud, Pierre Mallet, Valerio Olevano, Xavier Blase, M. Orlita, P.Plochocka, C. Faugeras, G. Martinez, Marek Potemski, Cécile Naud, Laurent Lévy and Didier Mayou Institut Néel CNRS and UJF LCNMI 25 Avenue des Martyrs 38042 Grenoble Cedex 9 *Corresponding author: didier.mayou@grenoble.cnrs.fr Abstract: It has been known for almost 25 years that high temperature treatment of polar faces of SiC crystals lead to the graphitization of the surface as a consequence of Si preferential sublimation. The group of Walt de Heer in Atlanta has proposed to use this procedure to obtain macroscopic multilayer graphene samples. This has lead to a renewal of interest for the study of the graphitic layers grown on the SiC surfaces. Due to the polar nature of the material, the 6H(4H)--SiC substrate present two non--equivalent surfaces: the (0001) is known as the Si face, and the (000--1) one is the C face. It turns out that the structure and the properties of the multilayer graphene samples obtained on those two surfaces are rather different. In particular the multilayer graphene samples grown on the C--terminated face presents electronic properties that are rather similar to those of monolayer graphene with very high mobility. This is shown by transport and optical experiments under high magnetic field. This surprising result is explained by the rotational stacking sequence, observed experimentally. which leads to an electronic decoupling as demonstrated by ab--initio band structure calculations.