Geodesic
On necessary conditions for Fourier multipliers of weak type
Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 75-95.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain necessary conditions for a bounded function $\varphi$ to be a Fourier multiplier of weak type $\bigl(\psi(L)(G),\psi(L)(G)\bigr)$, where $G=\mathbb R^d$ or $G=\mathbb T^d$, provided that the Young function $\psi(t)$ grows slower than $t\ln^{1/2}t$ as $t$ tends to infinity. 
@article{IM2_2007_71_1_a5,
     author = {A. V. Rozhdestvenskii},
     title = {On necessary conditions for {Fourier} multipliers of weak type},
     journal = {Izvestiya. Mathematics },
     pages = {75--95},
     publisher = {mathdoc},
     volume = {71},
     number = {1},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a5/}
}
TY  - JOUR
AU  - A. V. Rozhdestvenskii
TI  - On necessary conditions for Fourier multipliers of weak type
JO  - Izvestiya. Mathematics 
PY  - 2007
SP  - 75
EP  - 95
VL  - 71
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a5/
LA  - en
ID  - IM2_2007_71_1_a5
ER  - 
%0 Journal Article
%A A. V. Rozhdestvenskii
%T On necessary conditions for Fourier multipliers of weak type
%J Izvestiya. Mathematics 
%D 2007
%P 75-95
%V 71
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a5/
%G en
%F IM2_2007_71_1_a5
A. V. Rozhdestvenskii. On necessary conditions for Fourier multipliers of weak type. Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 75-95. http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a5/

[1] L. Khermander, Otsenki dlya operatorov, invariantnykh otnositelno sdviga, IL, M., 1962 ; L. Hörmander, “Estimates for translation invariant operators in $L^p$ spaces”, Acta Math., 104 (1960), 93–140 | Zbl | DOI | MR | Zbl

[2] I. M. Stein, Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 ; I. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | Zbl | MR | Zbl

[3] A. V. Rozhdestvenskii, “Ob additivnom kogomologicheskom uravnenii i zamene vremeni v lineinom potoke na tore s diofantovym vektorom chastot”, Matem. sb., 195:5 (2004), 115–156 | MR | Zbl

[4] M. Wojciechowski, “A necessary condition for multipliers of weak type $(1,1)$”, Canad. Math. Bull., 44:1 (2001), 121–125 | MR | Zbl

[5] K. Woźniakowski, “A new proof of the restriction theorem for weak type $(1,1)$ multipliers on $\mathbb R^n$”, Illinois J. Math., 40:3 (1996), 479–483 | MR | Zbl

[6] N. Asmar, E. Berkson, J. Bourgain, “Restrictions from $\mathbb R^n$ to $\mathbb Z^n$ of weak type $(1,1)$ multipliers”, Studia Math., 108:3 (1994), 291–299 | MR | Zbl

[7] A. Zigmund, Trigonometricheskie ryady, t. 1, 2, Mir, M., 1965 ; A. Zygmund, Trigonometric series, v. 1, 2, 2nd ed., Cambridge Univ. Press, New York, 1959 | MR | Zbl | Zbl | MR | Zbl

[8] A. Yu. Karlovich, L. Maligranda, “On the interpolation constant for Orlicz spaces”, Proc. Amer. Math. Soc., 129:9 (2001), 2727–2739 | DOI | MR | Zbl

[9] S. A. Pichugov, “Translation invariant operators in linear metric spaces”, Anal. Math., 18:3 (1992), 237–248 | DOI | MR | Zbl

[10] W. Rudin, Fourier analysis on groups, Tracts in Pure and Applied Mathematics, 12, Interscience Publ., New York, 1962 | MR | Zbl

[11] A. V. Rozhdestvenskii, “O neravenstvakh Dzheksona i multiplikatorakh v $L_p$”, Matem. zametki, 57:4 (1995), 551–579 | MR | Zbl

[12] A. Pełczyń ski, K. Senator, “On isomorphisms of anisotropic Sobolev spaces with “classical Banach spaces” and a Sobolov type embedding theorem”, Studia Math., 84 (1986), 169–218 | MR | Zbl

[13] E. Berkson, J. Bourgain, A. Pełczynski, M. Wojciechowski, “Canonical Sobolev projections of weak type $(1,1)$”, Mem. Amer. Math. Soc., 150:714 (2001), viii+75 pp. | MR | Zbl

[14] T. Tao, J. Wright, “Endpoint multiplier theorems of Marcinkiewicz type”, Rev. Mat. Iberoamericana, 17:3 (2001), 521–558 | MR | Zbl

[15] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, M., Izd-vo AFTs, 1999 ; B. S. Kashin, A. A. Saakyan, Orthogonal series, Translations of Mathematical Monographs, 75, Amer. Math. Soc., Providence, RI, 1989 | MR | Zbl | MR | Zbl

[16] R. Coifman, J. L. Rubio de Francia, S. Semmes, “Multiplicateurs de Fourier de $L^p(\mathbb R)$ et estimations quadratiques”, C. R. Acad. Sci. Paris. Sér. I. Math., 306:8 (1988), 351–354 | MR | Zbl