Geodesic
Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$
Izvestiya. Mathematics, Tome 26 (1986) no. 2, pp. 223-262 
I. L. Bloshanskii. Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$. Izvestiya. Mathematics, Tome 26 (1986) no. 2, pp. 223-262. http://geodesic.mathdoc.fr/item/IM2_1986_26_2_a0/
@article{IM2_1986_26_2_a0,
     author = {I. L. Bloshanskii},
     title = {Two criteria for weak generalized localization for multiple trigonometric {Fourier} series of functions in $L_p$, $p\geqslant1$},
     journal = {Izvestiya. Mathematics},
     pages = {223--262},
     year = {1986},
     volume = {26},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1986_26_2_a0/}
}
TY  - JOUR
AU  - I. L. Bloshanskii
TI  - Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$
JO  - Izvestiya. Mathematics
PY  - 1986
SP  - 223
EP  - 262
VL  - 26
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_1986_26_2_a0/
LA  - en
ID  - IM2_1986_26_2_a0
ER  - 
%0 Journal Article
%A I. L. Bloshanskii
%T Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$
%J Izvestiya. Mathematics
%D 1986
%P 223-262
%V 26
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_1986_26_2_a0/
%G en
%F IM2_1986_26_2_a0

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

The concept of weak generalized localization almost everywhere is introduced. For the multiple Fourier series of a function $f$, weak generalized localization almost everywhere holds on the set $E$ ($E$ is an arbitrary set of positive measure $E\subset T^N=[-\pi,\pi]^N$) if the condition $f(x)\in L_p(T^N)$, $p\geqslant1$, $f=0$ on $E$ implies that the indicated series converges almost everywhere on some subset $E_1\subset E$ of positive measure. For a large class of sets $\{E\}$, $E\subset T^N$, a number of propositions are proved showing that weak localization of rectangular sums holds on the set $E$ in the classes $L_p$, $p\geqslant1$, if and only if the set $E$ has certain specific properties. In the course of the proof the precise geometry and structure of the subset $E_1$ of $E$ on which the multiple Fourier series converges almost everywhere to zero are determined. Bibliography: 13 titles.

[1] Bloshanskii I. L., “Obobschennaya lokalizatsiya pochti vsyudu i skhodimost dvoinykh ryadov Fure”, DAN SSSR, 242:1 (1978), 11–13 | MR

[2] Bloshanskii I. L., “Generalised localisation and convergence tests for double trigonometric Fourier series of functions from $L_p$, $p>1$”, Analysis Math., 7:1 (1981), 3–36 | DOI

[3] Carleson L., “On convergence and growth of partial sums of Fourier series”, Acta Math., 116 (1966), 135–157 | DOI | MR | Zbl

[4] Hunt R., “On the convergence of Fourier series”, Ortogonal Expansions and their Continuous Analogues (Proc. Conf. Edwardsville I11., 1967), Southern Illinouis Univ. Press, Carbondale I11., 1968, 235–255 | MR | Zbl

[5] Bloshanskii I. L., “O ravnoskhodimosti razlozhenii v kratnyi trigonometricheskii ryad Fure i integral Fure”, Matem. zametki, 18:2 (1975), 153–168

[6] Bloshanskii I. L., “O kriteriyakh slaboi obobschennoi lokalizatsii v $N$-mernom prostranstve”, DAN SSSR, 271:6 (1983), 1294–1298 | MR

[7] Bloshanskii I. L., “Obobschennaya lokalizatsiya dlya kratnykh ryadov Fure i geometriya izmerimykh mnozhestv v $N$-mernom prostranstve”, DAN SSSR, 266:4 (1982), 780–783 | MR

[8] Bloshanskii I. L., “O geometrii mnozhestv v N-mernom prostranstve, na kotorykh spravedliva obobschennaya lokalizatsiya dlya kratnykh trigonometricheskikh ryadov Fure funktsii iz $L_p$, $p>1$”, Matem. sb., 121:1 (1983), 87–110 | MR

[9] Zigmund A., Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR

[10] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[11] Whitney H., “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36 (1934), 63–89 | DOI | MR | Zbl

[12] Bloshanskii I. L., “O skhodimosti dvoinykh ryadov Fure funktsii iz $L_p$, $p>1$”, Matem. zametki, 21:6 (1977), 777–788 | MR

[13] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR