Geodesic
On $(H,k)$-summability of multiple trigonometric Fourier series
Izvestiya. Mathematics , Tome 11 (1977) no. 4, pp. 889-908.

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

A theorem is proved from which, in particular, it follows that if $f\in L(\ln^+L)^{N-1}$ on $T^N\equiv[-\pi,\pi]^N$, then the multiple trigonometric Fourier series of $f$ and all conjugate series are $(H,k)$-summable almost everywhere on $T^N$ for every $k>0$. In the case where $f\in L(\ln^+L)^{N+1}$ this result was obtained by Marcinkiewicz (Collected papers, PWN, Warsaw, 1964). That it is unimprovable, in a certain sense, follows from a result of Saks (On the strong derivatives of functions of intervals, Fund. Math. 25 (1935), 235–252). Bibliography: 15 titles. 
@article{IM2_1977_11_4_a9,
     author = {L. D. Gogoladze},
     title = {On $(H,k)$-summability of multiple trigonometric {Fourier} series},
     journal = {Izvestiya. Mathematics },
     pages = {889--908},
     publisher = {mathdoc},
     volume = {11},
     number = {4},
     year = {1977},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1977_11_4_a9/}
}
TY  - JOUR
AU  - L. D. Gogoladze
TI  - On $(H,k)$-summability of multiple trigonometric Fourier series
JO  - Izvestiya. Mathematics 
PY  - 1977
SP  - 889
EP  - 908
VL  - 11
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1977_11_4_a9/
LA  - en
ID  - IM2_1977_11_4_a9
ER  - 
%0 Journal Article
%A L. D. Gogoladze
%T On $(H,k)$-summability of multiple trigonometric Fourier series
%J Izvestiya. Mathematics 
%D 1977
%P 889-908
%V 11
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1977_11_4_a9/
%G en
%F IM2_1977_11_4_a9
L. D. Gogoladze. On $(H,k)$-summability of multiple trigonometric Fourier series. Izvestiya. Mathematics , Tome 11 (1977) no. 4, pp. 889-908. http://geodesic.mathdoc.fr/item/IM2_1977_11_4_a9/

[1] Hardy G. H., Littlwood J. E., “On the strong summability of Fourier series”, Proc. London Math. Soc., 1926, 273–286

[2] Marcinkiewicz J., “Sur la sommability forte des series de Fourier”, J. London Math. Soc., 14 (1939), 162–168 | DOI | MR | Zbl

[3] Zygmund A., “On the convergence and summability of power series on the circle of convergence, (I)”, Fund. Math., 30 (1938), 170–196 ; “(II)”, Proc. London Math. Soc., 47 (1942), 326–350 | Zbl | DOI | MR | Zbl

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

[5] Marcinkiewicz J., Collected papers, PWN, Warsawa, 1964 | MR

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

[7] Zigmund A., Trigonometricheskie ryady, t. 1, Mir, M., 1965 | MR

[8] Stein E. M., “A maximal function with application to Fourier series”, Ann. Math., 68:3 (1958), 584–603 | DOI | MR | Zbl

[9] Stein E. M., “On limits of sequences of operators”, Ann. Math., 74 (1961), 140–170 | DOI | MR | Zbl

[10] Sunouchi G., “On the strong summability of power series and Fourier series”, Tohoku Math. J., 6 (1954), 220–226 | DOI | MR

[11] Sokol-Sokolowski K., “On trigonometric series conjugate to Fourier series of two variables”, Fund. Math., 34 (1947), 166–182 | MR | Zbl

[12] Zygmund A., “On the boundary values of functions of several complex variables”, Fund. Math., 1949, 207–235 | MR | Zbl

[13] Zhizhiashvili L. V., Sopryazhennye funktsii i trigonometricheskie ryady, TGU, Tbilisi, 1969 | MR

[14] Khardi G. G., Littlvud Dzh., Polia G., Neravenstva, IL, M., 1948

[15] Gogoladze L. D., “O $(H,k)$-summiruemosti trigonometricheskikh ryadov Fure”, Dokl. AN SSSR, 200:6 (1971), 1266–1268 | MR | Zbl