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Convergence almost everywhere of multiple Fourier series over cubes

Authors: Mieczysław Mastyło and Luis Rodríguez-Piazza
Journal: Trans. Amer. Math. Soc. 370 (2018), 1629-1659
MSC (2010): Primary 43A50
Published electronically: September 7, 2017
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Abstract: We study convergence almost everywhere of multiple trigonometric Fourier series over cubes defined on the $ d$-dimensional torus $ \mathbb{T}^d$. We provide a new approach which allows us to prove the novel interpolation estimates for the Carleson maximal operators generated by the partial sums of the multiple Fourier series and all its conjugate series. Combining these estimates we show that these operators are bounded from a variant of the Arias-de-Reyna space $ Q\!A^d$ to the weak $ L^1$-space on $ \mathbb{T}^d$. This implies that the multiple Fourier series of every function $ f\in Q\!A^d$ and all its conjugate series converge over cubes almost everywhere. By a close analysis of the space $ Q\!A^d$ we prove that it contains a Lorentz space that strictly contains the Orlicz space $ L(\log \,L)^{d} \log \log \log \,L(\mathbb{T}^d)$. This yields a significant improvement of a deep theorem proved by Antonov which was the best known result on the convergence of multiple Fourier series over cubes.

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  • [1] N. Yu. Antonov, Convergence of Fourier series, Proceedings of the XX Workshop on Function Theory (Moscow, 1995), 1996, pp. 187-196. MR 1407066
  • [2] N. Yu. Antonov, On the convergence almost everywhere of multiple trigonometric Fourier series over cubes, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 2, 3-22; English transl., Izv. Math. 68 (2004), no. 2, 223-241. MR 2057997,
  • [3] Juan Arias de Reyna, Pointwise convergence of Fourier series, Lecture Notes in Mathematics, vol. 1785, Springer-Verlag, Berlin, 2002. MR 1906800
  • [4] J. Arias-de-Reyna, Pointwise convergence of Fourier series, J. London Math. Soc. (2) 65 (2002), no. 1, 139-153. MR 1875141,
  • [5] Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 0199631,
  • [6] M. J. Carro, M. Mastyło, and L. Rodríguez-Piazza, Almost everywhere convergent Fourier series, J. Fourier Anal. Appl. 18 (2012), no. 2, 266-286. MR 2898729,
  • [7] Charles Fefferman, On the divergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 191-195. MR 0279529,
  • [8] Charles Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 744-745. MR 0435724,
  • [9] Charles Fefferman, Pointwise convergence of Fourier series, Ann. of Math. (2) 98 (1973), 551-571. MR 0340926,
  • [10] Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235-255. MR 0238019
  • [11] N. J. Kalton, Convexity, type and the three space problem, Studia Math. 69 (1980/81), no. 3, 247-287. MR 647141
  • [12] A. N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout, Fundamental Math. 4 (1923), 324-328.
  • [13] S. V. Konyagin, On the divergence everywhere of trigonometric Fourier series, Mat. Sb. 191 (2000), no. 1, 103-126; English transl., Sb. Math. 191 (2000), no. 1-2, 97-120. MR 1753494,
  • [14] S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
  • [15] Victor Lie, On the boundedness of the Carleson operator near $ L^1$, Rev. Mat. Iberoam. 29 (2013), no. 4, 1239-1262. MR 3148602,
  • [16] Per Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series., Ark. Mat. 7 (1969), 551-570. MR 0241885,
  • [17] Per Sjölin, Convergence almost everywhere of certain singular integrals and multiple Fourier series, Ark. Mat. 9 (1971), 65-90. MR 0336222,
  • [18] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. MR 0241685,
  • [19] N. R. Tevzadze, The convergence of the double Fourier series at a square summable function, Sakharth. SSR Mecn. Akad. Moambe 58 (1970), 277-279. MR 0298338
  • [20] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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Additional Information

Mieczysław Mastyło
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Umultowska 87, 61-614 Poznań, Poland

Luis Rodríguez-Piazza
Affiliation: Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Aptdo. de correos 1160, 41080 Sevilla, Spain

Keywords: Multiple trigonometric Fourier series, almost everywhere convergence, Lorentz spaces, Orlicz spaces, Banach envelope, Hilbert transform, multipliers
Received by editor(s): May 21, 2016
Published electronically: September 7, 2017
Additional Notes: The first-named author was supported by the National Science Centre (NCN), Poland, project no. 2011/01/B/ST1/06243
The second-named author was supported by research project MTM2015-63699-P (Spanish MINECO and European FEDER funds)
Article copyright: © Copyright 2017 American Mathematical Society

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