Cesàro summability of double Walsh-Fourier series
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- by F. Móricz, F. Schipp and W. R. Wade PDF
- Trans. Amer. Math. Soc. 329 (1992), 131-140 Request permission
Abstract:
We introduce quasi-local operators (these include operators of Calderón-Zygmund type), a hybrid Hardy space ${{\mathbf {H}}^\sharp }$ of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type $(\sharp ,1)$. As an application, we show that Cesàro means of the double Walsh-Fourier series of a function $f$ converge a.e. when $f$ belongs to ${{\mathbf {H}}^\sharp }$. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all $f \in {L^1}$ provided summability takes place in some positive cone.References
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
- N. J. Fine, Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 588–591. MR 70757, DOI 10.1073/pnas.41.8.588 J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
- F. Móricz and F. Schipp, On the integrability and $L^1$-convergence of double Walsh series, Acta Math. Hungar. 57 (1991), no. 3-4, 371–380. MR 1139331, DOI 10.1007/BF01903688
- F. Schipp, Über gewisse Maximaloperatoren, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18 (1975), 189–195 (1976). MR 430665
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- M. H. Taibleson, Fourier analysis on local fields, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR 0487295
- W. R. Wade, A growth estimate for Cesàro partial sums of multiple Walsh-Fourier series, A. Haar memorial conference, Vol. I, II (Budapest, 1985) Colloq. Math. Soc. János Bolyai, vol. 49, North-Holland, Amsterdam, 1987, pp. 975–991. MR 899590
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 329 (1992), 131-140
- MSC: Primary 42C10; Secondary 42B08
- DOI: https://doi.org/10.1090/S0002-9947-1992-1030510-8
- MathSciNet review: 1030510