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Cesàro Summability of
Two-dimensional Walsh-Fourier Series


Author: Ferenc Weisz
Journal: Trans. Amer. Math. Soc. 348 (1996), 2169-2181
MSC (1991): Primary 42C10, 43A75; Secondary 60G42, 42B30
DOI: https://doi.org/10.1090/S0002-9947-96-01569-3
MathSciNet review: 1340180
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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce p-quasi-local operators and the two-dimensional
dyadic Hardy spaces $H_{p}$ defined by the dyadic squares. It is proved that, if a sublinear operator $T$ is p-quasi-local and bounded from $L_{\infty }$ to $L_{\infty }$, then it is also bounded from $H_{p}$ to $L_{p}$ $(0<p \leq 1)$. As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from $H_{p}$ to $L_{p}$ $(1/2<p \leq \infty )$ and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function $f \in L_{1}$ converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from $H_{p}$ to $L_{p}$ for every $0<p \leq \infty $.


References [Enhancements On Off] (What's this?)

  • 1. Bennett, C., Sharpley, R., Interpolation of operators, Pure and Applied Mathematics, vol. 129, 1988. MR 89e:46001
  • 2. Bergh, J., Löfström, J., Interpolation spaces, an introduction, Berlin, Heidelberg, New York: Springer, 1976. MR 58:2349
  • 3. Coifman, R.R., Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 56:6264
  • 4. Fine, N.J., Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. USA 41 (1955), 558-591. MR 17:31f
  • 5. Fine, N.J., On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. MR 11:352b
  • 6. Fujii, N., A maximal inequality for $H^{1}$-functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc. 77 (1979), 111-116. MR 81b:42070
  • 7. Garsia, A.M., Martingale inequalities, Lecture notes series, New York: Benjamin Inc., 1973. MR 56:6844
  • 8. Marcinkievicz, J., Zygmund, A., On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
  • 9. Móricz, F., Schipp, F., Wade, W.R., Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc. 329 (1992), 131-140. MR 92j:42028
  • 10. Neveu, J., Discrete-parameter martingales, North-Holland, 1971. MR 53:6729
  • 11. Schipp, F., Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest Sect. Math. 18 (1975), 189-195. MR 55:3670
  • 12. Schipp, F., Simon, P., On some $(H,L_{1})$-type maximal inequalities with respect to the Walsh-Paley system, Coll. Math. Soc. J. Bolyai, vol. 35, Functions, Series, Operators, Budapest (Hungary), 1980, North Holland (Amsterdam, 1981), pp. 1039-1045. MR 86a:42032
  • 13. Schipp, F., Wade, W.R., Simon, P., Pál, J.:, Walsh series: An introduction to dyadic harmonic analysis, Adam Hilger, Bristol-New York, 1990. MR 44:7280
  • 14. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 44:7280
  • 15. Wade, W.R., A growth estimate for Cesàro partial sums of multiple Walsh-Fourier series, Coll. Math. Soc. J. Bolyai 49, Alfred Haar Memorial Conference, Budapest (Hungary), 1985, North Holland (Amsterdam, 1986), pp. 975-991. MR 89f:42026
  • 16. Weisz, F., Cesàro summability of double trigonometric-Fourier series, F. Math. Anal. Appl., preprint.
  • 17. Weisz, F., Interpolation between martingale Hardy and BMO spaces, the real method, Bull. Sc. Math. 116 (1992), 145-158. MR 93h:46028
  • 18. Weisz, F., Martingale Hardy spaces and their applications in Fourier-analysis, Lecture Notes in Math., vol. 1568, Berlin, Heidelberg, New York: Springer, 1994. CMP 95:09
  • 19. Weisz, F., Martingale Hardy spaces for $0<p \leq 1$, Probab. Th. Rel. Fields 84 (1990), 361-376. MR 91d:60107
  • 20. Zygmund, A., Trigonometric series, Cambridge Press, London, 1959. MR 21:6498
  • 21. Gát, Gy., Pointwise convergence of the Cesàro means of double Walsh series, Ann. Univ. Sci. Budapest Eötvös, Sect. Comp., preprint.

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Additional Information

Ferenc Weisz
Affiliation: Department of Numerical Analysis, Eötvös L. University, H-1088 Budapest, Múzeum krt. 6-8 , Hungary
Email: weisz@ludens.elte.hu

DOI: https://doi.org/10.1090/S0002-9947-96-01569-3
Keywords: Martingale Hardy spaces, p-atom, atomic decomposition, p-quasi-local operator, interpolation, Walsh functions, Cesàro summability
Received by editor(s): June 28, 1994
Additional Notes: This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No. F4189.
Article copyright: © Copyright 1996 American Mathematical Society

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