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## Colloquium Mathematicum

1997 | 74 | 1 | 123-133
Tytuł artykułu

### Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_∞$ to $L_∞$ then it is also bounded from the classical Hardy space $H_p(T)$ to $L_p$ (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p(T)$ to $L_p$ (3/4 < p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_{1}^{♯}(T^2)$ and verify that the maximal operator of the Cesàro means of a two-dimensional function is of weak type $(H_{1}^{♯}(T^2),L_1)$. So we deduce that the two-parameter Cesàro means of a function $f ∈ H_1^{♯}(T^2) ⊃ Llog L$ converge a.e. to the function in question.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
123-133
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-06-18
poprawiono
1996-12-11
Twórcy
autor
• Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary
Bibliografia
• [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129 Academic Press, New York, 1988.
• [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
• [3] D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal function characterization of the class $H^p$, Trans. Amer. Math. Soc. 157 (1971), 137-153.
• [4] R. R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 269-274.
• [5] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
• [6] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
• [7] R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 1, Springer, Berlin, 1982.
• [8] R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, Berlin, 1982.
• [9] C. Fefferman, N. M. Rivière and Y. Sagher, Interpolation between $H^p$ spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75-81.
• [10] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-194.
• [11] N. J. Fine, Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 558-591.
• [12] N. Fujii, A maximal inequality for $H^1$-functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc. 77 (1979), 111-116.
• [13] B. S. Kashin and A. A. Saakyan, Orthogonal Series, Transl. Math. Monographs 75, Amer. Math. Soc. 75, Providence, R.I., 1989.
• [14] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
• [15] F. Móricz, F. Schipp and W. R. Wade, Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc. 329 (1992), 131-140.
• [16] N. M. Rivière and Y. Sagher, Interpolation between $L^∞$ and $H^1$, the real method, J. Funct. Anal. 14 (1973), 401-409.
• [17] F. Schipp, Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18 (1975), 189-195.
• [18] F. Schipp and P. Simon, On some $(H,L_1)$-type maximal inequalities with respect to the Walsh-Paley system, in: Functions, Series, Operators, Budapest, 1980, Colloq. Math. Soc. János Bolyai 35, North-Holland, Amsterdam, 1981, 1039-1045.
• [19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
• [20] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
• [21] F. Weisz, Cesàro summability of one- and two-dimensional Walsh-Fourier series, Anal. Math. 22 (1996), 229-242.
• [22] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
• [23] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
Typ dokumentu
Bibliografia
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