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

Ferenc Weisz

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary

Abstract

We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from

L_{\infty}

to

L_{\infty}

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 \in H_{1}^{♯} (T^{2}) \supset L \log L

converge a.e. to the function in question.

Keywords

ENG

p-atom, Hardy spaces, Cesàro summability, atomic decomposition, p-quasilocal operator, interpolation

C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129 Academic Press, New York, 1988.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
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.
R. R. Coifman, A real variable characterization of $H^{p}$ , Studia Math. 51 (1974), 269-274.
R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
P. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970.
R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 1, Springer, Berlin, 1982.
R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, Berlin, 1982.
C. Fefferman, N. M. Rivière and Y. Sagher, Interpolation between $H^{p}$ spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75-81.
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), 137-194.
N. J. Fine, Cesàro summability of Walsh-Fourier series, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 558-591.
N. Fujii, A maximal inequality for $H^{1}$ -functions on a generalized Walsh-Paley group, Proc. Amer. Math. Soc. 77 (1979), 111-116.
B. S. Kashin and A. A. Saakyan, Orthogonal Series, Transl. Math. Monographs 75, Amer. Math. Soc. 75, Providence, R.I., 1989.
J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
F. Móricz, F. Schipp and W. R. Wade, Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc. 329 (1992), 131-140.
N. M. Rivière and Y. Sagher, Interpolation between $L^{\infty}$ and $H^{1}$ , the real method, J. Funct. Anal. 14 (1973), 401-409.
F. Schipp, Über gewissen Maximaloperatoren, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 18 (1975), 189-195.
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.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
F. Weisz, Cesàro summability of one- and two-dimensional Walsh-Fourier series, Anal. Math. 22 (1996), 229-242.
F. Weisz, Martingale Hardy Spaces and their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.

Title

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

Affiliations

Abstract

Keywords

Bibliography