Trigonometric Fourier Series with Characterization of Weighted Hardy Spaces

International Journal of Biology, Physics & Matematics
ISSN: 2721-3757, Volume 1, Issue 1, page 135 – 147
Date: 12 October 2018
© Copyright International Journal of Zambrut

Ali Mohamed Abu Oam

Ali Mohamed Abu Oam
Department of Mathematics and Computer, Faculty of Pure and Applied Sciences
International University of Africa
Suand (Sudan)

Abstract
We introduce p-quasilocal operators and show 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 Cesaro 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 give a characterization of weighted local Hardy spaces h_ω^1 (R^n) associated with local weights by using the truncated Reisz transforms. Then we established a result for each case.

Keywords: P-atom, Interpolation, Decomposition, P-quasilocal Operator, Characterization, Hardy Spaces, Local Weights.

1. Introduction
We discuss in section two it can be found in Zygmund [1] that the Cesaro means σ_n f of a function f ∈ L_1(T) converge a.e. to f as n → ∞ and that if f ∈ L 〖log〗^+ L(T^2) then the two-parameter Cesaro summability holds.
Analogous results for Walsh–Fourier series are due to Fine [2] and Moricz, Schipp and Wade [3].
The Hardy–Lorentz spaces H_(p,q ) of distributions on the unit circle are introduced with the L_(p,q ) Lorentz norms of the non-tangential maximal function. Of course, H_(p ) = H_(p,p ) are the usual Hardy spaces (0 < p ≤ ∞). In the one-dimensional case it is known (see Zygmund [1] ) that the maximal operator of the Cesaro means 〖sup〗_(n∈N) |σ_n | is of weak type (L_1, L_1), i.e. 〖sup〗_(γ>0) γλ(〖sup〗_(n∈N) |σ_n |>γ)≤C‖f‖_1 (f ∈ L_1 (T) )
( for the Walsh case see Schipp[4] ). Also, for Walsh–Fourier series, the bounded-ness of the operator 〖sup〗_(n∈N) |σ_n | from H_p to L_p was shown by Fujii [5] (p = 1) and by Weisz [6] (1/2 < p ≤ 1).
The theory of local Hardy space plays an important role in various fields of analysis and partial differential equations, Bui [15] studied the weighted version 〖 h〗_ω^p of the local Hardy space 〖 h〗^p considered by Goldberg [16], where the weight ω is assumed to satisfy the condition (A_∞) of Muckenhoupt.
The main purpose of section 3 is to give a characterization of weighted local Hardy spaces h_ω^1 (R^n) associated with local weights by using the truncated Reisz transforms.

2. Summability of One and Two-Dimensional
In this section we generalize the results for trigonometric-Fourier series with the help of the so-called p-quasilocal operators. An operator T is p-quasilocal ( 0 < p ≤ 1 ) if for all p-atoms a the integral of 〖|T_a |〗^p over T\ I is less than an absolute constant where I is the support of the atom a. We shall verify that a sublinear, p-quasilocal operator T which is bounded from L_∞ to L_∞ is also bounded from H_p to L_p ( 0 < p ≤ 1 ). By interpolation we find that T is bounded from H_(p,q) to L_(p,q) as well (0 < p < ∞, 0 < q ≤ ∞) and is of weak type (L_1, L_1).
It will be shown that 〖sup〗_(n∈N) |σ_n | is p-quasilocal for each 3/4 < p ≤ 1. Conseque-ntly, 〖sup〗_(n∈N) |σ_n | is bounded from H_(p,q) to L_(p,q) for 3/4 < p < ∞ and 0 < q ≤ ∞ and is of weak type (L_1, L_1). We will extend this result also to (C,β) means.
For two-dimensional trigonometric-Fourier series we will ……….

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