Published on International Journal of Biology, Physics & Matematics

ISSN: 2721-3757, Volume 1, Issue 1, page 135 – 147

Publication Date: 12 October 2018

**Ali Mohamed Abu Oam**

Department of Mathematics and Computer, Faculty of Pure and Applied Sciences

International University of Africa

Suand (Sudan)

Journal Full Text PDF: Trigonometric Fourier Series with Characterization of Weighted Hardy Spaces.

**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 ……….