Vector-Valued Inequalities in the Morrey Type Spaces

Hua Wang

doi:10.1155/2014/738125

Abstract

We will obtain the strong type and weak type estimates for vector-valued analogues of classical Hardy-Littlewood maximal function, weighted maximal function, and singular integral operators in the weighted Morrey spacesLp,κ(w)when1≤p<∞and0<κ<1, and in the generalized Morrey spacesLp,Φfor1≤p<∞, whereΦis a growth function on(0,∞)satisfying the doubling condition.

Highlights

The classical Hardy-Littlewood maximal function Mf(x) is defined for a locally integrable function f on Rn by Mf (x) = sup x∈B 1 |B| ∫ B 󵄨󵄨󵄨󵄨f (y)󵄨󵄨󵄨󵄨 dy, (1)where the supremum is taken over all balls B containing x
It is well known that the maximal operator M maps Lp(Rn) into Lp(Rn) for all 1 < p ≤ ∞ and L1(Rn) into WL1(Rn)
The classical Morrey spaces Lp,λ were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations

Summary

Introduction

The classical Hardy-Littlewood maximal function Mf(x) is defined for a locally integrable function f on Rn by. This nonlinear operator was introduced by Fefferman and Stein in [1], and since it has played an important role in the development of modern harmonic analysis In this remarkable paper [1], Fefferman and Stein extended the classical maximal theorem to the case of vector-valued functions. A weight w is a nonnegative, locally integrable function on Rn; B = B(x0, rB) denotes the ball with the center x0 and radius rB. Let us consider the vector-valued singular integral operators. 2F,o.r. sequence of Rn, as above, locally integrable the vector-valued singular integral operators with kernels Ω ∈ (Ds) can be defined as TΩ(f)⃗ = (TΩ(f1), TΩ(f2), . In [4], Andersen and John derived the weighted strong and weak type estimates for vector-valued singular integral operators with kernels Ω ∈ (D∞) (see [6, 7]).

Our Main Results

Boundedness in the Weighted Morrey Spaces

Boundedness in the Generalized Morrey Spaces