Boundedness of Vector-Valued Intrinsic Square Functions in Morrey Type Spaces

Hua Wang

doi:10.1155/2014/923680

Abstract

We will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the weighted Morrey spacesLp,κ(w)when1≤p<∞,0<κ<1, and in the generalized Morrey spacesLp,Φfor1≤p<∞, whereΦis a growth function on(0,∞)satisfying the doubling condition.

Highlights

Introduction and Main ResultsThe intrinsic square functions were first introduced by Wilson in [1, 2]; they are defined as follows
For 0 < α ≤ 1, let Cα be the family of functions φ defined on Rn such that φ has support containing in {x ∈ Rn : |x| ≤ 1}, ∫Rn φ(x)dx = 0, and, for all x, x󸀠 ∈ Rn
The classical Morrey spaces Lp,λ were originally introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations

Summary

Introduction and Main Results

The intrinsic square functions were first introduced by Wilson in [1, 2]; they are defined as follows. Let f⃗ = functions on(fR1, nf.2,F.o.r.)anbye a x sequence of locally integrable ∈ Rn, Wilson [2] defined the vector-valued intrinsic square functions of f⃗ by (f)⃗. For any given weight function w and λ > 0, there exists a constant C > 0 independent of f⃗ = The classical Morrey spaces Lp,λ were originally introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations. For the boundedness of vector-valued intrinsic square functions in the weighted Morrey spaces Lp,κ(w) for all 1 ≤ p < ∞ and 0 < κ < 1, we will prove the following.

Notations and Definitions

Proofs of Theorems 1 and 2

Proofs of Theorems 3 and 4