L p Estimates for Marcinkiewicz integral operators and extrapolation

Mohammed Ali

doi:10.1186/1029-242x-2014-269

Abstract

In this article, we establish estimates for parametric Marcinkiewicz integral operators with rough kernels. These estimates and extrapolation arguments improve and extend some known results on Marcinkiewicz integrals. MSC:40B20, 40B15, 40B25.

Highlights

1 Introduction Throughout this article, let Sn, n ≥ be the unit sphere in Rn which is equipped with the normalized Lebesgue surface measure dσ = dσ (·)
Let K,h = (u )h(|u|)|u|ρ–n, where ρ = a + ib (a, b ∈ R with a > ), h is a measurable function on R+ and is a function on Sn– with ∈ L (Sn– ) and (u) dσ (u) =
The study of parametric Marcinkiewicz integral operator Mρ,h was initiated by Hörmander in [ ] in which he showed that Mρ, is bounded on Lp(Rn) for < p < ∞ when ρ > and ∈ Lipα(Sn– ) with α >

Summary

Introduction

For a suitable mapping φ : R+ → R, a measurable function h on Sn– and an satisfying Let us recall the definition of the space L(log L)α(Sn– ) For α > , let L(log L)α(Sn– ) denote the class of all measurable functions on Sn– that satisfy

Results

Conclusion