Limiting case of the boundedness of fractional integral operators on nonhomogeneous space

Yoshihiro Sawano,Hitoshi Tanaka,Takuya Sobukawa

doi:10.1155/jia/2006/92470

Abstract

We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp.

Highlights

Harmonic analysis on Rd with nondoubling measures has been developed very rapidly; here, by a doubling measure, we mean a Radon measure μ on Rd satisfying μ(B(x, 2r)) ≤ c0μ(B(x, r)), x ∈ supp(μ), r > 0
Many function spaces and many linear operators for such measures stem from their works
We define the fractional integral operator Iα associated with the growth measure μ as f (y) Iα f (x) := Rd |x − y|nα dμ(y), 0 < α < 1

Summary

Introduction

Harmonic analysis on Rd with nondoubling measures has been developed very rapidly; here, by a doubling measure, we mean a Radon measure μ on Rd satisfying μ(B(x, 2r)) ≤ c0μ(B(x, r)), x ∈ supp(μ), r > 0. We deal with measures which do not necessarily satisfy the doubling condition. Many function spaces and many linear operators for such measures stem from their works. Tolsa has defined the Hardy space H1(μ) [11]. Han and Yang have defined the Triebel-Lizorkin spaces [3]. We mainly deal with the fractional integral operators. Μ is a Radon measure on Rd with μ B(x, r) ≤ c0rn for some c0 > 0, 0 < n ≤ d. A growth measure is a Radon measure μ satisfying (1.1). We define the fractional integral operator Iα associated with the growth measure μ as f (y) Iα f (x) := Rd |x − y|nα dμ(y), 0 < α < 1.

Limiting case of the fractional integral operators

Extrapolation theorem on the Morrey spaces

Precise estimate of the fractional integrals

Sharpness of the results