Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

Hendra Gunawan,Yoshihiro Sawano,Idha Sihwaningrum,Denny Ivanal Hakim

doi:10.1155/2013/809704

Abstract

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.

Highlights

We prove that the theory of generalized Morrey spaces can be staged on the nondoubling setting on Rd, so that we assume that μ is a positive Borel measure on Rd satisfying the growth condition; that is, there exist n ∈ [0, d]
We will always assume the Dini condition, that is, ∫01(ρ(t)/t)dt < ∞ and we assume that ρ satisfies the so-called growth condition; namely, there exist constants C > 0 and 0 < 2k1 < k2 < ∞ such that sup ρ (s)
Tf (x) = ∫ K (x, y) f (y) dμ (y) ∀x ∉ supp (f) . (61). As for this singular integral operator T, the following result is due to Nazarov, Treil, and Volberg

Summary

Introduction

We prove that the theory of generalized Morrey spaces can be staged on the nondoubling setting on Rd, so that we assume that μ is a positive Borel measure on Rd satisfying the growth condition; that is, there exist n ∈ [0, d]. The study of the boundedness of the fractional integral operator Iα on generalized Morrey spaces was initiated in [15, Theorem 3]. The following theorem presents the weak type inequalities for Iα on generalized nonhomogeneous Morrey spaces. Note that we can obtain the weak type inequalities for Iα on nonhomogeneous Lebesgue spaces which are proved in [17, 18] by taking φ(r) = r−n/p and 1/q = (1/p) − (α/n) in Theorem 1. We denote by Ck (k ∈ N) the fixed constants that satisfy certain conditions

Weak Type Inequalities for Iρ via the Chebyshev Inequality

Boundedness of Singular Integral Operators

Generalized Nonhomogeneous Orlicz-Morrey Spaces