Bilinear strongly generalized fractional integrals and their commutators over non-homogeneous metric spaces

Abstract

The aim of this paper is to establish the boundedness of the bilinear strongly generalized fractional integral operator T˜α and its commutator T˜α,b1,b2 formed by b1,b2∈RBMO˜(μ) and T˜α on non-homogeneous metric measure spaces. Under assumption that the dominating function λ and the Lebesgue measure function u(⋅,⋅) defined on X×(0,∞) satisfy certain conditions, we prove that the bilinear generalized fractional integral operator T˜α is bounded from the product of Lebesgue spaces Lp1(μ)×Lp2(μ) into spaces Lq(μ), bounded from the product of Morrey spaces Mq1p1(μ)×Mq2p2(μ) into spaces Mts(μ), and it is also bounded from the product of spaces Lp1,u1(μ)×Lp2,u2(μ) into spaces Lq,u(Rn), where u1×u2=u, 1q=1p1+1p2−2α, 1t=1q1+1q2−2α and 1s=1p1+1p2−2α for 1<q1≤p1<1α and 1<q2≤p2<1α. Furthermore, the boundedness of the T˜α,b1,b2 on spaces Lp(μ), on spaces Mqp(μ) and on spaces Lp,u(μ) is obtained.