Bilinear θ-type generalized fractional integral operator and its commutator on some non-homogeneous spaces

Abstract

Let (X,d,μ) be a non-homogeneous metric measure space which satisfies the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that the dominating function λ satisfies weak reverse doubling condition, the authors prove that bilinear θ-type generalized fractional integral BT˜α,θ is bounded from the product of spaces Lp1(μ)×Lp2(μ) into space Lq(μ), and bounded from the product of spaces Mq1p1(μ)×Mq2p2(μ) into space Mqp(μ). Moreover, via establishing the sharp maximal estimate for the commutator [b1,b2,BT˜α,θ] which is generated by b1,b2∈RBMO˜(μ) and BT˜α,θ, the boundedness of [b1,b2,BT˜α,θ] on space Lp(μ) and on space Mqp(μ) is also obtained.