Endpoint estimates for multilinear fractional integral operators on metric measure spaces

Abstract

Let (X,d,μ) be a metric measure space such that, for any fixed x∈X, μ(B(x,r)) is a continuous function with respect to r∈(0,∞). In this paper, we prove endpoint estimates for the multilinear fractional integral operators Im,α from the product of Lebesgue spaces L1(μ)×⋯×L1(μ)×Lpk+1(μ)×⋯×Lpm(μ) into the Lebesgue space Lq(μ), where k∈[1,m)∩N, α∈[k,m), pi∈(1,∞) for i∈{k+1,…,m} and 1/q=k+∑i=k+1m1/pi−α. We furthermore prove that Im,α is bounded from Lp1(μ)×⋯×Lpm(μ) into L∞(μ), where pi∈(1,∞) for i∈{1,…,m} and ∑i=1m1/pi=α∈[1,m).