Limiting weak type behavior for multilinear fractional integrals. (August 2020) - Tang, Hanli; Wang, Guanglan
Abstract: In this paper, we establish the limiting weak type behavior for multilinear fractional integrals I α, A and I α, A, which were introduced by Kenig and Stein (see the definition in section 1). Although Kenig and Stein proved that these two kinds of multilinear fractional operators have the same weak and strong type estimates, we provide that their limiting behaviors are totally different when p 1 = p 2 = ⋯ = p k = 1 . Specifically, motivated by Janakiraman's result, we establish when f i ≥ 0, 1 ≤ i ≤ k, there holds lim λ → 0 λ | { x ∈ R n : | I α, k ( f 1, …, f k ) ( x ) | > λ } | k n − α n = k α − k n v n k n − α n ∏ i = 1 k ‖ f i ‖ L 1 ( R n ), which extend the first author's result about linear fractional integral to multilinear case. But for I α, A, there holds lim λ → 0 λ | { x ∈ R n : | I α, A ( f 1, …, f k ) ( x ) | > λ } | k n − α n = 0 . Limiting behaviors for the other indexes are also presented and at last we set up limiting weak type behaviors for multilinear fractional integrals when λ → ∞ .