度量测度空间上多线性分数次积分算子的有界性

Boundedness of multilinear fractional integrals on metric measure spaces

设(X, d,μ)为一个度量测度空间,满足对于任意的x∈X,μ(B(x, r))关于r在(0,∞)上连续,或者设(X, d,μ)是Hyt?nen意义下满足上双倍条件和几何双倍条件的度量测度空间.在此两种背景条件下,本文建立多线性分数次积分算子Im,α在乘积Lebesgue空间上的端点估计、在乘积Morrey空间上的有界性以及弱型端点估计.

Let(X, d, μ) be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hyt?nen or 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 the above two settings, we prove that the multilinear fractional integrals map from L1(μ) × · · · × L1(μ) into Lq,∞(μ), where 0 < α < m, 1/q = m-α.We also establish the boundedness of the multilinear fractional integrals on the products of Morrey spaces and some weak type endpoint estimates.