A note on generalized fractional integral operators on generalized Morrey spaces.

Sawano, Yoshihiro; Sugano, Satoko; Tanaka, Hitoshi

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Let $m$ be a positive integer, $0 < α < m n$ , $\vec{b} = (b_{1}, \dots, b_{m}) \in {BMO}^{m}$ . We give sufficient conditions on weights for the commutators of multilinear fractional integral operators $ℐ_{α}^{\vec{b}}$ to satisfy a weighted endpoint inequality which extends the result in D. Cruz-Uribe, A. Fiorenza: Weighted endpoint estimates for commutators of fractional integrals, Czech. Math. J. 57 (2007), 153–160. We also give a weighted strong type inequality which improves the result in X. Chen, Q. Xue: Weighted estimates for a class of multilinear fractional...

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In this paper we study integral operators with kernels $K (x, y) = k_{1} (x - A_{1} y) \dots k_{m} (x - A_{m} y),$ $k_{i} (x) = {Ω_{i} (x) / |x|}^{n / q_{i}}$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.