By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.
Given a metric measure space $(X,d,\mu)$, a weight $w$ defined on $(0,\infty)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,d\mu(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,d\mu(y)$ on Lebesgue spaces $L^p(\mu)$ and generalized Lipschitz spaces $\mathrm{Lip}_\phi$, respectively, for certain range of the parameters depending on the $n$-dimension of $\mu$ and some indices associated to the weight $w$.
Abstract
We establish necessary and sufficient conditions for the compactness of fractional integral operators from Lp(X, μ) to Lq(X, μ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X. As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝn with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane.
We introduce one-sided Cohen’s commutators of singular integral operators and fractional integral operators, respectively. Using the extrapolation of one-sided weights, we establish the boundedness of these operators from weighted Lebesgue spaces to weighted one-sided Triebel-Lizorkin spaces.
The Boundedness of Multilinear Fractional Integral Operators with Variable Kernel on Variable Exponent Lebesgue Spaces
