A note on fractional integral operators defined by weights and non-doubling measures - Blasco, Oscar; Casanova, Vicente; Motos, Joaquín (MATHEMATICA SCANDINAVICA, 2010)
Given a metric measure space $(X,d,μ)$, a weight $w$ defined on $(0,∞)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,dμ(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,dμ(y)$ on Lebesgue spaces $L^p(μ)$ and generalized Lipschitz spaces $Lip_φ$, respectively, for certain range of the parameters depending on the $n$-dimension of $μ$ and some indices associated to the weight $w$.