Integral operators of potential type in spaces of homogeneous type (Q1595614)

The authors formulate a number of estimates for integral operators with kernel of potential type defined on spaces of homogeneous type. Let \((X,\rho,\mu)\) be a space of homogeneous type in the sense of Coifman and Weiss, i.e., \(\rho\) is a pseudo-metric and \(\mu\) is a measure satisfying the doubling condition. The kernel of the operator \(T_\gamma\) is given by the formula \(\rho(x,y)^\gamma\), \(-1<\gamma<0\). The continuity of the operators \(T_\gamma\) is investigated in weighted \(L_p\)-spaces \(L_{p,w}(X,\mu)\) and the spaces \(\Gamma_{p\theta,\phi}(X,\mu)\) \((\Gamma^*_{p\theta,\phi}(X,\mu))\). The last spaces are generalization of the weighted \(L_p\)-spaces introduced earlier by V. S. Guliev. No proofs are given.