Singular integral operators with non-necessarily bounded kernels on spaces of homogeneous type
A space of homogeneous type is a triple \((X,d,\mu)\), where \(d(x,y)\) is a quasi-distance and the measure \(\mu\) satisfies the condition: there exists a finite constant \(A>0\) such that \[ \mu\bigl(B_ d(x,2Kr)\bigr)\leq A\mu\bigl(B_ d(x,r)\bigr) \] holds for every \(x\in X\) and \(r>0\), where \(B_ d(x,r)\) is a ball. The properties of a space of homogeneous type are established. Singular integral operators \(T\) with non-necessarily bounded kernels on spaces of homogeneous type are studied. A criterion for \(T\) to be weakly bounded is obtained. An application to operators defined by principal value integrals gives a priori Lipschitz estimates for some parabolic differential equations.
scientific article; zbMATH DE number 62679