Herz spaces and integral operators on Morrey spaces

This paper establishes an approach for extending the mapping properties of integral operators from Lebesgue spaces to Morrey spaces. The main result shows that if the operator is bounded on the Lebesgue space and the kernel of the operator satisfies a condition involving a certain Herz space, then we can extend this operator to be a bounded operator on the corresponding Morrey space. We also obtain a new operator norm estimate of the integral operator on Morrey space in terms of the operator norm on Lebesgue space and the Herz space norm of the kernel of the integral operator. The main result applies to several integral operators including singular integral operators, fractional integral operators and strongly singular integral operators.