Singular integrals and fractional powers of operators

Abstract:

Recently R. Wheeden studied a class of singular integral operators, the hypersingular integrals, as operators from $L_p^\alpha (H)$ to ${L_p}(H);L_p^\alpha (H)$ is the range of the $\alpha$th order Bessel potential operator acting on ${L_p}(H)$ with the inherited norm. The purposes of the present paper are to extend the known results on hypersingular integrals to complex indices, to extend these results to operators defined over a real separable Hilbert space, and to use Komatsu’s theory of fractional powers of operators to show that the hypersingular integral operator ${G^\alpha }$ is ${\smallint _H}{( - {A_y})^\alpha }f d\mu (y)$ when $\operatorname {Im}(\alpha ) \ne 0$ or when $\Re (\alpha )$ is not a positive integer where ${A_y}g$ is the derivative of g in the direction y. The case where $\operatorname {Im} (\alpha ) = 0$ and $\Re (\alpha )$ is a positive integer is treated in a sequel to the present paper.