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 α ( H ) L_p^\alpha (H) to L p ( H ) ; L p α ( H ) {L_p}(H);L_p^\alpha (H) is the range of the α \alpha th order Bessel potential operator acting on L p ( H ) {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 α {G^\alpha } is ∫ H ( − A y ) α f d μ ( y ) {\smallint _H}{( - {A_y})^\alpha }f\,d\mu (y) when Im ⁡ ( α ) ≠ 0 \operatorname {Im}(\alpha ) \ne 0 or when ℜ ( α ) \Re (\alpha ) is not a positive integer where A y g {A_y}g is the derivative of g in the direction y. The case where Im ⁡ ( α ) = 0 \operatorname {Im} (\alpha ) = 0 and ℜ ( α ) \Re (\alpha ) is a positive integer is treated in a sequel to the present paper.