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Introduction. Let H denote a real separable Hilbert space and let 1

exists as a bounded operator on L(H) as o tends to zero and p tends to infinity. A theorem of this type extends the Calderon-Zygmund theory of singular integral operators on En to infinite dimensions. For if fc(#)||x||-" is a Calderon-Zygmund kernel and if £ is a bounded Borel set which is disjoint from a neighborhood of the origin then v{E) =fEk(x)\\x\\~ dx satisfies v(tE) = v(E) for £>0; if g(x) is an integrable radial function on En with support in a bounded annulus disjoint from a neighborhood of the origin, then fEng(x)k(x)\\x\\~ dx = 0. When fx satisfies a smoothness condition and n(H) = 0 , the set function v(E) —fâ[i{E/i)dt/t has these properties. The results in this paper are taken from the author's Cornell doctoral dissertation. The author wishes to express his most hearty thanks to his thesis advisor Professor Leonard Gross for his interest, advice, and encouragement during the preparation of the thesis.

Introduction. Let 1

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.