Hardy-Space Multipliers for Integral Operators with Sonine Kernels

We study integral operators obtained by averaging right translations on the Hardy space of the right half-plane. For a nonnegative kernel with finite Laplace transform, we determine the closed operator represented by this average, its domain, and the conditions for boundedness. The Laplace transform identifies the operator with multiplication on L²(0,∞ ) L 2 ( 0 , ∞ ) and yields its exact norm, spectrum, and resolvent. A gauge identity relates the same translation average to multiplication in the time variable. For nonnegative function kernels forming a Sonine pair, we establish inverse identities on the natural domains and identify the associated derivative with the reciprocal multiplier; this derivative is necessarily unbounded. We then use the established correspondence with conjugate special Bernstein functions to describe nonincreasing Sonine pairs, distinguishing function kernels from measures with an atom at zero and identifying the relevant Bernstein symbol.