Hausdorff-Young type theorems for the Laplace transform restricted to a ray or to a curve in the complex plane

Abstract

Let p and q be conjugate exponents, with p in [1,2]. It is shown that the Laplace transform acts boundedly between the Lp space with unit weight on the positive real semiaxis and the Lq space weighted by a well-projected measure (a term defined in the paper) in the right complex half-plane. The operator norm is uniformly bounded over classes of measures with the same "projection constants". Particular cases are arclength measures on some classes of rectifiable curves. In addition, an analog of the Hausdorff-Young inequality in Lorentz spaces is obtained in the case of "wrong" exponents p>2.