  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.
