LOWER BOUNDS FOR THE L(P)-NORM IN TERMS OF THE MELLIN TRANSFORM

Given a measurable function f on (0, infinity) with Mellin transform F (s), let \f\p denote the L(p)-norm of f with respect to the measure dx /x. We prove that under certain assumptions, for instance if f is real and non-negative and F(alpha) converges for alpha in an open interval and F(alpha) not-equal 0, then \f\p greater-than-or-equal-to c(p)\alp ha\1-1/p\F(alpha)exp(-alpha F'/F (alpha)\, where c(p) greater-than-or- equal-to (2e)-1. We derive similar inequalities for complex-valued f, for the L(p)-norm of the derivative off, and for the supremum of real- valued f and of its derivative. The lower bounds are eminently applica ble when f is a convolution product.