Differential Inequalities in Lp-Spaces

Abstract

In this article, we present inequalities related to the continuous representations of one-parameter groups. As an application, we obtain some differential inequalities of Bernstein type in L-p-spaces: We define the spectrum Sigma(f) of f is an element of L-p (R) (1 <= p < infinity), as Sigma(f) = boolean OR sp(B) {f * k} (1/P + 1/Q = 1), k is an element of L-q (R) where sp(B){.} is the Beurling spectrum. It is shown that if tau is an element of R satisfies the condition 0 <= tau sigma < pi, then f' is an element of L-p(R) and parallel to f'parallel to(p) <= sigma/2 sin tau sigma parallel to f(. + tau) - f(. - tau)parallel to(p), where sigma := sup{vertical bar lambda vertical bar: lambda is an element of Sigma(f)}. Some related problems are also discussed. (C) 2013 Elsevier Inc. All rights reserved.