Some new Stein and Hardy type inequalities

Abstract

We prove a generalization of the pointwise Stein inequality, considering two truncated versions. More generally than in the Stein inequality, we assume that the kernel is dominated by a radial function almost decreasing after the division by a power function with nonnegative exponent in the case of the truncation to the ball of the radius and almost increasing after the multiplication by a power function in the case of truncation to the exterior of this ball. We give some applications to a series of inequalities of Hardy type in norms of various function spaces, in particular, in the norm of variable exponent Lebesgue spaces $$ {L^{p\left( \cdot \right)}}\left( {{{\mathbb{R}}^n}} \right) $$ with weights. Bibliography: 40 titles.