Best constant of the critical Hardy-Leray inequality for curl-free fields in two dimensions

Sharp Hardy-leray Inequality for Axisymmetric Divergence-free Fields

appears for n = 3 in the pioneering Leray’s paper on the Navier-Stokes equations [2]. The constant factor on the right-hand side is sharp. Since one frequently deals with divergence-free fields in hydrodynamics, it is natural to ask whether this restriction can improve the constant in (1.1). We show in the present paper that this is the case indeed if n > 2 and the vector field u is axisymmetri...

The Best Constant in a Fractional Hardy Inequality

We prove an optimal Hardy inequality for the fractional Laplacian on the half-space. 1. Main result and discussion Let 0 < α < 2 and d = 1, 2, . . .. The purpose of this note is to prove the following Hardy-type inequality in the half-space D = {x = (x1, . . . , xd) ∈ R : xd > 0}. Theorem 1. For every u ∈ Cc(D), (1) 1 2 ∫

On Best Extensions of Hardy-hilbert’s Inequality with Two Parameters

This paper deals with some extensions of Hardy-Hilbert’s inequality with the best constant factors by introducing two parameters λ and α and using the Beta function. The equivalent form and some reversions are considered.

Determination of the Best Constant in an Inequality of Hardy, Littlewood, and Pólya

Best Constant in Sobolev Inequality

The equality sign holds in (1) i] u has the Jorm: (3) u(x) = [a + btxI,~',-'] 1-~1~ , where Ix[ = (x~ @ ...-~x~) 1⁄2 and a, b are positive constants. Sobolev inequalities, also called Sobolev imbedding theorems, are very popular among writers in part ial differential equations or in the calculus of variations, and have been investigated by a great number of authors. Nevertheless there is a ques...