p-Bessel Pairs, Hardy’s Identities and Inequalities and Hardy–Sobolev Inequalities with Monomial Weights

Abstract

In this paper, we first prove a general symmetrization principle for the Hardy type inequality with non-radial weights of the form $A\left( \left| x\right| \right) \left| x_{1}\right| ^{P_{1}}\ldots \left| x_{N}\right| ^{P_{N}}$ (Theorem 1.1). Using this symmetrization principle for Hardy’s inequalities, we can establish the improved $L^{p}$-Hardy inequalities with such non-radial monomial weights (Theorem 1.2). Second, we introduce the notion of p-Bessel pairs and give applications to $L^{p}$-Hardy identities with non-radial monomial weights (Theorem 1.3) and Hardy inequalities (see Theorem 1.4) and their virtual extremals (see Remark 1.2). (See Theorem 1.5 for the special case $p=2$ where we derive $L^{2}$-Hardy identities and inequalities with monomial weights which have not been studied in the literature). In the special case when $P=(0,\ldots ,0)$, they imply the $L^{p}$-Hardy identities and Hardy inequalities with remainder terms on any finite balls and the entire space ${\mathbb {R}}^{N}$ (Theorem 1.6), while in the special case $P=(0,\ldots ,0,\alpha )$, $\alpha >0$, our results provide the $L^{p}$-Hardy identities and Hardy inequalities on half balls and the half spaces (Theorem 1.7). By taking special examples of p-Bessel pairs, we establish some particular Hardy’s identities and weighted Sobolev inequalities which are of independent interest. We also establish the optimal Hardy inequalities with monomial weights and explicit forms of extremal functions. (See Corollaries 1.1, 1.2, 1.3, 1.4, 1.5.) Our above results sharpened earlier results in the literature even in the case of $L^{2}$ Hardy inequalities. Finally, we establish the sharp constants and optimal functions of the $L^{p}$-Hardy–Sobolev inequalities with monomial weights (Theorem 1.8).