On $L^p$-Hardy inequalities for magnetic $p$-Laplacians

TL;DR

The paper addresses remainder terms in $L^p$ Hardy inequalities for magnetic $p$-Laplacians and identifies the sharp constant in a key algebraic inequality. It introduces an integral representation $R_p$ that yields a precise expression for $C_p(x,y)/|y|^p$ and reduces the problem to a one-variable minimization over $k=x/|y|$. For $p\ge2$ the sharp constant $c_p$ is positive and given by $c_p=(p-1)(1-k_0)^p+pk_0(1-k_0)^{p-1}+k_0^p$ with $k_0=r_0/(1+r_0)$ where $r_0$ solves $r^{p-1}-(p-1)r-(p-2)=0$, while for $1<p<2$ one has $c_p=0$. The paper also provides the explicit value $c_3=2- obreak\sqrt{2}$ and sketches the proofs of these claims, including equality cases when $x=\lambda y$.

Abstract

In this paper we revisit the remainder terms of $L^p$-Hardy inequalities for magnetic $p$-Laplacians. In particular, we will give an integral representation of the sharp constant for a crucial algebraic inequality established by C. Cazacu, D. Krejčiřík, N. Lam, and A. Laptev.

Theorems & Definitions (9)

• Definition 1
• Theorem 2: Cazacu2024, Theorem 1.2
• Theorem 3
• Remark 4
• Remark 5
• Theorem 6
• Remark 7
• proof : Proof of Theorem \ref{['exrep']}
• proof : Proof of Theorem \ref{['solution']}