×

On continuous and discrete Hardy inequalities. (English) Zbl 1368.35010

The authors furnish a series of discrete and continuous Hardy inequalities (dHi and cHi for short), principally we state significant results of each section. Accordingly, the second section focuses on the continuous case, the authors state the generalized cHi with multiple singularities, i.e., \[ (*):\;\displaystyle\int_{\mathbb R^d}|\nabla u(x)|^2dx\geq\frac{(d-2)^2}{4} \int_{\mathbb R^d}\frac{G(x)}{[w(x)]^2}|u(x)|^2dx \] with \(G(x)=\displaystyle\sum_{j=1}^d\left(\sum_{k=1}^n\frac{x_j-a_{k_j}}{|x-a_k|^d}\right)^2\), \(u(x)\) is a smooth decaying function, \(a_k=(a_{k_j})_{1\leq j\leq d}\in\mathbb R^d\) for \(k\in\{1,\ldots,n\}\), and \(\displaystyle w=\sum_{k=1}^n\frac{1}{|x-a_k|^{d-2}}\), and \(d\geq 3\) (Theorem 2.1). The proof is based on using technical calculus. Then by assuming that \(u(x)=0\) for \(x\) belonging to \( \partial S_n\), the boundary of \(S_n=\left\{x\in\mathbb R^2:\displaystyle\prod_{k=1}^n|x-a_k|\leq 1\right\}\), the authors obtain the same inequality as \((*)\) on \(\mathbb R^2\setminus S_n\) with the coefficient \(\displaystyle\frac{1}{4}\) instead of \(\displaystyle\frac{(d-2)^2}{4}\) (Theorem 2.2). The third section deals with dHi on \(\mathbb Z^d\) for \(d\geq 3\). Precisely, for \(f\) a function defined on \(\mathbb Z^d\), they show \[ \displaystyle \sum_{n\in\mathbb Z^d}\frac{|f(n)|^2}{|n|^2}\leq C \sum_{n\in\mathbb Z^d}\sum_{j=1}^d|f(n)-f(n-{1}_j)|^2, \] where \({1}_j\) is the unit vector in the direction \(j\) and \(C\) is a constant bounded by an explicit function of \(d\) (Theorem 3.1). As regards to the proof, the authors use, essentially, the Parseval identity and cHi on the torus (Lemma 3.2). The fourth section concentrates on dHi on \(\mathbb Z^2\), and the authors state that there is a positive constant \(K\) such that \[ \displaystyle\sum_{x\in\mathbb Z^2,|x|\geq 2}\frac{|f(x)|^2}{|x|^2(\ln|x|)^2}\leq K\sum_{(x,y)\in A}|f(x)-f(y)|^2, \] such that \(A=\{(x,y)\in\mathbb Z^2:x\sim y\}\), where \(x\sim y\) means that there is an edge between \(x\) and \(y\), \(f\) is a function with compact support and \(f(x)=0\) if \(|x|\leq 1\) (Theorem 4.1). The proof is a little bit long and based on using new apparatuses as discrete polar coordinates on \(\mathbb Z^2\) and discrete Poincaré-Friedrichs type inequality (Lemmas 4.2 and 4.4).

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35P15 Estimates of eigenvalues in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.