Hardy, Sobolev, and CLR Inequalities

Abstract

The Hardy and Sobolev inequalities are of fundamental importance in many branches of mathematical analysis and mathematical physics, and have been intensively studied since their discovery. A rich theory has been developed with the original inequalities on \((0,\infty )\) extended and refined in many ways, and an extensive literature on them now exists. We shall be focusing throughout the book on versions of the inequalities in L p spaces, with \(1 < p < \infty \). In this chapter we shall be mainly concerned with the inequalities in \((0,\infty )\) or \(\mathbb{R}^{n}, n \geq 1\). Later in the chapter we shall also discuss the CLR (Cwikel, Lieb, Rosenbljum) inequality, which gives an upper bound to the number of negative eigenvalues of a lower semi-bounded Schrodinger operator in \(L^{2}(\mathbb{R}^{n})\). This has a natural place with the Hardy and Sobolev inequalities as the three inequalities are intimately related, as we shall show. Where proofs are omitted, e.g., of the Sobolev inequality, precise references are given, but in all cases we have striven to include enough background analysis to enable a reader to understand and appreciate the result.