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This volume presents advances that have been made over recent decades in areas of research featuring Hardy's inequality and related topics. The inequality and its extensions and refinements are not only of intrinsic interest but are indispensable tools in many areas of mathematics and mathematical physics.

Hardy inequalities on domains have a substantial role and this necessitates a detailed investigation of significant geometric properties of a domain and its boundary. Other topics covered in this volume are Hardy- Sobolev-Maz’ya inequalities; inequalities of Hardy-type involving magnetic fields; Hardy, Sobolev and Cwikel-Lieb-Rosenbljum inequalities for Pauli operators; the Rellich inequality.

The Analysis and Geometry of Hardy’s Inequality provides an up-to-date account of research in areas of contemporary interest and would be suitable for a graduate course in mathematics or physics. A good basic knowledge of real and complex analysis is a prerequisite.



Chapter 1. Hardy, Sobolev, and CLR Inequalities

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 Schrödinger 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.
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis

Chapter 2. Boundary Curvatures and the Distance Function

Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set
$$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}}$$
for the distance of \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega \). Our main objective in this chapter is to gather information about the regularity properties of δ.
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis

Chapter 3. Hardy’s Inequality on Domains

Let \(\Omega \) be a domain (an open, connected set) in \(\mathbb{R}^{n}\) with non-empty boundary, \(1 < p < \infty \), and denote by δ(x) the distance from a point \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega,\) i.e.,
$$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}.}$$
The basic inequality to be considered in this chapter is
$$\displaystyle{ \int _{\Omega }\vert \nabla f(\mathbf{x})\vert ^{p}d\mathbf{x} \geq c(n,p,\Omega )\int _{ \Omega }\frac{\vert f(\mathbf{x})\vert ^{p}} {\delta (\mathbf{x})^{p}} d\mathbf{x},\ \ \ f \in C_{0}^{\infty }(\Omega ); }$$
equivalently, the inequality is to hold for all \(f \in W_{0}^{1,p}(\Omega ).\)
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis

Chapter 4. Hardy, Sobolev, Maz’ya (HSM) Inequalities

From the Hardy and Sobolev inequalities
$$\displaystyle{\|\nabla u\|_{p,\Omega }^{p} \geq C_{ H}\|u/\delta \|_{p,\Omega }^{p},\ \ \ \|\nabla u\|_{ p,\Omega }^{p} \geq C_{ S}\|u\|_{p^{{\ast}},\Omega }^{p},\ \ \ u \in D_{ 0}^{1,p}(\Omega ),}$$
where \(\delta (\mathbf{x}) =\mathrm{ dist}(\mathbf{x},\partial \Omega ),C_{H},C_{S}\) are the optimal constants and p  = np∕(np), it follows that for 0 < α ≤ C H , 
$$\displaystyle\begin{array}{rcl} \|\nabla u\|_{p,\Omega }^{p} -\alpha \| u/\delta \|_{ p,\Omega }^{p}& \geq & \left (1 -\alpha /C_{ H}\right )\|\nabla u\|_{p,\Omega }^{p} \\ & \geq & \left (1 -\alpha /C_{H}\right )C_{S}\|u\|_{p^{{\ast}},\Omega }^{p}.{}\end{array}$$
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis

Chapter 5. Inequalities and Operators Involving Magnetic Fields

In classical mechanics the motion of charged particles depends only on electric and magnetic fields E, B which are uniquely described by Maxwell’s equations:
$$\displaystyle{\nabla \cdot \mathbf{E} = 4\pi \rho,}$$
$$\displaystyle{\nabla \cdot \mathbf{B} = 0,}$$
$$\displaystyle{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t},}$$
$$\displaystyle{\nabla \times \mathbf{B} = 4\pi \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}.}$$
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis

Chapter 6. The Rellich Inequality

In lectures delivered at New York University in 1953, and published posthumously in the proceedings [128] of the International Congress of Mathematicians held in Amsterdam in 1954, Rellich proved the following inequality which bears his name: for n ≠ 2
$$\displaystyle{ \int _{\mathbb{R}^{n}}\vert \Delta u(\mathbf{x})\vert ^{2}d\mathbf{x} \geq \frac{n^{2}(n - 4)^{2}} {16} \int _{\mathbb{R}^{n}}\frac{\vert u(\mathbf{x})\vert ^{2}} {\vert \mathbf{x}\vert ^{4}} d\mathbf{x},\ \ u \in C_{0}^{\infty }(\mathbb{R}^{n}\setminus \{0\}), }$$
while for n = 2, the inequality continues to hold but for a restricted class of functions u; see Remark 6.4.4 below.
Alexander A. Balinsky, W. Desmond Evans, Roger T. Lewis


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