Function Spaces and Potential Theory

Abstract

Softcover reprint of the hanlcover lst edition 1996 Cover design: MetaDesign plus GmbH, Berlin Photocomposed from the authors' TEX files after editing and reformatting by Kurt Mattes, Heidelberg, using the MathTime fonts and a Springer TEX macro-package Printed on acid-free paper SPIN: ]]304074 41/3111 -5432. 1 Preface Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE).Potential theory, which grew out of the theory of the electrostatic or gravitational potential, the Laplace equation, the Dirichlet problem, etc., had a fundamental role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More recently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development.The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch.-J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940's. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L. Lions [122] in the early 1950's, this point of view, blended with the L. Schwartz theory of distributions, led to a new understanding of the function spaces of Hilbert type. According to the classical Dirichlet principle, one obtains the solution of Dirichlet's problem for the Laplace equation in a domain a by minimizing the Dirichlet integral, fg IVu(x)1 2 dx, over a certain class of functions taking given values on the boundary aa. The natural explanation is that solutions to the Laplace equation describe an equilibrium state, a state attained when the energy carried by the system is at a minimum.Also, the classical electrostatic capacity (or capacitance, as it is known to physicists) of a compact set K C R 3 differs only by a constant factor from C (K) = inf f g I vu (x) 1 2 dx, where the infimum is taken over all smooth compactly supported functions u, such that u :::: 1 on K.As emphasized by H. Cartan, taking these infima of the Dirichlet integral is equivalent to taking projections in a Hilbert space normed by the square root of the Dirichlet integral. VI PrefaceA slight modification of the nonn defined by the Oirichlet integral leads to the simplest of the function spaces treated in this work, the Sobolev space denoted by H' or W1,2. It is nonned by the square root of lIull~1 = fRN(lu12 + IVuI2)dx. This Hilbert space ...

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

“…, and moreover µ K (E) ≤ Cap 1, q q−p+1 (E) for all compact (and Borel) sets E. Here the last property follows, e.g., from [1,Proposition 6.3.13] and the capacitability of Borel sets.…”

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

“…See [1,Theorem 4.1.3] and [28,Chapter 3]. It is easy to see that every Lipschitz function with compact support belongs to B p (R n ).…”

A tracing of the fractional temperature field

“…Since (1), (2) and the left equivalence of (3) are contained in [7, Theorems 3.2&3.3] whose proofs depend on Lemma 2.1, it is enough to check the right equivalence of (3). Our approach is a fractional heat potential analogue of the Riesz potential treatment carried in [6, Theorem 2.1].…”