Certain Function Spaces, Capacities, and Potentials

Abstract

Section 10.1 is of an auxiliary nature. Here we collect (mostly without proofs) the results of function theory that are applied later or related to the facts used in the sequel. First we discuss the theorems on spaces of functions having “derivatives of arbitrary positive order” (Sect. 10.1). The theory of these spaces is essentially presented in monographs (cf. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970; Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Series, I. Duke University Press, Durham, 1976; Nikolsky, Approximation of Functions of Several Variables and Imbedding Theorems, Nauka, Moscow, 1977; Besov, Il’in, and Nikolsky, Integral Representations of Functions and Imbedding Theorems, Nauka, Moscow, 1975; Triebel, Spaces of Besov–Hardy–Sobolev type, Teubner, Leipzig, 1978, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978; and Runst and Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin, 1996) though in some cases the reader interested in the proofs will have to refer to the original papers.