Remez type inequalities and Morrey-Campanato spaces on Ahlfors regular sets

Abstract

This paper is dedicated to our friend Michael Cwikel with respect and sympathy.Abstract. The paper presents several new results on Remez type inequalities for real and complex polynomials in n variables on Ahlfors regular subsets of Lebesgue n-measure zero. As an application we prove an extension theorem for Morrey-Campanato spaces defined on such sets.

On covering numbers of sublevel sets of analytic functions

“…Using these estimates we prove strong Remez type inequalities for the restrictions of analytic functions to certain fractal sets. The existence of such inequalities was conjectured in [4] in connection with the study of traces of Morrey-Campananto spaces to Markov subsets of R N . Motivated by boundary value problems for PDEs, classical trace theorems characterize traces of spaces of generalized smoothness (e.g., Sobolev, Besov etc.)…”

“…But in many cases one needs similar results for subsets of a more complicated geometric structure (for instance, after the change of variables initial data may be situated on a Lipschitz surface). The general project of the authors of [4] is devoted to the characterization of traces of spaces of a given generalized smoothness to (polynomially) regular subsets of R N via local polynomial approximation. The Remez type E-mail address: albru@math.ucalgary.ca.…”

“…inequalities are among the main tools of this approach. The results presented in [4] deal with the technically simplest case of Morrey-Campanato spaces (more general results will appear in the forthcoming book [5]). …”

“…An inequality of the form (2.5), i.e., where the constant depends on the ratio of the corresponding measures polynomially, is called in [4] a strong Remez type inequality. In turn, if the corresponding constant depends on the ratio of measures implicitly, then the corresponding inequality is called in [4] a weak Remez type inequality.…”

Remez-type inequality on sets with cusps

“…Define r(η) in the following way: = η −1 in (0, 1). 26 We leave it as an exercise to verify that, for t ≥ 0, ηt ≤ (1 + ηt)t r(η) . In particular,…”