Quantitative tame properties of differentiable functions with controlled derivatives

• Published in: Nonlinear analysis Vol. 237; p. 113372
• Main Author: Rainer, Armin
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2023
• Subjects: Bernstein inequality; Controlled derivatives; Differentiable functions; Level sets; Markov inequality; Quasianalyticity; Remez inequality; Reverse Hölder inequality; Sublevel sets; Zero sets; Quasianalyticity; Bernstein inequality; 58C25; Differentiable functions; Zero sets; Remez inequality; Controlled derivatives; 41A63; Markov inequality; 26E10; Sublevel sets; Level sets; Reverse Hölder inequality; 26D07; 41A17
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2023.113372

We show that differentiable functions, defined on a convex body K⊆Rd, whose derivatives are controlled by a suitable given sequence of positive real numbers share many properties with polynomials. The role of the degree of a polynomial is hereby played by an integer associated with the given sequence of reals, the diameter of K, and a real parameter linked to the C0-norm of the function. We give quantitative information on the size of the zero set, show that it admits a local parameterization by Sobolev functions, and prove an inequality of Remez-type. From the latter, we deduce several consequences, for instance, a bound on the volume of sublevel sets and a comparison of Lp-norms reversing Hölder’s inequality. The validity of many of the results only depends on the derivatives up to some finite order; the order can be specified in terms of the given data.