Bernstein-type Inequality and Lebesgue Classes of Constructible Functions in Weighted Bergman and Quasi-Banach Spaces

Abstract

We show the inequalities of Bernstein-type for derivatives of rational functions, inverse theorems of rational approximation, Kernels of Toeplitz operators, smooth functions and effective essential Hardy space interpolation constrained by weighted Hardy and Bergman norms. The Presburger Sets, -minimal fields, analytic -adic cell decomposition, integrals, and the classification of semi-algebraic -adic sets up to semi algebraic bijection are considered. We characterize the basic sequences and curves with zero derivative in -spaces and an -space with trivial dual where the Krein-Milman theorem holds. We discuss the asymptotic sharpness and application of a Bernstein-type inequality for rational functions and interpolation in Hardy, Dirichlet and weighted Bergman spaces. Methods of integration of positive constructible functions against Euler characteristic, dimension, loci of integrability, zero loci, stability under integration for constructible functions on Euclidean space with Lebesgue measure, Lebesgue classes and preparation of real constructible functions are studied. The existence and Lipschitz maps of primitives for continuous functions and the fundamental theorem of calculus with integration in quasi Banach spaces are established.