Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains

Abstract

In this work we introduce a new parameter,s≥1, in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) inequalities and show their validity (with an appropriates) for any compact subanalytic domain. The classical form of these SGN inequalities (s=1 in our formulation) fails for domains with outward pointing cusps. Our parameters measures the degree of cuspidality of the domain. For regular domainss=1. We also introduce an extension, depending on a parameter σ≥1, to several variables of a local form of the classical Markov inequality on the derivatives of a polynomial in terms of its own values, and show the equivalence of Markov and SGN inequalities with the same value of parameters, σ=s. Our extension of Markov's inequality admits, in the case of supremum norms, a geometric characterization. We also establish several other characterizations: the existence of a bounded (linear) extension ofC∞ functions with a homogeneous loss of differentiability, and the validity of a global Markov inequality. Our methods may broadly be classified as follows: 1. Desingularization and anLp-version of Glaeser-type estimates. In fact we obtain a bounds<-2d+1, whered is the maximal order of vanishing of the jacobian of the desingularization map of the domain. 2. Interpolation type inequalities for norms of functions and Bernstein-Markov type inequalities for multivariate polynomials (classical analysis). 3. Geometric criteria for the validity of local Markov inequalities (local analysis of the singularities of domains). 4. Multivariate Approximation Theory.