Summary: In this work we introduce a new parameter, \(s\geq 1\), in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) inequalities and show their validity (with an appropriate \(s\)) 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 parameter \(s\) measures the degree of cuspidality of the domain. For regular domains \(s=1\). We also introduce an extension, depending on a parameter \(\sigma\geq 1\), 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, \(\sigma=s\). 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 of \(C^\infty\) 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 an \(L_p\)-version of Glaeser-type estimates.
In fact we obtain a bound \(s\leq 2d +1\), where \(d\) 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.
Thus our approach brings together the calculus of Glaeser-type estimates from differential analysis, the algebra of desingularization, the geometry of Markov type inequalities and the analysis of Sobolev-Nirenberg type estimates. Our exposition takes into account this interdisciplinary nature of the methods we exploit and is almost entirely self-contained.