SOBOLEV-GAGLIARDO-NIRENBERG AND MARKOV TYPE INEQUALITIES ON SUBANALYTIC DOMAINS

In this work we introduce a new parameter, s greater than or equal to 1, in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) ine qualities and show their validity (with an appropriate s) for any comp act subanalytic domain. The classical form of these SGN inequalities ( s = 1 in our formulation) fails for domains with outward pointing cusp s. Our parameter s measures the degree of cuspidality of the domain. F or regular domains s = 1. We also introduce an extension, depending on a parameter sigma greater than or equal to 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 suprem um norms, a geometric characterization. We also establish several othe r characterisations: the existence of a bounded (linear) extension of C-infinity functions with a homogeneous loss of differentiability, and the validity of a global Markov inequality. Our methods may broadly b e classified as follows: 1. Desingularization and an L(p)-version of G laeser-type estimates. In fact we obtain a bound s less than or equal to 2d + 1, where d is the maximal order of vanishing of the jacobian o f the desingularization map of the domain. 2. Interpolation type inequ alities for norms of functions and Bernstein-Markov type inequalities for multivariate polynomials (classical analysis). 3. Geometric criter ia for the validity of local Markov inequalities (local analysis of th e singularities of domains). 4. Multivariate Approximation Theory. Thu s our approach brings together the calculus of Glaeser-type estimates from differential analysis, the algebra of desingularization, the geom etry of Markov type inequalities and the analysis of Sobolev-Nirenberg type estimates. Our exposition takes into account this interdisciplin ary nature of the methods we exploit and is almost entirely self-conta ined.