Lp. Bos et Pd. Milman, SOBOLEV-GAGLIARDO-NIRENBERG AND MARKOV TYPE INEQUALITIES ON SUBANALYTIC DOMAINS, Geometric and functional analysis, 5(6), 1995, pp. 853-923
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.