A sharp iteration principle for higher-order Sobolev embeddings

Andrea Cianchi, Luboš Pick, and Lenka Slavíková. "A sharp iteration principle for higher-order Sobolev embeddings." Banach Center Publications 101.1 (2014): 37-58. <http://eudml.org/doc/282498>.

@article{AndreaCianchi2014,
abstract = {We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving arbitrary rearrangement-invariant norms on open sets in ℝⁿ, possibly endowed with a measure density and satisfying an isoperimetric inequality of fairly general type, to considerably simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a direct application of the reduction principle we determine the optimal target space in the relevant Sobolev embeddings both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, the results apply to any-order Sobolev embedding on regular (John) domains, on Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and on families of product probability spaces, of which the Gauss space and the exponential measure space are classical instances.},
author = {Andrea Cianchi, Luboš Pick, Lenka Slavíková},
journal = {Banach Center Publications},
keywords = {isoperimetric function; higher-order Sobolev embeddings; rearrangement-invariant spaces; John domains; Maz'ya domains; product probability measures; Gaussian Sobolev inequalities; Hardy operator},
language = {eng},
number = {1},
pages = {37-58},
title = {A sharp iteration principle for higher-order Sobolev embeddings},
url = {http://eudml.org/doc/282498},
volume = {101},
year = {2014},
}

TY - JOUR
AU - Andrea Cianchi
AU - Luboš Pick
AU - Lenka Slavíková
TI - A sharp iteration principle for higher-order Sobolev embeddings
JO - Banach Center Publications
PY - 2014
VL - 101
IS - 1
SP - 37
EP - 58
AB - We survey results from the paper [CPS] in which we developed a new sharp iteration method and applied it to show that the optimal Sobolev embeddings of any order can be derived from isoperimetric inequalities. We prove thereby that the well-known link between first-order Sobolev embeddings and isoperimetric inequalities translates to embeddings of any order, a fact that had not been known before. We show a general reduction principle that reduces Sobolev type inequalities of any order involving arbitrary rearrangement-invariant norms on open sets in ℝⁿ, possibly endowed with a measure density and satisfying an isoperimetric inequality of fairly general type, to considerably simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a direct application of the reduction principle we determine the optimal target space in the relevant Sobolev embeddings both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, the results apply to any-order Sobolev embedding on regular (John) domains, on Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and on families of product probability spaces, of which the Gauss space and the exponential measure space are classical instances.
LA - eng
KW - isoperimetric function; higher-order Sobolev embeddings; rearrangement-invariant spaces; John domains; Maz'ya domains; product probability measures; Gaussian Sobolev inequalities; Hardy operator
UR - http://eudml.org/doc/282498
ER -