30,083 research outputs found
The Morse-Sard theorem revisited
Let be positive integers with . We establish an abstract
Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous
result of De Pascale's for Sobolev functions with and, on the other hand, also the following
new result: if satisfies
for every
(that is, is a Stepanov function), then the set
of critical values of is Lebesgue-null in . In the case that
we also show that this limiting condition holding for every
, where is a set of zero
-dimensional Hausdorff measure for some , is
sufficient to guarantee the same conclusion.Comment: We corrected some misprints and made some changes in the introductio
Calder\'{o}n commutators and the Cauchy integral on Lipschitz curves revisited III. Polydisc extensions
This article is the last in a series of three papers, whose scope is to give
new proofs to the well known theorems of Calder\'{o}n, Coifman, McIntosh and
Meyer. Here we extend the results of the previous two papers to the polydisc
setting. In particular, we solve completely an open question of Coifman from
the early eighties.Comment: 25 page
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