284 research outputs found
Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian
We establish a family of sharp Sobolev trace inequalities involving the
-norm. These inequalities are closely related
to the realization of fractional powers of the Laplacian on
as generalized Dirichlet-to-Neumann
operators associated to powers of the weighted Laplacian in upper half space,
generalizing observations of Caffarelli--Silvestre and of Yang.Comment: 25 page
Conformal invariants measuring the best constants for Gagliardo-Nirenberg-Sobolev inequalities
We introduce a family of conformal invariants associated to a smooth metric
measure space which generalize the relationship between the Yamabe constant and
the best constant for the Sobolev inequality to the best constants for
Gagliardo-Nirenberg-Sobolev inequalities . These invariants are constructed via a minimization
procedure for the weighted scalar curvature functional in the conformal class
of a smooth metric measure space. We then describe critical points which are
also critical points for variations in the metric or the measure. When the
measure is assumed to take a special form --- for example, as the volume
element of an Einstein metric --- we use this description to show that
minimizers of our invariants are only critical for certain values of and
. In particular, on Euclidean space our result states that either
or , giving a new characterization of the GNS inequalities whose
sharp constants were computed by Del Pino and Dolbeault.Comment: 20 page
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