17,226 research outputs found
On Sobolev spaces and density theorems on Finsler manifolds
Let be a Finsler manifold, a real number, a
positive integer and a certain Sobolev space determined by a
Finsler structure .
Here, it is shown that the set of all real functions with
compact support on is dense in the Sobolev space .
This result permits to approximate certain solution of Dirichlet problem
living on by functions with compact support on .
Moreover, let be a regular domain with the boundary
, then the set of all real functions in is dense in , where . This work is an
extension of some density theorems of T. Aubin on Riemannian manifolds.Comment: 13 page
-regularity of Laplace equation with singular data on boundary
In this paper we are interested in regularity and existence of
solutions for Laplace equation on the upper half-space with nonlinear boundary
condition with singular data in Morrey-type spaces. To overcome lack of real
interpolation property and trace theorems, we introduce a new functional space
in order to show existence and regularity. To this end, we prove sharp
estimates for Riesz potential . As a byproduct, in particular, we
get -regularity of solutions with
singular data, covering known results in and Morrey space
.Comment: We have changed the title of the article and update introduction and
sections, inspired by anonymous referee comment
L\sp p-L\sp q regularity of Fourier integral operators with caustics
The caustics of Fourier integral operators are defined as caustics of the
corresponding Schwartz kernels (Lagrangian distributions on ). The
caustic set of the canonical relation is characterized as the
set of points where the rank of the projection is smaller
than its maximal value, . We derive the L\sp p(Y)\to L\sp
q(X) estimates on Fourier integral operators with caustics of corank 1 (such
as caustics of type A\sb{m+1}, ). For the values of and
outside of certain neighborhood of the line of duality, , the L\sp p\to
L\sp q estimates are proved to be caustics-insensitive.
We apply our results to the analysis of the blow-up of the estimates on the
half-wave operator just before the geodesic flow forms caustics.Comment: 24 pages, 1 figur
Regularity for the CR vector bundle problem II
We derive a \mathcal C^{k+\yt} H\"older estimate for
, where is either of the two solution operators in Henkin's local
homotopy formula for on a strongly pseudoconvex real
hypersurface in , is a -form of class
on , and is an integer. We also derive a estimate for , when is of class and
is a real number. These estimates require that be of class
, or , respectively. The explicit
bounds for the constants occurring in these estimates also considerably improve
previously known such results. These estimates are then applied to the
integrability problem for CR vector bundles to gain improved regularity. They
also constitute a major ingredient in a forthcoming work of the authors on the
local CR embedding problem
-Paracompact and -Metrizable Spaces
Let be a directed set and a space. A collection of
subsets of is \emph{-locally finite} if where (i) if then and (ii) each is locally finite.
Then is \emph{-paracompact} if every open cover has a -locally finite
open refinement. Further, is \emph{-metrizable} if it has a -locally finite base. This work provides the first detailed study
of -paracompact and -metrizable spaces, particularly in the case when
is a , the set of all compact subsets of a separable metrizable
space ordered by set inclusion
Inhomogeneous Partition Regularity
We say that the system of equations , where is an integer matrix
and is a (non-zero) integer vector, is partition regular if whenever the
integers are finitely coloured there is a monochromatic vector with .
Rado proved that the system is partition regular if and only if it has a
constant solution. Byszewski and Krawczyk asked if this remains true when the
integers are replaced by a general ring . Our aim in this note is to answer
this question in the affirmative. The main ingredient is a new `direct' proof
of Rado's result
Semiclassical functional calculus for -dependent functions
We study the functional calculus for operators of the form within
the theory of semiclassical pseudodifferential operators, where denotes a family of -dependent
functions satisfying some regularity conditions, and is either an
appropriate self-adjoint semiclassical pseudodifferential operator in
or a Schr\"odinger operator in , being a closed
Riemannian manifold of dimension . The main result is an explicit
semiclassical trace formula with remainder estimate that is well-suited for
studying the spectrum of in spectral windows of width of order
, where .Comment: v2: minor corrections, 33 page
Estimates for the complex Green operator: symmetry, percolation, and interpolation
Let be a pseudoconvex, oriented, bounded and closed CR submanifold of
of hypersurface type. We show that Sobolev estimates for the
complex Green operator hold simultaneously for forms of symmetric bidegrees,
that is, they hold for --forms if and only if they hold for
--forms. Here equals the CR dimension of plus one.
Symmetries of this type are known to hold for compactness estimates. We further
show that with the usual microlocalization, compactness estimates for the
positive part percolate up the complex, i.e. if they hold for --forms,
they also hold for --forms. Similarly, compactness estimates for the
negative part percolate down the complex. As a result, if the complex Green
operator is compact on --forms and on --forms (), then it is compact on --forms for . It
is interesting to contrast this behavior of the complex Green operator with
that of the --Neumann operator on a pseudoconvex domain.Comment: Added a reference to related work, removed a reference that was not
quoted. To appear in Transactions of the American Mathematical Societ
On regularity lemmas and their algorithmic applications
Szemer\'edi's regularity lemma and its variants are some of the most powerful
tools in combinatorics. In this paper, we establish several results around the
regularity lemma. First, we prove that whether or not we include the condition
that the desired vertex partition in the regularity lemma is equitable has a
minimal effect on the number of parts of the partition. Second, we use an
algorithmic version of the (weak) Frieze--Kannan regularity lemma to give a
substantially faster deterministic approximation algorithm for counting
subgraphs in a graph. Previously, only an exponential dependence for the
running time on the error parameter was known, and we improve it to a
polynomial dependence. Third, we revisit the problem of finding an algorithmic
regularity lemma, giving approximation algorithms for several co-NP-complete
problems. We show how to use the weak Frieze--Kannan regularity lemma to
approximate the regularity of a pair of vertex subsets. We also show how to
quickly find, for each , an -regular partition
with parts if there exists an -regular partition with parts.
Finally, we give a simple proof of the permutation regularity lemma which
improves the tower-type bound on the number of parts in the previous proofs to
a single exponential bound.Comment: Erratum added at the end. See also arXiv:1801.0503
C-image partition regularity near zero
In \cite{dehind1}, the concept of image partition regularity near zero was
first instigated. In contrast to the finite case , infinite image partition
regular matrices near zero are very fascinating to analyze. In this regard the
abstraction of Centrally image partition regular matrices near zero was
introduced in \cite{biswaspaul}. In this paper we propose the notion of
matrices that are C-image partition regular near zero for dense subsemigropus
of .Comment: 13 pages. One Theorem remove
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