1,004 research outputs found
A Weighted Estimate for the Square Function on the Unit Ball in \C^n
We show that the Lusin area integral or the square function on the unit ball
of \C^n, regarded as an operator in weighted space has a linear
bound in terms of the invariant characteristic of the weight. We show a
dimension-free estimate for the ``area-integral'' associated to the weighted
norm of the square function. We prove the equivalence of the classical
and the invariant classes.Comment: 11 pages, to appear in Arkiv for Matemati
Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets
In this paper we study the boundary limit properties of harmonic functions on
, the solutions to the Poisson equation where is a p.c.f. set
and its Laplacian given by a regular harmonic structure. In
particular, we prove the existence of nontangential limits of the corresponding
Poisson integrals, and the analogous results of the classical Fatou theorems
for bounded and nontangentially bounded harmonic functions.Comment: 22 page
A Sobolev Poincar\'e type inequality for integral varifolds
In this work a local inequality is provided which bounds the distance of an
integral varifold from a multivalued plane (height) by its tilt and mean
curvature. The bounds obtained for the exponents of the Lebesgue spaces
involved are shown to be sharp.Comment: v1: 27 pages, no figures; v2: replaced citations of the author's
dissertation by proofs, material of sections 1 and 3 reorganised, slightly
more general results in section 2, some remarks, some discussion and some
references added, 40 pages, no figure
Ground State Energy of the One-Dimensional Discrete Random Schr\"{o}dinger Operator with Bernoulli Potential
In this paper, we show the that the ground state energy of the one
dimensional Discrete Random Schroedinger Operator with Bernoulli Potential is
controlled asymptotically as the system size N goes to infinity by the random
variable \ell_N, the length the longest consecutive sequence of sites on the
lattice with potential equal to zero. Specifically, we will show that for
almost every realization of the potential the ground state energy behaves
asymptotically as in the sense that the ratio of
the quantities goes to one
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Fine properties of self-similar solutions of the Navier-Stokes equations
We study the solutions of the nonstationary incompressible Navier--Stokes
equations in , , of self-similar form , obtained from small and homogeneous initial
data . We construct an explicit asymptotic formula relating the
self-similar profile of the velocity field to its corresponding initial
datum
Sobolev Inequalities for Differential Forms and -cohomology
We study the relation between Sobolev inequalities for differential forms on
a Riemannian manifold and the -cohomology of that manifold.
The -cohomology of is defined to be the quotient of the space
of closed differential forms in modulo the exact forms which are
exterior differentials of forms in .Comment: This paper has appeared in the Journal of Geometric Analysis, (only
minor changes have been made since verion 1
Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
We prove local and global well-posedness for semi-relativistic, nonlinear
Schr\"odinger equations with
initial data in , . Here is a critical
Hartree nonlinearity that corresponds to Coulomb or Yukawa type
self-interactions. For focusing , which arise in the quantum theory of
boson stars, we derive a sufficient condition for global-in-time existence in
terms of a solitary wave ground state. Our proof of well-posedness does not
rely on Strichartz type estimates, and it enables us to add external potentials
of a general class.Comment: 18 pages; replaced with revised version; remark and reference on blow
up adde
Slicing Sets and Measures, and the Dimension of Exceptional Parameters
We consider the problem of slicing a compact metric space \Omega with sets of
the form \pi_{\lambda}^{-1}\{t\}, where the mappings \pi_{\lambda} \colon
\Omega \to \R, \lambda \in \R, are \emph{generalized projections}, introduced
by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: assuming that
\Omega has Hausdorff dimension strictly greater than one, what is the dimension
of the 'typical' slice \pi_{\lambda}^{-1}{t}, as the parameters \lambda and t
vary. In the special case of the mappings \pi_{\lambda} being orthogonal
projections restricted to a compact set \Omega \subset \R^{2}, the problem
dates back to a 1954 paper by Marstrand: he proved that for almost every
\lambda there exist positively many such that \dim
\pi_{\lambda}^{-1}{t} = \dim \Omega - 1. For generalized projections, the same
result was obtained 50 years later by J\"arvenp\"a\"a, J\"arvenp\"a\"a and
Niemel\"a. In this paper, we improve the previously existing estimates by
replacing the phrase 'almost all \lambda' with a sharp bound for the dimension
of the exceptional parameters.Comment: 31 pages, three figures; several typos corrected and large parts of
the third section rewritten in v3; to appear in J. Geom. Ana
The shrinkage of hardening cement paste and mortar
This paper is an abstract from the report of the commission B10: "The influence of the shrinkage of cement on the shrink-age of concrete", of the Netherlands Committee for Concrete Research. Measurements of pulse velocity, volume shrinkage and heat of hydration on hardening portland cement support the idea that the formation of ettringite is an important link in the mechanism of shrinkage in the plastic stage of cement paste and mortar. Mechanical tests on prisms of 4x4x16 cm3 gave some information about the difference in sensitivity to surface corrosion of different types of cement
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