16 research outputs found
Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation
We obtain new oscillation inequalities in metric spaces in terms of the
Peetre functional and the isoperimetric profile. Applications provided
include a detailed study of Fractional Sobolev inequalities and the
Morrey-Sobolev embedding theorems in different contexts. In particular we
include a detailed study of Gaussian measures as well as probablity measures
between Gaussian and exponential. We show a kind of reverse Polya-Szego
principle that allows us to obtain continuity as a self improvement from
boundedness, using symetrization inequalities. Our methods also allow for
precise estimates of growth envelopes of generalized Sobolev and Besov spaces
on metric spaces. We also consider embeddings into and their connection
to Sobolev embeddings.Comment: 114 pages, made some editorial changes and made corrections to
chapters 3, 4 and
Interpolation of Morrey-Campanato and Related Smoothness Spaces
In this article, the authors study the interpolation of Morrey-Campanato
spaces and some smoothness spaces based on Morrey spaces, e.\,g., Besov-type
and Triebel-Lizorkin-type spaces. Various interpolation methods, including the
complex method, the -method and the Peetre-Gagliardo method, are studied
in such a framework. Special emphasize is given to the quasi-Banach case and to
the interpolation property.Comment: Sci. China Math. (2015
Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications
Let and be a ball
Banach function space satisfying some mild assumptions. Assume that
or is an
-domain for some . In this article,
the authors prove that a function belongs to the homogeneous ball Banach
Sobolev space if and only if and where is
related to . This result is of wide generality and can be
applied to various specific Sobolev-type function spaces, including Morrey,
Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or
global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice) Sobolev spaces,
which is new even in all these special cases; in particular, it is still new
even when with . The novelty
of this article exists in that, to establish the characterization of
, the authors provide a machinery via using a
generalized Brezis--Seeger--Van Schaftingen--Yung formula on ,
an extension theorem on , a
Bourgain--Brezis--Mironescu-type characterization of the inhomogeneous ball
Banach Sobolev space , and a method of extrapolation to
overcome those difficulties caused by that might be neither
the rotation invariance nor the translation invariance and that the norm of
has no explicit expression.Comment: arXiv admin note: text overlap with arXiv:2304.0094
Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains
Let be a bounded -domain
with , a ball Banach function space
satisfying some mild assumptions, and with
a -radial decreasing approximation of the identity
on . In this article, the authors establish two extension
theorems, respectively, on the inhomogeneous ball Banach Sobolev space
and the homogeneous ball Banach Sobolev space
for any . On the other hand, the
authors prove that, for any , where is
the Gamma function and is related to . Using
this asymptotics, the authors further establish a characterization of
in terms of the above limit. To achieve these, the authors
develop a machinery via using a method of the extrapolation, two extension
theorems on weighted Sobolev spaces, and some recently found profound
properties of to overcome those difficulties caused by
that the norm of has no explicit expression and that
might be neither the reflection invariance nor the
translation invariance. This characterization has a wide range of generality
and can be applied to various Sobolev-type spaces, such as Morrey
[Bourgain--Morrey-type, weighted (or mixed-norm or variable), local (or global)
generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, all of
which are new.Comment: arXiv admin note: substantial text overlap with arXiv:2307.10528,
arXiv:2304.0094
Recent Developments of Function Spaces and Their Applications I
This book includes 13 papers concerning some of the recent progress in the theory of function spaces and its applications. The involved function spaces include Morrey and weak Morrey spaces, Hardy-type spaces, John–Nirenberg spaces, Sobolev spaces, and Besov and Triebel–Lizorkin spaces on different underlying spaces, and they are applied in the study of problems ranging from harmonic analysis to potential analysis and partial differential equations, such as the boundedness of paraproducts and Calderón operators, the characterization of pointwise multipliers, estimates of anisotropic logarithmic potential, as well as certain Dirichlet problems for the Schrödinger equation
Elliptic Boundary Value Problems with Fractional Regularity Data: The First Order Approach
We study well-posedness of boundary value problems of Dirichlet and Neumann
type for elliptic systems on the upper half-space with coefficients independent
of the transversal variable, and with boundary data in fractional
Besov-Hardy-Sobolev (BHS) spaces. Our approach uses minimal assumptions on the
coefficients, and in particular does not require De Giorgi-Nash-Moser
estimates. Our results are completely new for the Hardy-Sobolev case, and in
the Besov case they extend results recently obtained by Barton and Mayboroda.
First we develop a theory of BHS spaces adapted to operators which are
bisectorial on , with bounded functional calculus on their
ranges, and which satisfy off-diagonal estimates. In particular, this
theory applies to perturbed Dirac operators . We then prove that for a
nontrivial range of exponents (the identification region) the BHS spaces
adapted to are equal to those adapted to (which correspond to
classical BHS spaces).
Our main result is the classification of solutions of the elliptic system
within a certain region of exponents. More
precisely, we show that if the conormal gradient of a solution belongs to a
weighted tent space (or one of their real interpolants) with exponent in the
classification region, and in addition vanishes at infinity in a certain sense,
then it has a trace in a BHS space, and can be represented as a semigroup
evolution of this trace in the transversal direction. As a corollary, any such
solution can be represented in terms of an abstract layer potential operator.
Within the classification region, we show that well-posedness is equivalent to
a certain boundary projection being an isomorphism. We derive various
consequences of this characterisation, which are illustrated in various
situations, including in particular that of the Regularity problem for real
equations.Comment: Changed title and fixed some minor typos. To appear in the CRM
Monograph Serie
Second order elliptic operators with complex bounded measurable coefficients in , Sobolev and Hardy spaces
Let be a second order divergence form elliptic operator with complex
bounded measurable coefficients. The operators arising in connection with ,
such as the heat semigroup and Riesz transform, are not, in general, of
Calder\'on-Zygmund type and exhibit behavior different from their counterparts
built upon the Laplacian. The current paper aims at a thorough description of
the properties of such operators in , Sobolev, and some new Hardy spaces
naturally associated to .
First, we show that the known ranges of boundedness in for the heat
semigroup and Riesz transform of , are sharp. In particular, the heat
semigroup need not be bounded in if . Then we provide a complete description of {\it all}
Sobolev spaces in which admits a bounded functional calculus, in
particular, where is bounded.
Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces
associated to , that serves the range of beyond .
It includes, in particular, characterizations by the sharp maximal function and
the Riesz transform (for certain ranges of ), as well as the molecular
decomposition and duality and interpolation theorems