Young Ja Park
The existence of extremal functions for the Sobolev trace inequalities is studied using the concentration compactness theorem. The conjectured extremal, the function of conformal factor, is considered and is proved to be an actual extremal function with extra symmetry condition on functions. One of the limiting cases of the Sobolev trace inequalities is investigated and the best constant for this case is computed.
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We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.
The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.
We prove an endpoint version of the uniform Sobolev inequalities in Kenig-Ruiz-Sogge [8]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [2]. We also prove restricted weak type Stein-Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.
Let $Ω$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $Ω$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $Ω$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half…
We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].
We study limiting trace inequalities in the style of Maz'ya and Meyers--Ziemer for Sobolev martingales. We develop the Bellman function approach to such estimates, which allows to provide sufficient and almost necessary conditions on the martingale space and the martingale transform under which the trace inequalities hold true
We prove fractional Sobolev-Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.
In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…
We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, $s$-John domains and Hölder-$α$ domains, and discuss their optimality.
The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…
A homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of Singular Integrals and Sobolev inequalities
本文扩展了线性与非线性交换子的经典Sobolev型不等式。
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
Pointwise estimates for rough operators with applications to Sobolev inequalities
本文研究粗糙算子的Sobolev不等式，证明其满足被Riesz势作用于梯度的逐点界，并由此推导出左侧带算子的新Sobolev型不等式。
The aim of this paper is to provide Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, some basic facts about Sobolev spaces with respect to certain vector measures are stated.
Our main purpose is to establish Gagliardo-Nirenberg type inequalities using fractional homogeneous Sobolev spaces, and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in [1, 2, 3, 7, 16, 21].
In this article we generalize the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces instead the classical Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality. We consider for simplicity only the case of whole space.
We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a `separation property', is a John domain.
We investigate local variants of Nash inequalities in the context of Dunkl operators. Pseudo-Poincaré inequalities are first established using pointwise gradient estimates of the Dunkl heat kernel. These inequalities allow to obtain relative Nash-type inequalities which are used to derive mean value inequalities for subsolutions of the heat equation on orbits of balls not necessarily centered on the origin.
We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.
The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $Γ$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{ö}dinger problem yields a variations interpretation…
We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces.
On fractals, different measures (mutually singular in general) are involved to measure volumes of sets and energies of functions. Singularity of measures brings difficulties in (especially non-linear) analysis on fractals. In this paper, we prove a type of Sobolev inequalities, which involve different and possibly mutually singular measures, on product Sierpinski spaces. Sufficient and necessary conditions for the validity of these Sobolev inequalities are given. Furthermore, we compute the sharp exponents which appears in the sufficient and necessary conditions for the product Kusuoka measure…
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