Limits of higher-order Besov spaces and sharp reiteration theorems

We compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem. Our results extend to higher-order spaces the recent work by Brézis-Bourgain-Mironescu and Maz’ya-Shaposhnikova. The interpolation methods we develop are of interest on their own and could have applications to related inequalities.     

Sharpness of some properties of weighted modulation spaces

We obtain some optimal properties on weighted modulation spaces. We find the necessary and sufficient conditions for product inequalities, convolution inequalities and embedding on weighted modulation spaces. Especially, we establish the analogue of the sharp Sobolev embedding theorem on weighted modulation spaces.     

Symmetrization inequalities and Sobolev embeddings

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.

     

Sharp Sobolev inequalities with interior norms

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain . Let and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If , there exist constants , such that and If , for any , there exist constants such that and Received March 15, 1997 / Accepted January 30, 1998     

An elementary proof of sharp Sobolev embeddings

We present an elementary unified and self-contained proof of sharp Sobolev embedding theorems. We introduce a new function space and use it to improve the limiting Sobolev embedding theorem due to Brézis and Wainger.

     

Embeddings for anisotropic Besov spaces

We prove embedding theorems for fully anisotropic Besov spaces. In particular, inequalities between modulus of continuity in different metrics and of Sobolev type are obtained. Our goal is to get sharp estimates for some anisotropic cases previously unconsidered.     

Orlicz–Sobolev boundary trace embeddings

We prove a sharp trace embedding theorem for Orlicz–Sobolev spaces into Orlicz spaces on the boundary. An optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established.     

Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group

The Heisenberg group is one of the simplest sub-Riemannian settings in which we can define non-elliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities.     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001     

Best constants for Sobolev inequalities for higher order fractional derivatives

We obtain sharp constants for Sobolev inequalities for higher order fractional derivatives. As an application, we give a new proof of a theorem of W. Beckner concerning conformally invariant higher-order differential operators on the sphere.     

A multiplicity result for a special class of gradient-type systems with non-differentiable term

We prove the existence of multiple solutions of certain systems of hemivariational inequalities, using a recent minimax result of B. Ricceri. We apply both our main theorem and the principle of symmetric criticality to obtain multiple solutions of systems of hemivariational inequalities defined on certain Sobolev spaces.     

Cone conditions and properties of Sobolev spaces

The Sobolev imbedding theorem and certain interpolation inequalities for Sobolev spaces are established for a wider class of domains than has been covered by earlier proofs. This class is defined by a weakened, measure theoretic version of the cone condition. The proofs are elementary.     

On Hardy–Rellich type inequalities in RN

In this note we introduce a new class of Hardy–Rellich type inequalities and explicitly obtain their corresponding sharp constants. Our approach suggests definitions of new Sobolev spaces and embedding results.     

Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces

We exhibit sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second order Sobolev space into Lorentz spaces. As a consequence, we derive a new proof of Adams?? inequality, which holds in the larger hypotheses of homogenoeous Navier boundary contidions.     

Two isoperimetric inequalities for the Sobolev constant

In this note, we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalizes an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.     

Generalized doubling meets Poincaré

We establish a modified segment inequality on metric spaces that satisfy a generalized volume doubling property. This leads to Sobolev and Poincaré inequalities for such spaces. We also give several examples of spaces that satisfy the generalized doubling condition.     

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Embedding of Sobolev space in Orlicz space for a domain with irregular boundary

In this paper, we establish the embedding of a weighted Sobolev space in an Orlicz space for a domain with irregular boundary. We find an estimate of the order of growth of the N-function (defining the Orlicz space) and show that, under certain additional constraints on the weights, this estimate is sharp. We also establish the embedding in the space of continuous functions.     

有限元空间的嵌入性质和紧致性

本文将Sobolev嵌入定理和Rellich-Kondrachov紧致定理推广到多套函数有限元空间.特殊地,在非协调元,杂交元和拟协调元空间等情形建立了这两个定理.