Herz-type Sobolev and Bessel potential spaces and their applications

The Herz-type Sobolev spaces are introduced and the Sobolev theorem is established. The Herz-type Bessel potential spaces and the relation between the Herz-type Sobolev spaces and Bessel potential spaces are discussed. As applications of these theories, some regularity results of nonlinear quantities appearing in the compensated compactness theory on Herz-type Hardy spaces are given. Project supported by the National Natural Science Foundation of China.     

Axiomatic theory of Sobolev spaces

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajłasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic spaces are then discussed.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

We present a definition of general Sobolev spaces with respect to arbitrary measures, Wh,p (Ω,μ) for 1 ≤p≤∞. In [RARP] we proved that these spaces are complete under very light conditions. Now we prove that if we consider certain general types of measures, then Cc∞ (R) is dense in thee spaces. As an application to Sobolev orthogonal polynomials, we study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials.     

Bessel (Riesz) potentials on banach function spaces and their applications I theory

In this paper, we shall introduce the concept of the Bessel (Riesz) potential Köthe function spacesX ^s (X ^s ) and give some dual estimates for a class of operators determined by a semi-group in the spacesL ^q (?T, T; X ^s ) (L ^q (?T, T; X ^s )). Moreover, some time-spaceL ^p ?L ^p′ estimates for the semi-group exp(it(-Δ) ^m/2) and the operatorA:=∫ ₀ ^t exp(i(t-τ)(-Δ) ^m/2)·dτ in the Lebesgue-Besov spacesL ^q (?T,T;B _p,2 ^s are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.     

Ordinary differential equations,transport theory and Sobolev spaces

Summary We obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.     

A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear Stochastic PDEs

We prove a relative compactness criterion in Wiener–Sobolev space which represents a natural extension of the compact embedding of Sobolev space H¹ into , at the level of random fields. Then we give a specific statement of this criterion for random fields solutions of semi-linear stochastic partial differential equations with coefficients bounded in an appropriate way. Finally, we employ this result to construct solutions for semi-linear stochastic partial differential equations with distribution as final condition. We also give a probabilistic interpretation of this solution in terms of backward doubly stochastic differential equations formulated in a weak sense.     

Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold

We introduce Sobolev spaces and capacities on the path space P _{m 0} (M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P _{m 0} (M). Moreover, we prove the tightness of (r, p)-capacities on P _{m 0} (M), \(\), which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Hölder continuous path space \(\).     

Regularity criteria for almost every function in Sobolev spaces

In this paper we determine the multifractal nature of almost every function (in the prevalence setting) in a given Sobolev or Besov space according to different regularity exponents. These regularity criteria are based on local Lp regularity or on wavelet coefficients and give a precise information on pointwise behavior.     

Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups

It is known that a function on $\text{R}$ⁿ which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense. In this paper an analogous theorem is proved in which $\text{R}$ⁿ is replaced by a set X, the averages over balls are replaced by a family of sublinear operators satisfying certain axioms, and the polynomials are replaced by a class of functions having certain regularity properties with respect to the averaging operators. Applications are given to function theory on domains in $\text{C}$ⁿ, to nilpotent Lie groups, and to the classical Euclidean case. The first application provides a characterization of the duals of Hardy spaces on the ball in $\text{C}$ⁿ.     

Nonlinear potential theory for Sobolev spaces on Carnot groups

Considering Bessel kernels on a Carnot group, we establish the main facts of nonlinear potential theory: a Wolff-type inequality, capacity estimates, and a strong capacity inequality. Deriving corollaries, we give an inequality of Sobolev-Adams type and relations between the capacity and Hausdorff measure, as well as lower bounds on the Teichmüller capacity. These yield the continuity of monotone functions of a Sobolev class and some estimates applicable to studying the fine properties of functions.     

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

各向异性函数空间上的多线性算子的估计及其应用

设A是Rn上的各向异性伸缩,L是由各向异性Calderón-Zygmund算子生成的一般的多线性算子.本文得到L从加权Lebesgue空间Lp(Rn)到无权的各向异性Hardy空间HpA(Rn)的有界性.另外,对各向异性Hardy空间H1(Rn)和加权各向异性BMO空间BMOwA(Rn)得到包含关系:BMOX(Rn)(...     

Sobolev weak solutions for parabolic PDEs and FBSDEs

This Note is devoted to the representation of Sobolev weak solutions to quasi-linear parabolic PDEs with monotone coefficients via FBSDEs. One distinctive character of this result is that the forward component of the FBSDE is coupled with the backward variable. To cite this article: F. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).