Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Estimates of difference norms for functions in anisotropic Sobolev spaces

We investigate the spaces of functions on ?ⁿ for which the generalized partial derivatives Dequation/tex2gif-sup-2.gif_kf exist and belong to different Lorentz spaces Lequation/tex2gif-sup-3.gif . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non‐increasing rearrangements. These methods enable us to cover also the case when some of the p_k's are equal to 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains

In this paper, the existence of (L²(Rn),L²(Rn))-pullback attractors and (L²(Rn),H¹(Rn))-pullback attractors are proved for reaction-diffusion equation in unbounded domains.     

Embeddings of Sobolev spaces on unbounded domains

Summary Piecewise polynomial and Fourier approximation of functions in the Sobolev spaces on unbounded domains Θ ⊂ Rⁿ are applied to the study of the type of compact embeddings into appropriate Lebesgue and Orlicz spaces. It is shown that if Θ and s satisfy certain conditions, the embeddings , m/n+1/q−1/p>0 and , Φ being an Orlicz function subordinate to both φ(t)=|t|^p exp |t|^n/(n−m) and Φ^σ(t)=exp |t|^σ−1, σ ⩾ 1, m/n>1/p, are of type l^s. One result dealing with multiplications maps from into L^q(Θ) is also obtained. Entrata in Redazione il 14 ottobre 1976.     

Sobolev functions on infinite-dimensional domains

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.     

Degree-independent Sobolev extension on locally uniform domains

We consider the problem of constructing extensions , where is the Sobolev space of functions with k derivatives in Lp and Ω⊂Rn is a domain. In the case of Lipschitz Ω, Calderón gave a family of extension operators depending on k, while Stein later produced a single (k-independent) operator. For the more general class of locally-uniform domains, which includes examples with highly non-rectifiable boundaries, a k-dependent family of operators was constructed by Jones. In this work we produce a k-independent operator for all spaces on a locally uniform domain Ω.     

Embeddings of anisotropic Sobolev spaces on unbounded domains

Summary Integral representations of junctions in the anisotropic Sobolev spaces on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain,     

On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first-order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.     

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in ${\mathbb{R}}^n$, with weights that are positive powers of the distance to the boundary.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

The de Rham complex on unbounded domains with Sobolev space topology

This paper studies the operator dd^∗+d^∗d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d^∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d^∗ is determined and an expression for d^∗ is obtained. The operator dd^∗+d^∗d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.     

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.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

Growth properties of Sobolev space functions over unbounded domains

Summary We study Sobolev space functions with prescribed growth properties on large spheres. In particular, we prove a weighted Poincaré type inequality for such functions. An extension to weighted Sobolev spaces is sketched.

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Sunto Si studiano funzioni che appartengono a certi spazi di Sobolev e che hanno crescita assegnata su sfere di grande raggio. Per tali funzioni si dimostra una disuguaglianza di tipo Poincaré con peso. Si indica inoltre anche una estensione di tale risultato a funzioni in spazi di Sobolev con peso.

     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

Nontrivial solutions for perturbations of a Hardy-Sobolev operator on unbounded domains

In this paper we study the existence of nontrivial solution of the problem −Δ_pu−(μ/[d(x)]^p)|u|^p−2u=f(u) in Ω and u=0 on ∂Ω, where is a bounded domain with smooth boundary in Existence is established using mountain-pass lemma and concentration of compactness principle.     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.