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1.
We prove that Stein's extension operator preserves Sobolev–Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n‐dimensional Euclidean space. 相似文献
2.
《Mathematische Nachrichten》2017,290(1):37-49
We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space. 相似文献
3.
Toni Heikkinen 《Journal of Mathematical Analysis and Applications》2010,368(2):508-524
We study the extension properties of Orlicz-Sobolev functions both in Euclidean spaces and in metric measure spaces equipped with a doubling measure. We show that a set E⊂R satisfying a measure density condition admits a bounded linear extension operator from the trace space W1,Ψ(Rn)E| to W1,Ψ(Rn). Then we show that a domain, in which the Sobolev embedding theorem or a Poincaré-type inequality holds, satisfies the measure density condition. It follows that the existence of a bounded, possibly non-linear extension operator or even the surjectivity of the trace operator implies the measure density condition and hence the existence of a bounded linear extension operator. 相似文献
4.
A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR
d
,B=AB
D
andB
D
is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL
2() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization. 相似文献
5.
Earl Berkson 《Bulletin des Sciences Mathématiques》2011,135(5):488
After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded. 相似文献
6.
Christian Le Merdy 《Integral Equations and Operator Theory》2000,37(1):72-94
This paper is devoted to dual operator algebras, that isw
*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak* similarity property: for every Hilbert spaceH, anyw
*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras,
in analogy with some recent work of Pisier on the similarity problem on operator algebras. 相似文献
7.
Lixin Yan 《Journal of Fourier Analysis and Applications》2000,6(6):559-582
The purpose of this paper is to present constructions of wavelet frames on a Lipschitz curve Γ. As applications, we obtain
characterizations of the Besov and Triebel-Lizorkin spaces on Lipschitz curves, and the trace theorem on Γ of the Besov spaces
onR
2. 相似文献
8.
Jana Björn 《Calculus of Variations and Partial Differential Equations》2009,35(4):481-496
We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear
elliptic equations on R
n . 相似文献
9.
We prove that the space BV (ℝ
n) of functions with bounded variation on ℝ
n has the bounded approximation property. 相似文献
10.
Jie Miao 《Monatshefte für Mathematik》1998,125(1):25-35
We study reproducing kernels for harmonic Bergman spaces of the unit ball inR
n
. We establish some new properties for the reproducing kernels and give some applications of these properties. 相似文献
11.
《Quaestiones Mathematicae》2013,36(2):241-256
Abstract Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C(m)(U) → C (p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables. 相似文献
12.
We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes
the result of Hansson and Brezis-Wainger for W
n
k/k
as a special case. We deal with generalized Sobolev spaces W
A
k
, where instead of requiring the functions and their derivatives to be in Ln/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to Ln/k.
We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces
are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an,
earlier optimality result obtained by Hansson with respect to the Riesz potential operator.
In memory of Gene Fabes.
Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund. 相似文献
13.
Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L∞(γ) to BLOa(γ). 相似文献
14.
On the setting of the half-spaceR
n–1×R
+, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL
p
-Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable. 相似文献
15.
16.
The standard problem of radiation transfer in a bounded regionG
n
can be reformulated as a weakly singular integral equation with an unknown functionu: GC(S
n–1) and a kernelK: ((G × G }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with (E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL
p
(G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem. 相似文献
17.
Christoph Hamburger 《Mathematische Annalen》2006,334(4):775-782
We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of Rn.
相似文献
相似文献
18.
We introduce the notion of a two-wavelet multiplier, and give a trace formula for a two-wavelet multiplier as a bounded linear operator in the trace class fromL
2(R
n
) intoL
2(R
n
) in terms of the symbol and the two admissible wavelets.This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562. 相似文献
19.
《Quaestiones Mathematicae》2013,36(1):105-110
Abstract Let A be a non-empty bounded subset of a locally convex space E. We show that if all the separable subsets of A are weakly metrisable, then the weak*-compact subsets of E1 satisfy geometrical conditions which are similar to the concept of “dentability” used to characterise the Radon-Nikodý Property in dual Banach spaces. 相似文献
20.
Pavel Shvartsman 《Journal of Geometric Analysis》2002,12(2):289-324
We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is
formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family.
Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most
2m+1 points, the restriction F|M′ of F to M′ has a selection fM′ (i. e., fM′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒM′(x) − ƒM′(y)‖X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper
bound of the number of points in M′, 2m+1, is sharp.
If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s
extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition
for a function defined on a closed subset of
R
2
to be the restriction of a function from the Sobolev space W
∞
2
(R
2).A similar result is proved for the space of functions on
R
2
satisfying the Zygmund condition. 相似文献