Completely 1-complemented subspaces of the Schatten spaces
Jean Roydor (2007)
Banach Center Publications
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We describe the subspaces of (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.
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Jean Roydor (2007)
Banach Center Publications
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We describe the subspaces of (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.
Sergey V. Astashkin, Lech Maligranda (2015)
Studia Mathematica
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The structure of the closed linear span of the Rademacher functions in the Cesàro space is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of if 1 < p < ∞.
Yasushi Hirata, Nobuyuki Kemoto (2003)
Fundamenta Mathematicae
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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.
Abdellah Youssfi (1995)
Annales de l'institut Fourier
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In this paper we consider the regularity problem for the commutators where is a locally integrable function and are the Riesz transforms in the -dimensional euclidean space . More precisely, we prove that these commutators are bounded from into the Besov space for and if and only if is in the -Triebel-Lizorkin space . The reduction of our result to the case gives in particular that the commutators are bounded form into the Sobolev space if and only if ...
Antonio Aizpuru, Francisco J. García-Pacheco (2007)
Bollettino dell'Unione Matematica Italiana
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In this paper, we show a necessary and sufficient condition for a real Banach space to have an infinite dimensional subspace which is hilbertizable and complemented using techniques related to -summand vectors.
Dorothee D. Haroske, Leszek Skrzypczak (2013)
Studia Mathematica
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We study embeddings of spaces of Besov-Morrey type, , where is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of . This continues our earlier studies relating to the case of . Moreover, we also characterise embeddings into the scale of spaces or into the space of bounded continuous functions.
Imed Feki, Ameni Massoudi, Houda Nfata (2018)
Czechoslovak Mathematical Journal
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The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces ; , and is the open unit disk or the annulus of the complex space .
Simon Lücking (2014)
Studia Mathematica
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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form or and which quotients of the form or have the Daugavet property. We show that is a rich subspace of C(G) if and only if is a semi-Riesz set. If is a rich subspace of L¹(G), then is a rich subspace of C(G) as well. Concerning quotients, we prove that has the Daugavet property if Λ is a Rosenthal set, and that is a poor subspace of L¹(G)...
Douadi Drihem (2023)
Commentationes Mathematicae Universitatis Carolinae
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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces and with suitable assumptions on the parameters and , and the pair of weights .
Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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We give new characterizations of the analytic Besov spaces on the unit ball of in terms of oscillations and integral means over some Euclidian balls contained in .
Maciej Paluszyński (2010)
Colloquium Mathematicae
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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family , which is equivalent to the weight function being > 0 a.e.
Shuichi Sato (2019)
Czechoslovak Mathematical Journal
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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted spaces, , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).
Dale E. Alspach, Simei Tong (2003)
Studia Mathematica
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Many of the known complemented subspaces of have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...
Karlheinz Gröchenig, Christopher Heil (2001)
Studia Mathematica
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It is known that Gabor expansions do not converge unconditionally in and that cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in -norm.
Hans Triebel (2010)
Studia Mathematica
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The paper deals with spaces of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.
Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)
Studia Mathematica
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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...
G. Kyriazis (2003)
Studia Mathematica
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Let be a decomposition system for indexed over D, the set of dyadic cubes in , and a finite set E, and let be the corresponding dual functionals. That is, for every , . We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients , e ∈ E, I ∈ D. Typical examples of such decomposition...
H. Begehr, Yu. Dubinskiĭ (2001)
Colloquium Mathematicae
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Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the -solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form. In the second kind decomposition the...
Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)
Journal of the European Mathematical Society
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We define a new function space , which contains in particular BMO, BV, and , . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving norms of integer-valued functions in . We introduce a significant closed subspace, , of , containing in particular VMO and , . The above mentioned estimates imply in particular that integer-valued functions belonging to are necessarily constant. This framework provides a “common roof”...
Fernando Cobos, Óscar Domínguez (2014)
Studia Mathematica
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This paper deals with Besov spaces of logarithmic smoothness formed by periodic functions. We study embeddings of into Lorentz-Zygmund spaces . Our techniques rely on the approximation structure of , Nikol’skiĭ type inequalities, extrapolation properties of and interpolation.
Liguang Liu, Dachun Yang (2009)
Studia Mathematica
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Let s ∈ ℝ, p ∈ (0,1] and q ∈ [p,∞). It is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from the Triebel-Lizorkin space to a quasi-Banach space ℬ if and only if sup: a is an infinitely differentiable (p,q,s)-atom of < ∞, where the (p,q,s)-atom of is as defined by Han, Paluszyński and Weiss.
Ron Kerman, Luboš Pick (2008)
Studia Mathematica
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We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space be compactly imbedded into the rearrangement-invariant space , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from into . The results are illustrated with examples in which ϱ and σ are both...
M.S. Shahrokhi-Dehkordi (2017)
Communications in Mathematics
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Let be a bounded starshaped domain and consider the -Laplacian problem where is a positive parameter, , and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
Ondrej Kováčik, Jiří Rákosník (1991)
Czechoslovak Mathematical Journal
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