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Displaying similar documents to “The characterization of radial subspaces of Besov and Lizorkin-Triebel spaces by differences”

Structure of Rademacher subspaces in Cesàro type spaces

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 C e s is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in C e s , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of C e s if 1 < p < ∞.

Separating by G δ -sets in finite powers of ω₁

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 G δ -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.

Regularity properties of commutators and B M O -Triebel-Lizorkin spaces

Abdellah Youssfi (1995)

Annales de l'institut Fourier

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In this paper we consider the regularity problem for the commutators ( [ b , R k ] ) 1 k n where b is a locally integrable function and ( R j ) 1 j n are the Riesz transforms in the n -dimensional euclidean space n . More precisely, we prove that these commutators ( [ b , R k ] ) 1 k n are bounded from L p into the Besov space B ˙ p s , p for 1 &lt; p &lt; + and 0 &lt; s &lt; 1 if and only if b is in the B M O -Triebel-Lizorkin space F ˙ s , p . The reduction of our result to the case p = 2 gives in particular that the commutators ( [ b , R k ] ) 1 k n are bounded form L 2 into the Sobolev space H ˙ s if and only if b ...

L 2 -Summand Vectors and Complemented Hilbertizable Subspaces

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 L 2 -summand vectors.

Embeddings of Besov-Morrey spaces on bounded domains

Dorothee D. Haroske, Leszek Skrzypczak (2013)

Studia Mathematica

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We study embeddings of spaces of Besov-Morrey type, i d Ω : p , u , q s ( Ω ) p , u , q s ( Ω ) , where Ω d is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of i d Ω . This continues our earlier studies relating to the case of d . Moreover, we also characterise embeddings into the scale of L p spaces or into the space of bounded continuous functions.

The Daugavet property and translation-invariant subspaces

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 C Λ ( G ) or L ¹ Λ ( G ) and which quotients of the form C ( G ) / C Λ ( G ) or L ¹ ( G ) / L ¹ Λ ( G ) have the Daugavet property. We show that C Λ ( G ) is a rich subspace of C(G) if and only if Γ Λ - 1 is a semi-Riesz set. If L ¹ Λ ( G ) is a rich subspace of L¹(G), then C Λ ( G ) is a rich subspace of C(G) as well. Concerning quotients, we prove that C ( G ) / C Λ ( G ) has the Daugavet property if Λ is a Rosenthal set, and that L ¹ Λ ( G ) is a poor subspace of L¹(G)...

Complex interpolation of function spaces with general weights

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 F ˙ p 0 , q 0 s 0 ( ω 0 ) and F ˙ , q 1 s 1 ( ω 1 ) with suitable assumptions on the parameters s 0 , s 1 , p 0 , q 0 and q 1 , and the pair of weights ( ω 0 , ω 1 ) .

Mobius invariant Besov spaces on the unit ball of n

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 B p on the unit ball 𝔹 of n in terms of oscillations and integral means over some Euclidian balls contained in 𝔹 .

A note on integer translates of a square integrable function on ℝ

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 ψ ( · - n ) n = - , which is equivalent to the weight function n = - | ψ ̂ ( · + n ) | ² being > 0 a.e.

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

Shuichi Sato (2019)

Czechoslovak Mathematical Journal

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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on n . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted L p spaces, 1 < p < , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Subspaces of L p , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of L p 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 L p . 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...

Gabor meets Littlewood-Paley: Gabor expansions in L p ( d )

Karlheinz Gröchenig, Christopher Heil (2001)

Studia Mathematica

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It is known that Gabor expansions do not converge unconditionally in L p and that L p cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that L p can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in L p -norm.

Sobolev-Besov spaces of measurable functions

Hans Triebel (2010)

Studia Mathematica

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The paper deals with spaces L p s ( ) of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces B p , q s ( ) of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.

Second derivatives of norms and contractive complementation in vector-valued spaces

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 p ( X ) , 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 p ( X ) admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

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Let Θ : = θ I e : e E , I D be a decomposition system for L ( d ) indexed over D, the set of dyadic cubes in d , and a finite set E, and let Θ ̃ : = Θ ̃ I e : e E , I D be the corresponding dual functionals. That is, for every f L ( d ) , f = e E I D f , Θ ̃ I e θ I e . 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 f , Θ ̃ I e , e ∈ E, I ∈ D. Typical examples of such decomposition...

Some orthogonal decompositions of Sobolev spaces and applications

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 W - 1 for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the Δ k -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...

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

Journal of the European Mathematical Society

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We define a new function space B , which contains in particular BMO, BV, and W 1 / p , p , 1 < p < . We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving L p norms of integer-valued functions in B . We introduce a significant closed subspace, B 0 , of B , containing in particular VMO and W 1 / p , p , 1 p < . The above mentioned estimates imply in particular that integer-valued functions belonging to B 0 are necessarily constant. This framework provides a “common roof”...

Embeddings of Besov spaces of logarithmic smoothness

Fernando Cobos, Óscar Domínguez (2014)

Studia Mathematica

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This paper deals with Besov spaces of logarithmic smoothness B p , r 0 , b formed by periodic functions. We study embeddings of B p , r 0 , b into Lorentz-Zygmund spaces L p , q ( l o g L ) β . Our techniques rely on the approximation structure of B p , r 0 , b , Nikol’skiĭ type inequalities, extrapolation properties of L p , q ( l o g L ) β and interpolation.

Boundedness of sublinear operators in Triebel-Lizorkin spaces via atoms

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 p , q s ( ) to a quasi-Banach space ℬ if and only if sup | | T ( a ) | | : a is an infinitely differentiable (p,q,s)-atom of p , q s ( ) < ∞, where the (p,q,s)-atom of p , q s ( ) is as defined by Han, Paluszyński and Weiss.

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

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 W m , ϱ ( Ω ) be compactly imbedded into the rearrangement-invariant space L σ ( Ω ) , 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 L ϱ ( 0 , | Ω | ) into L σ ( 0 , | Ω | ) . The results are illustrated with examples in which ϱ and σ are both...

On a class of ( p , q ) -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

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Let Ω n be a bounded starshaped domain and consider the ( p , q ) -Laplacian problem - Δ p u - Δ q u = λ ( 𝐱 ) | u | p - 2 u + μ | u | r - 2 u where μ is a positive parameter, 1 < q p < n , r p and p : = n p n - p is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the ( p , q ) -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.