Displaying similar documents to “On limiting embeddings of Besov spaces”

Sharp embedding results for spaces of smooth functions with power weights

Martin Meyries, Mark Veraar (2012)

Studia Mathematica

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We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on d , equipped with power weights w ( x ) = | x | γ , γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

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.

A new characterization of the Sobolev space

Piotr Hajłasz (2003)

Studia Mathematica

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The purpose of this paper is to provide a new characterization of the Sobolev space W 1 , 1 ( ) . We also show a new proof of the characterization of the Sobolev space W 1 , p ( ) , 1 ≤ p < ∞, in terms of Poincaré inequalities.

Sharp L 1 estimates for singular transport equations

Sergiu Klainerman, Igor Rodnianski (2008)

Journal of the European Mathematical Society

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We provide L 1 estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest.

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

Hidemitsu Wadade (2014)

Studia Mathematica

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We establish the embedding of the critical Sobolev-Lorentz-Zygmund space H p , q , λ , . . . , λ n / p ( ) into the generalized Morrey space Φ , r ( ) with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in H p , q , λ , . . . , λ n / p + 1 ( ) . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

The Bohr-Pál theorem and the Sobolev space W 1 / 2

Vladimir Lebedev (2015)

Studia Mathematica

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The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space W 1 / 2 ( ) . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if...

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 Sobolev type inequality and its applications

Witold Bednorz (2006)

Studia Mathematica

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Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball T : = B | | · | | ( 0 , r ) , r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, s u p s , t T | f ( s ) - f ( t ) | 6 A B ( 0 r ψ ( 1 / A ε n - 1 ) ε n - 1 d ε + 1 / ( n | B | | · | | ( 0 , 1 ) | ) T φ ( 1 / B | | f ( u ) | | ) d u ) , where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on...

A subelliptic Bourgain–Brezis inequality

Yi Wang, Po-Lam Yung (2014)

Journal of the European Mathematical Society

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We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space N L ˙ 1 , Q by L functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group n .

Variable exponent trace spaces

Lars Diening, Peter Hästö (2007)

Studia Mathematica

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The trace space of W 1 , p ( · ) ( × [ 0 , ) ) consists of those functions on ℝⁿ that can be extended to functions of W 1 , p ( · ) ( × [ 0 , ) ) (as in the fixed-exponent case). Under the assumption that p is globally log-Hölder continuous, we show that the trace space depends only on the values of p on the boundary. In our main result we show how to define an intrinsic norm for the trace space in terms of a sharp-type operator.

Composition operator and Sobolev-Lorentz spaces W L n , q

Stanislav Hencl, Luděk Kleprlík, Jan Malý (2014)

Studia Mathematica

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Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator T f : u u f maps the Sobolev-Lorentz space W L n , q ( Ω ' ) to W L n , q ( Ω ) for some q ≠ n then f must be a locally bilipschitz mapping.

Strong density for higher order Sobolev spaces into compact manifolds

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2015)

Journal of the European Mathematical Society

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Given a compact manifold N n , an integer k * and an exponent 1 p < , we prove that the class C ( Q ¯ m ; N n ) of smooth maps on the cube with values into N n is dense with respect to the strong topology in the Sobolev space W k , p ( Q m ; N n ) when the homotopy group π k p ( N n ) of order k p is trivial. We also prove density of maps that are smooth except for a set of dimension m - k p - 1 , without any restriction on the homotopy group of N n .

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function f L l o c p ( ) the notion of p-mean variation of order 1, p ( f , ) is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space W 1 , p ( ) in terms of p ( f , ) is directly related to the characterisation of W 1 , p ( ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

Concentration-Compactness Principle for embedding into multiple exponential spaces on unbounded domains

Robert Černý (2015)

Czechoslovak Mathematical Journal

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Let Ω n be a domain and let α < n - 1 . We prove the Concentration-Compactness Principle for the embedding of the space W 0 1 L n log α L ( Ω ) into an Orlicz space corresponding to a Young function which behaves like exp ( t n / ( n - 1 - α ) ) for large t . We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle...

What is a Sobolev space for the Laguerre function systems?

B. Bongioanni, J. L. Torrea (2009)

Studia Mathematica

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We discuss the concept of Sobolev space associated to the Laguerre operator L α = - y d ² / d y ² - d / d y + y / 4 + α ² / 4 y , y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials ( L α ) - s . An appropriate Laguerre-Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function f W k , p ( Ω ) possesses an L p derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space W k , p ( Ω ) . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

Rémi Arcangéli, Juan José Torrens (2013)

Studia Mathematica

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We collect and extend results on the limit of σ 1 - k ( 1 - σ ) k | v | l + σ , p , Ω p as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and | · | l + σ , p , Ω is the intrinsic seminorm of order l+σ in the Sobolev space W l + σ , p ( Ω ) . In general, the above limit is equal to c [ v ] p , where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

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The estimate D k - 1 u L n / ( n - 1 ) A ( D ) u L 1 is shown to hold if and only if A ( D ) is elliptic and canceling. Here A ( D ) is a homogeneous linear differential operator A ( D ) of order k on n from a vector space V to a vector space E . The operator A ( D ) is defined to be canceling if ξ n { 0 } A ( ξ ) [ V ] = { 0 } . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous...

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ( L q , l p ) α is a complex Banach space which is continuously included in the Wiener amalgam space ( L q , l p ) and contains the Lebesgue space L α . We study the closure ( L q , l p ) c , 0 α in ( L q , l p ) α of the space of test functions (infinitely differentiable and with compact support in d ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W ¹ ( ( L q , l p ) α ) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

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.