Displaying similar documents to “Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces”

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.

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman, Luboš Pick (2011)

Studia Mathematica

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We study imbeddings of the Sobolev space W m , ϱ ( Ω ) : = u: Ω → ℝ with ϱ ( α u / x α ) < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, τ ϱ and σ ϱ , that are optimal with respect to the inclusions W m , ϱ ( Ω ) W m , τ ϱ ( Ω ) L σ ϱ ( Ω ) . General formulas for τ ϱ and σ ϱ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

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.

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.

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 .

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

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The role of the second critical exponent p = ( n + 1 ) / ( n - 3 ) , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem Δ u + u p = 0 , u > 0 under zero Dirichlet boundary conditions, in a domain Ω in n with bounded, smooth boundary. Given Γ , a geodesic of the boundary with negative inner normal curvature we find that for p = ( n + 1 ) / ( n - 3 - ε ) , there exists a solution u ε such that | u ε | 2 converges weakly to a Dirac measure on Γ as ε 0 + , provided that Γ is nondegenerate in the sense of second...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation - Δ u + ϵ u - | u | p - 2 u + | u | q - 2 u = 0 , x N , where N 3 , q > p > 2 and when ϵ > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2 * = 2 N N - 2 . For p < 2 * the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2 * the solution asymptotically...

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.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

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We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of...

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...

Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces

Hidemitsu Wadade (2010)

Studia Mathematica

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We present several continuous embeddings of the critical Besov space B p n / p , ρ ( ) . We first establish a Gagliardo-Nirenberg type estimate | | u | | q , w r 0 , ν C ( 1 / ( n - r ) ) 1 / q + 1 / ν - 1 / ρ ( q / r ) 1 / ν - 1 / ρ | | u | | p 0 , ρ ( n - r ) p / n q | | u | | p n / p , ρ 1 - ( n - r ) p / n q , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function w r ( x ) = 1 / ( | x | r ) with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding B p n / p , ρ ( ) B Φ , w r 0 , ν ( ) , where the function Φ₀ of the weighted Besov-Orlicz space B Φ , w r 0 , ν ( ) is a Young function of the exponential type. Another point of interest is to embed B p n / p , ρ ( ) into the weighted Besov...

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...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

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.

Multiple conjugate functions and multiplicative Lipschitz classes

Ferenc Móricz (2009)

Colloquium Mathematicae

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We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class L i p ( α , . . . , α N ) for some 0 < α , . . . , α N < 1 and its marginal functions satisfy f ( · , x , . . . , x N ) L i p β , . . . , f ( x , . . . , x N - 1 , · ) L i p β N for some 0 < β , . . . , β N < 1 uniformly in the indicated variables x l , 1 ≤ l ≤ N, then f ̃ ( η , . . . , η N ) L i p ( α , . . . , α N ) for each choice of ( η , . . . , η N ) with η l = 0 or 1 for 1 ≤ l ≤ N.

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional k -dimensional Jacobian of a map u in the Sobolev space W 1 , k 1 which takes values in the sphere S k 1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k 1 . In case M is polyhedral, the...

Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Yoshihiro Mizuta, Tetsu Shimomura (2023)

Czechoslovak Mathematical Journal

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Our aim is to establish Sobolev type inequalities for fractional maximal functions M , ν f and Riesz potentials I , α f in weighted Morrey spaces of variable exponent on the half space . We also obtain Sobolev type inequalities for a C 1 function on . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents Φ ( x , t ) = t p ( x ) + ( b ( x ) t ) q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions, p ( x ) < q ( x ) for x , and b ( · ) is nonnegative and Hölder continuous of order θ ( 0 , 1 ] .

Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir, Gershon Wolansky (2005)

Journal of the European Mathematical Society

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We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere S 2 , on a bounded domain Ω 2 and on all of 2 . In some cases we also address the question of existence of minimizers.