Sobolev-Besov spaces of measurable functions

Hans Triebel

Displaying similar documents to “Sobolev-Besov spaces of measurable functions”

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 ${\dot{F}}_{p_{0}, q_{0}}^{s_{0}} (ω_{0})$ and ${\dot{F}}_{\infty, q_{1}}^{s_{1}} (ω_{1})$ with suitable assumptions on the parameters $s_{0}, s_{1}, p_{0}, q_{0}$ and $q_{1}$ , and the pair of weights $(ω_{0}, ω_{1})$ .

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K (t, f; L^{p} (C), L^{\infty} (C))$ to quasicontinuous functions f ∈ QC is equivalent to $K (t, f; L^{p} (C) \cap Q C, L^{\infty} (C) \cap Q C)$ . We apply this result to identify the interpolation space ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .

Interpolation of Cesàro sequence and function spaces

Sergey V. Astashkin, Lech Maligranda (2013)

Studia Mathematica

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The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $C e s_{p} (I)$ is an interpolation space between $C e s_{p ₀} (I)$ and $C e s_{p ₁} (I)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $C e s_{p} [0, 1]$ is not an interpolation space between Ces₁[0,1] and $C e s_{\infty} [0, 1]$ .

The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

Irina Asekritova, Natan Krugljak, Ludmila Nikolova (2005)

Studia Mathematica

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A complete description of the real interpolation space $L = {(L_{p ₀} (ω ₀), . . ., L_{p ₙ} (ω ₙ))}_{θ ⃗, q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$ , and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

On spaces $L^{p (x)}$ and $W^{k, p (x)}$

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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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 $ϱ (\partial^{α} u / \partial x^{α})$ < ∞ 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 $W^{m, ϱ} (Ω) \subset W^{m, τ_{ϱ}} (Ω) \subset L_{σ_{ϱ}} (Ω)$ . 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.

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 the equation x”(t) = F(t, x(t)) in the Sobolev space $H^{1} (R)$

Piotr Fijałkowski (1991)

Annales Polonici Mathematici

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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.

Real method of interpolation on subcouples of codimension one

S. V. Astashkin, P. Sunehag (2008)

Studia Mathematica

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We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N ₀, N ₁)}_{θ, q}$ and ${(X ₀, X ₁)}_{θ, q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_{i}$ is the normed space N with the norm inherited from $X_{i}$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{θ} = S - 2^{θ} I$ (S...

On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces

Wisam Alame (2005)

Banach Center Publications

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We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_{p}^{2 s + 2, s + 1} (Ω^{T})$ or to $B_{p, q}^{2 s + 2, s + 1} (Ω^{T})$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_{p}^{2 k + 2, k + 1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

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 $W^{1, 1} (ℝ ⁿ)$ . We also show a new proof of the characterization of the Sobolev space $W^{1, p} (ℝ ⁿ)$ , 1 ≤ p < ∞, in terms of Poincaré inequalities.

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 $T : = B_{| | \cdot | |} (0, r)$ , 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, $s u p_{s, t \in T} | f (s) - f (t) | \leq 6 A B (\int_{0}^{r} ψ (1 / A ε^{n - 1}) ε^{n - 1} d ε + 1 / (n | B_{| | \cdot | |} (0, 1) |) \int_{T} φ (1 / B | | \nabla f (u) | | ⁎) d u)$ , 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...

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 $T_{f} : u \mapsto u \circ f$ 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.

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 $f \in L_{l o c}^{p} (ℝ ⁿ)$ the notion of p-mean variation of order 1, $₁^{p} (f, ℝ ⁿ)$ 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 $W^{1, p} (ℝ ⁿ)$ in terms of $₁^{p} (f, ℝ ⁿ)$ is directly related to the characterisation of $W^{1, p} (ℝ ⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

The Bohr-Pál theorem and the Sobolev space $W ₂^{1 / 2}$

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 $W ₂^{1 / 2} ()$ . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if...

Interpolation theorem for the p-harmonic transform

Luigi D&#039;Onofrio, Tadeusz Iwaniec (2003)

Studia Mathematica

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We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces $ℒ^{s} (ℝ ⁿ)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation ${d i v | \nabla u |}^{p - 2 \nabla} u = d i v$ . In this example the p-harmonic transform is essentially inverse to $d i v (| \nabla |^{p - 2} \nabla)$ . To every vector field $\in ℒ^{q} (ℝ ⁿ, ℝ ⁿ)$ our operator $ℋ_{p}$ assigns the gradient of the solution, $ℋ_{p} = \nabla u \in ℒ^{p} (ℝ ⁿ, ℝ ⁿ)$ . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...

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, $i d_{Ω} :_{p ₁, u ₁, q ₁}^{s ₁} (Ω) ↪_{p ₂, u ₂, q ₂}^{s ₂} (Ω)$ , where $Ω \subset ℝ^{d}$ 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 $ℝ^{d}$ . Moreover, we also characterise embeddings into the scale of $L_{p}$ spaces or into the space of bounded continuous functions.

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 ${∥D^{k - 1} u∥}_{L^{n / (n - 1)}} \leq {∥A (D) u∥}_{L^{1}}$ 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 $ℝ^{n}$ from a vector space $V$ to a vector space $E$ . The operator $A (D)$ is defined to be canceling if $⋂_{ξ \in ℝ^{n} ∖ {0}} A (ξ) [V] = {0}$ . 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...

Some remarks on the interpolation spaces $A^{θ}, A_{θ}$

Mohammad Daher (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $(A_{0}, A_{1})$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{β}$ , we show that $A^{θ} = A_{θ}$ for every $θ \in (0, 1)$ . We also deal with assumptions on ${\bar{A}}^{β}$ , the closure of $A^{β}$ in the dual of ${(A_{0}^{*}, A_{1}^{*})}_{β}$ .

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 $k \in ℕ_{*}$ and an exponent $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ 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 $N^{n}$ .

$H^{\infty}$ functional calculus in real interpolation spaces

Giovanni Dore (1999)

Studia Mathematica

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Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $∥ λ {(λ I - A)}^{- 1} ∥$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

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Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1, p}$ -error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p = \infty$ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how...

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 $f \in W^{k, p} (Ω)$ 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.

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 ${\dot{N L}}^{1, Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group $ℍ^{n}$ .

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 $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p} (ℝ ⁿ)$ into the generalized Morrey space $ℳ_{Φ, r} (ℝ ⁿ)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p + 1} (ℝ ⁿ)$ . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

Interpolation and extension of Lipschitz-Hölder maps on $C_{p}$ spaces

Charles E. C Cleaver (1973)

Colloquium Mathematicae

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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 $ℝ^{d}$ , 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.

Modulus of dentability in $L ¹ + L^{\infty}$

Adam Bohonos, Ryszard Płuciennik (2008)

Banach Center Publications

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We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L ¹ + L^{\infty} .$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L ¹ + L^{\infty}$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L ¹ + L^{\infty}$ is empty.