A remark on the transport equation with b ∈ BV and $div_{x} b ∈ BMO$

Paweł Subko

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Displaying similar documents to “A remark on the transport equation with b ∈ BV and $d i v_{x} b \in B M O$ ”

A new function space and applications

Jean Bourgain, Haïm Brezis, Petru Mironescu (2015)

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We define a new function space $B$ , which contains in particular BMO, BV, and $W^{1 / p, p}$ , $1 < p < \infty$ . 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 $W^{1 / p, p}$ , $1 \leq p < \infty$ . The above mentioned estimates imply in particular that integer-valued functions belonging to $B_{0}$ are necessarily constant. This framework provides a “common roof”...

Grant acknowledgment to “A remark on the transport equation with b ∈ BV and $d i v_{x} b \in B M O$ ” (Colloq. Math. 135 (2014), 113-125)

Paweł Subko (2014)

Colloquium Mathematicae

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B. Bojarski (2011)

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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.

Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)

Bulletin de la Société Mathématique de France

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We establish in this paper a lower bound for the volume of a unit vector field $\vec{v}$ defined on $𝐒^{n} ∖ {\pm x}$ , $n = 2, 3$ . This lower bound is related to the sum of the absolute values of the indices of $\vec{v}$ at $x$ and $- x$ .

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Principalization algorithm via class group structure

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

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In [9], the author considers a sequence of invertible maps $T_{i} : S ¹ \to S ¹$ which exchange the positions of adjacent intervals on the unit circle, and defines as Aₙ the image of the set 0 ≤ x ≤ 1/2 under the action of Tₙ ∘ ... ∘ T₁, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) $\sum_{i = 1}^{n} (T o t . V a r . (T_{i} - I) + T o t . V a r . (T_{i}^{- 1} - I)) \geq C l o g 1 / h$ . We prove that (1) holds for general quasi incompressible invertible BV maps on ℝ, and that this estimate implies that the map Tₙ ∘ ... ∘ T₁ belongs to the Besov space $B^{0, 1, 1}$ , and its...

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A. Kruse (1967)

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Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

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Ralph McKenzie (1971)

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Janusz Matkowski (1989)

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Stephan Baier (2004)

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M. Đurić (1973)

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А.М. Вершик (1972)

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D. W. Hajek, I. Irizarry (1981)

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M. K. Sen (1971)

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On the regularity of the one-sided Hardy-Littlewood maximal functions

Feng Liu, Suzhen Mao (2017)

Czechoslovak Mathematical Journal

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In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators $ℳ^{+}$ and $ℳ^{-}$ . More precisely, we prove that $ℳ^{+}$ and $ℳ^{-}$ map $W^{1, p} (ℝ) \to W^{1, p} (ℝ)$ with $1 < p < \infty$ , boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $BV (ℤ) \to BV (ℤ)$ boundedly and map $l^{1} (ℤ) \to BV (ℤ)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$ , that is, $Var (M^{+} (f)) \leq Var (f) and Var (M^{-} (f)) \leq Var (f)$ if $f \in BV (ℤ)$ , where $Var (f)$ is the total variation of $f$ on $ℤ$ and $BV (ℤ)$ is the set of all functions $f : ℤ \to ℝ$ satisfying $Var (f) < \infty$ .

Displaying similar documents to “A remark on the transport equation with b ∈ BV and d i v x b ∈ B M O ”

Displaying similar documents to “A remark on the transport equation with b ∈ BV and $d i v_{x} b \in B M O$ ”