Sharp $L^1$ estimates for singular transport equations

Sergiu Klainerman; Igor Rodnianski

Displaying similar documents to “Sharp $L^{1}$ estimates for singular transport equations”

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

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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Variable exponent trace spaces

Lars Diening, Peter Hästö (2007)

Studia Mathematica

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The trace space of $W^{1, p (\cdot)} (ℝ ⁿ \times [0, \infty))$ consists of those functions on ℝⁿ that can be extended to functions of $W^{1, p (\cdot)} (ℝ ⁿ \times [0, \infty))$ (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.

Estimates of $\sum_{k = 1}^{N} k^{- s} 〈 k x 〉^{- t}$

A. Kruse (1967)

Acta Arithmetica

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A generalization to the Hardy-Sobolev spaces $H^{k, p}$ of an $L^{p}$ - $L^{1}$ logarithmic type estimate

Imed Feki, Ameni Massoudi, Houda Nfata (2018)

Czechoslovak Mathematical Journal

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The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k, p} (G)$ ; $k \in ℕ^{*}$ , $1 \leq p \leq \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$ .

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.

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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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.

A remark on the transport equation with b ∈ BV and $d i v_{x} b \in B M O$

Paweł Subko (2014)

Colloquium Mathematicae

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We investigate the transport equation $\partial_{t} u (t, x) + b (t, x) \cdot D_{x} u (t, x) = 0$ . Our result improves the classical criteria of uniqueness of weak solutions in the case of irregular coefficients: b ∈ BV, $d i v_{x} b \in B M O$ . To obtain our result we use a procedure similar to DiPerna and Lions’s one developed for Sobolev vector fields. We apply renormalization theory for BV vector fields and logarithmic type inequalities to obtain energy estimates.

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

A singular initial value problem for the equation $u^{(n)} (x) = g (u (x))$

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem $u^{(n)} (x) = g (u (x))$ , $u (0) = u^{'} (0) = . . . = u^{(n - 1)} (0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int ₀^{δ} 1 / s {[s / g (s)]}^{1 / n} d s < \infty$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

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 $N^{n}$ .

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...

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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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...

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.

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...

Universal embeddings of $l^{1} (α)$ into the space of continuous functions on a product space

N. Kalamidas, Th. Zachariades (1989)

Colloquium Mathematicae

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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 $Ω \subset ℝ^{n}$ be a bounded starshaped domain and consider the $(p, q)$ -Laplacian problem $\begin{matrix} - Δ_{p} u - Δ_{q} u = λ (𝐱) {| u |}^{p^{☆} - 2} u + μ {| u |}^{r - 2} u \end{matrix}$ where $μ$ is a positive parameter, $1 < q \leq p < n$ , $r \geq p^{☆}$ and $p^{☆} : = \frac{n p}{n - p}$ 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.

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 $M_{ℍ, ν} f$ and Riesz potentials $I_{ℍ, α} f$ in weighted Morrey spaces of variable exponent on the half space $ℍ$ . We also obtain Sobolev type inequalities for a $C^{1}$ function on $ℍ$ . 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 $p (\cdot)$ and $q (\cdot)$ satisfy log-Hölder conditions, $p (x) < q (x)$ for $x \in ℍ$ , and $b (\cdot)$ is nonnegative and Hölder continuous of order $θ \in (0, 1]$ .

Corrigendum to “Commutators on ${(\sum ℓ_{q})}_{p}$ ” (Studia Math. 206 (2011), 175-190)

Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)

Studia Mathematica

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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ${(\sum ℓ_{q})}_{p}$ ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

Embedding in $T_{3}$ - $F_{σ}$ spaces

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

Matematički Vesnik

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Displaying similar documents to “Sharp L 1 estimates for singular transport equations”

Displaying similar documents to “Sharp $L^{1}$ estimates for singular transport equations”