arXiv subjects
Publications and source records attributed to Laurent Moonens.
In this note we give an upper bound on the Hausdorff dimension of removable setsfor elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of Frostman's lemma in Euclidean space with an additional power decay.
In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.
We introduce the notion of \textit{Perron capacity} of a set of slopes $Ω\subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $Ω$ is finite then the directional maximal operator associated $M_Ω$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$Ω_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.
Consider $A(x,D):C^{\infty}(Ω,E) \rightarrow C^\infty(Ω,F)$ an elliptic and canceling linear differential operator of order $ν$ with smooth complex coefficients in $Ω\subset \mathbb{R}^{N}$ from a finite dimension complex vector space $E$ to a finite dimension complex vector space $F$ and $A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{*}(x,D)v=f$ (in the distribution sense) for a given distribution $f$; more precisely we show that any $x_0\inΩ$ is contained in a neighborhood $U\subset Ω$ in which its continuous solvability is characterized by the following condition on $f$: for every $ε>0$ and any compact set $K \subset \subset U$, there exists $θ=θ(K,ε)>0$ such that the following holds for all smooth function $φ$ supported in $K$: \begin{equation}\nonumber \left| f(φ) \right| \leq θ\|φ\|_{W^{ν-1,1}} + ε\|A(x,D) φ\|_{L^{1}}, \end{equation} where $W^{ν-1,1}$ stands for the homogenous Sobolev space of all $L^1$ functions whose derivatives of order $ν-1$ belongs to $L^{1}(U)$. This characterization implies and extends results obtained before for operators associated to elliptic complex of vector fields (see \cite{MP}); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator in [4] and [9].
In the following paper, one studies, given a bounded, connected open set $Ω$ $\subseteq$ R n , $κ$ > 0, a positive Radon measure $μ$ 0 in $Ω$ and a (signed) Radon measure $μ$ on $Ω$ satisfying $μ$($Ω$) = 0 and |$μ$| $κ$$μ$ 0 , the possibility of solving the equation div u = $μ$ by a vector field u satisfying |u| $κ$w on $Ω$ (where w is an integrable weight only related to the geometry of $Ω$ and to $μ$ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with measure data, and also construct a weight w that relies in a softer way on the geometry of $Ω$, improving its behavior (and hence the a priori behavior of the solution we construct) substantially in some instances. The method used in this paper follows a constructive approach of Bogovskii type.
In the present note, we examine the behavior of some homo\-thecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its \emph{shape}) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a \emph{shape-function}). Depending on the allowed angles and the corresponding shape-function, a basis may differentiate various Orlicz spaces. We here give some examples of shape-functions so that the corresponding basis differentiates $L\log L(\R^2)$, and show that in some `model' situations, a fast-growing shape function (whose speed of growth depends on $α>0$) does not allow the differentiation of $L\log^αL(\R^2)$.
In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.
In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset Ω$) to the divergence-type equation $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F,$$ where $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $Ω\subset \mathbb{R}^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\mathbb{R}^N$, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer.
We show that, given some lacunary sequence of angles $\mathbfθ=(θ_j)_{j\in\N}$ not converging too fast to zero, it is possible to build a rare differentiation basis $\mathcal{B}$ of rectangles parallel to the axes that differentiates $L^1(\mathbb{R}^2)$ while the basis $\mathcal{B}_{\mathbfθ}$ obtained from $\mathcal{B}$ by allowing its elements to rotate around their lower left vertex by the angles $θ_j$, $j\in\mathbb{N}$, fails to differentiate all Orlicz spaces lying between $L^1(\mathbb{R}^2)$ and $L\log L(\mathbb{R}^2)$.
In this work we investigate families of translation invariant differentiation bases $B$ of rectangles in $R^n$, for which $L\log^{n-1}L(R^n)$ is the largest Orlicz space that $B$ differentiates. In particular, we improve on techniques developed by A.~Stokolos 1988 and 2008.
Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincaré inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}\_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\_{1/w}$, $1\textless{}p\textless{}+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.