Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

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 $U \subset \Omega$) 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 $\Omega \subset \mathbb{R}^{N}$. 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

Averaging on n-dimensional rectangles

In this work we investigate families of translation invariant differentiation bases B of rectangles in Rn, 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 Stokolos in [11] and [13] (see the attached file)

AVERAGING ON n-DIMENSIONAL RECTANGLES

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 in [7] and [9]

REMOVABLE SINGULARITIES FOR div v = f IN WEIGHTED LEBESGUE SPACES

International audienceLet $w\in L^1_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e 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 $L^p_{1/w}$, $1<p<+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author

Differentiating Orlicz spaces with rare bases of rectangles

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

Removable singularities for the equation div v = 0

A compact subset S of R^N is removable for the equation div v = 0 if every bounded Borel vector field whose distributional divergence vanishes outside S, has a zero distributional divergence in the whole R^N. Here we establish that a compact subset of R^N is removable for the equation div v = 0 if and only if its (N−1)-dimensional Hausdorff measure vanishes