On the existence of vector fields with nonnegative divergence in rearrangement-invariant spaces
Abstract
We investigate the existence of solutions of
\begin{equation}
divF=\mu,\text{ on } \mathbf{R}^{d}. \tag{*}
\end{equation}
Here, $\mu\geq0$ is a Radon measure, and we look for
a solution $F\in X(\mathbf{R}^{d}\rightarrow\mathbf{R}^{d})$, where $X$ is a
rearrangement-invariant space.
We first prove the equivalence of the following
assertions:
(i) (*) has a solution for some nontrivial $\mu$;
(ii) the function $x\mapsto |x|^{1-d}1_{B^{c}}(x)$ belongs to $X$.
Here, $B$ is the unit ball in $\mathbf{R}^{d}$.
We next investigate the solvability of (*) when $\mu$
is fixed. A sufficient condition is that $I_{1}\mu\in X$, where $
I_{1}\mu$ is the 1-Riesz potential of $\mu$. This condition turns out to be
also necessary when the Boyd indexes of $X$ belong to $(0,1)$.
Our analysis generalizes the one of Phuc and Torres
(2008) when $X=L^{p}$.
Origin : Files produced by the author(s)
Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
Comment : Ce pdf est la version preprint de l'article (version soumise à l'éditeur, avant peer-reviewing)
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