Pointwise characterizations of Hardy-Sobolev functions

We establish pointwise characterizations of functions in the Hardy-Sobolev spaces $ H^{1,p}$ within the range $p\in (n/(n+1),1]$. In particular, a locally integrable function $u$ belongs to $ H^{1,p}(\real^n)$ if and only if $u\in L^p(\real^n)$ and it satisfies the Haj\l{}asz type condition $$ |u(x)-u(y)|\leq |x-y|(h(x)+h(y)),\quad x,y\in \real^n\setminus E, $$ where $E$ is a set of measure zero and $h\in L^p(\real^n)$. We also investigate Hardy-Sobolev spaces on subdomains and extend Hardy inequalities to the case $p\leq 1.$

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Published 1 January 2008