Pointwise characterizations of Hardy-Sobolev functions

By Pekka Koskela, Eero Saksman. 2006

99 citations recorded by the source corpus.

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 $p\leq 1.$

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/718f9e6ed2c43bb1a23797b4320d4b93942be041

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, background): “When s = 1, comparing this with the proof of Theorem 1 of [30], we see that the gradient on f appearing there is transferred to the vanishing moments of the test functions and the size conditions of the test functions and their first-order derivatives (see A) here.”
Multilinear analysis on metric spaces (2014, background): “Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved.”
Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces and Quasiconformal Mappings (2010, unknown): “For predecessors of such results, see [21, 31].”
Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces (2011, methodology): “The pointwise characterization for the Hardy-Sobolev spaces H1, p(Rn) for p ∈ (n/(n + 1), 1] was given by Koskela and Saksman [16].”
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients (2013, unknown): “Thus, by the criteri on of [39], we obtain immediately thatf ∈ Ḣ1,p(Rn), with ‖ f ‖Ḣ1,p(Rn) .”
New properties of Besov and Triebel-Lizorkin spaces on RD-spaces (2009, background): “Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.”
Sobolev Mappings between Manifolds and Metric Spaces (2009, background): “For the case p = 1, M 1,1(Rn) is not equivalent with W 1,1(Rn), [28], see, however, [ 57 ] and Theorem 3.16 below.”