On improved Sobolev embedding theorems

We present a direct proof of some recent improved Sobolev inequalities put forward by A. Cohen, R. DeVore, P. Petrushev and H. Xu [C-DV-P-X] in their wavelet analysis of the space $BV(\rr^2)$. These inequalities are parts of the Hardy-Littlewood-Sobolev theory, connecting Sobolev embeddings and heat kernel bounds. The argument, relying on pseudo-Poincaré inequalities, allows us to study the dependence of the constants with respect to dimension and to consider several extensions to manifolds and graphs.

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Published 1 January 2003