Mathematical Research Letters
Volume 10 (2003)
On improved Sobolev embedding theorems
Pages: 659 – 669
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.