On Improved Sobolev Embedding Theorems

{ We present a direct proof of some recent improved Sobolev inequalities put forward by A. in their wavelet analysis of the space BV (R 2). The argument, relying on pseudo-Poincar e inequalities, allows us to consider several extensions to manifolds and graphs. The classical Sobolev inequality indicates that for every function f on R n vanishing at innnity in some mild sense, kfk q C krfk 1 (1) where q = n n?1 and C > 0 only depends on n. The Sobolev inequality (1) is invariant under the ax + b (a > 0, b 2 R n) group action, but is not under the Weyl-Heisenberg group action. Namely, if f(x) = f ! (x) = e i!x '(x) where ' is in the Schwartz class, then krfk 1 = j!jk'k 1 + O(1) when j!j ! 1. In particular, (1) is not adapted to such modulated functions. In their study of the space BV (R 2), A. Cohen et al. C-DV-P-X] (see also CM -O]) improved the Sobolev inequality (1) into kfk q C krfk 1=q 1 kfk 1?(1=q) B (2) where B = B ?(n?1) 1;1 is the homogeneous Besov space of indices (?(n ? 1); 1; 1). This improved Sobolev inequality is easily seen to be sharper than (1). Furthermore, if f = f ! as above, then kfk B = j!j ?(n?1) k'k 1 + O(j!j ?n) so that (2) amounts in this case to the trivial bound kfk q k'k 1=q 1 k'k 1?(1=q) 1. The proof of (2) in C-DV-P-X] and CM -O] is based on wavelet decompositions together with weak-` 1 type estimates and interpolation results. The purpose of this note is to propose a direct semigroup argument without any use of wavelet decomposition. In particular, the approach we suggest emphasizes the use of pseudo-Poincar e inequalities (cf. SC]) for families of operators (heat kernels for example) and thus easily extends to more general frameworks including manifolds or graphs.