Search

A weak Sobolev inequality (WSI) is a weakened form of a Sobolev inequality associated to a Dirichlet form. It turns out that it is in fact equivalent to a Sobolev inequality. If there is a spectral gap and a (WSI) holds, then we can get a tight weake Sobolev quality (TWSI). Starting with a TWSI. we get upper and lower bounds on the density of the heat semigroup associated with the Dirichlet form, when t → 0 as well as when t → ∞. In the last chapter, we give a Γ2 criterium to get a TWSI on a manifold.

This is an expository paper about some recent works on the Riesz transforms associated with a general symmetric, elliptic second order differential operator. Except in the third part, there are no new results and the proofs are often ommited or just sketched.