  We present new proofs of two theorems of E.B. Davies and B. Simon about
ultracontractivity property for of semigroups of operators and logarithmic
Sobolev inequalities with parameter (LSIWP for short) satisfied by the
generator of the semigroup. In our proof, we use neither the $L^p$ version of
the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We
also present new results. In particular, we show that Nash type inequality and
LSIWP are equivalent.
