Generalizations of logarithmic Sobolev inequalities
Generalizations of logarithmic Sobolev inequalities
We generalize logarithmic Sobolev inequalities to logarithmic Gagliardo-Nirenberg inequalities, and apply these inequalities to prove ultracontractivity of the semigroup generated by the doubly nonlinear $p$-Laplacian <p align="center"> $\dot{u}=\Delta_p u^m.$ <p align="left" class="times"> Our proof does not use Moser iteration, but shows that the time-dependent Lebesgue norm $\||u(t)|\|_{r(t)}$ stays bounded for …