nash inequality 
Extremal functions for the sharp L2 Nash inequality


We give geometrical conditions under which there exist extremal functions for the sharp L2Nash inequality.


The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality.


This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality.


In this paper, we prove that, on Riemannian compact manifolds with boundary, there exists a second constant for trace Nash inequality with its first best constant.


By a method of cylindrical symmetrization for the functions belonging to , we give an estimate of the best constant in the trace Nash inequality on .


In this section, we consider homogeneous cases to illustrate that the local Nash Inequality is a natural extension of the classical Nash inequality.


Namely the local Nash inequality together with the volume doubling property and the exit time estimate implies an inhomogeneous heat kernel estimate.


This is based on a new inequality, analogous to the eNash inequality for Gaussian random variables.


The local Nash inequality is introduced as a natural extension of the classical Nash inequality yielding spacehomogeneous upper heat kernel estimate.


The local Nash inequality together with the exit time estimate suffices for an offdiagonal upper heat kernel estimate as follows.


The next lemma is obtained essentially by the classical argument from the Nash inequality to the heat kernel estimate.


Unlike the classical case, the local Nash inequality alone does not seem to give the upper ondiagonal estimate.

