  On doubling metric measure spaces endowed with a strongly local regular
Dirichlet form, we show some characterisations of pointwise upper bounds of the
heat kernel in terms of global scale-invariant inequalities that correspond
respectively to the Nash inequality and to a Gagliardo-Nirenberg type
inequality when the volume growth is polynomial. This yields a new proof and a
generalisation of the well-known equivalence between classical heat kernel
upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash
inequalities. We are able to treat more general pointwise estimates, where the
heat kernel rate of decay is not necessarily governed by the volume growth. A
crucial role is played by the finite propagation speed property for the
associated wave equation, and our main result holds for an abstract semigroup
of operators satisfying the Davies-Gaffney estimates.

Analysis of PDEs