Abstract
On a doubling metric measure space endowed with a “carré du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincaré inequalities (Pp). We show that the combination of (Gp) and (Pp) for p≥2 always implies two-sided Gaussian heat kernel bounds. The case p=2 is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in [37]. This relies in particular on a new notion of Lp Hölder regularity for a semigroup and on a characterisation of (P2) in terms of harmonic functions.
Highlights
On a doubling metric measure space endowed with a “carre du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincare inequalities (Pp)
Let M be a locally compact separable metrisable space equipped with a Borel measure μ, finite on compact sets and strictly positive on any non-empty open set
Crucial to our approach is Theorem 3.4 below where we prove the equivalence, under (VD) and (UE), between the Gaussian lower bound (LE) and the existence of some p ∈ [1, +∞) and η > 0 such that (Hpη,p) holds, a property which turns out to be independent of p ∈ [1, +∞)
Summary
We shall need a version of the Davies-Gaffney estimate (1.5) which includes the gradient, namely (2.1). The proof relies on the following inequality: for φ a non-negative cut-off function with support S,. − t∇e−tLf dμ , Bfor all t > 0 and f ∈ Lp(M, μ) By the doubling pr√operty, we may sum this inequality over a covering of the whole space at the scale t and deduce (V Ep,q), which is sup t>0. By interpolation [10, Proposition 2.1.5] between (V Ep,q) and (EVq ,p ), one obtains One may avoid the use of the highly non-trivial result from [43] by assuming directly (Gp) and (Pq) for some q ∈ (2, p)
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