Non ultracontractive heat kernel bounds by Lyapunov conditions

Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincaré) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples of reversible diffusion Markov semigroups in R, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, refining E. B. Davies’ original argument. The (Gagliardo-Nirenberg-) Nash inequality in Rd states that ‖f‖ 1+2/d L2(dx) ≤ C(d)‖∇f‖L2(dx) ‖f‖ 2/d L1(dx) (1) for functions f on Rd and is a powerful tool when studying smoothing properties of parabolic partial differential equations on Rd. In a general way, let (Pt)t≥0 be a symmetric Markov semigroup on a space E, with Dirichlet form E and (finite or not) invariant measure μ. Then the Nash inequality ‖f‖ 2+4/d L2(dμ) ≤ [ C1E(f, f) + C2‖f‖ 2 L2(dμ) ] ‖f‖ 4/d L1(dμ) (2) for a positive parameter d, or more generally Φ ( ‖f‖L2(dμ) ‖f‖2 L1(dμ) ) ≤ E(f, f) ‖f‖2 L1(dμ) (3) for an increasing convex function Φ, is equivalent, up to constants and under adequate hypotheses on Φ, to the ultracontractivity bound ‖Ptf‖L∞(dμ) ≤ n (t) ‖f‖L1(dμ), t > 0 where n(t) = ∫ +∞ t 1 Φ(x) dx; for instance n−1(t) ≤ Ct−d/2 for 0 < t ≤ 1 in the case of (2). We refer in particular to [6] and [13] in this case, and in the general case of (3) to the seminal work [9] by T. Coulhon where the equivalence was first obtained; see also [3, Chap. 6]. Let us observe that Nash inequalities are adapted to smoothing properties of the semigroup for small times, but can Date: July 25, 2014. 1 2 FRANÇOIS BOLLEY, ARNAUD GUILLIN, AND XINYU WANG also be useful for large times. This is in turn equivalent to uniform bounds on the kernel density of Pt with respect to μ, in the sense that for μ-almost every x in E one can write Ptf(x) = ∫ E f(y) pt(x, y) dμ(y) with pt(x, y) ≤ n (t) (4) for μ⊗μ-almost every (x, y) in E×E.Observe finally that (3) is equivalent to its linearised form ‖f‖L2(dμ) ≤ u E(f, f) + b(u) ‖f‖ 2 L1(dμ), u < u0 for a decreasing positive function b(u) related to Φ: this form was introduced by F.-Y. Wang [17] under the name of super-Poincaré inequality to characterize the generators L with empty essential spectrum. For certain b(u) it is equivalent to a logarithmic Sobolev inequality for μ, hence, to hypercontractivity only (and not ultracontractivity) of the semigroup. Moreover, relevant Gaussian off-diagonal bounds on the density pt(x, y) for x 6= y, such as pt(x, y) ≤ C(ε) t −d/2 e 2/(4t(1+ε)), t > 0 for all ε > 0, have first been obtained by E. B. Davies [12] for the heat semigroup on a Riemannian manifold E, and by using a family of logarithmic Sobolev inequalities equivalent to (2). Such bounds have been turned optimal in subsequent works, possibly allowing for ε = 0 and the optimal numerical constant C when starting from the optimal so-called entropy-energy inequality, and extended to more general situations: see for instance [3, Sect. 7.2], [6] and [13] for a presentation of the strategy based on entropy-energy inequalities, and [5], [10, Sect. 2] and [11] and the references therein for a presentation of three other ways of deriving offdiagonal bounds from on-diagonal ones (namely based on an integrated maximum principle, finite propagation speed for the wave equation and a complex analysis argument). In the more general setting where the semigroup is not ultracontractive, then the uniform bound (4) cannot hold, but only (for instance on the diagonal) pt(x, x) ≤ n (t)V (x) (5) for a nonnegative function V . Such a bound is interesting since it provides information on the semigroup : for instance if V is in L2(μ), then it ensures that Pt is Hilbert-Schmidt, and in particular has a discrete spectrum. It has been shown to be equivalent to a weighted super-Poincaré inequality ‖f‖L2(dμ) ≤ u E(f, f) + b(u) ‖fV ‖ 2 L1(dμ), u < u0 (6) as in [18], where sharp estimates on high-order eigenvalues are derived, and, as in [2], to a weighted Nash inequality Φ ( ‖f‖L2(dμ) ‖fV ‖2 L1(dμ) ) ≤ E(f, f) ‖fV ‖2 L1(dμ) · (7) The purpose of this note is twofold. First, to give simple and easy to check sufficient criteria on the generator of the semigroup for the weighted inequalities (6)-(7) to hold : for this, we use Lyapunov conditions, which have revealed an efficient tool to diverse functional inequalities (see [1] or [8] for instance) : we shall see how they allow to recover and extend examples considered in [2] and [18], in a straightforward way (see Example 8). Then, to derive offdiagonal bounds on the kernel density of the semigroup, which will necessarily be non uniform in our non ultracontractive setting. For this we refine Davies’ original ideas of [12]: indeed, NON ULTRACONTRACTIVE HEAT KERNEL BOUNDS BY LYAPUNOV CONDITIONS 3 we combine his method with the (weighted) super-Poincaré inequalities derived in a first step, instead of the families of logarithmic Sobolev inequalities or entropy-energy inequalities used in the ultracontractive cases of [3], [6] and [12]-[13]; we shall see that the method recovers the optimal time dependence when written for (simpler) ultracontractive cases, and give new results in the non ultracontractive case (extending the scope of [2] and [18]). Instead, we could have first derived on-diagonal bounds, such as (5), and then use the general results mentionned above (in particular in [11]) and giving off-diagonal bounds from on-diagonal bounds; but we will see here that, once the inequality (6)-(7) has been derived, the off-diagonal bounds come without further assumptions nor much more effort than the on-diagonal ones. To make this note as short and focused on the method as possible, we shall only present in detail the situation where U is a C2 function on Rd with Hessian bounded by below, possibly by a negative constant, and such that ∫ e−U dx = 1. The differential operator L defined by Lf = ∆f − ∇U · ∇f for C2 functions f on Rd generates a Markov semigroup (Pt)t≥0, defined for all t ≥ 0 by our assumption on the Hessian of U . It is symmetric in L2(μ) for the invariant measure dμ(x) = e−U(x) dx. We refer to [3, Chap. 3] for a detailed exposition of the background on Markov semigroups. Let us point out that the constants obtained in the statements do not depend on the lower bound of the Hessian of U , and that the method can be pursued in a more general setting, see Remarks 3, 7 and 11. We shall only seek upper bounds on the kernel, leaving lower bounds or bounds on the gradient aside (as done in the ultracontractive setting in [10, Sect. 2], [13] or [16] for instance). Let us finally observe that S. Boutayeb, T. Coulhon and A. Sikora [5, Th. 1.2.1] have most recently devised a general abstract framework, including a functional inequality equivalent to the more general bound pt(x, x) ≤ m(t, x) than (5), and to the corresponding off-diagonal bound. The derivation of simple practical criteria on the generator ensuring the validity of such (possibly optimal) bounds is an interesting issue, that should be considered elsewhere. Definition 1. Let ξ be a C1 positive increasing function on (0,+∞) and φ be a continuous positive function on Rd, with φ(x) → +∞ as |x| goes to +∞. A C2 map W ≥ 1 on Rd is a ξ-Lyapunov function with rate φ if there exist b, r0 ≥ 0 such that for all x ∈ R d LW ξ(W ) (x) ≤ −φ(x) + b 1|x|≤r0 . We first state our general result: Proposition 2. In the notation of Definition 1, assume that there exists a ξ-Lyapunov function W with rate φ. Then there exist C and s0 > 0 such that for any positive continuous function V on Rd