Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

We prove heat kernel bounds for the operator (1 + |x|^{\alpha})\Delta in R^N, through Nash inequalities and weighted Hardy inequalities.


Introduction and preliminary results
In this paper we prove heat kernel estimates for the operator L = m(x)(1 + |x| α )∆ in the whole space R N , under the assumption that m is a bounded, locally Hölder continuous function with inf m > 0. The solvability of the elliptic and parabolic problem associated with L, either in spaces of continuous functions or in L p spaces, has been widely investigated in literature. If α ≤ 2, L generates an analytic semigroup both in L p (R N ) and in C 0 (R N ), see [6]. If α > 2, the generation results depend upon the space dimension N . If N = 1, 2, L generates a semigroup in C b (R N ), the space of all continuous and bounded functions on R N , but C 0 (R N ) and L p (R N ) are not preserved. If N ≥ 3, the resolvent and the semigroup map C b (R N ) into C 0 (R N ) (see [10]) but L p (R N ) is preserved if and only if p > N/(N − 2). For N/(N − 2) < p < ∞ and under the additional assumption that m admits a finite limit at infinity, the semigroup is also analytic in L p (R N ) and, for m ≡ 1, it is contractive if and only if α ≤ (N − 2)(p − 1). We refer the reader to [8] for all these results, as well as for domain characterization and spectral properties of L. Here we only recall that the domain of L in L p (R N ) coincides with the maximal one D p,max (L) = {u ∈ W 2,p (R N ) : (1 + |x| α )∆u ∈ L p (R N )} and that the resolvents in L p (R N ) and L q (R N ) are consistent, provided that p, q > N/(N − 2). Finally, the resolvent is compact if and only if α > 2 and in this case the spectrum consists of a sequence of negative eigenvalues λ n diverging to −∞.
Due to the local regularity of the coefficients, the semigroup (T (t)) t≥0 generated by L admits an integral kernel p(x, y, t), with respect to the Lebesgue measure (see e.g. [10]), for which the following representation holds However, the operator L is symmetric with respect to the measure dµ(x) = (1 + |x| α ) −1 dx and it is more convenient to express T (t) through a kernel with respect to dµ, namely where p µ (x, y, t) = (1 + |y| α )p(x, y, t).
Our goal consists in obtaining upper bounds for the integral kernel p µ by working in L 2 µ spaces and then deducing upper bounds for p. This will be done by using the well-known equivalence between Nash inequalities and ultracontractivity for symmetric Markov semigroups, see [13, Section 6.1].
Throughout the paper the dimension N will be always assumed to be greater than or equal to 3 and α will be a positive real number.
We shall prove that for small t for α ≥ 4. Here φ is the first eigenfuncion of L and satifies the bounds We also show that the powers of t appearing in the bounds above are optimal. Estimates for large t easily follow from the semigroup law, since the semigroup dacays esponentially at infinity. Observe that for 2 < α < 4 both the first and the second estimate hold (see also Remark 2.16).

Definition of L via the quadratic form methods
Consider the Hilbert spaces L 2 µ , where dµ(x) = (m(x)(1 + |x| α )) −1 dx, endowed with its canonical inner product. Note that the measure µ is finite if and only if α > N . Consider also the Sobolev space H = {u ∈ L 2 µ : ∇u ∈ L 2 } endowed with the inner product and let V be the closure of C 1 c in H, with respect to the norm of H. Observe that Sobolev inequality holds in V but not in H (consider for example the case where α > N and u = 1). Here 2 * = 2N/(N − 2) and C 2 is the best constant for which the equality above holds. Next we introduce the continuous and weakly coercive symmetric form for u, v ∈ V and the self-adjoint operator L defined by Since a(u, u) ≥ 0, the operator L generates an analytic semigroup of contractions e tL in L 2 µ . An application of the Beurling-Deny criteria shows that the generated semigroup is positive and L ∞contractive. For our purposes we need that the resolvents and the semigroups generated by L and of (L, D p,max (L)) are coherent. This is stated in the following proposition. We refer to [8,Proposition 7.4] for its proof. and According with the above Proposition we write T (t) for the semigroup in L p with respect to the Lebesgue measure or in L 2 µ . It admits a positive integral kernel p(x, y, t) with respect to the Lebesgue measure, see [10,Theorem 4.4], for which the following representation holds Clearly we have also p µ (x, y, t) = m(y)(1 + |y| α )p(x, y, t).

Eigenfunctions and eigenvalues of L (α > 2)
Spectral properties of L have also been investigated in [8,Section 7] where the following result has been proved. To unify the notation, when p = ∞, L p stands for C 0 . We recall that the resolvent of (L, D p,max (L)) is compact in L p if and only if α > 2, see [8], a condition that we assume throughout this section.
Proof. Since the kernel p is positive, T (t) is irreducible and from [3,Proposition 1.4.3]) it follows that the eigenspace relative to the first eigenalue is one-dimensional and admists a strictly positive eigenfuncion φ. Therefore Next we find upper bounds for the other eigenfunctions. Since the spectrum is independent of p, the eigenfuncions φ i belong to C 0 and therefore are bounded.
The proof is based upon the following lemma whose proof can be found in [8, Lemma 6.1].
Then J is bounded in R N and has the following behaviour as |x| goes to infinity Proof (Proposition 1.4). As in Proposition 1.3 we write It follows that dy.
By Lemma 1.5, If α > N , we immediately deduce the claim. Otherwise, we iterate the procedure as follows. Assuming The claim then follows since at the step k + 1.

Kernel estimates
In this section we will prove kernel estimates trough weighted Nash inequalities involving suitable Lyapunov functions. The main tool will be [2, Theorem 2.5] whose original formulation is due to Wang, see [14,Theorem 3.3].

Definition 2.1 A Lyapunov function is a positive function V such that
for every x ∈ R N , t > 0 and some positive constant c (called a Lyapunov constant).
By following [2, Definition 2.2] we introduce weighted Nash inequalities. such that ψ(x) x is increasing. A Dirichlet form a on L 2 µ satisfies a weighted Nash inequality with weight V and rate function ψ if Next we state [2, Theorem 2.5], adapted to our situation.
3 Let (T (t)) t≥0 be a symmetric Markov semigroup on L 2 µ with generator L. Assume that there exists a Lyapunov function V with Lyapunov constant c ≥ 0 and that the Dirichlet form associated to L satisfies a weighted Nash inequality with weight V and rate function ψ, integrable near infinity and not integrable near zero. Then Here the function K is defined by From [2, Proposition 2.1], the following corollary follows.

Corollary 2.4
Under the assumptions ons of Theorem 2.3, (T (t)) t≥0 has a kernel p µ which satisfies Theorem 2.5], V is required to be in the domain of the operator and to satisfy LV ≤ cV . By the proof, however, it is evident that such an assumption is only used to ensure the validity of the inequality for all positive functions f ∈ L 2 µ such that f V ∈ L 1 µ . Observe that, if V is a Lyapunov function according to our Definition (2.1), then, for every positive f , then by Fubini theorem and the symmetry of p µ and hence (4) holds.

Intrinsic ultracontractivity
We show kernel estimates through weighted Nash inequalities with respect to the the first eigenfunction of L. More precisely we will prove that Definition (2.2) holds with Lyapunov function V given by the first eigenfunction φ of L and rate functions Theorem 2.6 Assume that α > 2. Then the kernel p µ of the semigroup generated by L satisfies or, equivalently, for every 0 < t ≤ 1, x, y ∈ R N .
Large time estimates follow from the semigroup law and small times estimates.

Proposition 2.7 There exists a positive constant C such that
for every x, y ∈ R N , t ≥ 1, where φ is the first eigenfunction of L and λ < 0 is the first eigenvalue.
By Theorem 2.6, we obtain Remark 2.8 By recalling that the kernel p with respect to the Lebesgue measure dx satisfies p µ (x, y, t) = (1 + |y| α )p(x, y, t), we can reformulate Theorem 2.6 as follows for 0 < t ≤ 1. A similar remark holds also for Proposition 2.7.

Remark 2.9 The estimate
which holds for α ≥ 4, is optimal among the estimates of the form c(t)ψ(x)ψ(y), since the space profile is that of the first eigenfunction and the factor t −N/2 cannot be improved (by the local arguments of the proof of Theorem 3.4). Concerning the estimate which holds for 2 < α ≤ 4, the same argument as before shows its optimality with respect to the space variable. The optimality with respect to time (among all estimates of the form t −β (1 + |x|) 2−N (1 + |y|) 2−N ) is proved by the following argument. Assume that holds for β > 0 and 0 < t ≤ 1. By the argument of Proposition 2.7 it holds for every t > 0 and then by [2, Theorem 2.10] the weighted Nash inequality is valid for every u ∈ V. By replacing u(x) with u(λx), λ > 0, in (6) we obtain for every u ∈ V and λ > 0. Letting λ to zero, such an inequality can be true only if the exponent of λ is nonnegative, that is if β ≥ (N + α − 4)/(α − 2). Observe also that (N + α − 4)/(α − 2) > N/2 if and only if 2 < α < 4.

Kernel estimates for α ≤ 4
In this subsection we find different kernel estimates which hold for 0 < α ≤ 4. In the case where 2 < α < 4, these estimates are better than those of the preceeding subsection with respect to the time variable, but worse with respect to the space variables. We emphasize that the case 0 < α ≤ 2 was not covered by the previous computations.
In the following proposition we show weighted Nash inequalities with respect to the weight Proof. Let u ∈ V. Then, by Hölder's inequality, , where 2 * = 2N/(N − 2). By Sobolev embedding, Next we show that V is a Lyapunov function. Proof. Observe that V ∈ C 0 (R N ). Let λ > 0 and set u = R(λ, L)V , where the resolvent is understood in C 0 (R N ), see [10] or [8,Section 6]. Since LV ≤ 0 then By the maximum principle it follows that V ≥ λR(λ, L)V . By iteration the last inequality implies V ≥ λ n R(λ, L) n V for every n ∈ N. Then From Proposition 2.10, Lemma 2.13 and [2, Corollary 2.8] the following result follows.
Theorem 2.14 If 0 < α ≤ 4 the semigroup generated by L has a kernel p µ with respect to the measure dµ that satisfies the following bounds for every t > 0, x, y ∈ R N .

Some consequences
In this section we assume α > 2 and deduce further properties of the eigenvalues and eigenfunctions of L from kernel estimates. Let us denote by the eigenvalues of L, repeated according to their multiplicity, and by φ n the corresponding eigenfunctions, which we assume to be normalized in L 2 µ . We write φ for φ 1 . for every x ∈ R N and 2 < α ≤ 4.
. Assume e.g. α ≥ 4. Theorem 2.6 and Proposition 2.7 show thatT (t) is a bounded by Ct −N/2 from (L 1 , V 2 dµ) into L ∞ and (4) shows that it is a contraction from (L 1 , V 2 dµ) into itself. ThenT (t) is bounded by Ct −N/4 from (L 2 , V 2 dµ) into L ∞ and hence for every x ∈ R N and t > 0. Minimizing over t we obtain for every x ∈ R N . The estimates for 2 < α ≤ 4 follow in a similar way.
For λ > 0 let N (λ) be the number of λ j such that |λ j | ≤ λ. The kernel estimates allow to deduce some information on the distribution of the eigenvalues. The following result is usually obtained as a corollary of the classical Mercer's Theorem. We refer to [9, Proposition 4.1] for a simple proof based on the semigroup property. The convergence of the integral below is easily verified using Theorem 2.6 for α ≥ 4 and Theorem 2.14 for 2 < α < 4.
The following proposition, which is a weaker (and elementary) version of Karamata's Theorem, allows to deduce informations on N (λ). For its proof we refer to [9,Proposition 7.2].
Moreover if (7) holds and lim inf t→0 t r n∈N e λnt ≥ C 2 (8) for some positive C 3 .
Theorem 3.4 Let N (λ) be defined as before. Then for some positive C 1 , C 2 .
Proof. By Theorems 2.6, 2.14 we deduce for some positive constant C 1 . On the other hand, for |x| ≤ 1 let Ω = B(x, 1) and denote by p Ω the heat kernel of the restriction of L to Ω with Dirichlet boundary conditions. Since we have (see [7,Remark 6] for 0 < t ≤ 1 and constant C independent of |x| ≤ 1, it follows that for some positive C 2 .

Appendix: some weighted Sobolev inequalities
The weighted Sobolev inequalities below have been a major tool to prove kernel estimates. For reader's convenience we provide a direct proof following the methods of [1]). We point out that such type of inequalities follow also by more general results about Sobolev inequalities with respect to weights satisfying Muckenhoupt-type conditions, sse [11], [12].
Then there exists a positive constant C such that for every u ∈ C ∞ c (R N ) First we prove an homogeneous version of the previous inequality.
Then there exists a positive constant C such that for every u ∈ C ∞ c (R N \ {0}) Proof. We follow [1]. By applying the divergence theorem to suitable vector fields, we prove (10) in correspondence of β = γ − 1 and p = q, then by classical embeddings theorems we prove (10) for γ = β and q = p * and finally, by interpolation, we deduce (10) in the general case. Let by the divergence theorem and Hölder's inequality, Let us now consider the case q = p * . Setting f (x) = |x| γ u(x), by Sobolev inequality we have Using (11) to estimate the first addendum in the right hand side of the previous expression, we arrive at Let now p ≤ q ≤ p * and γ − 1 ≤ β ≤ γ. There exists θ ∈ [0, 1] such that β = (1 − θ)(γ − 1) + θγ.