KERNEL ESTIMATES FOR ELLIPTIC OPERATORS WITH UNBOUNDED DIFFUSION, DRIFT AND POTENTIAL TERMS

. In this paper we prove that the heat kernel k associated to the operator for t > 0 , | x | , | y | ≥ 1, where b ∈ R , c 1 , c 2 are positive constants, λ 0 is the largest eigenvalue of the operator A , and γ = β − α +2 β + α − 2 , in the case where N > 2 , α > 2 and β > α − 2. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.

for t > 0, |x|, |y| ≥ 1, where b ∈ R, c 1 , c 2 are positive constants, λ 0 is the largest eigenvalue of the operator A, and γ = β−α+2 β+α−2 , in the case where N > 2, α > 2 and β > α − 2. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.

Introduction. Consider the following elliptic operator
where α > 2, β > α − 2, b ∈ R and c > 0. For simplicity we denote by A the operator A b,1 . The aim of this paper is to study the behaviour of the heat kernel associated to the operator A. As a consequence one obtains precise estimates for the eigenfunctions associated to A. In recent years the interest in second order elliptic operators with polynomially growing coefficients and their associated semigroups has increased considerably, see for example [3], [4], [5], [8], [14], [15], [12], [16], [17], [11] and the references therein.
Furthermore, since the coefficients of the operator A b,c are locally regular, we know that the semigroup T p (·) admits an integral representation through a heat kernel k(t, x, y) , i.e.
Our goal is to prove upper bounds for the operator A := A b,1 in the case where α > 2, β > α − 2 and any constant b ∈ R. Our method consists in providing upper and lower estimates for the ground state Φ of A corresponding to the largest eigenvalue λ 0 and adapting the arguments used in [6].
Taking here c = 1 is not a restriction. The case of more general c > 0 can be treated in the same way.
The paper is organized as follows. In Section 2 we show that the eigenfunction Φ of A associated to the largest eigenvalue λ 0 can be estimated from below and above by the function for |x| and |y| sufficiently large. In Section 3, by means of a suitable multiplication operator T u = ηu, we rewrite the operator A in the following form η + |x| β and use an associated positive, closed, symmetric form h(·, ·) defined on a domain D(h) in an appropriate weighted Hilbert space L 2 µ (R N ). This permits us to define the associated self-adjoint operator H µ and its corresponding semigroup (e tHµ ). It can be seen that (e tHµ ) is given by a heat kernel k µ . Adapting the arguments used in [6] and [12], we prove the following intrinsic ultracontractivity where c 1 , c 2 are positive constant and γ = β−α+2 β+α−2 , provided that N > 2, α > 2 and β > α − 2. Accordingly, we obtain upper bounds of the heat kernel for t > 0, |x|, |y| ≥ 1.
Notation. For x ∈ R N and r > 0 we set B r = {x ∈ R N : |x| < r}. We denote by < ·, · > the euclidean scalar product and by | · | the euclidean norm. We use as always a standard notation for function spaces. So, we denote by L p (R N ) and W 2,p (R N ) the standard L p and Sobolev spaces, respectively. The space of bounded and continuous functions on R N is denoted by C b (R N ). Finally, in the whole manuscript the notation φ ≈ ψ on a set Ω means that there are positive constants 2. Estimating the ground state Φ. For α > 2, β > α − 2 and N > 2, we denote by A p the realization in L p (R N ), 1 < p < ∞, of the operator A := A b,1 defined in (1). We recall, see [2,Theorem 3 and Theorem 4], that A p with domain generates a strongly continuous and analytic semigroup T p (·) in L p (R N ). Moreover, for t > 0, T p (t) maps L p (R N ) into C 1+ν b (R N ) for any ν ∈ (0, 1), see [2,Proposition 5], and the semigroup T p (·) is immediately compact, see [2,Proposition 6]. As a consequence one obtains that the spectrum σ(A p ) of A p consists of a sequence of negative real eigenvalues which accumulates at −∞, and σ(A p ) is independent of p. As in [3] and [12] one can see that T p (·) is irreducible, the eigenspace corresponding to the largest eigenvalue λ 0 of A p is one-dimensional and is spanned by a strictly positive function Φ, which is radial, belongs to C 1+ν b (R N )∩C 2 (R N ) for any ν ∈ (0, 1) and tends to 0 as |x| → ∞.
In this section we prove precise estimates for the eigenfunction Φ. The technique used here is inspired by the work [4].
Since Φ is radial, one has to analyze the asymptotic behavior of the solutions to an ordinary differential equation. In this context some ideas coming from the Wentzel-Kramers-Brillouin (or Liouville-Green) approximation will be of great help. For more details see [18]. Proposition 1. Assume that N > 2, α > 2 and β > α−2. Then, for the eigenfunction Φ corresponding to the largest eigenvalue λ 0 < 0 of A, the following estimate holds Proof. Let g α,β,λ be the function defined as

SALLAH EDDINE BOUTIAH, ABDELAZIZ RHANDI AND CRISTIAN TACELLI
where λ ∈ R, h(r) = r β 1+r α , and v λ is a smooth function to be chosen later on. If we set one obtains where and On the other hand, by computing directly the second derivative of (3) one has So, comparing with (4) we get That is To evaluate the function g 1 we note first that and by the same argument we have So, if we set ξ = β−α 2 + 1, which is positive thanks to the assumption β > α − 2, then (5) is reduced to where we obtain, as in the proof of [4, Proposition 2.2], where k ≥ 3 will be chosen later on. We take now c 1 , . . . , c k such that for i = 2, · · · , k − 1 and obtain

SALLAH EDDINE BOUTIAH, ABDELAZIZ RHANDI AND CRISTIAN TACELLI
For Φ we know that Since α − 2 > 0 and λ 0 < 0, for |x| large enough we have Thus, for all x ∈ R N \ B R for some R > 0. Comparing (8) and (9) for any constant C > 0. Since β > 0, we deduce that for |x| large enough. Note that both g α,β,2λ0 and Φ go to 0 as |x| → ∞ and since there exists C 2 such that Φ ≤ C 2 g α,β,2λ0 on ∂B R , we can apply the maximum principle to the problem where z := g α,β,2λ0 − C −1 2 Φ, to obtain that Φ ≤ C 2 g α,β,2λ0 in R N \ B R (and by continuity in R N \ B 1 ). Here we apply the classical maximum principle, since lim |x|→∞ z(x) = 0, cf. [9, Theorem 3.5]. Then, As regards lower bounds of Φ, we observe that . Note that |x| β + λ 0 is positive for |x| ≥ R and, arguing as above, by the maximum principle and using (10) we have , by modifying the constant C 1 , we can see that, the above lower estimate of Φ remain valid for 1 ≤ |x| ≤ R.
3. Intrinsic ultracontractivity and heat kernel estimates. In this section we prove heat kernel estimates for T p (·) through the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted L 2 -space. Consider the Hilbert space 2α and the multiplication operator T : L 2 η 2 µ → L 2 µ given by T u = ηu. The operator A defined above can be written in the following way A = T −1 HT, where H = (1 + |x| α )∆ − U and the potential U = (1 + |x| α ) ∆η η + |x| β . An easy computation gives us from which we can deduce that U is bounded from below, since β > α − 2.
Since, for every v ∈ C ∞ c (R N ), we have we can associate to H in L 2 µ the bilinear form h defined by µ is compact. Since the bilinear form h is densely defined, quasi-accretive, continuous, closed and symmetric, one can associate the self-adjoint, quasi-dissipative operator H µ defined by and Proof. Let us begin by proving (12). The first inclusion " ⊆ " is obtained by local elliptic regularity and (11).
For the second inclusion " ⊇ " let us take v ∈ D(h) ∩ W 2,2 loc (R N ) such that Integrating by parts we obtain By the density of C ∞ c (R N ) in D(h), (13) holds for every w ∈ D(h). This implies that v ∈ D(H µ ).
To prove the coherence of the resolvents, we consider, for a positive function f ∈ C ∞ c (R N ), the following elliptic problem for λ > d. Since the operator A is uniformly elliptic in the ball B n , it is known that (14) Setting v n = T u n and g = T f we have, by (14), Moreover v n ∈ W 2,2 (B n ) ∩ W 1,2 0 (B n ). Multiplying in (15) by wµ for w ∈ W 1,2 0 (B n ) and integrating by parts we obtain In particular we obtain Since Bn |∇v n | 2 dx + Bn (d + U )v 2 n dµ ≥ 0, it follows from (17) that . By the monotone convergence theorem, we deduce that lim n→+∞ v n = v in L 2 µ . Furthermore, by (17), for λ > d we have Choosing λ > λ := max{2d, 0}, we obtain |U | ≤ |λ+U |, and hence |U | µ , ∀λ > λ . It follows that there exists a suitable subsequence (v kn ) of (v n ) such that ∇v kn converges weakly. So, v ∈ H and v belongs to the closure in H of W 1,2 -functions with compact support, which implies that v ∈ D(h). Letting now n → +∞ in (16) we obtain h(v, w) = − λv − g, w L 2 µ , λ > λ , for all w ∈ W 1,2 having compact support, and hence for all w ∈ D(h). Thus, v ∈ D(H µ ) and λv − H µ v = g for all λ > λ . Therefore, since v = T u and g = T f , it follows that c (R N ) and λ > λ . So, the statement follows now from [2, Theorem 2].
Proof. The proof is based on the semigroup law and the symmetry of k µ (t, ·, ·) for t > 0. The semigroup law implies that By Hölder's inequality and (18), we deduce that for all t > 0 and x ∈ R N . Since β−α 2 + 1 > 0, it follows from Proposition 1 that ηΦ L 2 µ is finite. Thus, the assertion follows from (19) and (18). Now, in order to estimate k µ we use the techniques in [6,Chap 4].
Proposition 4. There exist positive constants C 1 , C 2 , C 3 , C 4 such that for any ε > 0 with γ = β−α+2 β+α−2 , and Proof. To prove (20), we apply the lower estimate of Φ obtained in Proposition 1 As a consequence, there are positive constants C 2 , C 3 such that Since ξ < β, γ = ξ β−ξ , by using Young's inequality, it follows that Taking into account that This proves (20). Concerning (21), by density, it suffices to show it for v ∈ C ∞ c (R N ). Using Hölder and Sobolev's inequalities we obtain So, one obtains (21) by observing that We give now the estimate of k µ (t, x, y). Theorem 1. Assume that N > 2, α > 2 and β > α − 2. Then there exist C 1 , C 2 positive constants such that Proof. It follows from Proposition 4 and Rosen's lemma, see Lemma A3, that for all 0 ≤ f ∈ L 2 (R N , (T Φ) 2 µdx) and ε > 0, we have So, applying Corollary A4, one obtains We observe that 0 < γ < 1. Then we can apply Proposition A2 and Proposition A1 to obtain that where k T Φ (·, ·, ·) is the heat kernel of the semigroup generated by the selfadjoint operator associated to the form h T Φ . The result follows now by taking into account that Now, we are ready to state and give the proof of the main result of this paper.
1. The heat kernel estimates k(·, ·, ·) in Theorem 2 is sharp in the space variables. This can be seen from Proposition 3.
If we denote by Φ j the eigenfunction of A 2 associated to the eigenvalue λ j , then T Φ j is the eigenfunction of H µ associated to λ j . Hence, for any t > 0 and any x ∈ R N , we have (22), we obtain the following estimates. for all j ∈ N, x ∈ R N \ B 1 and some constant C j > 0.