J. Evol. Equ. © 2017 Springer International Publishing AG DOI 10.1007/s00028-017-0408-0

Journal of Evolution Equations

Sharp kernel estimates for elliptic operators with second-order discontinuous coefficients G. Metafune, L. Negro and C. Spina

Abstract. We consider the second-order elliptic operator L =  + (a − 1)

N  xi x j x b Di j + c 2 · ∇ − , |x|2 |x| |x|2

i, j=1

a > 0, b, c ∈ R, and we prove sharp bounds for the heat kernel and the function.

1. Introduction In this paper, we prove sharp upper and lower bounds for the heat kernels of the operators N  xi x j x b L =  + (a − 1) Di j + c 2 · ∇ − 2 , (1) 2 |x| |x| |x| i, j=1

a > 0, b, c ∈ R. Observe that the leading coefficients are uniformly elliptic but discontinuous at 0, if a = 1, and singularities in the lower order terms appear when b or c is different from 0. In the special case b = c = 0, these operators have been introduced to provide counterexamples to the elliptic regularity (see for example [20,22]). Positive results have also been obtained by Manselli and Ragnedda, see [8–10], who proved existence and uniqueness results in Sobolev spaces in a bounded domain containing the origin and spectral properties in the two-dimensional case. In the special case a = 1, c = 0, the operator becomes the Schrödinger operator with inverse square potential b (2) L =− 2 |x| for which we recover sharp heat kernel bounds even in the critical case D := b +  N −2 2 = 0. 2 Concerning (2), Milman and Semenov prove in [17, Theorem 1] the same upper and lower bounds as in our Theorem 8.1 even with (almost) precise constants in the Mathematics Subject Classification: 47D07, 35B50, 35J25, 35J70 Keywords: Elliptic operators, Discontinuous coefficients, Analytic semigroups, Kernel estimates.

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Gaussian factor and including the critical case. We also refer to [6] for sharp bounds in bounded domains when the potential in (2) degenerates as the inverse of square of the distance from the boundary. Our methods work for the more general operators (1). This generalization is important to obtain precise bounds on the heat kernels of certain operators with unbounded coefficients, as shown in Sect. 8. Since upper bounds have already been proved in [16], the main contribution of this paper consists in proving lower bounds. We point out that generation properties and domain description for our operator have been previously investigated in [15]. If 1 < p < ∞, we define the maximal operator L p,max through the domain   2, p D(L p,max ) = u ∈ L p (R N ) ∩ Wloc (R N \{0}) : Lu ∈ L p (R N )    and L p,min ⊂ L p,max is defined as the closure, in L p (R N ) of L , Cc∞ R N \{0} . The equation Lu = 0 has radial solutions |x|−s1 , |x|−s2 where s1 , s2 are the roots of the indicial equation f (s) = −as 2 + (N − 1 + c − a)s + b = 0 given by s1 :=

N −1+c−a √ N −1+c−a √ − D, s2 := + D 2a 2a

where b D := + a



N −1+c−a 2a

(3)

2 .

(4)

The above numbers are real if and only if D ≥ 0. When D < 0, the equation u − Lu = f cannot have positive distributional solutions for certain positive f , see [15] and [14]. This fact constitutes an elliptic counterpart of a famous result due to Baras and Goldstein, see [2], in the case of the Schrödinger operator with inverse square potential where the above condition reads b + (N − 2)2 /4 ≥ 0. Assuming D ≥ 0, we have shown in [13,15] that there exists an intermediate operator L p,min ⊂ L p,int ⊂ L p,max which generates a semigroup in L p (R N ) if and only if Np ∈ (s1 , s2 + 2). In [16], we have also shown that the semigroup generated by L p,int is analytic of angle π/2 and that the spectrum of L p,int coincides with (−∞, 0], by proving complex upper estimates for the heat kernel. The main result of the paper consists in the following two-side estimates for the heat kernel p of L with respect to the measure |y|γ dy, see Theorem 6.2, p(t, x, y) c1 t

− N2

|x|

− γ2

|y|

− γ2



  c2 |x − y|2 exp − t

|x| 1

t2

  − N +1+√ D 2 |y| ∧1 ∧1 1 t2

where D ≥ 0 is defined in (4) and γ = (N − 1 + c)/a − N + 1. Here c1 , c2 are positive constants which may be different in the lower and upper bounds. A different but equivalent form of the above bounds is shown in Corollary 6.4, see also Remark 6.3.

Sharp kernel estimates for elliptic operators

As a consequence, we obtain sharp bounds for the function. For example, if N > 2 and D > 0, then (writing the kernel with respect to the measure |y|γ dy)  √ D− N −2 √ 2 γ |x||y| . (|x||y|) 2 G λ (x, y) e−c λ|x−y| |x − y|2−N 1 ∧ 2 |x − y| The critical cases N = 2 and D = 0 are also considered. As previously observed, we focus on lower estimates. We write L in spherical coordinates and observe that, on subspaces defined as tensor products of radial functions and spherical harmonics, it reduces to one-dimensional Bessel operator. This provides a decomposition of the kernel of L in terms of its one-dimensional counterparts and the main result follows by combining kernel estimates near the origin, obtained thanks to the explicit formula of one-dimensional Bessel operators, with gaussian estimates faraway from the origin already known for uniformly elliptic operators. In Sect. 2, we collect some preliminary results on spherical harmonics, and in Sect. 3, we construct the operator via form methods. In Sect. 4, we give an analytic proof of the explicit form of the heat kernel of Bessel operators. In Sect. 5, we decompose the heat kernel of L as the (infinite) sum of heat kernels of one-dimensional Bessel operators. Finally, in Sect. 6 we get the main result by proving sharp kernel estimates. Estimates of the Green functions are given in Sect. 7. Some special cases, including Schödinger operators and homogeneous operators with unbounded coefficients introduced in [11], are studied in Sect. 8. Notation. We denote by N0 = N ∪ {0} the natural numbers including 0. Often we use  for R N \{0}. When V is an open subset of R N , Cb (V ) is the Banach space of all continuous and bounded functions in V , endowed with the sup-norm, C0 (V ) its subspace consisting of functions vanishing at the boundary of V , including ∞ when V is unbounded. Cc∞ (V ) denotes the space of infinitely continuously differentiable functions with compact support in V . The unit sphere { x = 1} in R N is denoted by S N −1 . We adopt standard notation for L p and Sobolev spaces and the Lebesgue measure is understood, when no measure is explicitly written. We denote by C+ := {z ; Re z > 0}, Br := {x ∈ Rn : |x| < r } and B¯ r := {x ∈ Rn : |x| ≤ r }. Given a and b ∈ R, a ∧ b denotes the minimum between a and b. we write f (x) g(x) for x in a set I and positive f, g, if for some C1 , C2 > 0 C1 g(x) ≤ f (x) ≤ C2 g(x), x ∈ I. For x ∈ R N , we use spherical coordinates to write x = r ω, where r := |x|, ω := x N −1 . We denote by D , D the radial derivatives and by ∇ the tangential r rr τ |x| ∈ S component of the gradient. They are defined through the formulas Dr =

N  i=1

Di

xi , r

Drr =

N  i, j=1

Di j

xi x j , r2

∇ = Dr

∇τ x + . |x| r

With 0 , we denote the Laplace–Beltrami operator on S N −1 . Besides its geometric intrinsic definition, we can define it as follows. Given a function f ∈ C 2 (S N −1 ), we

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x consider its extension f˜ to R N \{0} given by f˜(x) := f ( |x| ). The gradient ∇τ and the N −1 Laplacian 0 on S are there given by

∇τ f (x) = ∇ f˜(x),

0 f (x) =  f˜(x),

for every x ∈ S N −1 .

Moreover,  = Drr +

N −1 0 Dr + 2 . r r

We refer to [23, Section 5, Chapter IX] for further details and the explicit formula of 0 in spherical coordinates. In order to help reading, we list below the main parameters and formulas we use systematically through the paper. •

The (nonnegative) discriminant of the indicial equation −as 2 + (N − 1 + c − a)s + b = 0 is denoted by D=

• •

b + a



N −1+c−a 2a

2 .

− N + 1. The reference measure is dμ = |x|γ dx, where γ = N −1+c a |x|−s1 , |x|−s2 are the radial solutions of Lu = 0 where s1 , s2 are the roots of the indicial equation given by N −1+c−a √ N −1+c−a √ − D, s2 := + D. 2a 2a √ Moreover, s1 = N2 − 1 − D + γ2 . s1 :=

2. Preliminaries: spherical harmonics We recall some results about spherical harmonics, referring to [21] for the proofs and other details. For fixed n ∈ N, let Hn denote the space of spherical harmonics of degree n, that is the restriction to the unit sphere of homogeneous harmonic polynomials of degree n. We recall that −λn = −n(n + N − 2) are the eigenvalues of the Laplace– Beltrami operator 0 on S N −1 and that the corresponding eigenspaces are the Hn . (n) Let Zω be the zonal harmonic of degree n with pole ω ∈ S N −1 defined by (n) Z(n) ω (η) := Z (ω, η) =

an 

Pin (ω)Pin (η)

(5)

i=1

where ω, η ∈ S N −1 and {Pin , i = 1, . . . , an } is an orthonormal basis of spherical   harmonics of degree n whose cardinality is given by an = dim (Hn ) = N +n−1 − n  N +n−3 n−2 . We collect in the following proposition some basic properties of zonal harmonics.

Sharp kernel estimates for elliptic operators

PROPOSITION 2.1. For fixed n ∈ N, the sum in (5) is independent of the choice of (n) the orthonormal basis of Hn . The zonal harmonic Zω is characterized by the relation  P(η)Z(n) P(ω) = ω (η)dη S N −1

valid for all P ∈ Hn and ω ∈ S N −1 . Moreover (n)

(n)

(i) for all ω, η ∈ S N −1 Zω (η) = Zη (ω) and if T is an orthogonal transformation, (n) (n) then Z T ω (T η) = Z ω (η); (ii) the following uniform estimates hold sup |Z ω(n) (η)| = Z ω(n) (ω) =

η∈S N −1

an |S N −1 |

and, from the asymptotic behaviour an ∼ n N −2 for n → ∞, we have sup ω,η∈S N −1

|Z ω(n) (η)| ≈

(iii) for all P ∈ Hn

sup |P(ω)| ≤

ω∈S N −1

n N −2 ; |S N −1 |

an

P L 2 (S n−1 ) . N |S −1 |

  Let us consider now, for a fixed α ∈ R, the weighted space L 2 R N , |x|α dx and let an

    L 2 (0, ∞), r α+N −1 dr ⊗ P jn L 2n = L 2 (0, ∞), r α+N −1 dr ⊗ Hn = i=1

=

an  i=1

L 2P n

(6)

i

  where Pin , i ∈ J is an orthonormal basis of Hn and L 2P = L 2 ((0, ∞), r α+N −1 dr ) ⊗ P. Zonal  harmonics  provide a simple way to describe the orthogonal projection of L 2 R N , |x|α dx onto L 2n . PROPOSITION 2.2. Let α ∈ R. The following properties hold.    (i) L 2 R N , |x|α dx = ∞ L2.  N n=0α n 2 (ii) for every u ∈ L R , |x| dx , the orthogonal projection on L 2n is given by Q n (u)(r, η) :=

an 

 Pin (η)

i=1

and moreover u =

∞

n=0

 S N −1

u(r, ω)Pin (ω)dω =

S N −1

  Q n (u) in L 2 R N , |x|α dx .

u(r, ω)Z ω(n) (η)dω

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Proof. Since the weight |x|α is radial, the assertion is a simple reformulation of [21, Lemma 4.2.18].  Note that, if P is a normalized spherical harmonic of degree n, the projection on is given by  Q P (u)(r, η) := P(η) u(r, ω)P(ω) dω.

L 2P

S N −1

  Moreover, Q n = i=1 Q Pin , Q P is symmetric and, for every u ∈ Cc∞ R N \{0} , Q P commutes with the radial derivative that is  (Q P u)r = P(η) u r (r, ω)P(ω) dω = Q p u r . an

S N −1

This follows since u r = ∇u ·

x |x| ,

by differentiating under the integral sign.

3. Definition of the operator via form methods Let L =  + (a − 1)

N  xi x j x b Di j + c 2 · ∇ − 2 |x|2 |x| |x|

i, j=1

N −1+c b − 0 Dr − , r r2

= a Drr +

where Dr , Drr denote radial derivatives and 0 is the Laplace–Beltrami on S N −1 . In what follows, we consider also the one-dimensional situation, by agreeing that 0 is not present and that R N stands for (0, ∞) when N = 1. Setting a = (a i j ) with a i j (x) = δi j + (a − 1)|x|−2 xi x j , x ∈ R N \{0}, γ = we can write L = |x|−γ div(|x|γ a∇) −

N −1+c − N + 1, (7) a

b . |x|2

(8)

We recall that the condition D=

b + a



N −1+c−a 2a

2 ≥0

is necessary and sufficient to get positive solutions, see [15], and we shall always assume it. Note that γ = 0 if and only if L is formally self-adjoint with respect to the Lebesgue measure. Using (8), we define L through a symmetric form in a weighted space. To this aim, we note that the matrix a(x) has eigenvalues a with eigenvector x and 1 with eigenspace the orthogonal complement of x.

Sharp kernel estimates for elliptic operators

DEFINITION 3.1. Consider the sesquilinear form a in L 2μ = L 2 (R N , dμ) with dμ = |x|γ dx defined by    b a∇u, ∇v + 2 uv dμ, D(a) := Cc∞ (R N \{0}). a(u, v) := N |x| R Denoting by ∇τ u the tangential component of the gradient and recalling that ∇u = x + ∇rτ u , the form a becomes u r |x| ∞ 

  

N −1+c ∇τ u ∇τ v b x x + , vr + + 2 uv r a dr dσ a ur |x| r |x| r r S N −1 0   ∞  N −1+c ∇ τ u ∇τ v b au r vr + + 2 uv r a dr dσ. = r2 r S N −1 0 

a(u, v) :=



We provide preliminary a simple proof of the Hardy inequality which can be found in [4, Lemma 5.3.1]. LEMMA 3.2. Let b, s ∈ R. For every u ∈ Cc∞ ((0, +∞)) setting v = ur has



  ∞  ∞ b (s − 1)2 v 2 |u  |2 + 2 u 2 r s dr = |v  |2 + b + r dr. r 4 r2 0 0

s−1 2

, one

Proof. We have

b 2 (s − 1)2 v 2 s−1  b 2 1−s  2 |v , vv u = r | + − + v r2 4 r2 r r2 ∞ and observing that 0 vv  dr = 0, we obtain |u  |2 +

∞

 0



   ∞ b 2 s (s − 1)2 v 2 s−1   2 |u | + 2 u r dr = |v | + b + vv r dr. − r 4 r2 r 0 

  ∞ (s − 1)2 v 2 = |v  |2 + b + r dr. 4 r2 0  2

 We immediately deduce from the last lemma Hardy’s inequality in R N . We recall that D is defined in (4). N −1+c−a

and v = COROLLARY 3.3. Given u, v ∈ Cc∞ (R N \{0}), let u = u 1 |x|− 2a − N −1+c−a 2a v1 |x| . Then,

 ∇τ u 1 ∇τ v1 aD 2−N a(u 1 )r (v1 )r + |x| + 2 u 1 v1 dx a(u, v) = |x|2 |x| RN

  ∞ ∇τ u 1 ∇τ v1 aD = r a(u 1 )r (v1 )r + + 2 u 1 v1 dr dσ. r2 r S N −1 0

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In particular,  a(u, u) =

 S N −1

|∇τ u 1 |2 aD 2 2 r a|(u 1 )r | + + 2 |u 1 | dr dσ ≥ 0 r2 r

∞

0

(9)

if D ≥ 0. Note that if a = 1, c = 0, then γ = 0 and D = b + (N − 2)2 /4 and we recover the classical Hardy’s inequality  a(u, u) =

RN

  (N − 2)2 |u|2 |∇u|2 − dx ≥ 0. 4 |x|2

The following lemma corresponds to the Friedrichs extension of a. We give a direct proof which is elementary and allows to describe the domain of the closure. LEMMA 3.4. If D ≥ 0, a is nonnegative, symmetric and closable in L 2μ = Denoting by a˜ the closure of a, the following properties hold:    N −1+c−a (i) if D = 0, D(˜a) = u ∈ L 2μ : u|x| 2a ∈ H01 R N \{0}, |x|2−N dx ,   where H01 R N \{0}, |x|2−N dx is the closure of Cc∞ (R N \{0}) with respect to the norm   

v 2H 1 R N ,|x|2−N dx = |∇v|2 + |v|2 |x|2−N dx; ) 0( Rn

L 2 (R N , dμ).

   γ (ii) if D > 0, then D(˜a) = u ∈ L 2μ : u|x| 2 ∈ H01 R N ; (iii) if Ms is the dilation defined by Ms u(x) = u(sx) then for every u, v ∈ D(˜a) and s > 0 we have Ms u, Ms v ∈ D(˜a) and a˜ (Ms u, Ms v) = s 2−γ −N a˜ (u, v); (iv) if Q is an orthogonal matrix in R N and M Q u(x) = u(Qx), then for every u, v ∈ D(˜a), M Q u, M Q v ∈ D(˜a) and a˜ (M Q u, M Q v) = a˜ (u, v).

Proof. Clearly, a is a nonnegative symmetric form in L 2 (R N , dμ), from the previous corollary. To prove its closability, by [19, Proposition 1.13], it is sufficient to show  that if (u n )n ⊂ Cc∞ R N \{0} satisfies u n → 0 in L 2μ as n → ∞ and a(u n − u m , u n − u m ) → 0 as n, m → ∞, then  a(u n , u n ) → 0 as n → ∞. Since a is locally uniformly 1 R N \{0} . Passing to a subsequence, we may assume that elliptic, u m → 0 in Hloc u m , ∇u m → 0 pointwise. From (9) setting u n = vn |x|−  a(u n , u n ) =

 S N −1

0

∞

N −1+c−a 2a

, we have

|∇τ vn |2 aD 2 2 r a|(vn )r | + + 2 |vn | dr dσ. r2 r

Sharp kernel estimates for elliptic operators

and then Fatou’s Lemma yields a(u n , u n ) ≤ lim inf a(u n − u m , u n − u m ). m→∞

This proves the closability of a.   Let now a˜ be the closure of a. By [19, Proposition 1.13] D a˜ is the closure of    Cc∞ R N \{0} with respect to the norm ||u||a := a(u, u) + ||u||2L 2 . μ  N  − N −1+c−a ∞ 2a If D = 0, let u ∈ Cc R \{0} and set u = v|x| . From (9)

 |∇τ v|2 a(u, u) = |x|2−N a|vr |2 + dx |x|2 RN x + ∇rτ v , we easily recognize that the last and recalling that a > 0 and ∇v = vr |x| integral is equivalent to ∇v 2L 2 R N ,|x|2−N dx . Since also the norms of u in L 2μ and v ( ) in L 2 (|x|2−N dx) coincide, we see that the norms ||u||a and ||v|| H 1 (R N ,|x|2−N dx ) are 0

equivalent on Cc∞ (R N \{0}) and (i) follows.   γ Suppose now that D > 0 and let u ∈ Cc∞ R N \{0} . Setting u = v|x|− 2 and proceeding as before, we obtain

   |∇τ v|2 (N − 2)2 v 2 2 a|vr | + dx +a D− a(u, u) = |x|2 4 |x|2 RN Then, from the Hardy inequality (9), ||u||a and ||v|| H 1 (R N ) are equivalent norms and 0 so

  γ D(˜a) = u ∈ L 2 (R N , dμ) : u|x| 2 ∈ H01 R N

since Cc∞ (R N \{0}) is dense in H01 (R N ), N ≥ 2, and clearly Cc∞ (0, ∞) is dense in H01 (0, ∞) when N = 1. Consider now the third statement. Let u ∈ D(˜a) and s > 0. From the definition of a˜ , there exists a sequence {u n }n ∈ Cc∞ R N \{0} such that u n → u in L 2μ and a(u n − u m , u n − u m ) → 0 as n, m → ∞. Therefore, Ms u n → Ms u in L 2μ , a(Ms u n − Ms u m , Ms u n − Ms u m )   s 2 a∇(u n − u m )(sx), ∇(u n − u m )(sx) = N R  b + 2 |(u n − u m )(sx)|2 |x|γ dx |x|   a∇(u n − u m )(y), ∇(u n − u m )(y) = s 2−γ −N RN  b + 2 |(u n − u m )(y)|2 |y|γ dy |y| = s 2−γ −N a(u n − u m , u n − u m ) → 0, and hence Ms u ∈ D(˜a).

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Finally,  Nlet u, v ∈ D(˜a) and s > 0. By approximating u and v with {u n }n , {vn }n ∈ ∞ Cc R \{0} as above and using a(Ms u n , Ms vn ) = s 2−γ −N a(u n , vn ) we obtain (iii) letting n to infinity. The proof of (iv) is similar.



If D ≥ 0, let −L be the operator associated with a˜ , that is    2 D(L) := u ∈ D(˜a) ; ∃v ∈ L μ s.t. a˜ (u, w) = vw dμ ∀w ∈ D(a) , RN

− Lu := v.

(10)

The basic properties of L are listed below, see also [15]. PROPOSITION 3.5. If D ≥ 0, the operator −L defined in (10) is nonnegative and self-adjoint. Moreover, (i) Cc∞ (R N \{0}) → D(L) and for every u ∈ Cc∞ (R N \{0}) Lu =

N 

a i j Di j u + c

i, j=1

x b · ∇u − 2 u 2 |x| |x|

2, p

(ii) D(L) → {u ∈ L 2μ ∩ Wloc (R N \{0}) ; Lu ∈ L 2μ }.     2n R N \{0} . (iii) Cc∞ R N \{0} → D(L n ) → Hloc (iv) s 2 L = Ms−1 L Ms , L = M Q ∗ L M Q if Q is an orthogonal matrix, hence the semigroup et L generated by L in L 2μ satisfies es

2t L

= Ms−1 et L Ms ,

et L = M Q ∗ et L M Q

t, s > 0.

(v) The semigroup is represented by a kernel p with respect to the measure dμ = |y|γ dy which satisfies   x y − N +γ p(t, x, y) = p(t, Qx, Qy) (11) p(t, x, y) = t 2 p 1, √ , √ , t t for t > 0, x, y ∈ R N \{0}. Proof. (i) is clear by construction and (ii) follows from interior elliptic regularity. (iii) follows from (i) and (ii), by induction. Concerning (iv), let u ∈ D(L) and v ∈ D(˜a). Then,  (Lu)Ms −1 v dμ a˜ (Ms u, v) = s 2−γ −N a˜ (u, Ms −1 v) = −s 2−γ −N RN  = −s 2 (Ms Lu)v dμ RN

hence Ms u ∈ D(L) and L Ms u = s 2 Ms Lu. Similarly for M Q . The existence of the kernel p is proved in [15]. Finally, properties (v) are a simple consequence of (iv),  taking into account that the reference measure is |y|γ dy.

Sharp kernel estimates for elliptic operators

In the following sections, we prove that the resolvent of L and the generated semigroup are the direct sum of the corresponding resolvents and semigroups generated by these one-dimensional Bessel operators. 4. The one-dimensional case: Bessel operators In this section, we find an explicit formula for the heat kernel of the operator c b Lu = u rr + u r − 2 u r r considered in L 2 ((0, ∞), dr ). L is the one-dimensional version of the operator defined in (10) choosing a = 1 and restricted to the positive half-line (0, ∞). As before, we 2 ≥ 0, which is necessary and sufficient for assume the condition D = b + (c−1) 4 the existence of a positive resolvent. In fact, when D < 0, through the change of variable r = es and the Sturm Comparison Theorem, one sees that every solution of the homogeneous equation λu − Lu = 0, λ > 0, oscillates near zero and therefore there is no way to construct a positive resolvent. We refer the reader to [11] for the proof and to [12,14] for an investigation of the generation properties of L when D < 0 and for uniqueness problems. The heat kernel of the Bessel is usually deduced by probabilistic tools. We refer, however, to [7] where the author uses the Weyl–Kodaira theory for Sturm–Liouville problems. We give a purely analytic proof of the heat kernel formula which has also the advantage to appear consistent with the construction of the operator, see also Remark 4.8. 4.1. Definition of the operator Let us consider now b, c ∈ R such that D = b +

 c−1 2 2

≥ 0 and

d  c  b c b Lu = u rr + u r − 2 u = r −c r ur − 2 . r r dr r c

Note that the parameter γ in (7) coincide with c. If u = r − 2 v and setting ν 2 = 2  , then b + c−1 2       c c−1 2 1 v − 2c Lu = r − vrr − b + = r − 2 L ν v. 2 2 4 r As in the N -dimensional case, L can be defined through a symmetric form in the weighted space L 2 ((0, ∞), r c dr ). Let   ∞ uv u r vr + b 2 r c dr, b(u, v) := D(b) := Cc∞ ((0, ∞)) r 0 The following Lemma is the one-dimensional version of Lemma 3.4.

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LEMMA 4.1. b is a nonnegative, symmetric and closable form in L 2 ((0, ∞), r c dr ). Denoting by b˜ the closure of b, the following proprieties hold:  2    ˜ = u ∈ L 2 ((0, ∞), r c dr ) : r c−1 2 u ∈ H 1 (0, ∞), r dr ; = 0, D( b) (i) if b+ c−1 0 2    c−1 2   c 1 ˜ = u ∈ L 2 ((0, ∞), r c dr ) : r 2 u ∈ H (0, ∞), dr ; (ii) if b + 2 > 0, D(b) 0 ˜ and (iii) if Ms is the dilation defined by Ms u(r ) = u(sr ), then for every u, v ∈ D(b) ˜ s > 0 we have Ms u ∈ D(b) and ˜ ˜ s u, v) = s 1−c b(u, Ms −1 v), b(M

˜ s u, Ms v) = s 1−c b(u, ˜ b(M v).

˜ that is Let −L be the operator associated with b,    ˜ ∃v ∈ L 2 (0, ∞), r c dr s.t. b(u, ˜ D(L) := u ∈ D(b); w) 

∞

= 0

 vw r c dr ∀w ∈ Cc∞ ((0, ∞)) , −Lu := v.

The next proposition shows the basic properties of L. PROPOSITION 4.2. −L is nonnegative and self-adjoint. Moreover, (i) Lu = u rr + rc u r − rb2 u for every u ∈ Cc∞ ((0, ∞)). (ii) s 2 L = Ms−1 L Ms , s > 0, where Ms is the dilation defined by Ms u(r ) = u(sr ). If c = 0, the operator L becomes a Schrödinger operator with inverse square potential. Since we have assumed b ≥ − 41 in order to get positive solutions, we may write our operator in the form   1 u . L ν u = u rr − ν 2 − 4 r2 It follows from the previous results that L ν is the operator associated with the Friedrichs extension a˜ν of the sesquilinear form aν defined in L 2 (0, ∞) by     ∞ 1 uv 2 u r vr + ν − dr, aν (u, v) := 4 r2 0 D(aν ) := Cc∞ ((0, ∞)) . We investigate now the relation between the operators L and L ν . Let us set ν 2 = 2  and consider, for this reason, the map b + c−1 2   c  : u ∈ L 2 (0, ∞), r c dr → r 2 u ∈ L 2 (0, ∞) . Obviously,  is an isometry which preserves Cc∞ (0, ∞). Moreover, integrating by parts we easily obtain aν (u, v) = b(u, v), for u, v ∈ Cc∞ (0, ∞). This relation between the two forms translates into a similar relation between the associated operator which we point out in the next proposition.

Sharp kernel estimates for elliptic operators

PROPOSITION 4.3. Let L and L ν be the operators defined above. Then, L = −1 L ν . Proof. We have already observed that aν (u, v) = b(u, v), for u, v ∈ Cc∞ ((0, ∞)). ˜ v) = a˜ν (u, v). For The same relation is inherited by the extensions and so b(u, ∞ every u ∈ D(b) and w ∈ Cc (0, +∞),  ∞  ∞  c −1 L ν u w r c dr = − − (L ν u) w r 2 dr 0

0

˜ = a˜ν (u, w) = b(u, w) This proves L = −1 L ν .



4.2. The resolvent and the heat kernel of L We first consider the case c = 0 corresponding to the operator L ν and then apply Proposition 4.3. We recall some well-known facts about the modified Bessel functions Iν and K ν which constitute a basis of solutions of the modified Bessel equation r2

d2 v dv − (r 2 + ν 2 )v = 0, r > 0. +r 2 dr dr

We recall that ∞

r 2m

r ν  1 , Iν (r ) = 2 m! (ν + 1 + m) 2

K ν (r ) =

m=0

π I−ν (r ) − Iν (r ) , 2 sin π ν

where limiting values are taken for the definition of K ν when ν is an integer. The basic properties of these functions we need are collected in the following lemma, see, e.g., [1, 9.6 and 9.7]. LEMMA 4.4. For every ν ≥ 0, Iν is increasing, K ν is decreasing and, the following asymptotic behaviour satisfies. If r → ∞, 1

1

1

1

|Iν (r )| ≈ r − 2 er , |Iν (r )| ≈ r − 2 er , |K ν (r )| ≈ r − 2 e−r , |K ν (r )| ≈ r − 2 e−r . Moreover, if ν > 0, then as r → 0, |Iν (r )| ≈ r ν , |Iν (r )| ≈ r ν−1 , |K ν (r )| ≈ r −ν , |K ν (r )| ≈ r −ν−1 , and |I0 (r )| ≈ 1, |I0 (r )| → 0, |K 0 (r )| ≈ | log r |, |K 0 (r )| ≈ r −1 . Note that

r r 1 e 1 e C1 (1 ∧ r )ν+ 2 √ ≤ Iν (r ) ≤ C2 (1 ∧ r )ν+ 2 √ r r for suitable C1 , C2 > 0. Let us compute the resolvent operator of L ν .

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PROPOSITION 4.5. Let λ > 0. Then, for every f ∈ L 2 ((0, ∞)),  ∞ (λ − L ν )−1 f = G(λ, r, s) f (s)ds 0

with

√ √ √ r s Iν ( λ r )K ν ( λ s) G(λ, r, s) = √ √ √ r s Iν ( λ s)K ν ( λ r )

r ≤s r ≥ s.

(13)

Proof. Let us first consider the case λ = 1. The homogeneous equation   1 u 2 −u =0 u rr − ν − 4 r2 transforms into vr − vrr + r



 ν2 +1 v =0 r2

√ √ √ ) . Therefore, u 1 (r ) = by setting v(r ) = u(r r Iν and u 2 (r ) = r K ν constitute a r basis of solutions. Since the Wronskian of K ν , Iν is 1/r , see [1, 9.6 and 9.7] that of √ √ r K ν , r Iν is 1. It follows that every solution of   1 u 2 −u = f u rr − ν − 4 r2 is given by  u(r ) = 0

∞

√ √ G(1, r, s) f (s) ds + c1 r Iν (r ) + c2 r K ν (r ),

(14)

with c1 , c2 ∈ R and √ r s Iν (r )K ν (s) G(1, r, s) = √ r s Iν (s)K ν (r )

r ≤s r ≥s

Elementary computations using Lemma 4.4 show that  ∞ G(1, r, s)ds < +∞. sup r ∈(0,+∞) 0

By the symmetry of the kernel and Young’s inequality, the integral operator T defined by G is therefore bounded in L 2 ((0, ∞)). Let f ∈ Cc∞ ((0, ∞)) with support in (a, b) and u = (1 − L ν )−1 f ∈ D(L ν ). Then, u is given by (14) with c1 = 0, since T is bounded in L 2 ((0, ∞)), K ν is exponentially decreasing and Iν is exponentially increasing near ∞. Since  r  b √ √ √ u(r ) = r s K ν (r )Iν (s) f (s) ds + r s K ν (s)Iν (r ) f (s) ds + c2 r K ν (r ), 0

r

Sharp kernel estimates for elliptic operators

we have for r < a  b √ √ √ √ r s K ν (s)Iν (r ) f (s) ds + c2 r K ν = c r Iν (r ) + c2 r K ν (r ) u(r ) = a

  1 for some c ∈ R. If c2 = 0, by Lemma 4.4 u  (r ) ≈ c2 21 − ν r − 2 −ν when ν = 0 and √u = cIν (r ) + c2 K ν (r ) for r < a when ν = 0. In both cases, by Lemma 4.1 (i), (ii), r

with c = 0, u ∈ D(a˜ν ). Therefore, c2 = 0 and, by density, (I − L ν )−1 = T , since both operators are bounded and coincide on compactly supported functions. Finally, let us compute the resolvent for a general λ > 0. Recalling that M√λ L ν M√ −1 λ = λL ν ,    √ 1 ∞ s −1 −1 √ −1 √ ds (λ − L ν ) f = λ M λ (I − L ν ) M −1 f = G(1, r λ, s) f √ λ λ 0 λ  ∞ √ √ 1 =√ G(1, r λ, s λ) f (s) ds λ 0 

which gives (13).

)−1

REMARK 4.6. Observe that the above proof shows the boundedness of (λ − L ν in L p ((0, ∞)) for every 1 ≤ p ≤ ∞, λ > 0. √ REMARK 4.7. We point out that the function r K ν does not belong to the domain of the form, but belongs to L 2 (0, ∞) if and only if ν < 1 or b < 3/4. In this range, other boundary conditions at r = 0 are possible and our choice, consistent with the rest of the paper, corresponds to a minimal resolvent, in the sense of positivity, see [14]. REMARK 4.8. When b ≥ 3/4, L ν is essentially self-adjoint on Cc∞ ((0, ∞)). Moreover, when b > 3/4, the domain of L v is given by D(L ν ) = {u ∈ H 2 (0, ∞) : u u , ∈ L 2 (0, ∞)}, see [11, Example 7.1]. This last coincides with H02 ((0, ∞)) = r2 r {u ∈ H 2 ((0, ∞)) : u(0) = u  (0) = 0}. In fact, the inclusion {u ∈ H 2 ((0, ∞)) : u u , ∈ L 2 ((0, ∞))} ⊂ H02 ((0, ∞)) is immediate since a function u ∈ H 2 ((0, ∞)) r2 r 2  has finite values u(0), u  (0) and these vanish when  ru/r , u /r are integrable  r near zero. 2 Conversely, if u ∈ H0 ((0, ∞)), then |u(r )| = | 0 (r − s)u (s) ds| ≤ r 0 |u  (s)| ds and u/r 2 ∈ L 2 ((0, ∞)), by Hardy inequality. A similar argument holds for u  /r . We denote now by pν (t, r, s) the heat kernel of the operator L ν . Its existence is well known, due to the local regularity of the coefficients. We show below a simple way to compute it, even without assuming its existence. We look for a smooth function p(t, r, s) such that, for every f ∈ L 2 ((0, ∞))  ∞ p(t, r, s) f (s) ds. et L ν f (r ) = 0

The function p should then satisfy pt (t, r, s) = prr (t, r, s) − p(0, r, s) = δs .

1 r2

 2 1 ν − 4 p(t, r, s)

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Since λ2 L ν = Mλ−1 L ν Mλ , we obtain etλ L ν = Mλ−1 et L ν Mλ . Rewriting this identity using the kernel p and setting λ2 t = 1, we obtain     1 1 r s r s p(t, r, s) = √ p 1, √ , √ := √ F √ , √ . t t t t t t 2

Then, (15) becomes      1 r r 1 s s 2 Frr √ , √ − 2 ν − tF √ , √ r 4 t t t t     1 1 r r r s s + F √ , √ + √ Fr √ , √ 2 2 t t t t t   1 s r s + √ Fs √ , √ = 0 2 t t t 

that is   1 1 1 1 2 Frr (r, s) − 2 ν − F (r, s) + F (r, s) + r Fr (r, s) r 4 2 2   r 1 s + s Fs √ , √ = 0. 2 t t Since for large r the operator L ν behaves like D 2 , having in mind the gaussian kernel, we look for a solution of the form   (r − s)2 1 H (r s) F(r, s) = √ exp − 4 4π with H depending only on the product of the variables. By straightforward computations, we deduce   1 1 s 2 Hrr (r s) + s 2 Hr (r s) − 2 ν 2 − H (r s) = 0 r 4 or Hx x (x) + Hx (x) −

1 x2

  1 ν2 − H (x) = 0. 4

x

Setting H (x) = u(x)e− 2 , u solves uxx

1 1 − u(x) − 2 4 x

  1 2 ν − u(x) = 0 4

and v(x) = u(2x) satisfies vx x

1 − v(x) − 2 x

  1 2 ν − u(x) = 0. 4

Sharp kernel estimates for elliptic operators

√ √ It follows that v(x) = c1 x Iν (x) + c2 x K ν (x). Since the function H captures the behaviour of the heat kernel near the origin (the behaviour at infinity is governed by the origin, the gaussian factor) and since the resolvent of L ν is constructed with Iν near   x  we choose c2 = 0 and write c instead of c1 . Therefore, u(x) = v 2 = c x2 Iν x2 ,    rs rs H (r s) = u(r s)e− 2 = c r2s Iν r2s e− 2 ,   r s r s  − rs c (r − s)2 Iν e 2 F(r, s) = √ exp − 4 2 2 4π  2   rs rs c r + s2 = √ exp − Iν 4 2 4π 2 and

  (r − s)2 p(t, r, s) = √ exp − H t 4t 4π t  2   r + s2 rs rs c exp − Iν . = √ 4t 2t t 4π 2 1

rs 

(16)

THEOREM 4.9. Let pν (t, r, s) be the heat kernel of L ν . Then  2 

rs  r + s2 1√ exp − . pν (t, r, s) = r s Iν 2t 2t 4t Proof. The Laplace transform of the right hand side of (16) is given by, see [5, p. 200], ⎧  √ √ ⎨ √2c r s Iν (r λ)K ν (s λ) r ≤s 4π  2 √ √ ⎩ √2c r s Iν (s λ)K ν (r λ) r ≥ s. 2 √

4π

For c = 2π , it coincides with the kernel √ G(λ, r, s) of the resolvent operator (λ − −1 L ν ) , see Proposition 4.5 (note that 2π appears in the asymptotic expansion at infinity of Iν ). Let S(t) be the operator defined through the kernel pν , that is p with c = √ 2π and let G(t) be the Gauss–Weierstrass semigroup in R. By (16) and since H (r ) =  r  −r r c 2 Iν 2 e 2 is bounded by Lemma 4.4, then |S(t) f | ≤ C G(t)| f |, pointwise and

S(t) ≤ C in L 2 ((0, ∞)). Given f ∈ Cc∞ ((0, ∞)), let u(t, r ) = S(t) f (r ). By the construction of the kernel p, we have u t = L ν u pointwise. Finally, for λ > 0,  ∞  ∞  ∞ e−λt u(t, r ) dt = e−λt dt p(t, r, s) f (s) ds 0 0 ∞ 0 ∞ f (s) ds e−λt p(t, r, s) dt = 0 0  ∞ G(λ, r, s) f (s) ds. = 0

It follows that the Laplace transform of S(t) f coincides with the resolvent of L ν , hence, by uniqueness, S(t) is the generated semigroup and p = pν its kernel, as in the statement. 

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REMARK 4.10. Observe that in the proof we have avoided the verification of the semigroup law and of the strong continuity, which hold a posteriori. The general case, now, immediately follows. PROPOSITION 4.11. Let λ > 0. Then, for every f ∈ L 2 ((0, +∞), r c dr ),  ∞ c −1 − 2c (λ − L) f = r G(λ, r, s)s − 2 f (s) s c ds 0

with G(λ, r, s) defined in (13). Proof. The result immediately follows by Proposition 4.5 by observing that (λ − L)−1 = −1 (λ − L ν )−1 .  The same argument gives the heat kernel of L. THEOREM 4.12. Let p be the heat kernel of L with respect to the measure s c ds. Then,  2 

rs  c 1 √ r + s2 − c exp − s 2 r s Iν p(t, r, s) = r − 2 2t 2t 4t that is for every f ∈ L 2 ((0, +∞), r c dr )  2   ∞

rs  √ c 1 r + s2 − c exp − s 2 f (s)s c ds. r s Iν et L f (r ) = r − 2 2t 0 2t 4t The asymptotic behaviour of Bessel functions allows to deduce explicit bounds for the heat kernel p. We need first the following elementary lemma. LEMMA 4.13. With C() := 1≤

 1+ 1+ 2 2

, we have for every r, s > 0

1 ∧ rs 2 ≤ C() e|r −s| . (1 ∧ r )(1 ∧ s)

Proof. We observe preliminarily that ⎧ ⎪ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨s, 1 ∧ rs = r1 , (1 ∧ r )(1 ∧ s) ⎪ ⎪ ⎪ ⎪r, ⎪ ⎪ ⎪ ⎩1, s It is easily seen from (17) that

if r, s ≤ 1 or r, s ≥ 1; if r s ≤ 1 and r ≤ 1 ≤ s; if r s ≥ 1 and r ≤ 1 ≤ s;

(17)

if r s ≤ 1 and s ≤ 1 ≤ r ; if r s ≥ 1 and s ≤ 1 ≤ r.

1∧ r s (1∧r )(1∧s)

≥ 1 in every case. To prove the other

inequality, we fix  > 0 and consider the function g(r, s) :=

1∧ r s −|r −s|2 . (1∧r )(1∧s) e

For

Sharp kernel estimates for elliptic operators −|r0 −s| has maximum in fixed 0 <  r0 ≤ 1, the function 0 < s  → f (s) := se 2

s0 =

r0 + r02 + 2 2

se

which gives

−|r0

−s|2

≤ f (s0 ) =

r0 +

 r02 +

2  −|r0 −s0 |2

2

e

≤

1+

Now, using (17), we distinguish three cases: (i) if r, s ≤ 1 or r, s ≥ 1 we have g(r, s) =

2 e−|r −s|

≤1≤

 1+

2 

2  1+ 1+ 2 2

.

;

 1+ 1+ 2 ≤ ; 2 2 se−|r −s| ≤

2 se−|r −s|

(ii) if r s ≤ 1 and r ≤ 1 ≤ s we get, recalling (18), g(r, s) = 2 (iii) similarly, if r s ≥ 1 and r ≤ 1 ≤ s, g(r, s) = r1 e−|r −s| ≤  1+ 1+ 2 2

.

The other cases follow by symmetry interchanging the role of r and s.



PROPOSITION 4.14. The heat kernel p of L, with respect to the measure dμ = satisfies

s c ds,

  ν+ 1    2 s r |r − s|2 1 − 2c 1∧ √ 1∧ √ exp −(1 − ) p(t, r, s) ≤ C() (r s) t 4t t t   ν+ 1    2 2 c s r |r − s| 1 1∧ √ 1∧ √ . exp − p(t, r, s) ≥ C (r s)− 2 t 4t t t Proof. From Theorem 4.12 and using (12), we have  2 

rs  1 r + s2 − 2c √ p(t, r, s) = (r s) exp − r s Iν 2t 2t 4t   1 

1 r s ν+ 2 |r − s|2 − 2c .

(r s) 1∧ exp − t t 4t Applying Lemma 4.13, we conclude the proof.



5. Decomposition of the N-dimensional operator In this section, we prove that on each L 2n , defined in (6), L coincides with a one 2 dimensional Bessel operator. Since L 2μ = ∞ n=0 L n , this provides us a complete decomposition of the N -dimensional resolvent and kernel in terms of its one-dimensional counterparts. We write L = a Drr +

N −1+c b − 0 Dr − r r2

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where 0 is the Laplace–Beltrami on S N −1 and recall that 0 P = −λn P if P is a spherical harmonic of degree n. If v(x) = u(r )P(ω) ∈ Cc∞ (R N \{0}) ∩ L 2n , then   N −1+c b + λn Lv = au rr + u P(ω) := (L n u)(r )P(ω). (18) ur − r r2 The one-dimensional Bessel operator L n is associated with the form   ∞ N −1+c b + λn uv r a dr an (u, v) = au r vr + 2 r 0 considered in Sect. 4. Observe that if u P, v P ∈ Cc∞ (R N \{0}) ∩ L 2n , with u, v ∈ Cc∞ (0, ∞), then (18) can be written in the equivalent form   ∞ N −1+c b + λn au r vr + a(u P, v P) = uv r a dr = an (u, v). 2 r 0 However, this is not yet sufficient to conclude that the part of L into L 2n is the Bessel operator L n , since domain questions arise (both at the level of the domains of the operators and of the closures of the forms). In the following, we fix a normalized spherical harmonic P of degree n. First, we show that L and the projection Q P , defined in Sect. 2, commute. LEMMA 5.1. Let u ∈ Cc∞ (R N \{0}). Then L Q P u = Q P Lu. Proof. Let w(r ω) = (Q P u)(r ω) = v(r )P(ω), where  v(r ) = u(r ω)P(ω) dω. S N −1

Since 0 is self-adjoint in L 2 (S N −1 ) and 0 P = −λn P, we get N −1+c b + λn vr − v avrr − r2 r   N −1+c b + λn au rr − ur − u P(ω) dω = r r2 S N −1    λn 0 u Lu − 2 u − 2 P(ω) dω = r r S N −1     λn 1 Lu − 2 u dω − 2 = u0 P dω r r S N −1 S N −1  = Lu dω. S N −1

Since

  N −1+c b + λn vr − Lw = P(ω) avrr − v , r r2

the claim follows.



Sharp kernel estimates for elliptic operators

We prove now the continuity of Q P with respect to the norm

u 2a = u 22 + a(u, u) = u 22 − (Lu, u) LEMMA 5.2. Let u ∈ Cc∞ (R N \{0}). Then, Q P u 2a ≤ u 2a . Proof. We write u = u 1 + u 2 where u 1 = Q P u and u 2 = (I − Q P )u. Then, a(u, u) = −(Lu, u) = −(Lu 1 , u 1 ) − (Lu 2 , u 2 ) − (Lu 1 , u 2 ) − (Lu 2 , u 1 ). By observing that, by Lemma 5.1, (−Lu 1 , u 1 ) = (−L Q P u, Q P u) = a(Q P u, Q P u); (−Lu 2 , u 2 ) = (−L(I − Q P )u, (I − Q P )u) = a((I − Q P )u, (I − Q P )u); (−Lu 1 , u 2 ) = (−L Q P u, (I − Q P )u) = −(Q P Lu, (I − Q P )u) = 0; (−Lu 2 , u 1 ) = (−L(I − Q P )u, Q P u) = −((I − Q P )Lu, Q P u) = 0, we get a(u, u) = a(Q P u, Q P u) + a((I − Q P )u, (I − Q P )u). The thesis follows from the positivity of the form and the boundedness of Q P in  L 2μ . REMARK 5.3. Observe that the proof above yields a(u, u) =

n 

a(u i Pi , u i Pi )

i=1

if u =

n

i=1 u i (r )Pi (ω)

with Pi spherical harmonics.

LEMMA 5.4. Let u, v ∈ D(˜a). Then, Q P u, Q P v ∈ D(˜a) and a˜ (Q P u, v) = a˜ (u, Q P v). Proof. If u, v ∈ Cc∞ (R N \{0}), the claim follows by Lemma 5.1. Let u, v ∈ D(˜a). There exist (u n )n∈N , (vn )n∈N ∈ Cc∞ (Cc∞ (R N \{0})) such that u n → u and vn → v in L 2μ and a(u n , u n ), a(vn , vn ) are Cauchy sequences. By Lemma 5.2, (Q P u n )n∈N is a Cauchy sequence in D(˜a). Since Q P u n → Q P u in L 2μ , we get Q P u in D(˜a) and Q P u n → Q P u in D(˜a). The same applies to v. Since, by Lemma 5.1, a˜ (Q P u n , vn ) = a˜ (u n , Q P vn ), the claim follows by letting n to infinity.



LEMMA 5.5. Let u ∈ D(L), then Q P u ∈ D(L) and L Q P u = Q P Lu. In particular, for f ∈ L 2μ , (λ − L)−1 Q P f = Q P (λ − L)−1 f , λ > 0, and et L Q P f = Q P et L f .

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Proof. By assumption a˜ (u, v) = −(Lu, v), ∀ v ∈ D(˜a). The above lemma yields a˜ (Q P u, v) = a˜ (u, Q P v) = −(Lu, Q P v) = −(Q P Lu, v). Therefore, Q P u ∈ D(L) and L Q P u = Q P Lu. The last assertion follows immediately.  Finally, let us prove that the part of L in L 2n is L n , by showing that the restriction of a˜ onto L 2n coincide with a˜ n . Note that this last form is defined on functions of one variable u = u(r ). However, for a fixed P, we identify u with u P and use a˜ n (u P, v P) for a˜ n (u, v). LEMMA 5.6. The forms a˜ and a˜ n coincide on L 2P . It follows that u(r )P(ω) belongs to D(L) if and only if u ∈ D(L n ) and, in this case, L(u(r )P(ω)) = P(ω)L n u(r ). Finally, (λ − L)−1 (u(r )P(ω)) = P(ω)(λ − L n )−1 u(r ) and et L (u(r )P(ω)) = P(ω) et L n u(r ). Proof. Let u, v ∈ Cc∞ (0, ∞). Then, as shown at the beginning of this section, a˜ n (u, v) = a˜ (u P, v P). Let now u, v ∈ D(˜an ). There exist (u k )k∈N , (vk )k∈N such that u k → u in L 2 (0, ∞) and an (u k , u k ) is a Cauchy sequence. Then, u k P → u P in L 2μ and a(u k P, u k P) is a Cauchy sequence. This implies that u P ∈ D(˜a). In similar way, we can argue for v. Since a˜ n (u k , vk ) = a˜ (u k P, vk P), the equality a˜ n (u, v) = a˜ (u P, v P) follows letting k to infinity. Conversely, let u P ∈ D(˜a) ∩ L 2P and let u k ∈ Cc∞ (R N \{0}) such that u k goes to u P in L 2μ and a(u k , u k ) is a Cauchy sequence. Then, Q P u k → u P in L 2μ and, by Lemma 5.2, a(Q P u k , Q P u k ) is a Cauchy sequence. Since a(Q P u k , Q P u k ) = a˜ n (Q P u k , Q P u k ), u P = Q p (u P) ∈ D(˜an ). The last statements now follow using also Lemma 5.4, since for v ∈ D(˜a), Q P v ∈ D(˜an ) and a˜ (u P, v) = a˜ (Q P (u P), v) = a˜ (u P, Q P v) = a˜ n (u P, Q P v).  The above two lemmas yield (λ − L)−1 ( f P) = P(λ − L n )−1 f for λ > 0 and P) = Pet L n f , when f = f (r ) and P is a normalized spherical harmonic ∞ 2 of degree n. This fact, together with the decomposition L 2μ = n=0 L n , allows to et L ( f

Sharp kernel estimates for elliptic operators

factorize et L as the direct sum of the one-dimensional semigroups et L n . We consider, using Proposition 2.2, the projection onto L 2n is given by Q n ( f )(r, ω) =

an 

 Pin (ω)

i=1

S N −1

f (r, η)Pin (η)dη.

an

an If f = i=1 f i (r )Pin (ω), then Lemma 5.6 gives et L f = Q n eet L f = i=1 Pin (ω) t L t L e n f i (r ) which we shorten to e n f , with a little abuse of notation. Then, we have  t Ln Q f . PROPOSITION 5.7. For f ∈ L 2μ , et L f = ∞ n n=0 Q n e Proof. Let f ∈ L 2μ , using Lemmas 5.5, 5.6 we obtain et L f = et L

∞  n=0

= =

∞ 

e

tL

Qn ( f ) = an 

∞ 

et L Q n ( f )

n=0





Pni (ω)

n=0 i=1 an ∞   Pni (ω)et L n n=0 i=1

S N −1

f (r, η)Pni (η)dη



S N −1

f (r, η)Pni (η)dη

=

∞ 

Q n et L n Q n f.

n=0



6. Kernel estimates In order to state and prove the main result of this paper, we recall that the formal adjoint of L, with respect to the Lebesgue measure, is given by L ∗ =  + (a − 1)

N  xi x j x Di j + c∗ 2 · ∇ − b∗ |x|−2 |x|2 |x|

i, j=1

where c∗ = 2(N − 1)(a − 1) − c and b∗ = b + (N − 2)(c − (N − 1)(a − 1)). Let us compute the numbers s1∗ , s2∗ , D ∗ defined as in (3), (4) and relative to L ∗ . We have   N − 1 + c∗ − a 2 b∗ + D := = D, a 2a N − 1 + c∗ − a √ ∗ (a − 1)(N − 1) − c ∗ s1,2 ∓ D = s1,2 + = N − 2 − s2,1 . := 2a a ∗

Recalling that γ = s1 =

N −1+c a

− N + 1 we have also

√ √ N γ N γ − 1 − D + , s1∗ := −1− D− . 2 2 2 2

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Since upper and lower bounds for N = 1 have been already deduced in Proposition 4.14, we assume N ≥ 2. We write f (x) g(x) if for some C1 , C2 > 0, C1 g(x) ≤ f (x) ≤ C2 g(x). Before stating the main result, we recall that in [16] we proved that the semigroup is analytic in the right half plane and satisfies the following complex upper bounds. THEOREM 6.1. ([16, Theorem 4.14]) Let  = R N \{0}. For every ε > 0, there exist Cε > 0 and m ε > 0 such that the heat kernel p of L, with respect to the measure dμ = |y|γ dy, satisfies for z ∈ C+ with | arg z| ≤ π2 − ε and (x, y) ∈  ×  | p(z, x, y)|  ≤ Cε |z|

− N2

|x|

− γ2

|y|

− γ2



|x|

∧1

1

|z| 2

|y| 1

|z| 2

− N +1+√ D 2

∧1

  |x − y|2 exp − m ε |z| (19)

The main result of this paper consists in showing that the above upper bound admits a lower bound for positive t. THEOREM 6.2. Let  = R N \{0}. The heat kernel p of L, with respect to the measure dμ = |y|γ dy, satisfies for (x, y) ∈  ×  p(t, x, y) t

− N2

− γ2

|x|

− γ2





|x|

|y|

∧1

1

t2

|y| 1

t2

∧1

 − N +1+√ D 2

  c|x − y|2 . exp − t (20)

The constant c > 0 may differ in the upper and lower bounds. Clearly, the upper bound follows from Theorem 6.1 REMARK 6.3. Using Lemma 4.13, one can replace 

|x| 1

t2

  − N +1+√ D 2 |y| ∧1 ∧1 1 t2

 with

− N +1+√ D 2 |x||y| ∧1 t

in the above Theorem, slightly changing the constant c. The previous kernel estimate can be rewritten in the following equivalent form, see [16, Corollary 4.15]. COROLLARY 6.4. N

p(t, x, y) t − 2 |y|−γ



|x| 1

t2

 −s1  −s ∗  1 |y| c|x − y|2 ∧1 ∧ 1 exp − 1 t t2

REMARK 6.5. We remark that the estimate in (20) becomes √

p(t, x, y) Ct −1−

D

|x|−s1 |y|−s1 ,

|x| |y| √ ≤ 1, √ ≤ 1 t t

Sharp kernel estimates for elliptic operators

and, using Corollary 6.4,   N m|x − y|2 , p(t, x, y) Ct − 2 |y|−γ exp − t

|x| |y| √ ≥ 1, √ ≥ 1. t t

We need some further preparation for the proof of the lower bound. Let pn (t, r, ρ) be the parabolic kernels of the Bessel operators L n with respect to N −1+c the measure ρ a dρ. Theorem 4.12 yields  2 

rρ  N −1+c−a r + ρ2 1 exp − , pn (t, r, ρ) = I √ Dn (rρ)− 2a 2at 2at 4at  N −1+c−a 2 n where Dn = b+λ and we write D for D0 . a + 2a In order to show that the heat kernel of L is the sum of the heat kernels of L n , we need the following lemma which will be also useful to prove lower bounds for small values of |x|, |y|. For this reason, we do not make any attempt to improve the bounds below for large r, ρ. LEMMA 6.6. There exists h ∈ C([0, ∞[), with h(0) = 0 such that % % %  % % % (n) %≤ % p (t, r, ρ)Z (η) pn (t, r, ρ)Z(n) n ω ω (ω) % % % n≥1 %n≥1

rρ √ D1 −√ D rρ  . h ≤ C p0 (t, r, ρ) 4at 4at  (n) In particular for every t > 0 the series n≥1 pn (t, r, ρ)Zω (η) converges uniformly on compact sets. (n)

(n)

Proof. We use Proposition 2.1 (ii) for the estimate |Z ω (η)| ≤ Z ω (ω) ≤ Cn N −2 . Then % % & % % % % 1 r 2 + ρ 2  √ rρ  (n) (n) − N −1+c−a % % 2a ≤ Zω (ω). p (t, r, ρ)Z (η) exp − I Dn (rρ) n ω % 2at % 4at 2at % %n≥1 n≥1

(21) √ √ √ We use (α+β) (α)(β) √ if α, β ≥ δ to obtain (m + D+1+ Dn − D) ≥ √ ≥ Cδ√ C(m + 1 + D)( Dn − D) for every n ≥ 1. Then, 

I √ Dn

n≥1

=

 n≥1

rρ  Z(n) ω (ω) 2at

Z(n) ω (ω)

∞  m=0

m!(m +

1 √

rρ 2m+√ Dn −√ D+√ D Dn + 1) 4at

∞

rρ √ Dn −√ D 

rρ 2m+√ D  1 = Z(n) √ ω (ω) 4at m!(m + Dn + 1) 4at n≥1 m=0

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≤C

 n≥1

= C I√ D

n

N −2

∞

rρ √ Dn −√ D

rρ 2m+√ D  1 1 √ √ √ 4at ( Dn − D) m=0 m!(m + 1 + D) 4at

rρ  rρ √ D1 −√ D 

rρ √ Dn −√ D1 1 n N −2 √ . √ 2at 4at 4at ( D − D) n n≥1

√ √ Since Dn ≈ cn, c = 1/ a as n → ∞, by the asymptotic of the Gamma function the series h(s) =



n N −2 s

√

√ Dn − D1

√

1

( Dn −

n≥1

√

D)

converges uniformly on compact sets of [0, ∞[ and does not vanish at 0. Then, (21) yields % % % % % % (n) % pn (t, r, ρ)Zω (η)%% % %n≥1 % √ √   2 C r + ρ 2 √ rρ  rρ  D1 − D rρ  − N −1+c−a 2a ≤ I D exp − h (rρ) 2at 4at 2at 4at 4at √ √

rρ  D1 − D rρ  = C p0 (t, r, ρ) . h 4at 4at  We can now obtain the announced kernel decomposition. PROPOSITION 6.7. Let p be the heat kernel of L with respect to the measure dμ(y) = |y|γ dx. Then, for x = r ω, y = ρη, r, ρ > 0, |ω| = |η| = 1 we have  ∞  2 N −1+c−a 1 r + ρ 2  √ rρ  (n) Zω (η) I Dn (rρ)− 2a exp − 2at 4at 2at n=0  pn (t, r, ρ)Z(n) = ω (η).

p(t, x, y) =

n≥0

Proof. We use Proposition 5.7. For f ∈ L 2μ et L f = = =

∞ 

Q n et L n Q n f

n=0 an ∞  

 Pin (ω)

n=0 i=1 ∞  ∞  n=0 0

S N −1

∞ 0

S N −1

pn (t, r, ρ)Pin (η) f (ρ, η)ρ

pn (t, r, ρ)Z(n) ω (η) f (ρ, η)ρ

N −1+c a

N −1+c a

dηdρ.

dηdρ

Sharp kernel estimates for elliptic operators N If f is continuous with  compact support in R \{0}, since by Proposition 6.6 the series φ(t, x, y) = pn (t, r, ρ)Z(n) ω (η) converges uniformly on compact sets, n≥0 we interchange the series with the integrals thus obtaining from above  ∞ N −1+c et L f (x) = φ(t, r ω, ρη) f (ρ, η)ρ a dηdρ S N −1 0 = φ(t, x, y) f (y)|y|γ dy. RN

On the other hand



et L f (x) = =

p(t, x, y) f (y)|y|γ dy

N R∞ 

0

S N −1

p(t, r η, ρω) f (ρ, η)ρ

N −1+c a

dηdρ.

For any fixed t > 0, x ∈ R N \{0} the L 1loc -function p(t, x, ·) − φ(t, x, ·) has integral zero against any continuous and compactly supported function. Therefore it vanishes.  √ We now start proving the lower estimate (20) near the origin, that is for |x|/ t, √ |y|/ t small, see also Remark 8.3. l. In this case, the behaviour of p is the same of radial part p0 . LEMMA 6.8. There exist δ > 0 such that if

|x| √ t

≤ δ and √

p(t, x, y) ≥ C p0 (t, r, ρ) ≥ Ct −1−

D

|y| √ t

≤ δ, then

|x|−s1 |y|−s1 .

Proof. By Proposition 6.7 p(t, x, y) = p0 (t, r, ρ)

1 |S N −1 |

+



pn (t, r, ρ)Z(n) ω (η).

n≥1

|x| |x| Next we choose δ > 0 such that if √ ≤ δ and √ ≤ δ, then C t t 1 1 2 |S N −1 |

 rρ √ D1 −√ D  rρ  h 4at ≤ 4at

and use Proposition 6.6 to infer that % % % % % 1 % 1 (n) % pn (t, r, ρ)Zω (η)%% ≤ p0 (t, r, ρ) N −1 . % |S | % 2 %n≥1

The proof is now completed by the explicit expression of p0 (t, r, ρ) =

 2 

rρ  N −1+c−a r + ρ2 1 exp − , I√ D (rρ)− 2a 2at 2at 4at

taking into account that the exponential term plays no role near the origin and using the behaviour of I√ D near 0. 

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REMARK 6.9. Observe that the above proof works if the product δ.

|x √y| t

is less than

|y| |x| The lower bound in (20) for large values of √ and √ is in the next Proposition. t t Note that if p is the heat kernel of L with respect to the measure dμ = |y|γ dy, then the heat kernel with respect to the Lebesgue measure is p(t, x, y)|y|γ . We prove, first, a preliminary lemma which shows a regularity property of the semigroup when applied to test functions.

LEMMA 6.10. Let  = R N \{0}, f ∈ Cc∞ () and set u(t, x) := et L f (x). Then u ∈ C 1,2 (]0, ∞[×) ∩ C ([0, ∞[×) and for every δ > 0 there exists Cδ > 0 s.t. |u(t, x)| ≤ Cδ for every t ≥ 0 and |x| ≥ δ. Proof. By using (20), we immediately have the required boundedness of u for t ≥ 0 and |x| ≥ δ. To prove the regularity properties, we preliminary observe that by 2n () and so fixing a sufficiently Proposition 3.5 we have Cc∞ () → D(L n ) → Hloc n 2 large n ∈ N we get D(L ) → C (). Let us consider now the semigroup in D(L n ); we have obviously u(t, ·) ∈ D(L n ) ⊆ C 2 () and since u is a solution of dtd u(t, ·) = Lu(t, ·) in D(L n ), the embedding D(L n ) → C 2 () yields that the time derivative is a classical derivative and we have also dtd u(t, x) = Lu(t, x) pointwise. This proves u ∈ C 1,2 (]0, ∞[×). Analogously u(t, ·) → f in D(L n ) as t → 0 and so pointwise and this implies u ∈ C ([0, ∞[×).  PROPOSITION 6.11. Let δ > 0 be fixed. Then, there exist positive constants C, c |y| |x| ≥ δ and √ ≥δ such that for √ t t   N |x − y|2 p(t, x, y) ≥ Ct − 2 |y|−γ exp −c . t Proof. Given δ > 0 let us set ω0 := 0 if b ≤ 0 and ω0 := 4b if b ≥ 0. We consider the δ 2

b N uniformly elliptic operator L 0 := L + |x|2 in Cb R \B δ with Dirichlet boundary 2

conditions. The generated the semigroup et L 0 is represented by a kernel q0 (t, x, y) satisfying      |x| − 2δ |y| − 2δ |x − y|2 − N2 (22) q0 (t, x, y) ≥ C 1 ∧ √ exp −c 1∧ √ t t t t for some positive constants C, c and t > 0, |x|, |y| ≥ 2δ , see [3, Theorem 3.8]. Given

 0 ≤ f ∈ Cc∞ Rn \ B¯ δ , let u(t, ·) = eω0 t et L f (·) and v(t, ·) = et L 0 f (·). Then, both 2

u and v are positive and satisfy d dt u(t, x)

= (L 0 −

u(0, x) = f (x)

b |x|2

+ ω0 )u(t, x)

t > 0, x ∈ , x ∈ ,

Sharp kernel estimates for elliptic operators

and

⎧ d ⎪ ⎪ ⎨ dt v(t, x) = L 0 v(t, x) v(0, x) = f (x) ⎪ ⎪ ⎩v(t, x) = 0

t > 0, |x| > 2δ , |x| ≥ 2δ ,

|x| = 2δ ,

respectively. Both u, v are bounded classical solution, continuous up to t = 0, see Lemma 6.10 for u. 

Now we observe that (∂t − L 0 )(u − v) = ω0 − |x|b 2 u ≥ 0 in ]0, 1] × R N \ B¯ δ , 2

u(0, x) − v(0, x) = 0 for |x| ≥ 2δ and u(t, x) − v(t, x) = u(t, x) ≥ 0 for 0 ≤ t ≤ 1, |x| = 2δ . By the maximum principle u(t, x) ≥ v(t, x) for 0 ≤ t ≤ 1, |x| ≥ 2δ , that is   eω0 t p(t, x, y)|y|γ f (y)dy ≥ q0 (t, x, y) f (y)dy. R N \Bδ



R N \Bδ

By the arbitrariness of f ∈ Cc∞ Rn \ B¯ δ and using (22), we get for t = 1, |x|, |y| ≥ 2 δ 

p(1, x, y)|y|γ ≥ q0 (1, x, y)e−ω0 ≥ C exp −c|x − y|2 . The scaling equality (11) now gives immediately the statement.



Finally, we can prove the lower bound in Theorem 6.2 Proof of Theorem 6.2. By the scaling property (11), we may assume that t = 1 and prove that   √ γ N p(1, x, y) ≥ C(|x||y|)− 2 ((|x| ∧ 1)(|y| ∧ 1))− 2 +1+ D exp −c|x − y|2 . (23) We use Proposition 6.7 to write p(1, x, y) =

 ∞  2 N −1+c−a 1 r + ρ 2  √ rρ  (n) Zω (η). I Dn (rρ)− 2a exp − 2a 4a 2a

(24)

n=0

Since by Lemma 6.8 and Proposition 6.11 inequality (23) holds if |x| ≤ δ, |y| ≤ δ or |x| ≥ δ, |y| ≥ δ, it holds whenever |x| = |y|. Let therefore x = r ω, y = r η. Then, we obtain  2  2  ∞ r r 1 − N −1+c−a √ a r Z(n) exp − I Dn ω (η) 2a 2a 2a n=0   √ ≥ Cr −γ (r ∧ 1)−N +2+2 D exp −cr 2 |ω − η|2 . Since (N − 1 + c − a)/a − γ = N − 2 we obtain  2  2 ∞   √  r r (n) N −2 −N +2+2 D √ I Dn exp Zω (η) ≥ Cr exp −cr 2 |ω − η|2 (r ∧ 1) 2a 2a n=0

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and, changing r 2 to rρ,

rρ   Z(n) I √ Dn ω (η) 2a n≥0

≥ C(rρ)

N −2 2

1

((rρ) 2 ∧ 1)−N +2+2

√

D

exp

 rρ  2a

  exp −crρ|ω − η|2 .

By putting the last estimate in (24), we deduce N −2

N −1+c−a

1

p(1, x, y) ≥ C (rρ) 2 − 2a ((rρ) 2 ∧ 1)−N +2+2    rρ  exp −crρ|ω − η|2 exp 2a

√

D

  2 r + ρ2 exp − 4a

  √ N −2 N −1+c−a 1 (r − ρ)2 = C (rρ) 2 − 2a ((rρ) 2 ∧ 1)−N +2+2 D exp − 4a   exp −crρ|ω − η|2   √ γ 1 ≥ C (rρ)− 2 ((rρ) 2 ∧ 1)−N +2+2 D exp −m|x − y|2 ,

1 with m ≥ c, 4a . To complete the proof of (23) it suffices now to apply Lemma 4.13. 

7. Green function estimates In this section, we prove sharp upper and lower bounds for the Green function G λ defined by  ∞ e−λt p(t, x, y) dt, x, y ∈ R N \ {0} . G λ (x, y) := 0

For λ > 0 the integral converges pointwise, due to Theorem 6.2, and defines the resolvent of λ − L (the kernel being written with respect to the measure |y|γ dy). However, we consider also the case λ = 0 when the integral converges, that is when D > 0. For l ≥ 1, m ≥ 1, α ≥ 0, β ∈ R let   −l+m

α β2 ∧1 . exp − F(t) := t −l t t We write also F(l, m, α, β, t) := F(t) in order to emphasize the explicit dependence on the parameters. With this notation, the estimates in Theorem 6.2, see Remark 6.3, take the form   √ N − γ2 , 1 + D, |x||y|, c|x − y|, t , F p(t, x, y) (|x||y|) 2 with the understanding that the constant c may differ in the upper and lower bounds. Defining  α  ∞  ∞ 2 β2 −λt −λt −l − βt −l+m e F(t)dt = e t e dt + α e−λt t −m e− t dt I (α, β) := 0

0

α

(25)

Sharp kernel estimates for elliptic operators

we have for l =

N 2

and m = 1 +

√

D γ

G λ (x, y) (|x||y|)− 2 I (|x||y|, c|x − y|) .

(26)

We treat separately the cases λ = 0 and λ > 0 for clarity and also because of technical details. 7.1. The Green function G 0 Our main result is the following. THEOREM 7.1. Let  = R N \{0} and let us suppose that D > 0. For (x, y) ∈ ×  with x = y, the Green function G 0 of L, with respect to the measure dμ = |y|γ dy, satisfies the estimates if N > 2  √ D− N −2 2 γ |x||y| (27) (|x||y|) 2 G 0 (x, y) |x − y|2−N 1 ∧ |x − y|2 and if N = 2

⎧ √ D ⎪ (|x||y|) ⎪ ⎪ √ , if ⎨ γ 2 D (|x||y|) 2 G 0 (x, y) |x − y|   ⎪ |x − y|2 ⎪ ⎪ , if ⎩1 − log |x||y|

|x−y|2 |x||y|

≥ 1;

|x−y|2 |x||y|

≤ 1.

REMARK 7.2. We remark that (27) becomes |x − y|2 ≤1 |x||y|

γ

(|x||y|) 2 G 0 (x, y) |x − y|2−N , and

√

D− N 2−2 √ y|2 D

|x − y|2 ≥ 1. |x||y| |x − √ The asymptotic behaviour of G 0 depends on the sign of D − (N − 2)/2, see Remark 7.4 below. γ 2

(|x||y|) G 0 (x, y)

(|x||y|)

,

The proof is an immediate consequence of (26) and the lemma below, recalling that α = |x||y| and β = |x − y|. We use the incomplete Gamma functions defined by  r  ∞ −t a−1 e t dt, γ (a, r ) := e−t t a−1 dt. (a, r ) := r

0

Clearly (a, r ) + γ (a, r ) = (a), moreover (a, r ) r a−1 e−r as r → ∞ and a (0, r ) − log r , γ (a, r ) ≈ ra as r → 0. In particular, (a, r ) γ (a, r )

r a−b , for r ≥ 1, a, b ≥ 0

r a−b , for r ≤ 1, a, b > 0. (b, r ) γ (b, r ) (28)

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LEMMA 7.3. Let l ≥ 1, m > 1, α, β > 0 and F(t) = t

−l

  −l+m β2 ∧1 . exp − t t

α

Then if l > 1 

∞ 0

and if l = 1



∞ 0

  α m−l F(t) dt β 2−2l 1 ∧ 2 β

⎧ 1−m ⎨ β2 , α

 F(t) dt

⎩1 − log β 2 , α

Proof. By the change of variables s = 

β2 t ,

if if

β2 α β2 α

≥ 1; ≤ 1.

we have

∞

F(t) dt 0

=β

−2l+2



∞ β2 α

s

l−2

exp (−s) ds + α

−(l−m) −2m+2



β

β2 α

s m−2 exp (−s) ds

0

    β2 β2 = β −2l+2  l − 1, + α −l+m β −2m+2 γ m − 1, . α α If

β2 α

≥ 1, (28) yields    2 l−m   β2 β β2  l − 1,  .  m − 1, α α α

Then 

∞ 0

 β2 F(t) dt α β γ m − 1, α  2 l−m   β β2 −2l+2  m − 1, +β α α   

 2 β2 β −l+m −2m+2 +  m − 1, =α γ m − 1, β α α −l+m −2m+2



= α −l+m β −2m+2  (m − 1) . The case

β2 α

≤ 1 and l > 1 is similar. Using    2 m−l   β2 β β2 γ m − 1,

, γ l − 1, α α α

(29)

Sharp kernel estimates for elliptic operators

we get  2 m−l      ∞ β β2 β2 + β −2l+2  l − 1, F(t) dt  α −l+m β −2m+2 γ l − 1, α α α 0   

 2 2 β β = β −2l+2 γ l − 1, +  l − 1, = β −2l+2  (l − 1) . α α Finally, if γ and  ∞

β2 α

≤ 1 and l = 1, then

  2 m−1   2   β β β2 β2

− log ,  0, m − 1, α α α α

F(t) dt c1 α −1+m β −2m+2



0

β2 α

m−1

 − c2 log

β2 α





1 − log

β2 α

 . 

∞

REMARK 7.4. Observe that when m ≥ l > 1, then 0 F(t) dt is bounded from above by β 2−2l and tends to 0 as α m−l for α → 0. When m < l, instead, it blows up as α → 0 like α m−l and is bounded if α, β ≥ δ > 0. If l = 1, the integral behaves as log α for α → ∞. 7.2. The Green function G λ , λ > 0 Let us consider now λ > 0. Using again (26), we look for estimates of I (α, β). √ We observe that if M√λ is the dilation defined by M√λ u(x) = u( λ x), the scaling property M√λ L M√ −1 = Lλ implies that 

λ

  1 √ M λ (I − L)−1 M√ −1 f (x) (λ − L)−1 f (x) = λ λ   √ w 1 |w|γ dw G 1 ( λ x, w) f √ = λ RN λ  √ √ γ +N −2 2 G 1 ( λ x, λ y) f (y) |y|γ dy. =λ RN

This proves that G λ (x, y) = λ

γ +N −2 2

√ √ G 1 ( λ x, λ y)

and allows us to treat only the case λ = 1 by estimating the integral  ∞  α β2 β2 e−t t −l e− t dt + α −l+m e−t t −m e− t dt. I (α, β) = α

0

For l, m ≥ 1, let h(t) := e−t t −l e−

β2 t

, g(t) := t l−m h(t)

(30)

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G. Metafune et al.

and



α

H (α, β) :=



α

h(t) dt, G(α, β) :=

0

g(t) dt. 0

We have the following identities, see [5], formula (29), page 146,  ∞ H (β) := h(t) dt = 2β 1−l K l−1 (2 β) , 0  ∞ g(t) dt = 2β 1−m K m−1 (2 β) G(β) :=

(31)

0

where the K ν are the modified Bessel functions. With this notation, the integral in (25) takes the form I (α, β) := H (α, β) + α m−l (G(β) − G(α, β)) . Let us observe that I (α, β) is decreasing with respect to β and, considered as a function of α, is increasing when m − l ≥ 0 and decreasing otherwise, since ∂ I (α, β) = (m − l) α m−l−1 (G(β) − G(α, β)) . ∂α We split the proof between the cases 0 ≤ D ≤ (N − 2)2 /4 and D > (N − 2)2 /4 and note that for Schrödinger operators the above conditions correspond to b ≤ 0, b > 0, respectively. 2  7.2.1. The case 0 ≤ D ≤ N 2−2 Since l = N /2 and m = 1 +

√

D, this corresponds to m ≤ l.

THEOREM 7.5. Let λ > 0,  = R N \{0} and let us suppose D ≥ 0. For (x, y) ∈  ×  with x = y, the Green function G λ , with respect to the measure dμ = |y|γ dy, satisfies the estimates (i) if N > 2 and D > 0 γ 2

(|x||y|) G λ (x, y) e

√ −c λ|x−y|

|x − y|

2−N

 √ D− N −2 2 |x||y| 1∧ . 2 |x − y|

(ii) If N = 2 and D = 0

⎧ √ ⎨e−c λ |x−y| , γ

√  (|x||y|) 2 G λ (x, y)

⎩1 − log λ |x − y| ,

√ λ |x − y| ≥ 1; √ if λ |x − y| < 1.

if

(iii) If N > 2 and D = 0 γ

(|x||y|) 2 G λ (x, y) ⎧

 2−N √ ⎪ 2 ⎨e−c λ|x−y| (|x||y|) ∧ |x−y| √ , λ

 √ 2−N ⎪ 2−N ⎩|x − y| ∨ (|x||y|) 2 (1 − log( λ |x − y|) ,

√ λ |x − y| ≥ 1; √ if λ |x − y| < 1.

if

Sharp kernel estimates for elliptic operators

All the constants appearing in the above estimates, including those hidden in the symbol , do not depend on λ; the generic constants c in the exponentials may differ in the upper and lower bounds. The logarithmic term is due either to the dimension N = 2 or to the degeneracy of the discriminant, D = 0. Note that when N > 2 but D = 0√the estimates are 2−N influenced both by the terms |x − y|2−N and (|x||y|) 2 (1 − log( λ |x − y|). The proof of the theorem follows immediately from the scaling property (30) and from the following lemma, by noticing that powers of |x − y| can be absorbed into the exponential, when |x − y| is large. LEMMA 7.6. If m ≤ l, we have     max H (β), α m−l G(β) ≤ I (α, β) ≤ 2 max H (β), α m−l G(β)

(32)

and therefore (i) if 1 < m ≤ l, we have ⎧

m−l ⎪ ⎨e−2β α m−l β 21 −m 1 ∧ β , α I (α, β)

m−l ⎪ ⎩β 2−2l 1 ∧ α2 , β

if β ≥ 1; if β < 1.

(ii) If m = l = 1, 1

I (α, β) 2K 0 (2β)

β − 2 e−2β

if β ≥ 1;

1 − log β,

if β < 1.

(iii) If 1 = m < l, we have 1

I (α, β)

Proof. We have  I (α, β) =

α

e−2β β − 2 (α ∧ β)1−l ,

if β ≥ 1;

β 2−2l ∨ α 1−l (1 − log β),

if β < 1.

 h(t)dt + α



∞



∞

g(t)dt ≤ h(t)dt + α 0 α 0   = H (β) + α m−l G(β) ≤ 2 max H (β), α m−l G(β) . m−l

On the other hand  α  I (α, β) = h(t)dt + α m−l 0

and

 I (α, β) =

∞ α

 t l−m h(t)dt ≥ 0



α

t 0

m−l

α

g(t)dt + α



α

≥ α m−l 0

m−l

g(t)dt +

α

 h(t)dt +

∞ α

g(t)dt 0

h(t)dt = H (β)

∞

g(t)dt

∞ g(t)dt = α m−l G(β). α



∞

m−l

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  It follows that I (α, β) ≥ max H (β), α m−l G(β) and this proves (32). It follows from (31) that     max H (β), α m−l G(β) = max 2β 1−l K l−1 (2β) , α m−l 2β 1−m K m−1 (2β) . If m = l, then I (α, β) = 2β 1−l K l−1 (2β) and Lemma 4.4 gives the result. The same Lemma applies in the other cases and we get for β ≥ 1   1 1 I (α, β) e−2β β − 2 max β 1−l , α m−l β 1−m = e−2β β 2 −m (α ∧ β)m−l . For β ≤ 1, we have, if m > 1, 

I (α, β) max β and if 1 = m,

2−2l

,α

β

m−l 2−2m



=β

2−2l

  α m−l 1∧ 2 β

  I (α, β) max β 2−2l , α 1−l (1 − log β) .

7.2.2. The case D >



 N −2 2 2

Since l = N /2 and m = 1 +

√

D, this corresponds to m > l.

2  THEOREM 7.7. Let λ > 0,  = R N \{0} and let us suppose D > N 2−2 . For (x, y) ∈  ×  with x = y, the Green function G λ , with respect to the measure dμ = |y|γ dy, satisfies the estimates (i) if N > 2, γ 2

(|x||y|) G λ (x, y) e

√ −c λ|x−y|

|x − y|

2−N

 √ D− N −2 2 |x||y| 1∧ . |x − y|2

(ii) If N = 2, γ

(|x||y|) 2 G λ (x, y) √ ⎧ √ −c λ|x−y| (1 ∧ λ |x||y|) D , ⎪ e ⎪ ⎪

√  ⎪ ⎪ ⎪ λ |x − y| , ⎨1 − log −√ D

|x−y|2 ⎪ ⎪ , ⎪ |x||y| ⎪  ⎪ 2 ⎪ |x−y| ⎩1 − log , |x||y|

√ λ |x − y| ≥ 1; √ if λ |x − y| < 1 ≤ λ |x||y|; √ 2 if λ |x||y|, λ |x − y| < 1, |x−y| |x||y| ≥ 1; √ 2 if λ |x||y|, λ |x − y| < 1, |x−y| |x||y| ≤ 1.

if

All the constants appearing in the above estimates, including those hidden in the symbol , do not depend on λ; the generic constants c in the exponentials may differ in the upper and lower bounds.

Sharp kernel estimates for elliptic operators

As before, the proof follows from some elementary but tedious lemmas on the integrals I (α, β). LEMMA 7.8. If m > l, we have   I (α, β) ≤ min H (β), α m−l G(β) . In particular, (i) if l > 1, I (α, β) ≤ C e

  α m−l 1∧ 2 . β

−β 2−2l

β

(ii) If l = 1, ⎧

m−1 1 ⎪ ⎨e−2β α m−1 β 2 −m 1 ∧ βα ,    I (α, β) ≤ C 1−m 2 ⎪ ⎩min βα , 1 − log β ,

if β ≥ 1; if β < 1.

Proof. We have  I (α, β) =



α



∞

h(t)dt + α g(t)dt = 0 α  ∞ + α m−l g(t)dt ≤ α m−l G(β) m−l

α

t m−l g(t)dt

0

α

and



α

I (α, β) = 0

 h(t)dt + α m−l

∞ α

t l−m h(t)dt ≤ H (β).

It follows that   I (α, β) ≤ min H (β), α m−l G(β)   = min 2β 1−l K l−1 (2β) , α m−l 2β 1−m K m−1 (2β) . Using Lemma 4.4, we have for β ≥ 1   1 I (α, β) ≤ Ce−2β β − 2 min β 1−l , α m−l β 1−m    m−l 1 β m−l β2 = Ce−2β α m−l β 2 −m 1 ∧ ≤ C1 e−β α m−l β 2−2m 1 ∧ . α α For β ≤ 1, by Lemma 4.4 again we have, if l > 1,  m−l   β2 2−2l m−l 2−2m m−l 2−2m 1∧ I (α, β) ≤ C min β = Cα ,α β β α

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and if l = 1, 

I (α, β) ≤ C min 1 − log β, β

α

2−2m m−1



 = C min 1 − log β,

β2 α

1−m & . 

The lower bound for I (α, β), in the case l > 1, is proved in the following Lemma. LEMMA 7.9. If m > l > 1, we have for some constant C = C(λ, m, l), c > 0   α m−l −cβ 2−2l 1∧ 2 I (α, β) ≥ C e β . β Proof. 1. Case α, β ≤ 1. Assume first β 2 ≤ 2α. Then  α  2 −t −l − βt −α 2−2l e t e dt ≥ e β I (α, β) ≥ 0

≥ e−α β 2−2l

α β2

1

s −l e− s ds

0



1 2

s −l e

− 1s

ds ≥ Cβ 2−2l .

0

Assume now β 2 ≥ 2α. Then,  I (α, β) ≥

β2 β2 2

e−t α m−l t −m e−

≥ e−β α m−l e−2 2



β2 β2 2

β2 t

dt ≥ e−β α m−l 2



β2 2

I (α, β) ≥

β

α

≥α

e−t α m−l t −m e−

m−l −β

e



β β 2

t −m e−

β2 t

dt

t −m dt ≥ Cα m−l β 2−2m .

2. Case α ≤ 1, β ≥ 1. Since I (α, β) is decreasing in β, we may assume that 

β2

β2 t

dt ≥ e−β α m−l



β 2 β β 2

≥ 1 ≥ α. Then, t −m e−

β2 t

dt

t −m dt ≥ Ce−2β α m−l .

3. Case β ≤ 1, α ≥ 1.  I (α, β) ≥

β2 β2 2

e−t t −l e−

β2 t

dt ≥ e−β e−2 2



β2 β2 2

t −l dt ≥ Cβ 2−2l .

3. Case α ≥ 1, β ≥ 1. Since I (α, β) is increasing with respect to α, we have I (α, β) ≥ I (1, β) ≥ Ce−2β . 

Sharp kernel estimates for elliptic operators

Finally, we treat the case l = 1. LEMMA 7.10. If m > l = 1, we have for some constant c > 0 ⎧ ⎪ e−cβ (1 ∧ α)m−1 , ⎪ ⎪ ⎪ ⎪ ⎨1 − log β, I (α, β) β 2 1−m ⎪ , ⎪ α ⎪

 ⎪ ⎪ ⎩1 − log β 2 , α

if β ≥ 1; if β < 1 ≤ α; if α, β < 1, β 2 ≥ α; if α, β < 1, β 2 ≤ α.

Proof. The upper estimates follow immediately by applying Lemma 33(ii) in the cases β ≥ 1 and β < 1 ≤ α and, for α, β < 1, by applying (29) after observing that ∞ I (α, β) ≤ 0 F(t) dt. Let us prove, now, the lower estimates. We write I (l, m, α, β) to make explicit the dependence on the parameters. With this notation, we have from Lemma 7.9 −

  α m−1 ∂ I (1, m, α, β) = 2β I (2, m + 1, α, β) ≥ C e−cβ β −1 1 ∧ 2 . ∂β β

(33)

1. Case β ≥ 1. This follows as in cases 2 and 4 of Lemma 7.9. 2. Case β < 1 ≤ α. This follows as in case 3 of Lemma 7.9, integrating between β and 1, instead of β 2 /2 and β 2 . 3. Case α, β < 1, β 2 ≥ α. Observing that limβ→∞ I (α, β) = 0, we integrate (33) between β and ∞. Then,  I (1, m, α, β) ≥ Cα

m−1

∞

β

e−cs s −2m+1 ds ≥ Cα m−1 β −2m+2 .

4. Case α, β < 1, β 2 ≤ α. √ Integrating (33) between β and α, we have √

I (1, m, α, β) ≥ I (1, m, α α) + C

 β

√

α

√  2 C β 1 α ds ≥ C log = − log . s β 2 α 

By comparing the estimates for G λ and G 0 , we obtain the following Corollary. COROLLARY 7.11. If N > 2 and D > 0, then G λ (x, y) e−c

√ λ|x−y| G

0 (x,

y).

Note that the above corollary does not hold for N = 2, D > 0, since G λ is bounded when |x||y| → ∞, whereas G 0 is not, see Remark 7.4. Using the estimates proved in the previous theorems, we obtain the following result γ (note that (|x||y|) 2 G λ (x, y) is symmetric).

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COROLLARY 7.12. Assume that λ, D ≥ 0 and that D > 0 when λ = 0. For any fixed y = 0, the following asymptotic relations hold. (i) As x → 0 γ

(|x||y|) 2 G λ (x, y) |x|

√

D− N 2−2

.

(ii) As |x| → ∞ γ 2

(|x||y|) G λ (x, y)

e−c |x|

√

λ|x| , √ − D− N 2−2

if λ > 0; ,

if λ = 0.

(iii) As x → y γ

(|x||y|) 2 G λ (x, y)

|x − y|2−N ,

if N > 2;

log |x − y|,

if N = 2.

7.3. Resolvent and spectrum of L We start by proving that the spectrum of L coincides with (−∞, 0]. PROPOSITION 7.13. The operator L generates a bounded positive analytic semigroup of angle π/2 in L 2μ , and its spectrum is the half-line (−∞, 0]. Proof. The analyticity of angle π/2 and the inclusion σ (L) ⊂ (−∞, 0] follow from the self adjointness in L 2μ proved in Sect. 3. To prove that the equality holds, let us assume that the resolvent set ρ(L) contains a point in the negative real axis. If Ms u(x) = u(s x), the scaling property Ms L Ms −1 = s −1 L implies that (s 2 λ − L)−1 = s −2 Ms (λ − L)−1 Ms −1

(34)

It follows that the resolvent set contains the point −1, hence the unit circle S 1 and the resolvent estimate (λ − L)−1 ≤ C|λ|−1 hold for every λ = 0. This implies that λ(λ − L)−1 is a bounded entire function and then it coincides with a constant operator A. Letting λ → ∞, we get A = I and hence L = 0, which is a contradiction.  Next, we prove that, for λ = 0, the resolvent (λ − L)−1 is given by an integral kernel K λ (x, y), which we still call the Green function. Clearly, K λ (x, y) = G λ (x, y) whenever the integral defining G λ converges. This happens if Re λ > 0, since % ∞ %  ∞ % % |G λ (x, y)| = %% e−λt p(t, x, y) dt %% ≤ e−(Reλ)t p(t, x, y) dt = G Re λ (x, y). 0

0

By using the upper estimates for the Green function G λ for positive λ proved in the previous section and the kernel estimates of the semigroup for complex z provided by Theorem 6.1, we show bounds for the Green function K λ for every λ = 0.

Sharp kernel estimates for elliptic operators

THEOREM 7.14. Let  = R N \{0}. For every λ ∈ C\(−∞, 0], the resolvent (λ − L)−1 can be represented trough an integral kernel K λ (x, y) with respect to the measure dμ = |y|γ dy. Moreover, for every ε > 0, there exist Cε , m ε > 0 such that, for λ ∈ C\(−∞, 0] with | arg λ| ≤ π − ε and (x, y) ∈  × , |K λ (x, y)| ≤ Cε G m ε |λ| (x, y). Proof. Let ε > 0, and let us consider the sector π − := {λ ∈ C | | arg λ| ≤ π − }. Let us fix λ ∈ π − \{0} and let θ be the angle defined by θ = π2 − 2 if arg λ ≥ 0 and by θ = − π2 + 2 if arg λ < 0. Let us denote by T (eiθ t) the semigroup generated by eiθ L in L 2μ . Setting μ := e−iθ λ = |λ|ei(arg λ−θ) , since | arg μ| ≤ π2 − 2 , we have obviously Reμ > 0. Let us define  ∞ e−μt p(eiθ t, x, y) dt. G θ,μ (x, y) = 0

We observe, preliminary, that by Theorems 6.1 and 6.2 there exist Cε , C˜ ε , m ε > 0 such that, after a suitable choice of a constant m˜ ε > 0 in the argument of the real kernel, we have | p(eiθ t, x, y)| ≤ Cε |x|

− γ2

|y|

− γ2 − N2

t

≤ C˜ ε p(m˜ ε t, x, y).



|x| 1

t2

   − N +1+√ D  2 |y| |x − y|2 dt ∧1 ∧ 1 exp − 1 mεt t2

From now on let Cε and m ε be different at each occurrence. After a change of variable in the integral and observing that cos(arg μ) ≥ cos( π2 − 2 ) = sin( 2 ), it follows from the previous relation that G θ,μ (x, y) satisfies  ∞ % % %G θ,μ (x, y)% ≤ Cε e−Reμ t p(m˜ ε t, x, y) dt 0 ∞  e−m ε |λ| sin( 2 ) t p(t, x, y) dt ≤ Cε 0

= Cε G m ε sin( 2 )|λ| (x, y). Since (μ − e L) iθ

−1



∞

=

(35)

e−μt T (eiθ t) dt

0

it follows that G θ,μ (x, y) is the integral kernel of (μ − eiθ L)−1 (the kernel being written with respect to the measure |y|γ dy). By multiplying by eiθ , we deduce that the Green function K λ (x, y) of (λ − L)−1 satisfies K λ (x, y) = eiθ G θ,μ (x, y) and the proof is concluded using (35).



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8. Applications and examples 8.1. Schrödinger operators with inverse square potential  2 If a = 1, c = 0, then γ = 0, D = b + N 2−2 ≥ 0 and L =  − |x|b 2 is the Schrödinger operator with inverse square potential. Kernel estimates for the Schrödinger operator have already been widely investigated in the literature. Using Theorem 6.2, we can obtain the sharp bounds of [17] including 2  the critical case D = b + N 2−2 = 0. THEOREM 8.1. The heat kernel p of L, with respect to the Lebesgue measure, satisfies p(t, x, y) t

− N2



|x| 1

t2

   − N +1+√ D  2 |y| c|x − y|2 ∧1 ∧1 exp − 1 t t2

Using the results of Sect. 7, we obtain sharp bounds for the Green function (also in  2 the critical case D = b + N 2−2 = 0) which we state for N > 2. THEOREM 8.2. For N > 2, the Green function G λ , with respect to the Lebesgue measure, satisfies the estimates (i) if D > 0, λ ≥ 0, G λ (x, y) e

√ −c λ|x−y|

|x − y|

2−N

 √ D− N −2 2 |x||y| 1∧ . 2 |x − y|

(36)

(ii) If D = 0, and λ > 0, G λ (x, y) ⎧

 2−N √ ⎪ ⎨e−c λ|x−y| (|x||y|) ∧ |x−y| 2 , λ

 √ 2−N ⎪ ⎩|x − y|2−N ∨ (|x||y|) 2 (1 − log( λ |x − y|) ,

√ λ |x − y| ≥ 1; √ if λ |x − y| < 1. if

REMARK 8.3. Note that G λ → 0, ∞ in (36) as |x| → 0 and y = 0 fixed (or conversely), according to D > (N − 2)2 /4 or D < (N − 2)2 /4, that is when b > 0 or b < 0. We refer also to [18, Theorem 3.11] for the local behaviour of the Green function when D > 0. The above estimates in the critical case D = 0 seem to be new. 8.2. Purely second-order operators If b = c = 0, then D =

 N −1−a 2 2a

,γ =

L =  + (a − 1)

(N −1)(1−a) a

and

N  xi x j Di j . |x|2

i, j=1

Using Theorem 6.2, we deduce the following kernel estimates.

Sharp kernel estimates for elliptic operators

THEOREM 8.4. Let  = R N \{0}. The heat kernel p of L, with respect to the (N −1)(1−a) a measure dμ = |y| dy, satisfies N

p(t, x, y) t − 2 |x|

− (N −1)(1−a) 2a

|y|

− (N −1)(1−a) 2a



  c|x − y|2 . exp − t

%

%

  − N +1+%% N −1−a %% 2 2a |x| |y| ∧1 ∧1 1 1 t2 t2

Using Theorems 7.1, 7.5 and 7.7 we obtain sharp bounds for the Green function and the resolvent operator.  2 Observe that the condition D ≤ N 2−2 becomes a ≥ 1 if N ≥ 3 and a = 1 if N = 2. We omit the case N = 2 and λ > 0 to focus the estimates near the origin but let λ ≥ 0 if N ≥ 3 in order to treat the critical case a = N − 1 (when N = 2 the critical case a = 1 corresponds to the Laplacian). THEOREM 8.5. Let  = R N \{0}. For (x, y) ∈  ×  with x = y, the Green (N −1)(1−a) a function G λ , with respect to the measure dμ = |y| dy, satisfies the estimates (i) If N > 2, λ ≥ 0 and a = N − 1 (|x||y|)

(N −1)(1−a) 2a

e−c

√

λ|x−y|

G λ (x, y)

|x − y|2−N

%

%

 % N −1−a % N −2 |x||y| % 2a %− 2 1∧ . |x − y|2

(ii) If N > 2, a = N − 1 and λ > 0 2−N

(|x||y|) 2 G λ (x, y) ⎧

 2−N √ ⎪ 2 ⎨e−c λ|x−y| (|x||y|) ∧ |x−y| √ , λ

 √ 2−N ⎪ ⎩|x − y|2−N ∨ (|x||y|) 2 (1 − log( λ |x − y|) ,

√ λ |x − y| ≥ 1; √ if λ |x − y| < 1.

if

(iii) If N = 2, a = 1, λ = 0

(|x||y|)

1−a 2a

G 0 (x, y)

⎧

⎪ ⎪ ⎨

% % % 1−a % |x||y| % 2a % , |x−y|2 

 y|2

|x − ⎪ ⎪ ⎩1 − log |x||y|

,

if

|x−y|2 |x||y|

≥ 1;

if

|x−y|2 |x||y|

≤ 1.

The Green function with respect to the Lebesgue measure is G λ (x, y) =  G λ(x, y) |x| |y|γ G λ (x, y). It follows that

2−N +

N −1 a −1 +

1− N a−1

+

as |x| → 0 for y = 0 fixed

and G λ (x, y) |y| as |y| → 0 for x = 0 fixed. The joint behaviour in |x||y| is more complicated and when N = 2 or a = N − 1 logarithmic terms also appear.

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8.3. Operators with unbounded coefficients Consider operators of the form ˜ α−1 S = |x|α  + c|x|

x ˜ α−2 , x ∈ R N \{0} · ∇ − b|x| |x|

with α = 2, c, ˜ b˜ ∈ R. Note that α can be positive or negative, but the case α = 2 is special and easier, see [11], and will be not treated here. Generation properties and domain characterization have already been studied in [11]. Here, we follow the same method as in [16], where upper bounds are proved, to deduce lower bounds. Since the proof are similar, we do not repeat them and refer the reader to the above paper. The crucial point consists in observing that the operators previously studied and the last ones are related by an isometry in L p (R N ). LEMMA 8.6. Let 1 ≤ p ≤ ∞, J p : L p (R N ) → L p (R N ) given by %1 %α

 α % p − Nα % J p u(x) := % − 1% |x| 2 p u |x|− 2 x . 2 Then, J p is an isometry in L p (R N ) and   −1 α α−1 x α−2 ˜ · ∇ − b|x| |x|  + c|x| Jp Jp ˜ |x| =  + (a − 1)

N  xi x j x Di j + c 2 · ∇ − b|x|−2 , 2 |x| |x|

i, j=1

where   2 Nα Nα − 1 ; b = b˜ + N − 2 + c˜ − ; 2 2p 2p  

  2 Nα 2 α α −1 −1 + 1− c˜ − α + 1− . c = (N − 1) 2 2 2 p

a=

α

From Theorem 6.2, we deduce sharp kernel estimates for the heat kernel pS associated with S which we write this time with respect to the Lebesgue measure. THEOREM 8.7. pS (t, x, y) 

t

− N2

|x|

˜ Nα − c−α 2 − 4

|y|

c−α ˜ Nα 2 − 4

|x|

2−α 2 1

t2  % α α %2  c%|x|− 2 x − |y|− 2 y % × exp − . t

 ∧1

|y|

2−α 2 1

t2

− N2 +1+|1− α2 |−1 ∧1

√

D˜

Sharp kernel estimates for elliptic operators

Similarly, by Theorems 7.1, 7.1, and 7.7, we can deduce the estimates for  ∞7.5 S −λt the Green function G λ (x, y) = 0 e pS (t, x, y) dt. We state them when D˜ = 2

c˜ > 0, N > 2. b˜ + N −2+ 2 THEOREM 8.8. Let λ > 0,  = R N \{0} and let us suppose D˜ > 0, N > 2. For (x, y) ∈  ×  with x = y, the Green function G λ , with respect to the Lebesgue measure, satisfies the estimates G λ (x, y)S

e

√ α α −c λ |x|x|− 2 −y|y|− 2 |

 × 1∧

|x|

− α2Np

|y|

− 2αpN +γ (1− α2 )



s α −2 − α2 +1 − α2 +1 − α2 − α2 2 (1− 2 ) D˜ |x| |y| |x|x| − y|y| | 

|x|x| |x|

− α2

−

− α2 +1

α y|y|− 2 |2

 (1− α )−2 D− ˜ N −2 2 2

α

|y|− 2 +1

where

α −1 s = 1− 2



and γ=

N − 1+(N − 1)

N − 2 + c˜ − 2



α 2

Nα p



% α %%−1 % − %1 − % 2

' b˜ +

  2  − 1 − 1 + 1 − α2 c˜ − α + α 2 2 −1

Nα 2



N − 2 + c˜ 2

1−

2 p

2

 − N +1.

Proof. The proof follows by the equality

α α α  − α2Np − 2αpN +γ (1− α2 ) |x| GS |y| G λ (x|x|− 2 , y|y|− 2 ) λ (x, y) = 1 − 2 and by observing that, since N − 2 + c˜ − Npα N −1+c−a   = , 2a 2 1 − α2 we get

α −2 ˜ D = 1− D 2 and the number s1 for the operator L becomes '     Nα % %−1

α α −1 N − 2 + c˜ − p N − 2 + c˜ 2 % % s = 1− . b˜ + − %1 − % 2 2 2 2 

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8.4. A special case Theorem 6.7 provided us a complete decomposition of the N -dimensional kernel in terms of its one-dimensional counterparts  ∞  2 N −1+c−a 1 r + ρ 2  √ rρ  (n) Zω (η) p(t, x, y) = I Dn (37) (rρ)− 2a exp − 2at 4at 2at n=0

where p is the heat kernel of L with respect to the measure dμ = |y|γ dy, x = r ω, y = ρη, r, ρ > 0, |ω| = |η| = 1, (−λn )n∈N0 are the eigenvalues of 0 and  N −1+c−a 2 n . Dn = b+λ a + 2a Setting a = 1, and b = c = 0, the operator L becomes the Laplace operator and 2   2 Dn = N 2−2 + λn = N 2−2 + n . Inserting in (37) the expression of the Gauss– Weierstrass kernel of , we find ∞

rρ   rρ   1 rρ  N 2−2 Z(n) I N −2  (ω) = exp ω·η (38) η N 2t 2t 2 +n (4π ) 2 4t n=0 Formula (38) allows us to write explicitly the heat kernel of L in some other special cases. PROPOSITION 8.9. Assume that a = 1 and b + and p(t, x, y) =

c2 4

+ 2c (N − 2) = 0. Then γ = c

  |x − y|2 − 2c − 2c |x| |y| exp − N 4t (4π t) 2 1

where p is the heat kernel of L with respect to the measure dμ = |y|c dy. Proof. It is enough to observe that, under the assumption on the parameters a, b, c,  2   N −2 2 N −2 +n + λn = Dn = 2 2 is the same as for the Laplacian and therefore (38) holds. Inserting this in (37), the proof follows.  Note that the parameter c is unrestricted but b ranges from −∞ to (N − 2)2 /4, attained when c = 2 − N . We point out that the same result can be proved in a more direct way. With this c c choice of parameters, given u, v ∈ D(˜a) and setting u = u 1 |x|− 2 and v = v1 |x|− 2 the form becomes    ∇ τ u ∇τ v b u r vr + + 2 uv |x|c dx a˜ (u, v) = r2 r RN  2   b + c4 + 2c (N − 2) ∇τ u 1 ∇τ v1 = + u 1 v1 dx (u 1 )r (v1 )r + |x|2 |x|2 RN

  ∇τ u 1 ∇τ v1 (u 1 )r (v1 )r + dx = ∇u 1 ∇v1 dx. = |x|2 RN RN

Sharp kernel estimates for elliptic operators c

This shows that, with the isometry J : L 2 (R N , dx) → L 2μ , J v = v|x|− 2 , satisfies J −1 L J = , and hence J −1 et L J = et and the heat kernel of L is readily obtained by that of the Laplacian. Acknowledgements The authors thank Prof. M. Choulli for several discussions on the topic and for pointing out reference [3]. REFERENCES [1]

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N. N. Uraltseva: On impossibility of W p2 estimates for multidimensional elliptic equations with discontinuous coefficients, Zap. nauchn. seminarov LOMI im. Steklova, Vol. 5 (1967) (Russian), 250–254. N. J. Vilenkin: Special functions and the theory of group representations, Translation of mathematical monographs, Vol. 22, American Mathematical Society Providence, 1968. G. Metafune, L. Negro and C. Spina Dipartimento di Matematica “Ennio De Giorgi” Università Del Salento C.P. 193 73100 Lecce Italy E-mail: [email protected] L. Negro E-mail: [email protected] C. Spina E-mail: [email protected]