Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$-spaces

In this paper we give sufficient conditions on $\alpha \geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.


Introduction
Let us consider the elliptic operator where α ≥ 0. In this paper we want to study the perturbation of L with a singular potential. More precisely, we consider the operator L 0 = L + c |x| 2 and we look for optimal conditions on c ∈ R and α ensuring that L 0 with a suitable domain generates a positivity preserving C 0 -semigroup in L p (R N ).
Let us recall first some known results for Schrödinger operators with inversesquare potentials. It is known, see [19,Theorem 2], that the realization A 2 of the Schrödinger operator A = ∆ + c|x| −2 in L 2 (R N ) is essentially selfadjoint on C . Using perturbation techniques it is proved in [15,Theorem 6.8] that A 2 is selfadjoint provided that c < c 0 . These techniques were generalized to the L p -setting, 1 < p < ∞, and it is obtained that A p , the realization of A in L p (R N ), with domain W 2,p (R N ) generates a contractive and positive C 0 -semigroup in L p (R N ), and C ∞ c (R N ) is a core for A p , if N > 2p and c < (p − 1)(N − 2p)N p 2 =: β 0 , see [16,Theorem 3.11]. In the case where N ≤ 2p, it is proved that A p with domain D(A p ) = W 2,p (R N ) ∩ {u ∈ L p (R N ); |x| −2 u ∈ L p (R N )} is m-sectorial if c < β 0 , see [16,Theorem 3.6].
If one replaces the Laplacian by the Ornstein-Uhlenbeck operator similar results were obtained recently in [3,7].
In this paper we obtain similar results as in [16,Theorem 3.11] when replacing ∆ by L. We discuss also the generation of a C 0 -semigroup of the operator (1 + |x| α )∆ − η|x| β + c |x| 2 , where η is a positive constant, α ≥ 2 and β > α − 2. Now, let us recall some definitions. An operator (A, D(A)) on a Banach space X is called accretive if −A is dissipative. It is m-accretive if A is accretive and X = R(λ + A), the range of the operator (λ + A). An accretive operator (A, D(A)) is called essentially m-accretive if its closure A is m-accretive.
Our approach relies on the following perturbation result due to N. Okazawa, see [16,Theorem 1.7]. Theorem 1.1. Let A and B be linear m-accretive operators in L p (R N ), with p ∈ (1, +∞). Let D be a core of A. Assume that (i) there are constantsc, a ≥ 0 and k 1 > 0 such that for all u ∈ D and ε > 0 is m-accretive and any core of A is also a core for A + cB. Furthermore, A − kB is essentially m-accretive on D(A).
In order to apply the above theorem, we need some preliminary results on the operator L and some Hardy's inequalities.

Preliminary results
Let us begin with the generation results for suitable realizations L p of the operator L in L p (R N ), 1 < p < ∞. Such results have been proved in [6,9,11]. More specifically, the case α ≤ 2 has been investigated in [6] for 1 < α ≤ 2 and in [9] for α ≤ 1, where the authors proved the following result.
The case α > 2 is more involved and is studied in [11], where the following facts are established.
4. If N ≥ 3, p > N/(N − 2) and 2 < α < N (p−1) p the domain D max coincides with the space is a core for L. If we consider the operatorL := L − η|x| β with η > 0 and β > α − 2 then we can drop the above conditions on p, α and N , as the following result shows, see [1], where the quasi-contractivity can be deduced from the proof of Theorem 4.5 in [1].
If α > 2 and β > α − 2 then, for any p ∈ (1, ∞), the realizationL p ofL with domain generates a positive and strongly continuous quasi-contractive analytic semigroup. Moreover, C ∞ c (R N ) is a core forL p . From now on we assume N ≥ 3, α ≥ 0. We set The inequality holds true even if u is replaced by |u|.
Proof. By density, it suffices to prove (2.2) for u ∈ C 1 c (R N ). So, for every λ ≥ 0, let us consider the vector field F (x) = λ x |x| 2 |x| α , x = 0, and set dµ(x) = |x| α dx. Integrating by parts and applying Hölder and Young's inequalities we get In the computations above, we used the identity ∇|u| p = p|u| p−2 Re(u∇ū). Hence, By taking the maximum over λ of the function ψ(λ) = λ(N − 2 + α) − λ 2 p 2 /4, we get (2.2). We note here that the integration by parts is straightforward when p ≥ 2. For 1 < p < 2, |u| p−2 becomes singular near the zeros of u. Also in this case the integration by parts is allowed, see [10].
By using the identity ∇|u| p = p|u| p−1 ∇|u| in the computations above, the statement holds with u replaced by |u|.
Remark 2.6. Hardy's inequality (2.2) holds even if u is replaced by u + := sup(u, 0), As a consequence of Lemma 2.4 we have the following results.
So, using the identities |∇|u|| 2 ≤ |∇u| 2 and |u|∇|u| = Re(u∇u), we obtain if p ≥ 2. The case 1 < p < 2 can be handled similarly. Thus, by Hölder's inequality we have Taking the assumption on V into account we obtain α ≤ 0, thanks again to Lemma 2.4. The latter inequality is equivalent to α ≤ (N −2)(p−1), which is the assumption. This ends the proof.
Hence, in order to apply Theorem 1.1, we have established For more details on dispersive operators we refer to [14, C-II.1].
be real-valued and fix δ > 0. Replacing u by u + in the proof of Proposition 2.7 and since u + ∈ W 1,p (R N ), we deduce that Then, where here we take δ = 0 if p ≥ 2 and δ > 0 if 1 < p < 2. Thus, letting δ → 0 if 1 < p < 2, and applying Hölder's inequality we obtain As in the proof of Proposition 2.7, the assertion follows now by Lemma 2.4 and Remark 2.6.
The next proposition deals with the operator L − η|x| β .

Main results
In this section we state and prove the main results of this paper. In order to apply Theorem 1.1 to our situation we need the following lemma whose proof follows the same lines of [16,Lemma 3.4].
Proof. Let u ∈ C ∞ c (R N ) and set u δ = (R|u|) 2 + δ 1 2 , where R p := V p−1 ε . In the computations below, we have to take δ > 0 in the case 1 < p < 2, whereas we only take δ = 0 to deal with the case p ≥ 2. We have
Remark 3.2. We rewrite estimate (3.4) as follows The easiest case (see Lemma 3.1) is when µ ≥ 0. Now, we prove a similar estimate for the operator L = L − η|x| β .
Proof. We proceed as in the proof of Lemma 3.1. From Remark 3.2 and the inequality |x| 2 V ε ≤ 1 it follows that Thus the proof of the lemma is concluded.
Applying Corollary 2.9, Lemma 3.1 and Theorem 1.1 we obtain the following generation results. We distinguish the two cases α ≤ 2 and α > 2 since the hypotheses on the unperturbed operator L are different. 1) then, for every c < k the operator L + c |x| 2 endowed with the domain D p defined in Theorem 2.1 generates a contractive positive C 0 -semigroup in L p (R N ). Moreover, C ∞ c (R N ) is a core for such an operator. Finally, the closure of L + k |x| 2 , D p generates a contractive positive C 0 -semigroup in L p (R N ).
and α < N (p−1) p , then for every c < k the operator L + c |x| 2 endowed with the domain D p given in Theorem 2.2 generates a contractive positive C 0 -semigroup in L p (R N ). Moreover, C ∞ c (R N ) is a core for such an operator. Finally, the closure of L + k |x| 2 , D p generates a contractive positive C 0 -semigroup in L p (R N ).
The proofs of the two above theorems are identical. We limit ourselves in proving the latter.
Proof of Theorem 3.5. In order to apply Theorem 1.1, set A = −L, D(A) = D p , D = C ∞ c (R N ) and let B be the multiplicative operator by 1 |x| 2 endowed with the maximal domain D(|x| −2 ) = {u ∈ L p (R N ); |x| −2 u ∈ L p (R N )} in L p (R N ). We observe that the Yosida approximation B ε of B is the multiplicative operator by V ε = 1 |x| 2 +ε . Both A and B are m-accretive in L p (R N ). Then, Lemma 3.1 yields (i) in Theorem 1.1 with k 1 = β 0 ,c = 0 and a = 0. The second assumption (ii) in Theorem 1.1 is obviously satisfied. The last one, (iii), holds with k 2 = (p − 1)γ 0 thanks to Corollary 2.9. Then, we infer that for every c < k, −L − c If 2p ≥ N , then β 0 ≤ 0 and we cannot apply Theorem 1.1. However, if at least β α ≥ 0, that is 2p − N ≤ α, then we still have a generation result, relying on the following abstract theorem by Okazawa (see [16,Theorem 1.6]). Theorem 3.6. Let A and B be linear m-accretive operators in L p (R N ), 1 < p < +∞. Let D be a core of A. Assume that there are constantsc, a, b ≥ 0 such that for all u ∈ D and ε > 0, In our framework the above result leads to the following theorems. We recall that D(|x| −2 ) = {u ∈ L p (R N ); |x| −2 u ∈ L p (R N )}.
As before, we limit ourselves in proving the latter.