On vector-valued Schrödinger operators with unbounded diffusion in L p spaces

. We prove generation results of analytic strongly continuous semigroups on L p ( R d , R m ) (1 < p < ∞ ) for a class of vector-valued Schrödinger operators with unbounded coefﬁcients. We also prove Gaussian type estimates for such semigroups.


Introduction
Systems of elliptic and parabolic equations with (possibly) unbounded coefficients arise quite naturally in the study of mathematical models which describe physical phenomena such as Navier Stokes equations (see e.g. [17,19,20]). Differently from the scalar case, where the theory of elliptic and parabolic operators with unbounded coefficients is widely studied (see [24] and the reference therein), the interest on the vector-valued case is quite recent as the quoted references show.
In this paper, we consider realizations in L p -spaces of Schrödinger type vectorvalued elliptic operators defined on smooth functions u : R d → R m , (d, m ∈ N), by Au = div(Q∇u) − V u =: A 0 u − V u, where the diffusion matrix Q(x) is positive definite at each x ∈ R d and V is a measurable matrix-valued function. The first problem is to find conditions on Q and V and identify a realization of the operator A in L p (R d , R m ) which generates a strongly continuous semigroup (T p (t)) t≥0 ; afterwards one is interested in regularity properties of the semigroup, such as analyticity, ultracontractivity and kernel estimates.
This type of operators have been recently studied in different settings. For an analysis in the space of bounded and continuous functions we refer the reader to the papers [1,4,12]. Also the L p -context has been investigated. As shown by the scalar case, the L p -spaces related to the invariant measure represent the most suitable L p -spaces where to set problems associated with elliptic operators with unbounded coefficients, but, in the vector-valued case, only partial results in this direction are available so far (see [2,3]). Nevertheless, the classical L p -context is important in view of the applications. A perturbation method, due to Monniaux and Prüss, see [29], has been applied in the seminal paper [18], where generation results in L p (R d ; R m ) are proved for a class of nondegenerate weakly coupled elliptic operators containing also a first-order diagonal term, and more recently in [22], where a class of vector-valued Schrödingertype operators is considered, when the matrix-valued function Q is bounded and can degenerate neither at some x ∈ R d nor at infinity, while the potential V is quasi accretive and locally Lipschitz continuous on R d , and |D j V (−V ) −α | ∈ L ∞ (R d ) for some α ∈ [0, 1/2). In this case the domain of the L p -realization of A is the intersection of the domains of the diffusion and the potential part of the operator. By perturbing V with a scalar potential v ∈ W 1,∞ loc (R d ) satisfying the condition |∇v| ≤ cv on R d for some positive constant c (in the L 2 -setting, actually even with more general matrix-valued perturbation), a larger class of potentials has been considered in [5,25], by using perturbation results due to N. Okazawa [30]. In [23], assuming again strict ellipticity and boundedness for the diffusion coefficients, pointwise accretivity and local boundedness of the potential term, the authors prove that A, endowed with its maximal domain, generates a strongly continuous semigroup in L p (R d ; C m ).
A different approach has been adopted in [7] to deal with a class of vector-valued nondegenerate elliptic operators, coupled up to the first order. More precisely, using the semigroup, obtained in [1] in the space C b (R d ; R m ), the authors of [7] provide sufficient conditions which allow to extend this semigroup to the L p scale. It is worth mentioning that this approach does not provide any sharp description of the domain of the generator of the semigroup.
A class of vector-valued elliptic operators L, including also a first-order coupling term, has been considered very recently in [6]. This class contains the operator A, but the presence of the drift term causes technical problems which prevent us from both analyzing the L 1 -setting and proving an integral representation of the semigroup (T p (t)) t≥0 associated with the operator L in L p (R d ; C m ). For this reason, it is of interest to consider the operator A on its own. We introduce assumptions on its coefficients that can be made independent of p ∈ [1, ∞) and, as a consequence, allow to prove summability improving properties as ultracontractivity and the integral representation of the associated semigroup (T (t)) t≥0 in terms of a kernel K . A domination with a scalar semigroup associated to a suitable form, allows us also to deduce some Gaussian type estimates satisfied by the kernel K , which are expressed in terms of a distance associated with the diffusion coefficients. To our knowledge, this result seems to be new and of interest even in the scalar case.
Notation. Throughout the paper K = R or K = C. The Euclidean inner product in K m and the associated norm are denoted by ·, · and | · | respectively. Given a function u : Ω ⊆ R d → K m , we denote by u k its k-th component and we set is the classical vector-valued Sobolev space, namely the space of all functions u ∈ L p (R d , K m ) whose components have distributional derivatives up to the order k belonging to L p (R d , K). The norm of W k, p (R d , K m ) will be denoted by · k, p . The spaces W k, p loc (R d , K m ) are defined analogously. C(R d , K m ) stands for the space of functions u : R d → K m whose components belong to C(R d , K). A similar notation is used for the subspaces of C(R d , K m ). The subscript "b" (resp. "c") stands for "bounded" (resp. "compactly supported"). When m = 1 and K = R, we simply write . Finally, by B(r ) we denote the ball of R d centered at 0 and with radius r .

Generation results
We are going to consider the elliptic operator A acting on vector-valued smooth functions u as follows: where the coefficients Q = (q i j ) 1≤i, j≤d and V = (v hk ) 1≤h,k≤m satisfy the following assumptions: for each i, j = 1, . . . , d and Q(x)ξ, ξ is positive for every x ∈ R d and 0 = ξ ∈ R d ; (ii) v hk : R d → R are measurable functions for every h, k = 1, . . . , m; (iii) there exist a function v ∈ C 1 (R d ), with positive infimum v 0 , and positive constants c 1 , γ and C γ such that Hypothesis 1(iii) goes back to the papers [8,9], where the domain of the realization in L 2 (R d ) of the scalar Schrödinger operator −Δ + v is characterized under suitable assumptions on v. Afterwards the same type of assumptions, but for more general diffusion matrices, have been considered in [26,27]. On the other hand, Hypothesis 1(iv) allows us to apply to the operator A in (1) the following result which is a particular case of [ For every ξ, η ∈ C m , we set q(ξ, η) = Q(·)ξ, η and q(ξ ) = Q(·)ξ, ξ . We also recall that for every u ∈ W 1, p (R d , C m ) it holds that For each p ∈ (1, ∞), we consider the realization A p of the operator A in L p (R d ; R m ) endowed with the minimal domain On D p we consider the norm u D p = A 0 u p + vu p . By the assumptions on V and v, · D p is clearly equivalent to the norm u → u p + A 0 u p + V u p . Moreover, (D p , · D p ) is a Banach space, since A 0 is a closed operator.

Theorem 2. Under Hypotheses 1, if p < 1+4γ −2 , then the operator A p generates an analytic contraction semigroup
Proof. We assume first that C γ = 0 and prove that there exists a constant C = Au and, for every ε > 0, consider the function u ε introduced in Proposition 1. By using both the formulas in (2) and integrating by parts, we obtain By applying Young's inequality, we get that for every δ > 0 there exists a positive Letting ε tend to 0 yields and we get the assertion by the assumptions on γ and p, choosing δ sufficiently small. As a consequence of the estimate that we have proved, we can show that there exist positive constants M 1 and M 2 , depending on c 0 , c 1 and γ such that If C γ = 0, then we fix λ > 0 such that V + λI and v + λ satisfy Hypothesis 1(iii) with C γ = 0. Applying (6) to A − λI and using the dissipativity of A, we obtain while the other inequality follows as in (7).
Next, we show that D p = D p,max . Clearly, we just need to prove that D p,max ⊂ D p . We fix u ∈ D p,max and a sequence ( and such that Au n converges to Au in L p (R d ; R m ) as n tends to ∞ (see Theorem 1). By (6), (u n ) is a Cauchy sequence in the Banach space D p (endowed with the norm · D p ). Hence, u ∈ D p and the inclusion D p,max ⊂ D p follows.
In particular, C ∞ c (R d ; R m ) is dense in D p and, by Theorem 1, (A, D p ) generates a contraction semigroup. Theorem 3. Under Hypotheses 1, suppose that v(x) blows up as |x| tends to ∞. Then, for every p ∈ (1, 1 + 4γ −2 ), the operator A p has compact resolvent.
Proof. We have to show that the unit ball B of D p is totally bounded in L p (R d ; R m ). First let observe that for every ε > 0 there exists

Consistency and summability properties
Let us define As usually, we denote by · a = Re a(·, ·) + · 2 2 the norm of D(a) associated with the form a. The following properties hold true: and, up to a subsequence, we can assume that ∇u n converges to ∇u and u n converges to u almost everywhere in R d . Since q(∇u n ) 2 + V u n , u n 2 ≤ M for all n ∈ N and a positive constant M, by applying Fatou's lemma we get that u ∈ D(a) and lim n→∞ u n − u a = 0; , implies that u = 0. The proof of this statement runs analogously to that of [6, Theorem 2.2].
Since the diffusion part in A is diagonal, integrating by parts, we deduce that the operator B associated with a coincides with − A 2 is a core for A 2 (see Theorem 2), B = −A 2 follows from the closedness of B.
Based on these remarks, we can prove the consistency of the semigroups (T p (t)) t≥0 .
Proof. We begin the proof by observing that Hypotheses 1(ii) and (iii)(a) imply that, for every r ∈ (1, ∞), the realization in L r (R d ; R m ) of the multiplication operator for every t > 0, f ∈ L r (R d ; R m ) and almost every x ∈ R d . These semigroups are clearly consistent. Next, we denote by ( T (t)) t≥0 the semigroup associated with the form a in (10), where we take V = 0. This semigroup coincides with the semigroup (that we denote by (T 0 2 (t)) t≥0 ) associated with the operator A 0 , provided by Theorem 2, since their generators coincide. Since A 0 is diagonal, the semigroup (T 0 2 (t)) t≥0 is diagonal as well and it consists of m copies of the scalar semigroup (T 0 2 (t)) t≥0 associated with the scalar version a s of the form a (with V = 0). The form a s satisfies the Beurling-Deny criterion. Thus, the semigroup (T 0 2 (t)) t≥0 can be extended to the L r -scale (r ≥ 2) and these semigroups are consistent, see [11,Theorem 1.4.1]. As a byproduct, also the semigroup (T 0 2 (t)) t≥0 can be extended from to a strongly continuous semigroup ( T r (t)) t≥0 on L r (R d , R m ) for every r ∈ [2, ∞). These semigroups are consistent. We claim that T 0 r (t) = T r (t) for every r ∈ (2, ∞) and t > 0. For this purpose, we observe that [13, Lemma 1.11] (applied to the scalar semigroup . This fact and Theorem 2 imply that is a core both for A 0 r and A r and this is enough to infer that the semigroups (T 0 r (t)) t≥0 and ( T r (t)) t≥0 coincide. Now, we fix p, q as in the statement and we first assume that p, q ∈ [2, ∞). Applying Trotter product formula (see [14, Corollary III.5.8]), we conclude that for every t > 0 and r ∈ {p, q}. Fix t > 0 and f ∈ L p (R d ; R m ) ∩ L q (R d ; R m ). Then, from the above arguments we infer that where the limit is taken in L p (R d ; R m ). Then, up to a subsequence, we obtain that (T 0 q (t/n)• S q (t/n)) n f converges pointwise both to T p (t) f and to T q (t) f as n tends to ∞, so that the equality T p (t) f = T q (t) f follows.
Next, we assume that p, q ∈ (1, 2] and observe that T p (t) = T p (t) * and T q (t) = T q (t) * for every t > 0, where T p (t) and T q (t) denote the semigroup in L p (R d ; R m ) and L q (R d ; R m ), respectively, associated with the operator A * , adjoint to operator A. Since the potential V * satisfies the same assumptions as the potential V and p , q ≥ 2, from the results so far proved we conclude that T p (t) and follows for every t > 0.

Remark 1.
As a straightforward consequence of Theorems 3 and 4, we deduce that if, Hypotheses 1 hold true for every γ > 0 and Q(x)ξ, ξ ≥ q 0 |ξ | 2 for every x, ξ ∈ R d and some positive constant q 0 , then the spectrum of operator A p is independent of p and consists of eigenvalues only.
Based on Theorem 3 we can now prove the following result.
Theorem 5. Assume that Hypotheses 1 hold true for some γ > 0 and let 1 < p < 1+ 4γ −2 . Then the restriction of the semigroup are cores for the generator A 1 of (T 1 (t)) t≥0 and A 1 u = A q u = Au for every u ∈ X q and q < 1 + 4γ −2 . Finally, if Hypotheses 1 hold true for every γ < 2 and then, for every t > 0, for each 1 < q < p and, since the semigroups (T q (t)) t≥0 are consistent, we can estimate From this chain of inequalities and letting r tend to ∞, it follows that we can extend the restriction of the semigroup (T p (t)) t≥0 to L p (R d ; R m ) ∩ L 1 (R d ; R m ) with a contraction semigroup (T 1 (t)) t≥0 on L 1 (R d ; R m ). This semigroup is strongly continuous. Indeed, since it is uniformly bounded, it suffices to prove that T 1 Letting q tend to 1 and then r tend to ∞, we infer that T 1 (t) f converges to f in L 1 (R d ; R m ) as t tends to 0.
Next, we observe that Vector-valued Schrödinger operators for every u ∈ X q . Thus, the ratio t −1 (T 1 (t)u − u) converges to A q u in L 1 (R d ; R m ), as t tends to 0. This shows that X q ⊂ D( A 1 ) and A 1 u = A q u for u ∈ X q . To conclude that the set X q is a core for A 1 for every q < 1 + 4γ −2 , it suffices to apply [14,Proposition II.1.7]. Indeed, the semigroup (T 1 (t)) t≥0 leaves X q invariant, since it is consistent with the semigroup (T q (t)) t≥0 , and, clearly, X q is also dense in To prove the last part of the assertion, we assume that Hypotheses 1 hold true for every γ < 2 and recall that (T 2 (t)) t≥0 is the semigroup associated with the quadratic form a. By (11) there exists C > 0 such that a( f , f ) ≥ C f 2 1,2 for every f ∈ D(a). Since by a straightforward adaptation of Nash's inequality to the vector-valued case, f for some constant C 1 > 0 and for some constant K > 0 and, therefore, If Hypotheses 1 hold true for each γ > 0, then, by Theorem 4, the semigroups (T p (t)) t≥0 exist for each p ∈ [1, ∞) and are consistent. In this case, we will write (T (t)) t≥0 instead of (T p (t)) t≥0 . Theorem 6. Assume that Hypotheses 1 hold true for every γ > 0 and that estimate (11) is satisfied too. Then, for every t > 0 and 1 Proof. By observing that A * u = div(Q∇u) − V * u for each u ∈ C ∞ c (R d ; R m ) and that V * satisfies the same assumptions as V , we deduce that T * (t) is bounded from for every t > 0. Thus, by applying the usual duality argument and the semigroup law, it follows that T (t) is bounded from To establish the existence of the kernel, observe that , and therefore, by the Dunford-Pettis theorem, there exists We get the assertion by defining K (t, x, y) as the matrix with entries k i j (t, x, y) for every t > 0 and almost every x, y ∈ R d .
Finally, we establish a domination property of the semigroup (T (t)) t≥0 with the scalar semigroup generated by A 0 − v I . Proposition 2. Assume Hypotheses 1 with γ < 2 and that (11) holds true. Then, the semigroup (T (t)) t≥0 is dominated by the analogous scalar semigroup (T 2 (t)) t≥0 generated by the realization of the operator div( Proof. As in the proof of Theorem 4, we denote by a s the sesquilinear form a in the scalar case. Observe that for every u ∈ D(a), by the formulas in (2) and using also the formula ∇|u| = |u| −1 Re m j=1 (∇u j )u j χ {u =0} , we can show that |u| belongs to D(a s ) and m k=1 q(∇u k , ∇(|u| −1 u k )) ≥ 0. Moreover, for every f ∈ D(a s ), Thus, the assertion follows by applying [31, Theorem 2.30].

Gaussian estimates
In this section we mainly focus on the scalar operator A 0 − v I , where A 0 = div(Q∇), assuming that Hypotheses 1 hold true for some γ ∈ (0, 2) and that condition (11) is satisfied. We prove a sharp Gaussian estimate for the kernel of the semigroup generated by A 0 − v I , with explicit control on the constants appearing. Based on this result, we will prove a Gaussian estimate for the kernel of the semigroup (T (t)) t≥0 .
In the scalar case, we adapt the arguments in [31, Chapter 6] to our situation in which the matrix Q is allowed to be unbounded. As a consequence, the Gaussian estimate will be expressed in terms of a distance associated with the diffusion term, namely for every x, y ∈ R d .
It is well known (see e.g. [10,Theorem 7]), that this distance is equivalent to the Euclidean metric if there exist two positive constants q 0 and q 1 such that q 0 |ξ | 2 ≤ Q(x)ξ, ξ ≤ q 1 |ξ | 2 for every ξ, x ∈ R d . In our case, only the inequality holds true in general. We will provide an example at the end of the section.
for all t > 0 and almost every x, y ∈ R d .
Proof. Observe first that the semigroup generated by and observe that for every ψ ∈ W and every ρ ∈ R it holds that e ρψ u ∈ D(a) for every u ∈ D(a), where a denotes the sesquilinear form associated with −A 0 + v I , i.e., the scalar version of the form a in (9) and (10). Fix ρ ∈ R, ψ ∈ W and consider the form a ρ,ψ : D(a) × D(a) → C, defined by a ρ,ψ (u, w) = a(e −ρψ u, e ρψ w) for every u, w ∈ D(a). Straightforward computations show that for every u, w ∈ D(a). It follows that a ρ,ψ is continuous with respect to · a . Indeed, taking into account that a is a continuous form, we can estimate for every u, w ∈ D(a), where C is a positive constant not depending on u, w. Moreover, by observing that we get that a ρ,ψ is closed and quasi-accretive. Therefore, the opposite of the operator associated with a ρ,ψ generates a C 0 -semigroup (T ρ,ψ (t)) on L 2 (R d , C). An easy calculation shows that T ρ,ψ (t)(·) = e ρψ T 2 (t)(e −ρψ ·) for t ≥ 0, where (T 2 (t)) t≥0 is the (contraction) C 0 -semigroup associated with a on L 2 (R d , C). We need to estimate the norm T ρ,ψ (t) 1→∞ and we do this through several steps.
As a concrete example, we take a i (s) = 1 + s 2 for every s ∈ R and i = 1, . . . , d, and obtain that σ i (t) = log(t + √ 1 + t 2 ) for every t ∈ R.
As an immediate consequence of Proposition 2, Theorem 6 and Theorem 7, we can prove the following Gaussian type estimates for the matrix-valued kernel K which generalizes Theorem 5.4 in [25].  for every t > 0, i, j = 1, . . . , m and almost every x, y ∈ R d .