Generation results for vector-valued elliptic operators with unbounded coefficients in L^p spaces

We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,\infty). Sufficient conditions to prove generation results of an analytic C_0-semigroup T(t), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.


Introduction
In this paper, we consider vector-valued elliptic operators with unbounded coefficients acting on smooth functions f : R d → R m (m ≥ 2) as follows: where Q, B i (i = 1, . . . , d) and V are matrix-valued functions, and we study their realizations in the L p -spaces with respect to the Lebesgue measure. In recent years, the interest on systems of elliptic and parabolic equations with unbounded coefficients has considerably grown motivated by a wide variety of mathematical models for physical and financial problems where they appear (backwardforward stochastic differential equations in connection with Nash equilibria in the theory of games, Navier Stokes equations, etc., see e.g., [2,10,12,19,20,22,23]). Beside the analysis in spaces of bounded and continuous functions and the study of the so-called invariant measures (see e.g., [2,3,4,6,16]), the research on such systems of PDEs has been devoted to the L p -setting with respect to the Lebesgue measure. As it is known from the scalar case, the presence of an unbounded drift term produces additional difficulties to prove generation results of strongly continuous or analytic semigroups on the usual Lebesgue space L p (R d ). Usually, one has to require strong conditions on the growth of the drift term or, as an alternative, to assume the existence of a dominating potential term. The L p -theory of parabolic systems with unbounded coefficients is not well developed and the literature concerns essentially weakly coupled elliptic operators (i.e. the coupling between the equations is through a potential term), whose diffusion coefficients are assumed to be uniformly elliptic and bounded. Moreover, often the techniques used to study these problems and prove generation results are based on perturbation methods, see [21,24,25,27]. The assumptions in [21] allow the drift term to grow like |x| log(1 + |x|) and the potential term as log(1 + |x|) as |x| → ∞, and the generation result is proved via a Dore-Venni type theorem on sums of noncommuting operators, due to Monniaux and Prüss [31]. This method works also in the case considered in [24] to prove generation results in L p spaces for vector-valued Schrödinger operators of the form Au = div(Q∇u) − V u. In that paper the operator is nondegenerate and the diffusion matrix Q is symmetric, with entries which are bounded and continuously differentiable with bounded derivatives. The entries of the potential V are locally Lipschitz continuous on R d and satisfy the conditions V (x)ξ, ξ ≥ |ξ| 2 for every x ∈ R d , ξ ∈ R m , and |D j V (−V ) −α | ∈ L ∞ (R d ) for some α ∈ [0, 1/2). The last assumption allows for potentials V whose entries grow more than linearly at infinity. For instance the potential V (x) = (1 + |x| r )V 0 , for every x ∈ R d , where V 0 is an antisymmetric constant matrix and r ∈ [1, 2), is allowed. Under slightly different hypotheses on the potential V (pointwise accretivity and local boundedness), generation results for the operator A as above are proved in [25] but, differently from [24], where the domain of the L p -realization of A is characterized as the intersection of the domains of diffusion and the potential terms of the operator, in [25] only a weak characterization of the domain is provided (in fact, the generation result is proved in the maximal domain of the realization of the operator A in L p (R d ; R m )). A more general class of potentials, whose diagonal entries are polynomials of type |x| α or even |x| r log(1 + |x|) as well as e |x| , for α, r ≥ 1, is considered in [27] where the operator A is perturbed by a scalar potential v ∈ W 1,∞ loc (R d ) satisfying |∇v| ≤ cv for some positive constant c. A perturbation theorem (due to Okazawa [32] and used in [27]) works for a matrix-valued perturbation of V in the L 2 -setting (see [8]) allowing for different growth rates of the type of [27] for the diagonal entries of the potential matrix. In [8] the operator A is also perturbed by a diagonal first-order term that can grow at most linearly at infinity.
In this paper, using direct methods and suitable assumptions that depend on p ∈ (1, ∞), we prove that the realization of A in L p (R d ; R m ) with domain D p = {u ∈ L p (R d ; R m ) ∩ W 2,p loc (R d ; R m ) : div(Q∇u), V u ∈ L p (R d ; R m )} generates an analytic C 0 -semigroup on L p (R d ; R m ). Our results improve all the above mentioned results: the novelty of this paper relies on the form of the operator, where a coupling first-order term is allowed, and on the fact that the diffusion coefficients can be unbounded in R d . Actually systems of elliptic operators coupled at the first-order are considered also in [9] with a different approach. More precisely, in such a paper conditions to extrapolate the semigroup T (t), first generated in C b (R d ; R m ) in [2], to the L p -scale are provided, but with no characterization of the domain. In general, the determination of the domain is a quite complicate issue which requires more assumptions on the coefficients and which can be simplified considerably assuming that the diffusion coefficients are uniformly elliptic and bounded. Instead, in the case of unbounded diffusion coefficients, already in the scalar case there are only partial results (see [29] and the reference therein).
Our main assumptions are listed in Hypotheses 3.1 and are inspired by those considered in [29] where the scalar equation is studied. We introduce an auxiliary potential v that controls the matrix-valued function V and, through Hypothesis 3.1(iv), allows to prove the regular L p -dissipativity of the operator A. On the other hand, the assumption (3.1) and the oscillation condition in Hypothesis 3.1(v) are crucial to interpolate the term d i=1 B i D i u between A 0 u and V u (see Remark 3.3) and consequently, assuming further (3.4), if p ∈ (1, 2), and (3.5), if p ∈ [2, ∞), to identify the domain. Condition (3.2) together with a bound on γ, was already used in the scalar case in [13,14] to show that the domain of the Schrödinger operator  [30,Example 3.7]). We point out that, already in the case of Schrödinger vector-valued operators, our hypotheses allow for the entries of V to grow at infinity as e |x| β , for every β > 0, improving the growth-rate considered in [27], (see Example 3.9). In addition, our results, besides giving a precise description on the domain of the generator in L p (R d ; R m ), work under less restrictive assumptions than those considered in [9]. In that paper two different set of hypotheses are considered: the first one imposes a sign on the drift term, the second one forces the matrices B i to be bounded when the diffusion coefficients are themselves bounded. In our case, neither a sign on the drift term is assigned, nor the drift has to be bounded when Q is bounded. Moreover, to extrapolate the semigroup T (t) to the L p -scale (p ∈ [2, ∞)), the first set of assumptions of [9] forces the quadratic form associated with the matrix-valued function −2V − i D i B i to be bounded from above. Our hypotheses cover also cases in which the previous form is not bounded from above.
We then slightly change our main hypotheses, to make them independent of p in the range [p 0 , ∞), for some p 0 > 1 (see Remark 3.5). Under, this new set of assumptions, we prove the consistency of the semigroups T p (t) := T (t) for every p ≥ p 0 . We then show that each operator Theorem 4.6). This is done through a comparison between the semigroup T ∞ (t) generated by A in C b (R d ; R m ) (that actually coincides with that generated in L p (R d ; R m )) and the scalar semigroup associated to the Schrödinger operator A v = div(Q∇)−v. The analysis in L 1 (R d ; R m ) will be deferred to a future paper.
Notation. Let d, m ∈ N and let K = R or K = C. We denote by ·, · and by | · |, respectively, the Euclidean inner product and the norm in K m . Vector-valued functions are displayed in bold style. Given a function u : Ω ⊆ R d → K m , we denote by u k its k-th component. For every p ∈ [1, ∞), L p (R d , K m ) denotes the classical vector valued Lebesgue space endowed with the norm f p = ( R d |f (x)| p dx) 1/p . The canonical pairing between L p (R d , K m ) and L p ′ (R d , K m ) (p ′ being the index conjugate to p), i.e., the integral over R d of the function is the classical vector valued Sobolev space, i.e., the space of all functions u ∈ L p (R d , K m ) whose components have distributional derivatives up to the order k, which belong to L p (R d , K). The norm of W k,p (R d , K m ) is denoted by · k,p . If the matrices B i (i = 1, . . . , d) have differentiable entries, then we set the matrix whose entries are obtained differentiating with respect to the variable x i the corresponding entries of the matrix B i . Given a d × d-matrix-valued function Q, we denote by q(u, v) the function defined by x → Q(x)∇u(x), ∇v(x) on smooth enough functions u and v. We simply write q(u) when u = v. Finally, given a vector-valued function u and ε > 0, we denote by w ε the scalar valued function w ε = (|u| 2 + ε) 1/2 .

Cores
The aim of this section is to prove vector valued versions of results about cores for elliptic operators with unbounded coefficients, in the line of those proved in [5]. Throughout the section, we will consider the elliptic operator A in (1.1) assuming that the matrix B i are diagonal, i.e., B i = b i I for some functions b i : R d → R, and we set b = (b 1 , . . . , b d ).
In the following lemma we adapt some known results about scalar elliptic regularity to the vector valued case. Lemma 2.1. Suppose that Q is locally positive definite, i.e. Q(x)ξ, ξ ≥ µ(x)|ξ| 2 for every x, ξ ∈ R d and some positive function µ such that inf K µ > 0 for every for some r ∈ (1, ∞). Then, the following properties hold true.
By the arbitrariness of h we get both assertions from the scalar case, by applying [11, Corollary 2.10] and standard elliptic regularity (see [1], or, e.g., [ Further, assume that Q is locally uniformly elliptic and there exists a positive function ψ ∈ C 1 (R d ) such that lim |x|→∞ ψ(x) = ∞ and b, ∇ψ ψ log ψ ≥ −C 1 and Q∇ψ, ∇ψ for some positive constants C 1 , C 2 and that in R d for every ξ ∈ R m . Then, the operator (A, C ∞ c (R d ; R m )) is closable on L p (R d ; R m ) and its closure generates a strongly continuous semigroup.
Since u ∈ W 1,r loc (R d ; R m ) with r > d, it follows from the Sobolev embedding theorem that ϕ n = ζ 2 n uw p ′ −2 ε belongs to W 1,2 loc (R d ; R m ) and has compact support. Hence, writing (2.4) with ϕ = ϕ n , we get Taking into account that D i w ε = w −1 ε D i u, u for every i = 1, . . . , d, it is easy to check that Moreover, applying Hölder's inequality we can estimate for every δ > 0. Finally, observing that and integrating by parts we obtain Replacing (2.6) and (2.8) in formula (2.5) and taking (2.7) (with δ < p ′ − 2) and (2.3) into account, we get Note that the second to last integral in (2.9) converges to 0 as ε → 0. Indeed, for each ε ∈ (0, 1) we can estimate which vanishes as ε → 0 + since the function ζ n is compactly supported in R d and the functions divb, w 1 are, respectively, locally integrable and locally bounded on R d . Here, (p ′ − 2) + denotes the positive part of p ′ − 2. Hence, letting ε → 0 and using the dominated convergence theorem, we deduce that Since, for every n ∈ N, ζ ′ (n −1 log ψ(x)) = 0 only if 1 ≤ n −1 log ψ(x) ≤ 2, taking (2.2) into account we can estimate Moreover, since ζ ′ ≤ 0 on [0, ∞), it follows that Hence, by dominated convergence we can let n tend to ∞ in both sides of (2.10) and conclude that λ u p ′ ≤ 0, whence u = 0.
. . , m and v hk ∈ L ∞ loc (R d ) for every h, k = 1, . . . , m with h = k. Further, assume that Q is locally uniformly elliptic and that there exists a positive function ψ ∈ C 1 (R d ), which diverges to ∞ as |x| tends to ∞, such that b, ∇ψ ψ log ψ ≤ C 1 and Q∇ψ, ∇ψ (ψ log ψ) 2 ≤ C 2 for some constants C 1 , C 2 > 0. Finally, assume that condition (2.3) holds true. Then, the realization A p of the operator is a core for (A, D p,max ). Finally, if b identically vanishes on R d , then, the previous semigroups exist for every p ∈ (1, ∞) and are consistent.
Then, there exists a sequence (u n ) in C ∞ c (R d ; R m ) such that u n and Au n converge, respectively to some function u and g in L p (R d ; R m ), as n tends to ∞. Hence, for where A * is the formal adjoint to the operator A, which implies that Au = g = Au distributionally. By standard elliptic regularity results, we deduce that u ∈ D p,max .
Let us now prove that λI − A is injective on D p,max for some λ > 0. For this purpose, we fix u ∈ D p,max such that λu = Au. Then, for every Next, we fix a function u ∈ D p,max and set Finally, let us assume that b identically vanishes on R d . To prove that the semigroups generated by the operators A p and A q are consistent, one can take advantage of the Trotter product formula (see [17,Corollary III.5.8]) to write r is the scalar semigroup generated by the operator div(Q∇). Both the semigroups e tA 0 r and e −tVr are consistent on the L pscale. If p, q ∈ (1, 2], then we observe that the operator A * adjoint to A satisfies the same assumptions as the operator A. Therefore, for r ∈ {p, q}, the semigroup e tAr is the adjoint of the semigroup generated in and the equality e tAp f = e tAq f follows. Finally, if p < 2 and q > 2, then follows for every t > 0 also in this case.

The full operator A
In this section, we consider the elliptic operator A defined in (1.1) assuming that p ∈ (1, ∞) and that the coefficients Q = (q ij ), B i = (B i hk ), V = (v hk ) satisfy the following assumptions: with positive infimum c 0 and positive constants κ, c 1 and θ < p such that Q∇ψ, ∇ψ ≤ Cψ 2 log 2 ψ.
Proposition 3.2. Under Hypotheses 3.1(i)-(vi), assume further that γ 2 ≤ 4 p−1 , if 1 < p < 2. Then, for every ε > 0 there exists a positive constant K ε , depending on p and the constants κ, c 0 , c 1 , c 2 , γ, C γ , such that The proof follows from the analogous scalar inequality proved in [29, Lemma 2.5, Proposition 3.3]. This is the unique point of the paper where we need conditions on the oscillation of the function Q, in the case p > 2.
Remark 3.3. It is worth observing explicitly that, as already proved in the scalar case, condition (3.3) does not imply that the drift term is a small perturbation of Since A 0 is a closed operator, D p endowed with the norm · Dp is a Banach space.
We can now state the main generation result: if p ∈ (1, 2), and the condition if p ≥ 2. Then, the operator A p generates an analytic contraction semigroup T p (t) in L p (R d ; C m ). Moreover, if the above assumptions are satisfied also for some Remark 3.5. We point out that, if Hypothesis 3.1(v) is satisfied for every γ > 0, then conditions (3.4) and (3.5) reduce, respectively, to It is easy to check that (3.7) is satisfied for every p ≥ 2 for instance if θ < 1/2 and km < √ 8. On the other hand, condition (3.6) cannot be satisfied for every If ∆ > 0, then the third-order polynomial f , in the variable p, defined in (3.6) has a local maximum f (p 1 ) and a local minimum f (p 2 ) at some points 0 < p 1 < p 2 .

Finally, if inequality (3.1) is replaced by the new condition
for every x, η k ∈ R d (k = 1, . . . , m), ξ ∈ R m and some positive constants κ and C κ , then the generation result in Theorem 3.4 can be applied to the operator A − λ κ for a suitable λ κ > 0. In particular, the operator A p generates a strongly continuous analytic semigroup (not contractive, in general) in L p (R d ; R m ). Indeed, condition (3.8) implies that for every ε > 0 there exists a positive constant λ such that Hypotheses 3.1 are satisfied with V being replaced by V + λ, provided that ε is chosen sufficiently small such that condition (3.4) (resp. (3.5)) holds true with κ being replaced by κ + ε. In particular, if (3.8) is satisfied by every κ > 0, then conditions (3.4) and (3.5) reduce to θ < p. Hence, if θ < 1, then we get generation results of a family of consistent semigroups for every p ∈ (1, ∞).
In the proof of Theorem 3.4 we will take advantage of the following result.
Proof. Let us first prove (3.9). We fix u, η and ε as in the statement of the lemma and observe that for p = 2, formula (3.9) can be obtained just integrating by parts and using the symmetry of the matrices B i (i = 1, . . . , d).
In the case p = 2, we denote by I the left-hand side of (3.9) and set By integrating by parts and taking into account the symmetry of the matrices B i , we deduce that which immediately yields (3.9).
Step 1. Here, we prove that , an integration by parts shows that Hence, using formula (3.9), splitting the term Re and integrating by parts the second integral term, we deduce that (3.14) To ease the notation, we denote by I and J the third and fourth integral terms in the right-hand side of the previous formula. Taking (3.1) into account and applying Cauchy-Schwarz and Hölder's inequalities, for every ε 0 , ε 1 > 0 we get Therefore, On the other hand, since for every j = 1, . . . , m, we can estimate We now distinguish between the cases p ∈ (1, 2) and p ≥ 2. In the first case, the coefficient of the second term in the right-hand side of (3.15) is negative. Using inequality (3.10), we can continue estimate (3.15) and get −Re where for every x 1 , x 2 > 0, and, similarly, The supremum of function f 1 , subject to the constraint g 1 (x 1 , x 2 ) > 0, is , see Subsection A.3, which is positive thanks to condition (3.4). Then, we can choose ε 0 and ε 1 positive and such that the coefficients of the two terms in the right-hand side of (3.16) are both positive. Thus, we get .
Now, we address the case p ≥ 2. Here, the coefficient of the second term in the right-hand side of (3.15) can be made positive by choosing ε 1 small enough. Note that the supremum of the function f 1 , subject to the constraints x 1 ∈ (0, p/(κ(p − 2))) and x 2 ∈ (0, p/(κm)), is (see Subsection A.4), which is positive due to condition (3.5). Hence, we can determine ε 0 and ε 1 such that the coefficients of the three terms in the right-hand side of (3.15) are all positive. With this choice of the parameters, estimate (3.19) follows immediately with Finally, letting ε tend to 0 + in (3.19), by dominated convergence we get (3.12) in both cases.
Step 2. Here, we prove that there exists a constant C = C(m, p, γ, κ, θ) > 0 such that assuming that C γ = 0 in Hypothesis 3.1(v). Clearly, since the coefficients of the operator A are real-valued, we can limit ourselves to considering functions with values in R m . We fix u ∈ C ∞ c (R d ; R m ), ε > 0 and set f = −Au. Then, Integrating by parts, taking (3.13) into account, we deduce that Thus, applying Cauchy-Schwarz and Hölder's inequalities and taking Hypothesis 3.1(iv) into account, we get We now estimate the term J 2 . Using the same arguments as in the proof of (3.14) and applying Young's inequality, we get Summing up, we have proved that where for every (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 + . By applying Young's inequality, we can easily show that for every δ > 0 there exists a positive constant C = C(δ, p) such that where B(0, R) is any ball containing the support of the function f . By combining (3.21) and (3.23), we get As in Step 1, we distinguish between the cases p ∈ (1, 2) and p ≥ 2. In the first case, assuming that 1 − ε 1 − p −1 κ|p − 2|ε 3 > 0 and using (3.10), we can combine the first two terms in the left-hand side of (3.24) and obtain the inequality where the function g 2 : R 4 + → R is defined by for every (x 1 , . . . , x 4 ) ∈ R 4 + . It is easy to check that the supremum of f 2 , subject to the constrain g 2 (x 1 , . . . , x 4 ) > 0, is given by (see Subsection A.1 for further details). Due to condition (3.4), this supremum is positive. Thus, we can choose the parameters ε j (j = 1, . . . , 4) such that the coefficients of the two first integral terms in the left-hand side of (3.25) are both positive, so that Letting ε tend to 0 + , we can choose δ > 0 such that estimate (3.20) follows. If p ≥ 2, then the supremum of the function f 2 , subject to the constrains p(1 − (see Subsection A.2 for further details), which is positive due to condition (3.5). Now, we can argue as in the case p ∈ (1, 2) to obtain estimate (3.20).
Step 3. Here, we prove that there exist positive constants M 1 and M 2 depending on κ, c 0 , c 1 , c 2 , γ, C γ , θ and p, such that Also in this case, we can assume that u takes values in R m . We first assume that C γ = 0 and observe that condition (3.4) implies that γ 2 > 4(p − 1) −1 if p ∈ (1, 2). Then, for every u ∈ C ∞ c (R d ; R m ), taking into account Step 2 and (3.3), we can estimate so that using Step 1, which shows that the operator A is dissipative, we get u Dp ≤ 2K Au p ≤ 2K Au − u p + 2K u p , ≤ 4K Au − u p , where K = 1 + (1 + K 1/2 )c 1 C. The first part of (3.27) follows with M 1 = (4K) −1 . If C γ = 0, then we can determine a positive constant λ such that V + λI and v + λ satisfy Hypothesis 3.1(ii) with C γ = 0. In such a case, using again the dissipativity of A we obtain and the first part of (3.27) follows with M 1 = ((1 + λ) K) −1 and K is a positive constant, depending on c 1 , K 1/2 and C.
To prove the other part of (3.27) we argue similarly, observing that , so let us prove the other inclusion. We fix u ∈ D p,max (A 0 − V ) and observe that Theorem 2.3 guarantees the existence of a sequence (u Step 3, applied with B i = 0, shows that (u n ) is a Cauchy sequence in D p endowed with the norm · Dp . Since this latter is a Banach space, we conclude that u ∈ D p and the inclusion D p,max (A 0 − V ) ⊂ D p follows. This argument also shows that C ∞ c (R d ; R m ) is dense in D p .
Step 5. Here, we prove that the operator (A, D p ) generates a contraction semigroup in L p (R d ; R m ). In view of Step 1, it suffices to show that the operator I − A : D p → L p (R d ; R m ) is surjective. For this purpose, we apply the continuity method. For every t ∈ [0, 1], we introduce the operator Due to the density of C ∞ c (R d ; R m ) in D p , we can first extend (3.3) to every u ∈ D p and, then, using this inequality, we can extend (3.27) to every u ∈ D p . Thus, we can determine a positive constant K, independent of t ∈ [0, 1], such that L t u p ≥ K u Dp for every t ∈ [0, 1] and u ∈ D p . Moreover, Theorem 2.3 and Step 4 show that the operator (A 0 − V, D p ) generates a strongly continuous semigroup of contractions in L p (R d ; R m ). Hence, the operator L 0 is surjective on L p (R d ; R m ). Since the operator A 0 has real-valued coefficients, it follows that L 0 is surjective on L p (R d ; R m ). The continuity method applies showing that also the operator Step 6. Here, we complete the proof, showing that, if q is a different index in (1, ∞), which satisfies the assumptions of the theorem, then T p (t)f = T q (t)f for every t > 0 and f ∈ L p (R d ; C m ) ∩ L q (R d ; C m ). Since both T p (t) and T q (t) map functions with values in R m in functions with values in R m , we can limit ourselves to considering functions with values in R m . By Theorem 2.3, the semigroups generated by the closure of the operator . As a byproduct, writing the resolvent operators as the Laplace transform of the semigroups. we infer that the resolvent operators coincide on L p (R d ; R m ) ∩ L q (R d ; R m ). Therefore, for every f ∈ L p (R d ; R m ) ∩ L q (R d ; R m ) and λ ∈ R sufficiently large there exists a unique u ∈ D p ∩ D q which solves the equation λu − A 0 u + V u = f .
Next, we observe that all the computations in the previous steps can be performed replacing · Dp with · Dp + · Dq and by applying the method of continuity in the space L p (R d ; R m )∩L q (R d ; R m ) endowed with the norm · p + · q . It follows that λI − A : D p ∩D q → L p (R d ; R m )∩L q (R d ; R m ) is invertible for every λ > 0, and By the representation formula of semigroups in terms of the resolvents, we get the assertion.
Remark 3.7. We point out that if there exists µ 0 > 0 such that Q(x)ξ, ξ ≥ µ 0 |ξ| 2 for every x, ξ ∈ R d , then D 2 is continuously embedded in W 1,2 (R d ; R m ). Indeed, from (3.24) from which it follows immediately that Since C ∞ c (R d ; C m ) is a core for A 2 , the previous inequality extends to every u ∈ D 2 . To conclude this section we prove that D p coincides with the maximal domain of the realization A p of A in L p (R d ; R m ) and we provide some examples of operators A which satisfy our assumptions.  To prove that D max (A p ) ⊂ D p , it suffices to prove that λI − A is injective on D max (A p ) for some (hence all) λ > 0. So, let us consider u ∈ D max (A p ) such that λu − Au = 0. We have to show that u ≡ 0. To this aim, we prove that λ R d |u| 2 z p−2 dx ≤ 0, (3.28) where z = w ε and ε is any positive constant, if p ∈ (1, 2), whereas z = |u| if p ≥ 2. Once formula (3.28) is proved, the claim follows easily letting ε → 0. The argument used to prove (3.28) is similar to that already used in the proof of Theorems 2.2 and 3.4. For this reason we give a sketch of it. Note that (3.29) To ease the notation, we denote by I j , j = 1, . . . , 5 the last five integral terms in the right-hand side of (3.29). Using Hölder's inequality, we get for every ε 0 > 0. Moreover, using Hypothesis 3.1(iv) and again Hölder's inequality we deduce that for every ε 2 > 0 and for every ε 1 > 0. Finally, for every ε 3 > 0, In addition, Now, we distinguish between the cases p ≥ 2 and p ∈ (1, 2). In the first case, from all the above estimates we obtain where function f 1 is defined by (3.17). Now, thanks to condition (3.5), with γ = 0, we can choose ε i > 0 (i = 0, 1, 2, 3) to ensure that the first three terms in the right hand side of (3.30) are nonpositive. We refer the reader to Subsection A.3 for further details. Thus, we get for some positive constant C depending on m, p, κ. Then, arguing as in the proof of Theorem 2.2, letting n → ∞ we deduce (3.28).
In the second case, when p ∈ (1, 2) we use estimate (3.10) to deduce that where the functions g 1 and f 2 are defined by (3.18) and (3.22). Also in this case, using condition (3.4), with γ = 0, we can choose ε i > 0 (i = 0, 1, 2, 3) to make the first two terms in the right hand side of (3.31) nonpositive and then we can conclude as in the first case. We refer the reader to Subsection A.4 for further details.
On the other hand, if we set then, by applying the Cauchy-Schwarz inequality, we get for every x, η k ∈ R d . Denote by λ A i , Λ A i , respectively, the minimum and the maximum eigenvalues of A i and let λ ∈ R d the vector with entries λ i = |λ A i |∨|Λ A i | (i = 1, . . . , d). Then, again by the Cauchy-Schwarz inequality, we can estimate for every x ∈ R d and ξ ∈ R m , where ϑ := (2γ + β)|λ|c 0 and Therefore, Hypotheses 3.1 are satisfied for every p > c 0 (2γ + β)|λ|. If we assume further that 4(p 2 − θp)(p − 1) − A 2 0 (2 − p) 2 ( √ m + m) 2 > 0, then, by Theorem 3.4 and (see also Remark 3.5) and [18,Theorem 2.7], the operator generates an analytic contraction semigroup An example of function V satisfying (3.34) for m = 2 is In this section, under suitable assumptions on the coefficients of the operator A, we prove that we can associate a semigroup T ∞ (t) with A in C b (R d ; C m ) that, under the assumptions of Theorem 3.4, coincides with the semigroup T (t) generated in L p (R d ; C m ) for each p ∈ (1, ∞). Again, we can limit ourselves to considering functions with values in R m .
Proof. We first consider the case p ∈ (1, 2). A straightforward computation reveals that the function z ε,p = w p ε − S(·)(|f | 2 + ε) p 2 is a classical solution to the equation Using the hypotheses, Cauchy-Schwartz inequality which implies that q(|u| 2 ) ≤ 4w 2 ε m k=1 q(u k ) and Young's inequality, we infer that for every σ > 0. Choosing σ = p − 1 and taking Hypothesis 4.1(iii) into account, for every t > 0 and x ∈ R d , where we used the fact that, by Hypothesis 4.1(iv) and [26, (proof  Finally, the case p ∈ (2, ∞) follows easily from the case p = 2 if we recall that S(t) admits an integral representation with a kernel p : (0, ∞) × R d × R d → R satisfying the condition p(t, x, ·) L 1 (R d ) ≤ 1 for every t > 0 and x ∈ R d (see [26,Theorem 1.2.5]). Hence, by the Hölder's inequality we conclude that |u(t, ·)| p ≤ (S(t)|f | 2 ) p 2 ≤ S(t)|f | p in R d for every t > 0.
The following theorem is an immediate consequence of Proposition 4.2.
Proof. The proof of this result is standard. The uniqueness of the locally in time bounded classical solution u follows from Proposition 4.2, with p = 2. The existence can be obtained by compactness, considering the sequence (u n ) ⊂ C 1+α/2,2+α loc of functions such that D t u n = Au n on (0, ∞) × B(0, n), u n vanishes on (0, ∞) × ∂B(0, n) and equals function f on {0} × B(0, n) for every n ∈ N. Each function u n also satisfies the estimate u n (t, ·) ∞ ≤ e −c0t for every t > 0. We refer the reader to [2,Theorem 2.8] for the missing details.
Thanks to Theorem 4.3 we can associate a semigroup T ∞ (t) to A in C b (R d , R m ), by setting T ∞ (·)f := u, where u is the solution to the Cauchy problem (4.1) provided by Theorem 4.3. Clearly, T ∞ (t) L(C b (R d ;R m )) ≤ e −c0t for every t > 0. This semigroup can be easily extended to C b (R d ; C m ) in a straightforward way.
In order to show that the semigroups T p (t) (p ≥ p 0 ) are consistent also with T ∞ (t) on C b (R d ; R m ) and, hence, on C b (R d ; C m ) we show that the domain D(A) of the weak generator of T ∞ (t) coincides with the maximal domain of A in C b (R d ; R m ). The notion of weak generator A has been extended to vector-valued elliptic operators with unbounded coefficients in [16,4] mimicking the classical definition of infinitesimal generator of a strongly continuous semigroup. Its domain is the set of all functions u ∈ C b (R d ; R m ) such that the function t → t −1 (T ∞ (t)u − u) is bounded in (0, 1] with values in C b (R d ; R m ) and it pointwise converges on R d to a continuous function, which defines Au.
Proof. As we have already stressed, we can limit ourselves to proving that with compact support, and consider the resolvent equation λu − Au = f . By Theorem 3.4 and Proposition 4.4 the previous equation admits a unique solution u r ∈ D(A r ), for every r ∈ [p 0 , ∞), and a unique solution u ∞ ∈ D max (A). Since u r is the Laplace transform of the semigroup T r (t), the consistency of the semigroups T p (t) and T q (t) implies that u p = u q for p, q ∈ [p 0 , ∞). Hence, we can simply write u instead of u p , and u ∈ W 2,q loc (R d ; R m ) for every q ∈ [p 0 , ∞), so that it is continuous over R d . To prove that it is also bounded on R d , we fix r > 0, arbitrarily, and R > 0 such that supp(f ) ⊂ B(0, R). Since the operator A q is dissipative, we can write u L q (B(0,r);R m ) ≤ u q ≤ λ −1 f L q (B(0,R);R m ) . Letting q tend to ∞ and using the arbitrariness of r > 0, we conclude that u is bounded over R d , so that it belongs to C b (R d ; R m ). Since Au = λu − f , the function Au is bounded and continuous over R d . Thus, u ∈ D max (A) and the equality u = u ∞ follows.
Next, we fix a function ψ ∈ C ∞ c (R d ; R m ), write u, ψ = u ∞ , ψ and integrate both sides of this equality over R d . Taking (4.4) into account, we deduce that From the uniqueness of the Laplace transform, we conclude that from which the equality T p (t)f = T ∞ (t)f follows for every t > 0. To remove the condition on the support of f , we observe that every function and converge to f pointwise in R d . Taking the limit as n tends to ∞ in the equality T p (t)f n = T ∞ (t)f n , we complete the proof.
Theorem 4.6. Let the assumptions of Proposition 4.5 be satisfied. Then, the semigroup T (t) maps L p (R d ; C m ) into L q (R d ; C m ) for every t > 0 and p 0 ≤ p ≤ q ≤ ∞, and, for every T > 0, there exists a positive constant C = C(T, p, q) such that Proof. Also in this case we provide the assertion for functions with values in R m . For every p ∈ [p 0 , ∞), we consider the contraction semigroup S p (t) generated by the [29,Theorem 2.4]). The restriction of the semigroup S 2 (t) to L 2 (R d ) ∩ L 1 (R d ) can be extended to a contraction C 0 -semigroup S 1 (t) on L 1 (R d ) which is consistent with S p (t) for each p ∈ (1, ∞). Indeed, fix f ∈ L 2 (R d ) ∩ L 1 (R d ) and r > 0. Then, f ∈ L p (R d ) for every 1 < p < 2 and, since the semigroups S p (t) are consistent, we get By letting r tend to ∞, we conclude that the restriction of S 2 (t) to L 2 (R d )∩L 1 (R d ) extends, by density, to a contraction semigroup S 1 (t) on Hence for every p > 1 it follows that Letting first p tend to 1 and then r tend to ∞, we deduce that S 1 (t)f converges to f in L 1 (R d ) as t tends to 0. By density it follows that S 1 (t) is a strongly continuous semigroup.
On the other hand, S 2 (t) is the semigroup associated with the quadratic form Note that a(f ) ≥ min{c 0 , µ 0 } f 1,2 for every f ∈ D(a). Since by Nash's inequality f for some constant K > 0 , and therefore S 1 (t) is bounded from L 1 (R d ) into L 2 (R d ) (see [15,Theorem 2.4.6]). By observing that A v is self-adjoint, the usual duality argument proves that S 1 (t) is bounded from L 2 (R d ) into L ∞ (R d ) and, by applying the semigroup law, it follows that S 1 (t) is bounded from Using this property together with the estimate in the statement of Proposition 4.2 it is immediate to check that T p (t) maps L p (R d ; R m ) into L ∞ (R d ; R m ) for every p ≥ p 0 and (4.5) follows in this case.
Finally, applying Riesz-Thorin interpolation theorem, we conclude the proof.
Remark 4.7. It is worth noticing that, if the assumptions of Proposition 4.5 are satisfied, with (3.1) replaced by (3.8), to be satisfied by every κ > 0, and θ < 1, then the semigroup T p (t) is defined for every p > 1 (since, by Remark 3.5, the assumptions of Theorem 3.4 are satisfied by every p > 1) and estimate (4.5) holds true for every p, q ∈ (1, ∞), with p < q.
Example 4.8. Let A be the operator defined in Example 3.9 with a potential matrix V whose entries belong to C α loc (R d ) and satisfy (3.34). Assuming further that A 0 ≤ √ 2m −1 if p ≥ 2 and pA 2 0 m−4(p−1) 2 ≤ 0 if p ∈ (1, 2), all the assumptions in Hypotheses 4.1 are satisfied. Hypothesis 4.1(ii) is satisfied, for instance, by the function ϕ : R d → R, defined by ϕ(x) = 1 + |x| 2 for every x ∈ R d . Indeed, since diverges to −∞ as |x| → ∞, clearly we can find a positive constant λ such that for every x ∈ R d . Since ψ(x) diverges to −∞ as |x| → ∞, Hypothesis 4.1(iv) is satisfied too. Thus, Proposition 4.5 can be applied to deduce that T p (t)f = T ∞ (t)f for every f ∈ L p (R d ; C m ) ∩ C b (R d ; C m ).
Finally, Remark 4.7 infers that T (t) maps L p (R d ; C m ) into L q (R d ; C m ) for every t > 0 and 1 < p ≤ q ≤ ∞ and estimate (4.5) holds true.
A.4. On the inequality (3.31). Here, we prove in details that the free parameters in (3.31) can be chosen to guarantee that the first two terms in the right-hand side of (3.31) are all nonpositive. Of course, we just need to show that we can choose the positive parameters ε1 and ε3 in such a way that f2(ε1, ε3) and g1(ε1, ε3) are both nonnegative.