On improvement of summability properties in nonautonomous Kolmogorov equations

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided.


Introduction
Let A be an autonomous second order uniformly elliptic operator with unbounded coefficients defined in R d . It is well known that, under suitable assumptions on its coefficients, a Markov semigroup T (t) can be associated in C b (R d ) to the operator A. More precisely, for any f ∈ C b (R d ), T (t)f is the value at t of the (unique) bounded classical solution of the Cauchy problem D t u(t, x) = Au(t, x), (t, x) ∈ (0, +∞) × R d , u(0, x) = f (x), x ∈ R d .
Estimate (1.1) is equivalent to the occurrence of some functional inequalities satisfied by the invariant measure µ. We refer to [9], the pioneering work on such topics, where a characterization of the hypercontractivity and the supercontractivity of the semigroup T (t) is given in terms of some logarithmic Sobolev inequalities.
Ultraboundedness and ultracontractivity have been widely studied in the autonomous setting, mainly in the symmetric case (where they are equivalent). The first result in this direction is due to Davies and Simon [5,6] that, following the idea of Gross and requiring some additional integrability conditions, connect ultracontractivity with a family of logarithmic Sobolev inequalities.
Other different approaches to study ultracontractivity have been also suggested by [3] and, more recently, by [20].
On the other hand, to the best of our knowledge, results on summability improving have been not yet studied in the nonautonomous case.
In the recent paper [2] we have dealt with hypercontractivity and we have extended the connection with logarithmic Sobolev inequalities in a nonautonomous setting, where the semigroup T (t) and the invariant measure µ are replaced, respectively, by a Markov evolution operator G(t, s) and an evolution system of measures {µ t }.
In this paper we are interested in exploiting some regularizing properties, stronger than hypercontractivity, for the evolution operator G(t, s), and in characterizing them in terms of suitable inequalities satisfied by an evolution system of measures {µ t }.
Let I be an open right halfline and for every t ∈ I consider the nonautonomous second order differential operator A(t) defined on smooth functions ζ by (A(t)ζ)(x) = Tr(Q(t)D 2 ζ(x)) + b(t, x), ∇ζ(x) , x ∈ R d .
We assume some smoothness on Q = [q ij ] i,j=1,...,d and b = (b 1 , . . . , b d ), defined in I and I × R d , respectively. Moreover, we require that the coefficients q ij are bounded and that the operators A(t) are uniformly elliptic, i.e., there exists a positive constant η 0 such that Assuming the existence of a Lyapunov function, for every s ∈ I and f ∈ C b (R d ), the nonautonomous Cauchy problem D t u(t, x) = A(t)u(t, x), (t, x) ∈ (s, +∞) × R d , u(s, x) = f (x), x ∈ R d , admits a unique bounded classical solution u = G(·, s)f , where G(t, s) is a Markov evolution operator. The function G(·, s)f belongs to C 1+α/2,2+α loc ((s, +∞)×R d ) and admits the following representation formula (G(t, s)f )(x) = R d g t,s (x, y)f (y)dy, s < t, x ∈ R d , f ∈ C b (R d ), (1.2) where g t,s : R d × R d → R is a positive function such that g t,s (x, ·) L 1 (R d ) = 1 for any t, s ∈ I, with t > s, and any x ∈ R d . The existence of a Lyapunov function such that lim |x|→+∞ ϕ(x) = +∞ and (A(t)ϕ)(x) ≤ a − γ ϕ(x), (t, x) ∈ I × R d , for some positive constants a and γ, allows (see [12]) to prove the existence of tight evolution systems of measures {µ t : t ∈ I}, i.e., families of Borel probability measures such that µ t (B(0, R)) tends to 1 as R → +∞, uniformly with respect to t ∈ I, and The interest in evolution systems of measures is due to the good properties that the evolution operators enjoy in the L p -spaces related to these systems. Indeed, using (1.3) and the density of C ∞ c (R d ) in L p (R d , µ t ) for every t ∈ I, the evolution operator can be extended to a contraction (still denoted by G(t, s)) from L p (R d , µ s ) to L p (R d , µ t ) for every p ∈ [1, +∞).
In this context a generalization to the nonautonomous case of the definitions of hypercontractivity, supercontractivity, ultracontractivity and ultraboundedness (see Definition 2.6) and of their characterizations is significant and interesting.
As it has been already remarked, in [2] hypercontractivity of the evolution operator G(t, s) has been studied, assuming some stronger assumption than the minimal ones that guarantee the basic properties of G(t, s) and the existence of an evolution system of measures {µ t : t ∈ I}. In fact, if the dissipativity condition is satisfied for some r 0 < 0, then the logarithmic Sobolev inequality (in short LSI) for the unique tight evolution system of measures {µ s : s ∈ I} holds for any s ∈ I, f ∈ H 1 (R d , µ s ) and some positive constant C, independent of f and s. The hypercontractivity of G(t, s) in L p spaces related to the unique tight evolution system of measures, is obtained as a consequence of the (LSI). In general, evolution systems of measures are infinitely many (see e.g., [8]). Among all of them, the unique tight system as a prominent role. Indeed, it is related to the asymptotic behaviour of G(t, s) as t → +∞. As it has been proved in [2], under condition (1.4) uniformly with respect to f ∈ L p (R d , µ s ), p ∈ [1, +∞), where m s (f ) denotes the average of f with respect to the measure µ s .
In this paper, we assume that condition (1.4) holds true and consider the unique tight evolution system of measures {µ s : s ∈ I}.
We first prove that the supercontractivity property of the evolution operator G(t, s) is equivalent to the validity of the following family of logarithmic Sobolev inequalities (in short LSI ε ) for every s ∈ I, f ∈ H 1 (R d , µ s ), ε > 0 and some positive decreasing function β. We follow the method of [17] that, on a Riemann manifold M , deals with the diffusion semigroup P t generated by the autonomous operator L = ∆ + Z∇ with Neumann boundary conditions on ∂M , where Z is a C 1 -vector field satisfying a curvature condition. The condition on the curvature is used to deduce the following logarithmic Sobolev inequality satisfied by P t which holds for every f ∈ C ∞ 0 (M ), t > 0 and some positive constant K > 0. The starting point of our analysis is the analogue of (1.6) in the nonautonomous case; we prove a logarithmic Sobolev inequality satisfied by the probability measures g t,s (x, dy) = g t,s (x, y)dy defined in (1.2). More precisely, we show that for every f ∈ C 1 b (R d ) and t, s ∈ I such that t ≥ s. The key tool for the proof of estimate (1.7) (and of many results in the paper) is the pointwise gradient estimate (1.8) that has been proved in [12] under the assumption (1.4) (which is equivalent to the condition considered in [17]). Even in the autonomous case, (1.8) does not hold when the diffusion coefficients depend on x and they do not satisfy the condition in [21]. This is the reason why we confine ourself to the case of diffusion coefficients depending only on t.
Another important consequence of (1.8) is the Harnack type estimate satisfied by any f ∈ C b (R d ). Estimate (1.9) and LSI ε allow us to prove a second criterion for supercontractivity: we show that the integrability with respect to the measures {µ t : t ∈ I} (uniform in t) of the Gaussian functions ϕ λ (x) := e λ|x| 2 , for every λ > 0, is another condition equivalent to the supercontractivity of G(t, s). This second characterization is useful in order to provide a sufficient condition for the evolution operator G(t, s) to be supercontractive as stated in Theorem 3.9. The Harnack type estimate (1.9) is also the key tool to prove that, if G(t, s)ϕ λ ∈ L ∞ (R d ) for every t > s ∈ I and λ > 0, and sup s,t∈I t−s≥δ then G(t, s) is ultrabounded. We provide a sufficient condition for G(t, s)ϕ λ to be bounded for every t > s ∈ I and every λ > 0 (see Theorem 4.1). Actually, condition (1.10) is also necessary to get ultraboundedness. We prove the necessity of this condition using the characterization of the supercontractivity property in terms of the family of inequalities (1.5).
A quite sharp condition to get ultraboundedness of G(t, s) is given in terms of the inner product between the drift b(t, x) and x, which has to satisfy b(t, x), x ≤ −K 1 |x| 2 (log |x|) α , t ∈ I, |x| ≥ R, (1.11) for some positive constants K 1 , α > 1 and R > 1. Under some stronger condition than (1.11) on b(t, x), x , we prove that G(t, s) is bounded from L 1 (R d , µ s ) to L 2 (R d , µ t ), hence it is ultracontractive.
Then we extend supercontractivity, ultraboundedness and ultracontractivity to evolution operators associated to nonautonomous operators with non zero potential term.
Finally, we establish some consequences of the regularizing properties of G(t, s). More precisely, we get an L ∞ -estimate for the integral kernel g t,s of G(t, s) (see (1.2)) and some L 2 -uniform integrability properties of G(t, s).
The paper is organized as follows. First, in Section 2, we state our main assumptions, we collect some known results on the evolution operator G(t, s) and we give the definition of supercontractivity, ultraboundedness and ultracontractivity in our nonautonomous setting. Section 3 is devoted to prove two criteria for the supercontractivity property of G(t, s). In Section 4 we provide a characterization of ultraboundedness for G(t, s) in terms of the boundedness of the function G(t, s)ϕ λ . Section 5 concerns the L 1 -L 2 boundedness of G(t, s) and the consequent ultracontractivity property. Finally, in Section 6, we collect some consequences of the ultracontractivity of G(t, s).
Notations. Let k ∈ N ∪ {0, +∞}, we consider the usual space C k (R d ), as well as C k b (R d ), the subspace of C k (R d ) consisting of bounded functions with bounded derivatives up to the k-th order. We use the subscript "c" instead of "b" for the subsets of the above spaces consisting of functions with compact support.
If J ⊂ R is an interval and α ∈ (0, 1), C α/2,α (J × R d ) denotes the usual parabolic Hölder space. We use the subscript "loc" to denote the space of all f ∈ C(J × R d ) which are (α/2, α)-Hölder continuous in any compact set of J × R d .
Let µ be a probability measure on R d and 1 ≤ p < ∞. We denote by L p (R d , µ) the set of µ-measurable functions f : consists of all the functions which belong to L 2 (R d , µ) together with their first order distributional derivatives.
Let T be an operator mapping L p (R d , µ) to L q (R d , ν) for 1 ≤ p ≤ q ≤ +∞ where µ, ν are two probability measures on R d . If no confusion may arise, we denote by T p→q the operator norm T L(L p (R d ,µ),L q (R d ,ν)) .
About partial derivatives, the notations D t f := ∂f ∂t , D i f := ∂f ∂xi , D ij f := ∂ 2 f ∂xi∂xj are extensively used. About matrices and vectors, we denote by Tr(Q) and x, y the trace of the square matrix Q and the inner product of the vectors x, y ∈ R d , respectively.
The ball in R d centered at 0 with radius r > 0 is denoted by B(0, r). Finally, we set 0 log 0 = 0 by definition.

Assumptions, definitions and a review of some properties of G(t, s)
Let I be an open right halfline. For t ∈ I we consider linear second order differential operators A(t) defined on smooth functions ζ by under the following assumptions on their coefficients.
Hypotheses 2.1 yield the existence of a Markov evolution operator G(t, s) and a unique ([2, Rem. 2.8]) tight evolution system of measures {µ t : t ∈ I} associated to the evolution operator G(t, s) (where tight means that for any ε > 0 there exists R > 0 such that µ t (B(0, R)) ≥ 1 − ε for any t ∈ I). More precisely for every s ∈ I and f ∈ C b (R d ), G(·, s)f is the unique bounded classical solution of the Cauchy problem and it can be represented by From formula (2.5) the following result, which is extensively used in the paper, follows at once. Lemma 2.3. For any I ∋ s < t and any nonnegative and non identically van- By Lemma 2.3, formula (1.3) and the density of for every t > s, p ∈ [1, +∞) and f ∈ L p (R d , µ s ). Therefore, G(t, s) may be extended to a contraction (still denoted by The dissipativity condition (2.3) yields the pointwise gradient estimate . Other remarkable (smoothing) properties of the evolution operator G(t, s) and of the associated evolution system of measures {µ t : t ∈ I}, which are extensively used in this paper, are stated in the following two propositions and they can be proved assuming only Hypotheses 2.1(i)-(iii).

Proposition 2.4 ([12, Lemma 3.2])
. For any f ∈ C 2 (R d ), which is constant outside a compact set, and any t ∈ I, the function The aim of this paper, as already announced in the introduction, is to study the smoothing effects of the evolution operator G(t, s) on functions with a certain degree of summability. As in the autonomous case we can distinguish different levels of regularization as specified in the following definition.
Definition 2.6. The evolution operator G(t, s) is called: for any 1 < p < q < +∞ and t > s, and there exists a positive decreasing function C p,q : (0, +∞) → (0, +∞), such that lim r→0 + C p,q (r) = +∞ and for every p > 1 and t > s, and there exists a decreasing function C p,∞ : (0, +∞) → (0, +∞) such that lim r→0 + C p,∞ (r) = +∞ and states that G(t, s) maps L ∞ (R d ) into C b (R d ) for every t > s and it is a contraction, i.e., for every f ∈ L ∞ (R d ) Throughout the paper, if not otherwise specified, we assume that all the conditions in Hypotheses 2.1 are satisfied.

Supercontractivity and LSI ε
In this section we provide two criteria to characterize the supercontractivity of the evolution operator G(t, s) by means of a family of logarithmic Sobolev inequalities.
3.1. The first criterion. In this subsection we are devoted to prove the following result.
Theorem 3.1. The following properties are equivalent.
The proof of Theorem 3.1 is based on the following two propositions. In the first one, we prove a logarithmic Sobolev inequality satisfied by the evolution operator G(t, s), namely a LSI type estimate satisfied by the probability measures g t,s (x, y) dy in place of the invariant measures µ s .
Proof. We can limit ourselves to proving (3.1) for p = 2. Indeed, for every p > 2 and , the claim can be obtained applying (3.1) with p = 2 to the function |f | p/2 . Moreover, it is enough to prove (3.1), with p = 2, for nonnegative functions To this aim we introduce a standard sequence of cut-off functions and a nonnegative function f ∈ C 1 b (R d ) with f ∞ ≤ 1, and consider the function which is well defined by Lemma 2.3. For any s ≤ r ≤ t, F n (r) converges to F (r) = as n → +∞, by the monotone convergence theorem (see (2.5)). Moreover, since the function θ n (G(r, s)f ) 2 log(G(r, s)f ) 2 belongs to C 2 b (R d ) for every r > s and it vanishes outside B(0, 2n), by Proposition 2.4 and the formula A(r)(g 2 log g 2 ) = 2g(1 + log g 2 )A(r)g + 2(3 + log g 2 ) Q(r)∇g, ∇g , which holds for every positive function g ∈ C 2 (R d ) and every r ∈ I, we get for any r ∈ [s, t]. Using the dominated convergence theorem, we have Similarly, since ∇θ n vanishes uniformly in R d , as n → +∞, we easily conclude that I 3,n (r) tends to 0 as n → +∞. Now, let us consider the term I 2,n ; by (2.4), we can estimate Summing up, we have proved that for any r ∈ [s, t] and any x ∈ R d . Since the sequence F ′ n is bounded from below by a constant, from the Fatou lemma we can conclude that Using the gradient estimate (2.6) we get and (3.1) follows.
Next proposition shows that the boundedness of G(t, s) from L p (R d , µ s ) into L q (R d , µ t ), for any t > s, yields a family of logarithmic Sobolev inequalities satisfied by the system of invariant measures {µ t : t ∈ I}. The key tools used in the proof are estimate (3.1) and the Riesz-Thorin's interpolation theorem. Proposition 3.3. Assume that, for every s ∈ I, t > s and 1 < p < q < +∞, C p,q (t, s) := G(t, s) p→q < +∞. Then, Proof. The proof can be obtained adapting the arguments in the proof of [17, Thm.
For the reader's convenience we enter into details. We split the proof into two steps. In the first one we show that it suffices to prove (3.4) for functions f ∈ C 1 c (R d ) such that f 2,µs = 1. In the second step, we get estimate (3.4) for such functions.
Step 1. For notational convenience, we set We assume that inequality (3.4) holds for any function f ∈ C 1 c (R d ) such that f 2,µs = 1, and we show that it actually holds for any f ∈ H 1 (R d , µ s ). For this purpose, let f ∈ H 1 (R d , µ s ) satisfy f 2,µs = 1, and consider a sequence µs) tends to 0 as n → +∞ (see [2, Lemma 2.5]). Without loss of generality, we can assume that f n 2,µs = 1 for any n ∈ N.
Up to a subsequence, f n (x) converges to f (x) for almost every x ∈ R d as n → +∞ and Thus, by Fatou lemma we deduce that which leads immediately to (3.4).
Step 2. Let us prove the claim for (3.5) for any s, t ∈ I, with s < t and any f ∈ C 1 c (R d ). Integrating (3.5) in R d with respect to the measure µ t and using (1.3), we get Let us fix 1 < p < q < +∞. By assumptions, G(t, s) p→q =C p,q (t, s) < +∞, for every t, s ∈ I such that t > s. Since G(t, s) 1→1 ≤ 1, from the Riesz-Thorin's interpolation theorem we get that for every f ∈ L p (R d , µ s ) and h ∈ (0, 1 − 1/p), where Fix f ∈ C b (R d ) such that f 2,µs = 1. Then, from (3.7) and, since which holds also for h = 0. Consequently, The first and the last sides of (3.8) represent respectively the incremental ratio at h = 0 of the functions h → G(t, s)|f | 2(1−h) q h q h ,µs and h → (C p,q (t, s)) r h q h . Since these two functions are differentiable at h = 0, we immediately deduce that or, equivalently, since {µ t : t ∈ I} is an evolution system of measure, which, replaced into (3.6), yields and the claim is proved.
Proof of Theorem 3.1. "(i) ⇒ (ii)" By Proposition 3.3, if G(t, s) is supercontractive, then the following family of logarithmic Sobolev inequalities holds for every s ∈ I, t > s, and f ∈ C 1 b (R d ) with f 2,µs = 1. SinceC p,q (t, s) ≤ C p,q (t − s), in formula (3.9) we have β is a positive function defined in (0, ∞) and 2 ≤ p ≤ q.
"(ii) ⇒ (i)" Assume that estimate (LSI ε ) holds for every f ∈ H 1 (R d , µ s ), s ∈ I, ε > 0 and some positive decreasing function β : (0, +∞) → (0, +∞). Then, for every f ∈ C 1 c (R d ), p ∈ (1, +∞) and s ∈ I, writing (LSI ε ) for the function |f | p/2 , we have Using (3.10) we deduce supercontractivity of G(t, s). Indeed, let ε > 0, f ∈ C 1 c (R d ) be nonnegative and non identically vanishing in R d , p ∈ (1, +∞) and s ∈ I; we set for any t ≥ s. To prove that G(t, s) is supercontractive, we show that H is a non increasing function. We would like to differentiate the function H and show that its derivative is nonpositive in (s, +∞). Unfortunately, we can differentiate functions of the type t → R d ψdµ t only when ψ is constant outside a compact set, which, in general, is not our case. For this purpose we use an approximation argument and introduce the functions H n (n ∈ N) defined by where θ n is defined in (3.2). From Proposition 2.5, for every n ∈ N, the function H n is differentiable for t > s with derivative given by Using (3.3) we can show that lim sup n→+∞ ϕ n (t) ≤ ψ(t) for every t > s, where (3.12) Now, applying the logarithmic Sobolev inequality (3.10) with G(σ, s)f and q(σ) in place of f and p respectively, we get by the definition of q and m given in (3.11). Therefore, from (3.12) we deduce that H(t) ≤ H(s), so that H is nonincreasing, i.e, Now, for any q > p and t > s, we fix ε = 2η 0 (t − s)(log((q − 1)/(p − 1))) −1 . We thus deduce that q(t) = q and, from (3.13) we obtain which is a decreasing function since β is decreasing as well.
The density of C 1 c (R d ) in L p (R d , µ s ) allows us to complete the proof.

A second criterion.
Here we show that the integrability with respect to the measures {µ t : t ∈ I} (uniform in t) of the Gaussian functions ϕ λ (x) := e λ|x| 2 for every λ > 0 is another condition equivalent to the supercontractivity of G(t, s). To this aim we first prove some preliminary results. The first proposition, whose proof is an adaption of Ledoux's method [13] to our setting, yields some exponential integrability result. A more general result than next Proposition 3.4 has been proved in [10], in the autonomous setting still assuming the validity of the (LSI ε ), where the evolution system of measures is replaced by a unique invariant measure. Moreover, if the inequality (LSI ε ) holds, then ϕ λ ∈ L 1 (R d , µ s ) for every λ > 0 and Proof. For every n ∈ N, let ψ n : [0, +∞) → R be a smooth increasing function such that ψ n (t) = t for any t ∈ [0, n], ψ n (t) = n + 1 for any t ≥ n + 2 and 0 ≤ ψ ′ n (t) ≤ 1 for any t ≥ 0. The functions f n (x) := ψ n (|x|) are bounded and satisfy |∇f n | ∞ ≤ 1 for any n ∈ N. Moreover, f n (x) converges increasingly to f (x) := |x| for any x ∈ R d , as n → +∞. Fix s ∈ I, λ > 0 and n ∈ N. We set H n,λ (r) := R d e λrfn dµ s (x) for any r > 0, and observe that (3.14) Applying the logarithmic Sobolev inequality (LSI) to the function e λrfn/2 and using (3.14), we get for every n ∈ N. Now, dividing by r 2 H n,λ (r) we have Integrating (3.15) from 1 to 2 with respect to r we deduce that H n,λ (2) ≤ e Cλ 2 (H n,λ (1)) 2 . Hence, R d e λfn dµ s (x) ≤ 2e M for any s ∈ I, and letting n → +∞ we get the first part of the claim. In order to prove the second part of the claim assume that (LSI ε ) holds and, for brevity, we set H n := H n,1 . Arguing as before and applying (LSI ε ) to e rfn/2 , we get 1 r log H n (r)

Since the evolution system of measures {µ
for every ε > 0 and n ∈ N. Integrating (3.16) from γ to σ we deduce that Therefore, for every 0 < γ < σ and ε > 0, Now, we observe that Moreover, by (3.17) which is finite for every 0 < λ < 1 2ε and n ∈ N. By the arbitrariness of ε and observing that sup s∈I e γ|·| 1/γ 1,µs < +∞, by the first part of the proof, we deduce that for some positive constant K, independent of s. Finally, we get the claim by the monotone convergence theorem letting n → +∞ in (3.19). Next proposition is an Harnack-type estimate satisfied by the evolution operator G(t, s). The proof of this result is essentially based on the gradient estimates (2.6) and extends the method used in [19] to the nonautonomous case. Proposition 3.6 (An Harnack-type inequality). For every f ∈ C b (R d ), p > 1, t > s and x, y ∈ R d we have . (3.20) Proof. Since |G(t, s)f | ≤ G(t, s)|f | for every f ∈ C b (R d ) and t > s, it suffices to prove (3.20) for nonnegative functions f . We split the proof into two steps. In the first one we prove (3.20) for nonnegative functions f ∈ C 1 b (R d ). In the second step, by standard approximation arguments we extend (3.20) to every nonnnegative function f ∈ C b (R d ).
Step 1. Let f ∈ C 1 b (R d ) be a nonnegative function. Fix t > s, x, y ∈ R d and set Φ n (r) := {G(t, r)[θ n (G(r, s)f ) p ]}(ψ(r)), where θ n is the sequence of cut-off functions defined in (3.2) and By Lemma 2.3 and Proposition 2.4, the function log Φ n is well defined, it belongs to C 1 ((s, t)) for every n ∈ N and there exist n 0 ∈ N and a positive constant C Φ such that Φ n (r) ≥ C Φ for every n > n 0 and r ∈ [s, t]. This last assertion follows since Φ n (r) > 0 for every r < t and Φ n (t) = (θ n (G(t, s)f ) p )(x) for every n ∈ N. Hence, choosing n large enough such that x ∈ supp θ n we conclude. Differentiating the functions r → log Φ n (r) (n ∈ N) in (s, t) we get Let observe that, if g = G(·, s)f , then where in the last inequality we have used (2.6) and (2.1). From (3.21) we get where C s,t is the constant in (2.4), we can estimate Recalling that γ 2 − βγ ≥ −β 2 /4 for every β, γ ∈ R and G(t, s)g 1 ≥ G(t, s)g 2 for every t ≥ s if g 1 ≥ g 2 (see Lemma 2.3) from (3.22) we deduce that for every n > n 0 . Integrating with respect to r between s and t we get and (3.20) follows letting n → +∞.
Step 2. Let f ∈ C b (R d ) be a nonnegative function; we can consider a sequence (f n ) n ⊂ C 1 b (R d ) of nonnegative functions converging to f uniformly on compact sets of R d and such that f n ∞ ≤ f ∞ . Then, by Step 1 we have for every t > s ∈ I and x, y ∈ R d . Taking into account formula (2.5), this yields the claim by the dominated convergence theorem.
The second announced characterization of the supercontractivity of G(t, s) is given in the following theorem. Its proof is based on Propositions 3.4, 3.6 and also on the first criterion given in Theorem 3.1.  (3.20) with respect dµ t (y) and recalling that {µ t : t ∈ I} is an evolution system of measures, we get for every t > s, r > 0, x, y ∈ R d , and f ∈ C b (R d ). Hence, where R is such that µ t (B(0, R)) > 2 −p , for any t ∈ I. Let us now fix q > p and set λ 0 = (2η 0 (p − 1)(t − s)) −1 q. By (3.25) we can estimate for any I ∋ s < t. Now, it is clear the monotonicity of the function r → C p,q (r) and that, by density, we can extend the previous inequality to any f ∈ L p (R d , µ s ). This completes the proof.
Our aim is now to provide a sufficient condition for the supercontractivity of the evolution operator G(t, s). First we prove a preliminary lemma.
Proof. A straightforward computation shows that as |x| → +∞. Hence, the function in brackets tends to −∞ as |x| → +∞, if γ and λ are as in the statement of the lemma. It is now immediate to show that there exist two positive constants a = a(λ, δ) and γ = γ(λ, δ) such that A(t)ψ λ,δ ≤ a − γψ λ,δ for any t ∈ I and (2.2) holds.
Theorem 3.9. Assume that there exist K 1 > 0 and R > 1 such that Then, the evolution operator G(t, s) is supercontractive.
Proof. In view of [12,Thm. 5.4], the proof is an immediate consequence of Theorem 3.7 and Lemma 3.8.

Ultraboundedness
In this section we provide a condition equivalent to the ultraboundedness property of the evolution operator G(t, s). As in [4,17,19,22]), which deal with the autonomous case, we use the Harnack type estimate (3.20) satisfied by G(t, s) to get ultraboundedness of G(t, s). However, we need to strengthen assumption (2.2), as next theorem shows.
Proof. We prove the claim for p ∈ (1, 2]. For p > 2, estimate (2.7) will follow from the Hölder inequality. We split the proof into two steps. First, we consider the case p = 2 and, then, the case p ∈ (1, 2).
Step 1. An insight into the proof of [14,Thm. 3.3] (see also [1,Thm. 4.3] for further details) shows that, under our assumptions, the function t → (G(t, s)ϕ λ )(x) is well defined for each t > s and x ∈ R d , and G(t, s)ϕ λ ∈ L ∞ (R d ) for every t > s and λ > 0. More precisely, if for every δ, λ > 0 we set then M δ,λ turns out to be a positive constant independent of t and s. This is enough to establish (2.7) with p = 2. Indeed, integrating both sides of estimate (3.20) (with p = 2) with respect to dµ t (y) and arguing as in the proof of Theorem 3.7, we get where R is such that µ t (B(0, R)) > 1 4 . Hence we obtain for every f ∈ C b (R d ) and for λ 0 = 1 2η0(t−s) . Formulas (4.2) and (4.3) yield . (4.5) The monotonicity of the function r → C 2,∞ (r) is immediate consequence of the fact that M δ2,λ1 ≤ M δ1,λ2 , for every 0 < δ 1 ≤ δ 2 and 0 < λ 1 ≤ λ 2 , as it can be easily proved. Now, let f ∈ L 2 (R d , µ s ) and consider f n ∈ C b (R d ) converging to f in L 2 (R d , µ s ) as n → +∞. Since G(t, s) is a contraction from L 2 (R d , µ s ) to L 2 (R d , µ t ), G(t, s)f n converges to G(t, s)f in L 2 (R d , µ t ) as n → +∞. Moreover for every t > s, and n, m ∈ N. Formula (4.6) yields that the sequence G(t, s)f n converges uniformly in R d to some function g ∈ C b (R d ) and that g = G(t, s)f . Then, we conclude writing (4.4) for f n and letting n → +∞.
Step 2. To prove (2.7) when p ∈ (1, 2), we observe that for any λ > 0, any s ∈ I and any n ∈ N, where ϕ λ,n = min{ϕ λ , n}. Letting n → +∞ and using (4.2) with δ = 1, we obtain Hence, condition (3.23) is satisfied, and Theorem 3.7 shows that the evolution operator G(t, s) is supercontractive. Therefore, for any f ∈ L p (R d , µ s ) and any s, t ∈ I with s < t. This completes the proof.

Remark 4.2.
Each function ϕ λ , as in Theorem 4.1, satisfies Hypothesis 2.1(iii), i.e., it is a Lyapunov function for the nonautonomous elliptic operators A(t). Indeed, since h λ is a convex function which tends to +∞ as r → +∞, there exist a λ > 0 and b λ ∈ R such that h λ (r) ≥ a λ r + b λ for any r ≥ 0. From (4.1) it thus follows that (A(t)ϕ λ )(x) ≤ −a λ ϕ λ (x) + b λ for any t ∈ I and any x ∈ R d \ B(0, R). Up to replacing b λ with a larger constant, if needed, we can assume that the previous inequality is satisfied by any x ∈ R d , so that (2.2) is satisfied. From [12,Thm. 5.4], we deduce that sup s∈I ϕ λ 1,µs < +∞ for any λ > 0, and this gives an alternative proof of the first part of Step 2 in Theorem 4.1.
As a consequence of Theorem 4.1 we now provide a sufficient condition for G(t, s) to be ultrabounded.
Remark 4.4. The condition (4.7) is rather sharp. Indeed in [11], the authors consider the autonomous operator (Aζ)(x) = ∆ζ(x) − ∇Φ(x), ∇ζ(x) , where Φ is such that e −Φ ∈ L 1 (R d ), and prove that, if Φ(x) ∼ |x| 2 log |x| as |x| → +∞, then the semigroup T (t) associated to A in C b (R d ) is not ultrabounded in the Lebesgue spaces with respect the invariant measure dµ(x) = e −Φ −1 1 e −Φ(x) dx. The Harnack type estimate (3.20) and the fact that G(t, s)ϕ λ ∈ L ∞ (R d ) for every λ > 0 and t > s represent the key tools used in the proof of Theorem 4.1 to get ultraboundedness. Hypotheses 2.1 are enough to prove the Harnack formula (3.20). On the other hand, to prove that G(t, s)ϕ λ ∈ L ∞ (R d ) for every λ > 0 and t > s we have strengthened our assumptions requiring the additional condition (4.1). The condition G(t, s)ϕ λ ∈ L ∞ (R d ) for every λ > 0 and t > s is optimal to get ultraboundedness of G(t, s) for every t > s. The proof of this fact is based on the occurrence of the family of logarithmic Sobolev inequalities (3.4) and the consequent measure concentration result proved in Proposition 3.4.
Theorem 4.5. The evolution operator G(t, s) is ultrabounded if and only if, for every λ > 0 and t > s, the function G(t, s)ϕ λ belongs to L ∞ (R d ) and, for any δ, λ > 0, there exists a positive constant K δ,λ such that Proof. In view of the proof of Theorem 4.1 the "if" part of the statement is true. Conversely, if G(t, s) is ultrabounded, then it is bounded from L p (R d , µ s ) into L q (R d , µ t ) for every t > s and 1 < p < q < +∞, and G(t, s) p→q ≤ G(t, s) p→∞ < +∞.

Ultracontractivity
In this section we assume the following additional assumption on the drift term of the operators A(t).
We can now prove the boundedness of G(t, s) from L 1 (R d , µ s ) into L 2 (R d , µ t ) following the basic ideas in the proof of [15,Thm. 3.4] for the autonomous case. We stress that the nonautonomous setting gives rise to some additional technical difficulties.
Theorem 5.3. Under Hypotheses 2.1 and 5.1, for any s, t ∈ I, with s < t, the operator G(t, s) is bounded from L 1 (R d , µ s ) into L 2 (R d , µ t ).
Proof. As a first step we observe that, for any s ∈ I and any nonnegative g ∈ C b (R d ), 2 g 2 2,µs log g 2,µs − g 2 2,µs log g 1,µs ≤ R d It suffices to prove (5.3) for functions with g 1,µs = 1, which reduces to 2 g 2 2,µs log g 2,µs ≤ R d g 2 log g dµ s (x), (5.4) since (5.3) in the general case will follow from applying (5.4) to the function g −1 1,µs g. To prove estimate (5.4) we observe that the measure dν s (x) = gdµ s (x) is a probability measure and the function ψ(x) = x log x is convex in (0, +∞). Therefore, Jensen inequality yields which is (5.4).
Finally, by density we can extend (5.7) to any f ∈ L 1 (R d , µ s ) and complete the proof.
As a consequence of Theorems 4.3 and 5.3 we get the announced ultracontractivity property of G(t, s). Proof. It suffices to prove the claim for p = 1. For p > 1 the statement follows from the Hölder inequality.
We can now prove the announced heat kernel estimates.