On the domains of elliptic operators in L 1

On the domains of elliptic operators in L ¹

Alessandra Lunardi

Dipartimento di Matematica, Universit`a di Parma Via D’Azeglio 85/A, 43100 Parma, Italy

E-mail: lunardi@unipr.it, www: http://math.unipr.it/∼lunardi Giorgio Metafune

Dipartimento di Matematica “Ennio De Giorgi”, Universit`a di Lecce C.P.193, 73100, Lecce, Italy

E-mail: giorgio.metafune@unile.it

Abstract

We prove optimal embedding estimates for the domains of second order elliptic opera- tors in L¹spaces. Our procedure relies on general semigroup theory and interpolation arguments, and on estimates for ∇T (t)f in L¹, in L^∞, and possibly in fractional Sobolev spaces, for f ∈ L¹. It is applied to a number of examples, including some degenerate hypoelliptic operators, and operators with unbounded coefficients.

1 Introduction

It is well known that the domains of the realizations of elliptic operators in L¹ spaces are not as nice as in the case of L^p spaces with p ∈ (1, ∞). For instance, it is easy to see that the domain of the realization of the Laplacian in L¹(R^N) is not contained in W^2,1(R^N) if N > 1. However, it is continuously embedded in W^1+β,1(R^N) for each β ∈ (0, 1).

Therefore, if u and ∆u are in L¹(R^N), then any first order derivative D_iu, i = 1, . . . , N , belongs to W^β,1(R^N) for each β ∈ (0, 1), and by Sobolev embedding it belongs also to L^p(R^N) for each p < N/(N − 1), with

kD_iukL^p≤ C(p) kuk_L1 + k∆uk_L¹
. (1.1) Easy counterexamples show that we cannot take p = N/(N − 1). They show also that, in general, Diu does not belong to any Lorentz space L^{N/(N −1),q}_loc (R^N) with q < ∞, see Example 6.2. On the other hand, it is possible to prove that each derivative D_iu belongs to the Besov space B∞^1,1(R^N), and

kD_iuk_B^1,1

∞(R^N)≤ C kuk_L1 + k∆uk_L¹
. (1.2) See e.g. [5, Cor. 4.3.16]. Since B∞^1,1(R^N) is continuously embedded in the Lorentz space L^{N/(N −1),∞}(R^N), it follows that each derivative D_iu belongs to L^{N/(N −1),∞}(R^N), and

kD_iuk_LN/(N −1),∞(R^N)≤ C kuk_L1 + k∆uk_L¹
. (1.3) For definitions and properties of Lorentz and Besov spaces see Section 2. In this paper we show that similar embeddings hold for a large class of second order elliptic operators,

A =

N

X

i,j=1

a_ijD_ij+

N

X

i=1

b_iD_i+ c. (1.4)

We assume that the realization of A, endowed with a suitable domain D(A), is the in- finitesimal generator of a semigroup (T (t))t≥0 in L¹(Ω), Ω being an open set in R^N with empty or smooth boundary. Our approach is based on interpolation arguments, on esti- mates for the semigroup (T (t))t≥0 and on the well known representation formula for the resolvent

R(λ, A) = Z ∞

0

e^−λtT (t)dt, for λ large enough. The needed estimates are

(i) k∇T (t)f k_L¹ ≤ Ce^ωtt^−γ1kf k_L1, (ii) k∇T (t)f k∞≤ Ce^ωtt^−γ2kf k_L1, t > 0, f ∈ L¹(Ω), (1.5) for Lorentz regularity, with 0 ≤ γ1 < 1 < γ2, ω ∈ R, and

kT (t)f k_Wβ,1 ≤ Ce^ωtt^−γkf k_L1, t > 0, (1.6) for Besov regularity, with β > 0, γ > 1, ω ∈ R. Then we show that the map

∇ : D(A) → L^{γ2−γ1γ2−1,∞}(Ω, R^N) (respectively, D(A) → B_∞^β/γ,1(Ω, R^N))

is bounded, by a procedure which was used up to now to find optimal Schauder estimates.

See e.g. [17, Theorem 2.1], [20]. In the case of the Laplace operator in R^N, the heat semigroup (T (t))t≥0 is easily seen to satisfy both (1.5) with γ1 = 1/2, γ2 = (N + 1)/2, and (1.6) with β = 3, γ = 3/2, for each f ∈ L¹(R^N). Hence we get (1.2) and (1.3).

Embedding results in weaker norms are much easier. Estimates (1.5), assuming for simplicity that ω = 0, imply by interpolation k∇T (t)f k_L^p ≤ C(N )t⁻N (1−1/p)+1

2 kf k_L1, for every p ∈ (1, ∞). Let u ∈ D(A) and, for λ > 0, set f = λu − Au. Then u = R∞

0 e^−λtT (t)f dt, and taking p < N/(N − 1) we get

k∇uk_L^p ≤ C(N, p)λ^γ−1kf k_L1 ≤ C(N, p)(λ^γkuk_L1 + λ^γ−1kAuk_L1), where γ = N (1−1/p)+1

2 < 1. Taking the minimum for λ > 0 one obtains the inequality k∇uk_L^p ≤ C(N, p)kuk^1−γ₁ kAuk^γ₁,

which is well known for elliptic operators with bounded smooth coefficients, see e.g. [27, Theorem 5.8], [4, Theorem 8], [2, Proposition 9.2]. Taking p = N/(N −1) we get γ = 1 and this direct approach fails. To treat the limiting case we need the more refined argument used in the proofs of Theorems 3.1 and 4.1.

Estimates of the type (1.5) or (1.6) are satisfied by a number of elliptic operators, including elliptic operators in bounded domains with Dirichlet or Neumann boundary conditions, a class of elliptic operators with unbounded Lipschitz continuous coefficients in R^N, and degenerate elliptic operators such as Kolmogorov type operators, the Heisenberg Laplacian in R³, and more generally sublaplacians in nilpotent stratified Lie groups. For all these operators the application of Theorems 3.1 and 4.1 gives optimal embedding results.

See Sections 5 and 6. Some of these embeddings were already known, and ours is just an alternative approach. This is the case of elliptic operators with bounded and H¨older continuous coefficients in the whole R^N, and in regular bounded open sets in R^N with Dirichlet boundary condition. See [14, Thm. 1.7, Thm. 3.3], where a class of general 2m-order elliptic systems is considered. On the other hand, the study of elliptic operators with unbounded coefficients is much more recent and the general theory is still under developement. Here we consider operators of the type

A = ∆ +

N

X

i=1

F_iD_i,

with globally Lipschitz continuous coefficients Fi. It has been recently proved that the domain of the realization of A in L^p(R^N) is contained in W^2,p(R^N) for 1 < p < ∞ ([24]), and that the domain of the realization of A in C^α(R^N) is contained in C^2+α(R^N) for 0 < α < 1 ([21]). We prove here that the semigroup T (t) generated by the realization of A in L¹(R^N) satisfies (1.5) with γ1 = 1/2, γ2 = (N + 1)/2, like in the case of bounded regular coefficients. As a first step, ultracontractivity of T (t) is proved adapting to our situation a classical argument based on Nash inequality. This gives kT (t)kL(L¹,L^∞)≤ Ce^ωtt^{−(N +1)/2}. Then the estimate k∇T (t)k_L(L^∞₎≤ Ce^ωtt^−1/2, which is known for smooth F , is extended to our situation by a perturbation argument. Using the semigroup law, we arrive at (1.5)(ii). For (1.5)(i), we use the fact that the commutator T (t)∇−∇T (t) may be extended to a bounded operator from L^∞(R^N, R^N) to itself, and then we use a duality argument and again the estimate k∇T (t)kL(L^∞) ≤ Ce^ωtt^−1/2 to conclude. Theorem 4.1 about Lorentz regularity implies then that for each u ∈ D(A), D_iu belongs to L^{N/(N −1),∞}(R^N) for i = 1, . . . , N , and

kD_iuk_LN/(N −1),∞(R^N) ≤ C kuk_L1+ kAuk_L¹
.

2 Preliminaries, Besov and Lorentz spaces

In the next sections we shall consider the usual real interpolation spaces (X, D)_θ,q for couples of Banach spaces (X, D) with D ⊂ X. We refer to the book [28] for an extensive treatment of these spaces. We recall that if X, Y , D are Banach spaces such that D ⊂ Y ⊂ X with continuous embeddings, and 0 ≤ θ ≤ 1, Y is said to belong to the class Jθ(X, D) if there is C > 0 such that

kf k_Y ≤ Ckf k^1−θ_X kf k^θ_D, ∀f ∈ D.

The Reiteration Theorem (see e.g. [28, §1.10.2]) implies that if Y₁ belongs to J_θ1(X, D), and Y₂ belongs to J_θ2(X, D), then

(Y₁, Y₂)_θ,q ⊃ (X, D)_{(1−θ)θ1+θθ2,q} for all θ ∈ (0, 1), q ∈ [1, +∞], and the embedding is continuous.

Let Ω be an open set in R^N. By L^p(Ω), 1 ≤ p ≤ ∞, we mean L^p(Ω, dx) where dx is the usual Lebesgue measure. The norm of a function f in L^p(Ω) is denoted by kf k_p.

Now we recall briefly definitions and main properties of the Besov and Lorentz spaces B∞^s,1(Ω) and L^p,q(Ω). We refer the reader to [29, 28, 13] for a detailed treatment of these spaces. Let s = r + α with r ∈ N and 0 < α ≤ 1. A function f ∈ W^r,1(R^N) belongs to B∞^s,1(R^N) if

kf k_Bs,1

∞ = kf k_Wr,1(R^N)+ X

|β|=r

sup

h∈R^N

|h|^−αkD^βu(· + h) − 2D^βu + D^βu(· − h)k₁ < ∞.

B∞^s,1(R^N) is a Banach space with the above norm. If Ω 6= R^N, the space B∞^s,1(Ω) consists of the restrictions to Ω of the functions in B∞^s,1(R^N). The norm of f is the infimum of k ef k_B^s,1

∞(R^N) over all the extensions ef of f to the whole R^N.

Besov spaces arise naturally as real interpolation spaces between Sobolev spaces. We recall the definition of Sobolev spaces of noninteger order. Let s = r + α with r ∈ N and 0 < α < 1. A function f ∈ W^r,1(R^N) belongs to W^s,1(R^N) if

kf k_Ws,1(R^N)= kf k_Wr,1(R^N)+ X

|β|=r

Z

R^N×R^N

|D^βf (x + h) − D^βf (x)|

|h|^{N +α} dx dh < ∞.

If Ω 6= R^N, the space W^s,1(Ω) consists of the restrictions to Ω of the functions in W^s,1(R^N).

The norm of f is the infimum of k ef k_Ws,1(R^N) over all the extensions ef of f to the whole R^N. If Ω is the whole R^N, or a half space, or a bounded open set with C^ρboundary, then for 0 < θ < 1, 1 ≤ q ≤ ∞ we have

(L¹(Ω), W^s,1(Ω))θ,∞= B^θs,1_∞ (Ω),

for 0 < s < ρ, with equivalence of the respective norms, see e.g. [29, 13]. This is in fact the only property of Besov spaces that we will need in the sequel.

Let now Ω be any measurable subset of R^N. Let f : Ω 7→ C be a measurable function and define for σ ≥ 0

m(σ, f ) =

{x ∈ Ω : |f (x)| > σ}

,

where |E| denotes the Lebesgue measure of E. Next, we introduce the decreasing rear- rangement of f , that is

f^∗(t) = inf{σ : m(σ, f ) ≤ t}.

The function f^∗ is decreasing on (0, ∞), it satisfies f^∗(0) = sup ess|f | and, moreover, {t > 0 : f^∗(t) > σ}

=

{x ∈ Ω : |f (x)| > σ}

. The Lorentz spaces L^p,q(Ω) (1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞) are defined by

L^p,q(Ω) =n

f ∈ L¹(Ω) + L^∞(Ω) : kf k_p,q:=Z ∞ 0

(t^1/pf^∗(t))^qdt t

1/q

< ∞o for q < ∞, and

L^p,∞(Ω) =n

f ∈ L¹(Ω) + L^∞(Ω) : kf k_p,∞:= sup

t>0

t^1/pf^∗(t) < ∞o .

We remark that L^p,q1(Ω) ,→ L^p,q2(Ω) if q1 < q2 and that L^p,p(Ω) = L^p(Ω). We shall consider the Lorentz spaces mainly with q = ∞, using the following characterization in terms of the distribution function m(f, σ), see [28, Lemma 1.18.6].

Proposition 2.1 The following equality holds:

L^p,∞(Ω) =n

f : Ω → C : sup

σ>0

σ(m(f, σ))^1/p< ∞o . Moreover, kf k_p,∞:= sup_σ>0σ(m(f, σ))^1/p, for every f ∈ L^p,∞(Ω).

Also the Lorentz spaces arise naturally as real interpolation spaces between Lebesgue spaces. Indeed, for 0 < θ < 1, 1 ≤ q ≤ ∞ we have

(L¹(Ω), L^∞(Ω))θ,q = L^1/(1−θ),q(Ω), with equivalence of the respective norms. See [28, §1.18.6].

The following lemma allows us to treat the vector-valued integrals appearing in the proofs of Theorems 3.1 and 4.1. The space V^1,∞(Ω) is defined by

V^1,∞(Ω) =n

f ∈ L¹(Ω) : ∇f ∈ L^∞(Ω, R^N)o

(2.1) and it is a Banach space when endowed with the norm kf k_V^1,∞_(Ω) = kf k1+ k |∇f | k∞. Lemma 2.2 Let Ω be an open set in R^N. Then the following properties hold.

(i) Closed balls in W^1,1(Ω) ∩ V^1,∞(Ω) are closed in L¹(Ω).

(i) If s is not an integer and Ω = R^N, or Ω is a half space, or Ω is bounded with C^ρ boundary, ρ > s, then closed balls in W^s,1(Ω) are closed in L¹(Ω).

Proof. (i) Let (f_n) ⊂ W^1,1(Ω) ∩ V^1,∞(Ω) converge to f in L¹(Ω), and be such that kD_if_nk₁ ≤ C₁, kD_if_nk∞ ≤ C₂ for i = 1, . . . , N , with C₁, C₂ independent of n. Then kD_ifnk₂ ≤ C = max{C₁, C2} for each i, and hence (∇f_n) converges to ∇f weakly in L²(Ω, R^N). It follows that for every φ ∈ C₀^∞(Ω) and for each i = 1, . . . , N

   Z

Ω

D_if φ   = lim

n→∞

   Z

Ω

D_if_nφ   ≤







C1kφk_∞ C₂kφk₁

Hence kD_if k₁ ≤ C₁, kD_if k∞ ≤ C₂. (ii) Let (f_n) ⊂ W^s,1(Ω), kf_nk_Ws,1(Ω) ≤ C be convergent to f ∈ L¹(Ω). Write s = r + α with r ∈ N and 0 < α < 1. By interpolation, (f_n) is a Cauchy sequence in W^r,1(Ω) and hence (f_n) → f in W^r,1(Ω). Fatou’s Lemma yields easily kf k_Ws,1(Ω) ≤ C.

Observe that (ii) of Lemma 2.2 fails if s is an integer. For example, if s = 1, the function f in the proof above is just of bounded variation and it does not belong necessarily to W^1,1(Ω).

3 Besov regularity

Throughout the section we shall consider the following assumptions (C > 0, ω ∈ R).

(H1) (T (t))_t≥0 is a strongly continuous semigroup in L¹(Ω) with generator A : D(A) 7→

L¹(Ω), and

kT (t)f k₁ ≤ Ce^ωtkf k₁, f ∈ L¹(Ω).

(H2) There are γ > 1, β > 0 such that for t > 0 and f ∈ L¹(Ω), T (t)f belongs to W^β,1(Ω) and satisfies the estimate

kT (t)f k_Wβ,1≤ Ce^ωtt^−γkf k₁.

Moreover, either Ω = R^N, or Ω is a half space, or Ω is bounded with C^ρ boundary, ρ > β.

We shall adapt to the present situation the procedure of [20, Theorem 2.5], to get an optimal embedding result for D(A). To be more precise, we get an optimal embedding result for (L¹(Ω), D(A²))_1/2,∞, in which D(A) is continuously embedded (see [28, Thm.

1.13.2(b)]. Since we work in specific Sobolev spaces, we do not need the measurability assumption made in [20].

Theorem 3.1 Let (H1) and (H2) hold. Then

(L¹(Ω), D(A²))_1/2,∞ ⊂ B_∞^β/γ,1(Ω) with continuous embedding, and therefore

D(A) ⊂ B_∞^β/γ,1(Ω) with continuous embedding.

Proof. As a first step, we prove that the statement holds for β 6∈ N. Since D(A²) = D((A − ωI)²) with equivalence of the norms, possibly replacing A by A − ωI we may assume also that ω = 0. We shall show that

(L¹(Ω), D(A^m))_1/m,∞ ⊂ (L¹(Ω), W^β,1(Ω))_1/γ,∞,

with continuous embedding, for m large enough. Indeed, it is well known (see e.g. [28, Thm. 1.13.5]) that (L¹(Ω), D(A²))_1/2,∞ = (L¹(Ω), D(A^m))_1/m,∞ for every integer m ≥ 2.

Fix any integer m > γ. Let u ∈ D(A^m), λ > 0 and set (λI − A)^mu = f . Then u = (R(λ, A))^mf , so that

u = (−1)^m−1 (m − 1)!

d^m−1

dλ^m−1R(λ, A)f = 1 (m − 1)!

Z ∞ 0

t^m−1e^−λtT (t)f dt, (3.1) the integral being understood in L¹(Ω). However, u belongs to W^β,1(Ω), with

kuk_Wβ,1(Ω)≤ C (m − 1)!

Z ∞ 0

t^m−1−γe^−λtdt kf k1. (3.2) Indeed,

u = L¹− lim

a→0, b→∞

1 (m − 1)!

Z b a

t^m−1e^−λtT (t)f dt, and for fixed 0 < a < b < ∞,Rb

at^m−1−γe^−λtT (t)f dt = L¹− lim_n→∞I_n, where I_n= b − a

n

n

X

k=1

(a + k(b − a)/n)^m−1e−λ(a+k(b−a)/n)T (a + k(b − a)/n)f. (3.3)

Due to assumption (H2), I_n∈ W^β,1(Ω) and kI_nk_Wβ,1(Ω)≤ C(b − a)

n

n

X

k=1

(a + k(b − a)/n)^m−1−γe−λ(a+k(b−a)/n)kf k₁.

The right hand side goes to CRb

at^m−1−γe^−λtdtkf k_L¹ as n goes to +∞. By Lemma 2.2(ii) closed balls in W^β,1(Ω) are closed in L¹(Ω), and then Rb

a t^m−1e^−λtT (t)f dt ∈ W^β,1(Ω) and

Z b a

t^m−1e^−λtT (t)f dt W^β,1(Ω)

≤ C Z b

a

t^m−1−γe^−λtdtkf k₁.

Letting a → 0 and b → ∞ and using again Lemma 2.2(ii) it follows that u belongs to W^β,1(Ω) and that it satisfies (3.2). The proof now goes on as in [20]; we write it down for the reader’s convenience. From (3.2) we get

kuk_Wβ,1(Ω) ≤ C (m − 1)!

Z ∞ 0

e^−λtt^m−γ−1dtkf k1= CΓ(m − γ)

(m − 1)! λ^γ−mkf k₁

= CΓ(m − γ) (m − 1)! λ^γ−m

m

X

r=0

m r



λ^m−r(−1)^rA^ru 1

≤ C₁

m

X

r=0

λ^γ−rkA^ruk1, λ > 0.

We recall now the interpolation inequalities

kA^ruk1 ≤ C₂kuk^r/m_D(Am)kuk^1−r/m₁ , r = 1, . . . , m, u ∈ D(A^m) (see e.g.. [28, §1.14.3]). Using such inequalities and then ab ≤ (a²+ b²)/2 we get

kuk_Wβ,1(Ω)≤ C₃λ^γ(λ^−mkuk_D(Am)+ kuk_L¹), λ > 0,

so that, taking the minimum for λ > 0,

kuk_Wβ,1(Ω) ≤ C₄kuk^1−γ/m₁ kuk^γ/m_D(Am).

This means that W^β,1(Ω) ∈ J_γ/m(L¹, D(A^m)), and therefore, by the Reiteration Theorem, (L¹(Ω), D(A^m))_1/m,∞⊂ (L¹(Ω), W^β,1(Ω))_1/γ,∞= B_∞^β/γ,1(Ω),

with continuous embedding. So, the statement is proved if β /∈ N. If β ∈ N fix β⁰ ∈ (β − 1, β) such that γ⁰ = γβ⁰/β > 1. Using assumptions (H1) and (H2) we get by interpolation

kT (t)f k_Wβ0,1 ≤ Ce^ωtt^−γ0kf k₁, t > 0, f ∈ L¹. Since β⁰ is not integer, the first part of the proof yields

(L¹(Ω), D(A²))_1/2,∞ ⊂ B_∞^β0/γ0,1(Ω) = B_∞^β/γ,1(Ω).

In the following corollary we consider an important special case of Theorem 3.1.

Corollary 3.2 Let either Ω = R^N, or Ω be a half space, or Ω be a bounded open set with C^ρ boundary, ρ > 2. Let (H1) and (H2) hold, with 2 < β < ρ and γ = β/2. Then

D(A) ⊂ B^2,1_∞(Ω),

with continuous embedding. Therefore, the gradient of each function u ∈ D(A) belongs to B∞^1,1(Ω, R^N), which is continuously embedded in the space L^{N/(N −1),∞}(Ω, R^N).

In the case of the Laplacian and of other elliptic operators with smooth coefficients in R^N, (H2) is satisfied with β = 3, γ = 3/2 and we may apply Corollary 3.2. Therefore we get

D(A) ⊂ B^2,1_∞(R^N), (3.4)

with continuous embedding. So, the gradient of each function u ∈ D(A) belongs to B∞^1,1(R^N, R^N), which is continuously embedded in the space L^{N/(N −1),∞}(R^N, R^N). In other situations, the semigroup T (t) does not map L¹ into W^β,1 for β > 2, and the embedding (3.4) is not known. However, we are still able to get optimal Lorentz regularity provided a somewhat weaker condition than (H2) holds. See next section.

4 Lorentz regularity

Throughout the section we shall consider Assumption (H1) together with the following one (C > 0, ω ∈ R).

(H3) There are constants γ₁ ∈ (0, 1), γ₂ > 1 such that for t > 0 and f ∈ L¹(Ω) the function T (t)f belongs to W^1,1(Ω) ∩ V^1,∞(Ω) and satisfies

k∇T (t)f k₁≤ Ce^ωtt^−γ1kf k₁, k∇T (t)f k_∞≤ Ce^ωtt^−γ2kf k₁. By a method similar to the one of Theorem 3.1 we get the following result.

Theorem 4.1 Assume that (T (t))_t≥0 satisfies hypotheses (H1) and (H3). Then the map

∇ : (L¹, D(A²))_1/2,∞ → L^{γ2−γ1γ2−1,∞}(Ω, R^N) is bounded. In particular, the map ∇ : D(A) → L^{γ2−γ1γ2−1,∞}(Ω, R^N) is bounded.

Proof. We shall show that W^1,1(Ω) and V^1,∞(Ω) belong to J_γ1/m(L¹(Ω), D(A^m)) and to J_γ2/m(L¹(Ω), D(A^m)), respectively, for any integer m > γ2. By the Reiteration Theorem it will follow that (L¹(Ω), D(A^m))_1/m,∞ ⊂ (W^1,1(Ω), V^1,∞(Ω))_1−γ1

γ2−γ1,∞, with continuous embedding. The latter space consists of functions in W^1,1(Ω) + V^1,∞(Ω) whose first order derivatives belong to (L¹(Ω), L^∞(Ω))_1−γ1

γ2−γ1,∞ = L^{γ2−γ1γ2−1 ,∞}(Ω). See [28, §1.18.6]. This yields the statement because (L¹(Ω), D(A^m))_1/m,∞ = (L¹(Ω), D(A²))_1/2,∞, with equivalence of the respective norms ([28, Thm. 1.12.5]). We may assume that ω = 0 without loss of generality. Let u ∈ D(A^m) with m > γ₂, let λ > 0 and set (λI − A)^mu = f . Then (3.1) holds, the integral being well defined in L¹(Ω). We show now that u ∈ W^1,1(Ω) ∩ V^1,∞(Ω), and

kD_iuk1≤ C (m − 1)!

Z ∞ 0

t^m−1−γ1e^−λtdtkf k1,

kD_iuk∞≤ C (m − 1)!

Z ∞ 0

t^m−1−γ2e^−λtdtkf k₁.

(4.1)

As in the proof of Theorem 3.1, we remark that for 0 < a < b < ∞, the sums In defined in (3.3) belong to W^1,1(Ω) ∩ V^1,∞(Ω), and for each i = 1, . . . , N we have

kD_iI_nk₁ ≤ C(b − a) n

n

X

k=1

(a + k(b − a)/n)^m−1−γ1e−λ(a+k(b−a)/n)kf k₁,

kD_iI_nk_∞≤ C(b − a) n

n

X

k=1

(a + k(b − a)/n)^m−1−γ2e−λ(a+k(b−a)/n)kf k₁. The right hand sides go to CRb

at^m−1−γie^−λtdtkf k_L1, i = 1, 2, as n goes to +∞. It follows that the function Rb

a t^m−1e^−λtT (t)f dt belongs to W^1,1(Ω) ∩ V^1,∞(Ω) and the proof of Lemma 2.2(i) gives

kD_i Z b

a

t^m−1e^−λtT (t)f dtk1 ≤ C Z b

a

t^m−1−γ1e^−λtdtkf k1,

kD_i Z b

a

t^m−1e^−λtT (t)f dtk∞≤ C Z b

a

t^m−1−γ2e^−λtdtkf k₁,

for i = 1, . . . , N . Letting a → 0 and b → ∞ we find that u belongs to W^1,1(Ω) ∩ V^1,∞(Ω) and that it satisfies (4.1). Once (4.1) is established, arguing as in the proof of Theorem 3.1 we obtain that W^1,1(Ω) belongs to J_γ1/m(L¹, D(A^m)), and V^1,∞(Ω) belongs to J_γ2/m(L¹, D(A^m)), just what we needed.

Let us write down explicitly a particular but important case of Theorem 4.1.

Corollary 4.2 Assume that the hypotheses in Theorem 4.1 are fulfilled with γ1 = 1/2 and γ2 = (N + 1)/2. Then the map ∇ : (L¹(Ω), D(A²))_1/2,∞ → L^{N/(N −1),∞}(Ω, R^N) is bounded. In particular, the map ∇ : D(A) → L^{N/(N −1),∞}(Ω, R^N) is bounded.

A less sharp version of Theorem 4.1 has a simpler proof, that we write below.

Theorem 4.3 Assume that (T (t))t≥0 satisfies hypotheses (H1) and (H3). Then the map

∇ : D(A) → L^{γ2−γ1γ2−1 ,∞}(Ω, R^N) is bounded.

Proof. Let λ > ω and let u ∈ D(A). Setting λu − Au = f , then u is the Laplace transform of T (t)f , that is

u = Z ∞

0

e^−λtT (t)f dt.

Let us fix ξ > 0 and split u = a(ξ) + b(ξ) where a(ξ) =

Z ξ 0

e^−λtT (t)f dt, b(ξ) = Z ∞

ξ

e^−λtT (t)f dt.

The procedure of Theorem 4.1 gives a(ξ) ∈ W^1,1(Ω), a(ξ) ∈ V^1,∞(Ω), and k∇a(ξ)k₁ ≤ C₁ξ^1−γ1kf k₁, k∇b(ξ)k_∞≤ C₂ξ^1−γ2kf k₁.

Since {|∇u| > σ} ⊂ {|∇a(ξ)| > σ/2} ∪ {|∇b(ξ)| > σ/2} for every σ > 0, choosing ξ such that ξ^1−γ2 = σ/2C₂ we get {|∇b(ξ)| > σ/2} = ∅ and therefore

m(|∇u|, σ) ≤ |{|∇a(ξ)| > σ/2}| ≤ C₃σ^{−γ2−γ1γ2−1} kf k₁, with C₃ independent of f . Applying Proposition 2.1 concludes the proof.

Remark 4.4 Assume that the hypotheses in Theorem 4.1 are fulfilled with γ₁= 1/2, ω = 0. The first inequality in (4.1) with m = 1 yields

k∇uk₁≤ Kλ^−1/2kf k₁ ≤ K λ^1/2kuk₁+ λ^−1/2kAuk₁
for each u ∈ D(A). Minimizing over λ > 0, we get

k∇uk₁ ≤ Ckuk^1/2₁ kAuk^1/2₁ , that is, W^1,1(Ω) belongs to the class J_1/2(L¹(Ω), D(A)).

5 Operators with Lipschitz continuous coefficients

In this section we apply the results of Section 4 to operators in R^N of the form A = ∆ + F · ∇,

assuming that the drift vector field F is globally Lipschitz continuous. An important example is the Ornstein-Uhlenbeck operator, which may be reduced to this one by an obvious change of coordinates,

A =

n

X

i,j=1

q_ijD_ij +

n

X

i,j=1

b_ijx_jD_i,

where Q = (q_ij) is a real symmetric positive definite matrix, and B = (b_ij) is a nonzero real matrix. The Ornstein-Uhlenbeck semigroup in L¹(R^N) has the explicit representation

(T (t)f )(x) = 1

(4π)^n/2(det Qt)^1/2 Z

R^N

e^{−<Q−1t y,y>/4}f (e^tBx − y) dy, where

Q_t= Z t

0

e^sBQe^sB∗ds. (5.1)

Having an explicit representation formula, the hypotheses of Corollary 3.2 can be checked immediately. Therefore u ∈ B∞^2,1(R^N) and hence ∇u ∈ L^{N/(N −1),∞}(R^N, R^N), for every u ∈ D(A). We shall prove that the last result still holds for general F . This will be done in some steps; the first one consists in showing that a suitable realization of A in L¹(R^N)

generates a strongly continuous semigroup. For technical reasons we need to consider the realizations of the operator A in every L^p(R^N), hence we set for 1 < p < ∞

Dp = n

u ∈ W^2,p(R^N) : F · ∇u ∈ L^p(R^N) o

, (5.2)

and

D∞=n

u ∈ C₀(R^N) ∩ W_loc^2,p(R^N) for every p < ∞ : Au ∈ C₀(R^N)o

. (5.3)

Here C₀(R^N) is the space of the continuous functions on R^N vanishing as |x| → +∞, endowed with the sup norm. Observe that C₀^∞(R^N) is dense in Dp for 1 < p < ∞. We set

k = sup

x∈R^N

−div F (x)
. (5.4)

Proposition 5.1 For 1 < p < ∞ the operator (A, Dp) generates a strongly continuous positive semigroup (Tp(t))t≥0 in L^p(R^N) satisfying

kT_p(t)f k_p ≤ e^kt/pkf k_p, f ∈ L^p(R^N).

The operator (A, D∞) generates a strongly continuous semigroup of positive contractions in C₀(R^N).

Proof: The proof is in [24] for 1 < p < ∞. The statement about C0(R^N) follows from [25, Corollary 3.18], since F has at most linear growth.

Note that for q 6= p, T_p(t)f = T_q(t)f for each f ∈ L^p(R^N) ∩ L^q(R^N). Letting p → 1 it follows that Tp(t) may be extended to a bounded operator T (t) in L¹(R^N), and

kT (t)f k₁ ≤ e^ktkf k₁, f ∈ L¹(R^N). (5.5) Actually more is true, as the next proposition shows.

Proposition 5.2 (T (t))t≥0 is strongly continuous in L¹(R^N) and its infinitesimal gener- ator is the L¹-closure of A : C₀^∞(R^N) 7→ L¹(R^N).

Proof. Let B = A + div F and u ∈ C₀^∞(R^N) be real valued. Since Z

R^N

Au signu dx ≤ 0, (5.6)

the operator B : C₀^∞(R^N) 7→ L¹(R^N) is dissipative. Next we show that (λI −B)(C₀^∞(R^N)) is dense in L¹(R^N) for λ > 0. Let f ∈ L^∞(R^N) be such that

Z

R^N

(λu − Bu)f dx = 0

for every u ∈ C₀^∞(R^N). By local elliptic regularity (see [1, Theorem 6.2]), f belongs to W_loc^1,2(R^N) and it is a weak solution of the equation λf − B^∗f = 0, where B^∗ = ∆ − F · ∇ is the formal adjoint of B. Since the coefficients of B^∗ are locally bounded, the classical method of difference quotients yields f ∈ W_loc^2,2(R^N), hence λf − B^∗f = 0 a.e. in R^N and, finally, f ∈ W_loc^2,p(R^N) for all p < ∞, by [12, Lemma 9.16]. Taking λ > 0 and observing that the coefficients of B^∗ have at most linear growth, we may apply the maximum principle in R^N (see e.g. [19, Proposition 2.2]) to conclude that f = 0. Since A = B − div F , the Lumer-Phillips Theorem implies that the closure of A : C₀^∞(R^N) 7→ L¹(R^N) generates a strongly continuous semigroup (S(t))t≥0 satisfying kS(t)f k1 ≤ e^ktkf k₁. On the other hand, since the generator of (S(t))_t≥0coincides with the generator of (T (t))_t≥0on C₀^∞(R^N) which is a core both in L¹ and in L^p, it follows that the Laplace transform of S(t)f and T (t)f coincide for f ∈ L¹(R^N)∩L^p(R^N). Therefore S(t)f = T (t) for f ∈ L¹(R^N)∩L^p(R^N) and the proof is complete.

In some arguments below we need the adjoint semigroup (T (t)^∗)t≥0 of (T (t))t≥0. Let us consider the formal adjoint A^∗ of A defined by

A^∗= ∆ − F · ∇ − div F. (5.7)

Since div F is bounded, it follows from Proposition 5.1 that A^∗: D_p 7→ L^p(R^N) generates a strongly continuous semigroup in L^p(R^N) for 1 < p < ∞.

Lemma 5.3 Let 1 < p < ∞. Then the adjoint semigroup (T (t)^∗)t≥0 in L^p0(R^N) is generated by (A^∗, D_p⁰).

Proof. The adjoint semigroup in L^p0(R^N) is generated by the dual operator of (A, Dp), let it be C : D(C) 7→ L^p0(R^N). An elementary integration by parts yields Cφ = A^∗φ for each φ ∈ C₀^∞(R^N). Since C₀^∞(R^N) is a core for (A^∗, D_p⁰), by Proposition 5.1 we obtain that (C, D(C)) is an extension of (A^∗, Dp⁰), hence they do coincide because both of them are generators in L^p0(R^N).

Next we prove an ultracontractivity property of (T (t))_t≥0, showing that T (t) maps L¹(R^N) into L^∞(R^N) with a behavior similar to the one of the heat semigroup near t = 0.

We adapt a classical argument based on the so called Nash inequality

kgk^2+4/N₂ ≤ Ckgk²_W1,2kgk^4/N₁ , g ∈ W^1,2(R^N) ∩ L¹(R^N). (5.8) See e.g. [7, Theorem 2.4.6].

Proposition 5.4 There exist constants M, ω such that for every f ∈ L¹(R^N), t > 0 kT (t)f k∞≤ M e^ωt

t^N/2 kf k₁. (5.9)

Moreover, for every t > 0, T (t) maps L¹(R^N) into C₀(R^N).

Proof. Let λ ≥ 1 + k/2, where k is given by 5.4. For each u ∈ C₀^∞(R^N) we have Z

R^N

u(λu − Au)dx = Z

R^N



|∇u|²+ (λ +1

2div F )|u|²

 dx ≥

Z

R^N



|∇u|²+ |u|²



dx. (5.10) Since C₀^∞(R^N) is dense in D2, (5.10) is true for each u ∈ D2.

Let f ∈ C₀^∞(R^N), f 6≡ 0, and set

v(t) = ke^−λtT (t)f k²₂, t ≥ 0.

Since T (t)f goes to f as t → 0, then v(t) > 0 at least for t in a right neighborhood I of 0.

Moreover v is differentiable, and v⁰(t) = 2e^−2λt

Z

R^N

T (t)f ((A − λ)T (t)f )dx, t ≥ 0.

Since T (t)f ∈ D2∩ L¹ for each t, we may apply estimate (5.10) and then estimate (5.8), getting for t ∈ I

v⁰(t) ≤ −2ke^−λtT (t)f k²_W1,2(R^N) ≤ −Cke^−λtT (t)f k^2+4/N₂ ke^−λtT (t)f k^−4/N₁ . Therefore, for each t ∈ I,

d

dt(v(t)^−2/N) ≥2C

N ke^−λtT (t)f k^−4/N₁ ≥ C₁e^4(λ−k)t/Nkf k^−4/N₁ .

(We used (5.5) to get the last inequality). Integrating between 0 and t we get v(t)^−2/N ≥ C₂kf k^−4/N₁

Z t 0

e^4(λ−k)s/Nds ≥ C2tkf k^−4/N₁ , t ∈ I, provided λ ≥ k. This gives

kT (t)f k₂ ≤ C₃e^λtt^−N/4kf k₁, t > 0. (5.11) Thanks to Lemma 5.3, the same argument may be applied to T (t)^∗, and it yields for g ∈ C₀^∞(R^N) and λ ≥ max{1 + k/2, 0}

kT (t)^∗gk₂ ≤ C₄e^λtt^−N/4kgk₁, t > 0. (5.12) Now we use a standard duality argument to get a bound for kT (t)f k∞. We have

kT (t)f k_∞= sup

g∈C₀^∞(R^N), kgk1=1

Z

R^N

T (t)f g dx = sup

g∈C₀^∞(R^N), kgk1=1

Z

R^N

f T (t)^∗g dx,

and for each g ∈ C₀^∞(R^N) with kgk₁ = 1, inequality (5.12) yields Z

R^N

f T (t)^∗g dx ≤ kf k2kT (t)^∗gk2≤ kf k₂C4e^λtt^−N/4, t > 0 hence

kT (t)f k_∞≤ C₄e^λtt^−N/4kf k₂, t > 0. (5.13) Since C₀^∞(R^N) is dense in L¹(R^N), inequalities (5.11) and (5.13) hold for every f ∈ L¹(R^N) and we get (5.9) using the semigroup law. Finally, let us show that T (t) maps L¹(R^N) into C0(R^N). If f ∈ C₀^∞(R^N), then T (t)f ∈ C0(R^N), since (T (t))t≥0 preserves C0(R^N) (see Proposition 5.1). The general case where f ∈ L¹(R^N) follows by approximation, using estimate (5.9).

Now we turn to gradient estimates. We recall that inequalities like

k∇T (t)f k_∞≤ e^ωtt^−1/2kf k_∞, (5.14) with f ∈ C_b(R^N), are valid under a dissipativity condition on the drift F , that is when the quadratic form defined by the Jacobian matrix of F is bounded from above. Clearly this condition is satisfied in our case. We refer to [3] and to the references therein for a discussion of inequalities of the type (5.14). The paper [3] deals with a general class of second order elliptic operators with coefficients in C_loc^1+α(R^N). For this reason we assume for the moment that F ∈ C_loc^1+α(R^N); this extra assumption will be removed later.

Corollary 5.5 Assume that F ∈ C_loc^1+α(R^N) for some α > 0. Then the semigroup (T (t))t≥0 satisfies

k∇T (t)f k_∞≤ Ce^ωtt^{−(N +1)/2}kf k₁, f ∈ L¹(R^N), for suitable C > 0, ω ∈ R.

Proof. Since k∇F k∞ < ∞, the assumptions of [3] are satisfied, and (5.14) holds for each f ∈ Cb(R^N). The statement then follows using Proposition 5.4 and the semigroup law.

Now we estimate the operator norm of ∇T (t) from L¹(R^N) to (L¹(R^N))^N. It will be deduced from the analogous result for p = ∞ via the following lemma.

Lemma 5.6 Assume that F ∈ C_loc^1+α(R^N) for some α > 0. Then the operator ∇T (t) − T (t)∇ may be extended to a bounded operator Γ(t) from (C₀(R^N))^Ninto itself, satisfying kΓ(t)f k_∞≤ Ce^ωt√

tkf k∞ for suitable C > 0, ω ∈ R.

Proof. Let f ∈ C0^∞(R^N). The function u(t, x) = T (t)f (x) is in C_loc^1+α/2,3+α((0, ∞) ×R^N).

Moreover, ∇u is bounded in [0, ∞) × R^N and it satisfies ∇u(t, x) → ∇f (x) as t → 0, see [3]. Set

v(t, ·) = ∇T (t)f − T (t)∇f, t > 0.

Then v ∈ C_loc^1+α/2,2+α((0, ∞) × R^N) ∩ L^∞((0, ∞) × R^N) and v is continuous up to t = 0.

A straightforward computation shows that v satisfies

 D_tv − Av = F_x∇u, t > 0, x ∈ R^N, v(0, x) = 0, x ∈ R^N,

where Fx denotes the Jacobian matrix of F . Since Fx is bounded, using estimates (5.14) we get

|D_tv(t, x) − Av(t, x)| ≤ C₁e^ωtt^−1/2kf k_∞, t > 0, x ∈ R^N. For i = 1, . . . , N let w_i = v_i−2C₁e^ωt√

tkf k∞. Then w_i ∈ C^1,2((0, T ]×R^N)∩C_b([0, T ]×R^N) for every T < ∞ and it satisfies

 Dtwi− Aw_i≤ 0, t > 0, x ∈ R^N, w_i(0, x) = 0, x ∈ R^N.

The maximum principle (in the form stated e.g. in [15, Exercise 8.1.22] for operators with unbounded coefficients) yields wi ≤ 0. The same argument applied to −w_i eventually gives

k∇T (t)f − T (t)∇f k_∞= kv(t, ·)k∞≤ C₂e^ωt√ tkf k∞,

for f ∈ C₀^∞(R^N), hence for f ∈ C₀(R^N), by density. Finally we show that Γ(t)f ∈ C₀(R^N).

If f ∈ C₀^∞(R^N) then T (t)f ∈ D_p ⊂ W^2,p(R^N) for each p > 1, and taking p > N , we get

∇T (t)f ∈ C₀(R^N) by Sobolev embedding. Since T (t)∇f also belongs to C0(R^N) we deduce that Γ(t)f ∈ C₀(R^N). The case f ∈ C₀(R^N) easily follows by approximation, using the continuity of Γ(t) with respect to the sup norm.

Corollary 5.7 Assume that F ∈ C_loc^1+α(R^N) for some α > 0, and that F is Lipschitz continuous. Then T (t)f ∈ W^1,1(R^N) for every f ∈ L¹(R^N), t > 0 and

k∇T (t)f k₁ ≤ Ce^ωtt^−1/2kf k₁, t > 0, (5.15) for suitable C > 0, ω ∈ R.

Proof. Let f, g ∈ C₀^∞(R^N). Then T (t)f , T (t)g ∈ D₂ and therefore

−h∇T (t)f, gi = hf, T (t)^∗∇gi = hf, ∇T (t)^∗gi + hf, (T (t)^∗∇ − ∇T (t)^∗)gi.

Using estimate (5.14) and Lemma 5.6 we obtain h∇T (t)f, gi

≤ Ce^ωt(t^−1/2+ t^1/2)kf k₁kgk_∞

and the statement follows for f ∈ C₀^∞(R^N). By density, (5.15) holds for every f ∈ L¹(R^N).

Now we are ready to remove the extra regularity assumption on F and to prove gradient regularity for functions belonging to the domain D(A) of A in L¹(R^N).

Theorem 5.8 Assume that F is globally Lipschitz continuous in R^N. Then the map

∇ : D(A) → L^{N/(N −1),∞}(R^N, R^N) is bounded.

Proof. As a first step, assume that F is in C_loc^1+α(R^N) for some α > 0. Corollaries 5.5 and 5.7 show that (T (t))t≥0 satisfies hypotheses (H1), (H3) of Section 4, and Theorem 4.1 yields the statement. The general case can be handled by perturbation. Let 0 ≤ ρ ∈ C₀^∞(R^N),R

R^Nρ = 1 and define F₀ = F ? ρ. Set moreover A0= ∆ + hF0, ∇i

and let D₀ be the domain of A₀ in L¹(R^N). Since F₀ is smooth and Lipschitz continuous, for each u ∈ C₀^∞(R^N) we have

k∇uk_{N/(N −1),∞} ≤ C kuk₁+ kA₀uk₁

(5.16) Since F is globally Lipschitz continuous, F − F0 is bounded and therefore

kAu − A₀uk1 = khF − F0, ∇uik1≤ Ck∇uk₁. (5.17) Moreover, using Remark 4.4, for each ε > 0 we have

k∇uk₁ ≤ εkA₀uk₁+C

εkuk₁ ≤ εkAuk₁+ Cεk∇uk₁+C εkuk₁, and, taking ε small enough,

k∇uk₁ ≤ C kuk₁+ kAuk₁
. (5.18)

Using (5.16), (5.17), and (5.18) we get

k∇uk_{N/(N −1),∞} ≤ C kuk₁+ kA₀uk₁
≤ C kuk₁+ k∇uk₁+ kAuk₁
≤ Ckuk_D(A) for u ∈ C₀^∞(R^N), and by density for u ∈ D(A).

6 Remarks and other examples

In this section we give further examples to show how Theorems 3.1, 4.1 and their corollaries may be applied to a wide class of second order elliptic operators in L¹(Ω).

Example 6.1 Assume that the operator A in (1.4) is uniformly elliptic in R^N with bounded and H¨older continuous coefficients. It is well-known that the realization of A in L^p(R^N) generates an analytic semigroup T_p(t), 1 < p < ∞, which can be expressed through a kernel G(t, ·, ·) independent of p, i.e.

(Tp(t)f )(x) = Z

R^N

G(t, x, y)f (y) dy. (6.1)

Since

|G(t, x, y)| ≤ Ce^ωtt^−N/2e^{−b|x−y|2/t} |∇G(t, x, y)| ≤ Ce^ωtt^{−(N +1)/2}e^{−b|x−y|2/t} (6.2) for suitable C, b > 0, ω ∈ R (see [9, Ch. 1, Sect. 6]), it is easy to see that Tp(t) may be extended to a strongly continuous semigroup T (t) in L¹(R^N) which satisfies (H1), (H3)