Journal of Functional Analysis 255 (2008) 2732–2759
www.elsevier.com/locate/jfa
On maximal regularity of type Lp –Lq under minimal
assumptions for elliptic non-divergence operators
Peer Christian Kunstmann
Institut für Analysis, Universität Karlsruhe, Englerstr. 2, D-76128 Karlsruhe, Germany
Received 28 November 2007; accepted 23 September 2008
Available online 16 October 2008
Communicated by N. Kalton
Abstract
We prove maximal regularity of type Lp –Lq for operators in non-divergence form with complex-valued
measurable coefficients on Rn . For a certain range of q, which depends on dimension and the order of the
operators, this is done under the sole assumption that they generate an analytic semigroup in Lq . Thus we
give, for this class of operators and this range of q, a positive answer to Brézis’ question whether generation
of an analytic semigroup entails maximal Lp -regularity. For other values of q we give several additional assumptions. The proof relies on a result on maximal regularity under the assumption of suitable off-diagonal
bounds due to S. Blunck and the author, which we improve here. These off-diagonal bounds are obtained
by a modification of Davies’ technique suited to cope with operators in non-divergence form, and they also
imply new results on the scale of Lq -spaces in which a given operator generates an analytic semigroup. We
thus obtain a completely new approach to maximal Lp -regularity for operators of this kind, in particular
for those whose highest order coefficients belong to VMO. Moreover, we obtain extensions of recent results
due to Kim and Krylov for operators with measurable coefficients depending on one coordinate.
© 2008 Elsevier Inc. All rights reserved.
Keywords: Non-divergence form operators; Elliptic operators; Maximal regularity of type Lp –Lq ; Analytic
semigroups; Davies–Gaffney estimates; Off-diagonal bounds; Elliptic regularity
1. Introduction and main results
For elliptic differential operators A, maximal regularity of type Lp –Lq is an important property in the study of parabolic equations which, as a tool, has many applications including
E-mail address: peer.kunstmann@math.uni-karlsruhe.de.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2008.09.017
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2733
existence and uniqueness of solutions of semi- and quasi-linear parabolic equations (see, e.g.,
[1,18,27,41]). Let −A be the generator of an analytic semigroup in a Banach space X = Lq (Ω)
where q ∈ (1, ∞), let p ∈ [1, ∞] and T ∈ (0, ∞]. Then A is said to have maximal regularity
of type Lp –Lq on [0, T ) if, for any right-hand side f ∈ Lp ([0, T ), Lq (Ω)), the unique mild
solution to
u (t) + Au(t) = f (t),
t ∈ [0, T );
u(0) = 0,
satisfies u , Au ∈ Lp ([0, T ), Lq (Ω)). We also say that A has maximal Lp -regularity on [0, T ) in
Lq (Ω). If A has maximal Lp -regularity on [0, T ) in Lq (Ω) then, by the closed graph theorem,
there exists a constant C > 0 such that
u Lp ([0,T ),Lq (Ω)) + AuLp ([0,T ),Lq (Ω)) Cf Lp ([0,T ),Lq (Ω)) ,
and, in case T < ∞, also
uLp ([0,T ),Lq (Ω)) Cf Lp ([0,T ),Lq (Ω)) .
It is easy to see that maximal regularity of type Lp –Lq on [0, T ) does not depend on T ∈ (0, ∞).
Moreover, by considering ν + A in place of A where ν 0 is sufficiently large, we may always
resort to the case T = ∞, in which case the semigroup generated by −A is necessarily bounded.
It is well known that the property of maximal regularity of type Lp –Lq is independent of p ∈
(1, ∞) (cf., e.g., [30] for this consequence of a result from [11]). It is also well known that
analyticity of the semigroup (e−tA ) generated by −A in Lq (Ω) is a necessary condition for
maximal Lp -regularity in Lq (Ω) (cf., e.g., [24]). In the sequel we shall thus simply say that A
has maximal regularity in Lq (Ω) if, for some p ∈ (1, ∞) and some T ∈ (0, ∞], A has maximal
Lp -regularity on [0, T ) in Lq (Ω). We recall the following question by H. Brézis (presented in
[20] for the more general class of UMD-spaces X):
Let q ∈ (1, ∞) and let −A generate an analytic semigroup in Lq (Ω); does A have the property
of maximal Lp -regularity on Lq (Ω) where p ∈ (1, ∞)?
On L2 (Ω), an application of Plancherel’s theorem yields that maximal L2 -regularity holds for
T ∈ (0, ∞) whenever −A generates an analytic semigroup. Hence one always has maximal
Lp -regularity, p ∈ (1, ∞), on L2 (Ω) for negative generators of analytic semigroups. Brézis’
question has been answered in the negative by G. Lancien and N.J. Kalton (for the more general class of Banach spaces with an unconditional Schauder basis, cf. [31]): Among the spaces
Lq (Ω), q ∈ (1, ∞), the only space, in which any negative generator of an analytic semigroup
has maximal regularity, is L2 (Ω). The proof is indirect and relies on a deep result from Banach
space geometry. Up to now, no concrete example of a negative generator of an analytic semigroup
in Lq (Ω), q ∈ (1, ∞), in particular no differential operator, is known to fail maximal regularity. On the other hand, for the class of UMD-spaces X, L. Weis has characterized the operators
A having maximal regularity on [0, ∞) as those being R-sectorial of type < π/2 (cf. [50,51]).
In a space X = Lq (Ω) where q ∈ (1, ∞), an operator A has maximal regularity on [0, ∞) for
p ∈ (1, ∞) if and only if the set of operators {e−zA : z = 0, |arg z| δ} is R2 -bounded in Lq (Ω)
for some δ > 0 (cf. [41,50,51] and Theorem 2.1 below). Here we recall that, for s ∈ [1, ∞], a set
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
T of (sub-)linear operators is called Rs -bounded in a space X = Lq (Ω), q ∈ [1, ∞], if there is a
constant C > 0 such that, for all n ∈ N, all T1 , . . . , Tn ∈ T and all f1 , . . . , fn ∈ X, we have
n
n
1/s 1/s |Tj fj |s
|fj |s
C
,
j =1
j =1
X
(1)
X
where for s = ∞, we replace ( j |gj |s )1/s by supj |gj |. Brézis’ question has been answered positively [14] for elliptic operators in divergence form with, e.g. bounded measurable coefficients
(cf. [14] for further classes of coefficients). In this paper we study Brézis’ question for elliptic non-divergence operators of order m ∈ 2N with bounded measurable coefficients in spaces
X = Lq (Rn , C), q ∈ (1, ∞). These operators have the following form
Aq =
aα (x)D α , D(Aq ) = Wqm Rn , C ,
(2)
|α|m
where aα : Rn → C are bounded measurable and complex-valued functions, i.e., aα ∈ L∞ (Rn , C).
We use the usual multi-index notation α = (α1 , . . . , αn ) ∈ Nn0 , |α| = α1 + · · · + αn , and
D α = D1α1 . . . Dnαn where Dj := −i∂/∂xj for j = 1, . . . , n. The question then reads:
Suppose that −Aq generates an analytic semigroup in Lq (Rn , C); does Aq have maximal
regularity of type Lp –Lq , p ∈ (1, ∞)?
We want to emphasize that D(Aq ) = Wqm (Rn , C), and −Aq with this domain is required to
generate a semigroup. If we denote Σδ := {z ∈ C \ {0}: |arg z| < δ} for δ ∈ (0, π), then, by
L. Weis’ characterization cited above, the question becomes:
Suppose that −Aq generates an analytic semigroup (T (z)) in Lq (Rn , C); is the set
{e−νz T (z): z ∈ Σδ } an R2 -bounded subset of Lq (Rn , C) for some δ > 0, ν 0?
This is the question we shall attack in this paper. Thus we require as assumption on the operator
Aq the one in Brézis’ question above, namely
(Hq ) −Aq with D(Aq ) = Wqm (Rn ) is the generator of an analytic semigroup Tq (·) in Lq (Rn ).
By what was said above, this is clearly a minimal assumption. Our main result on Brézis’ question
2n
2n
, 2] ∩ (1, ∞). For q ∈ (1, ∞) \ [ 2m+n
, 2]
states that (Hq ) implies maximal regularity if q ∈ [ 2m+n
we need an additional assumption which may take different forms. In any case, the proofs rely
heavily on generalized Gaussian estimates. For 1 q0 q1 ∞ we say that an analytic semigroup T (·), defined on a dense subspace of Lq0 , satisfies generalized Gaussian (q0 , q1 )-estimates
(of order m > 1) if there are constants C, b > 0, ν 0, and δ ∈ (0, π/2) such that, for all z ∈ Σδ ,
1
B(x,|z|1/m )
T (z)1B(y,|z|1/m ) q
0 →q1
m
1
n 1
−m
( q − q1 ) ν Re z −b( |x−y| ) m−1
Re z
0
1
C|z|
e
e
,
x, y ∈ Rn , (3)
where we write Sq→r for the operator norm of a linear operator S : D → Lr , which is defined
on a subspace D of the normed space Lq . If we modify the zero order term in (2) and consider ν +
A in place of A we can arrange for ν = 0 in (3). We want to remark that a generalized Gaussian
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2735
(q0 , q1 )-estimate of order m > 1 is known to be a powerful tool to establish further properties
for semigroups or their generators. For the technique of generalized Gaussian estimates we refer
in particular to [43, Section 2], [14,17,39], and the recent series of papers [5–8]. In case q0 = q1
bounds of the form (3) for z = t real are referred to as Gaffney–Davies estimates. A first version
of our main result now reads as follows.
2n
Theorem 1.1. Let q ∈ (1, ∞) and assume that (Hq ) holds. If q ∈
/ [ 2m+n
, 2] assume in addi2n
) or
tion that the semigroup Tq (·) satisfies generalized Gaussian (q, 2)-estimates (if q < 2m+n
q
generalized Gaussian (2, q)-estimates (if q > 2). Then Aq has maximal regularity in L (Rn ).
The proof of Theorem 1.1 uses three main ingredients. First we prove the following amelioration of a result in [14], which shows that generalized Gaussian bounds imply maximal regularity.
This shall be proved in Section 2.
Theorem 1.2. Assume that Sq (·) is an analytic semigroup on Lq (Rn ) satisfying generalized
Gaussian (q, 2)-estimates (if q < 2) or generalized Gaussian (2, q)-estimates (if q > 2). Then,
for r ∈ (q, 2] or r ∈ [2, q), respectively, the semigroup Sq (·) extends to an analytic semigroup
Sr (·) in Lr (Rn ) whose negative generator Br has maximal regularity on Lr (Rn ).
Secondly, we show that (Hq ) implies generalized Gaussian (q, q ∗ )-estimates for Tq (·) where
nq
:= n−mq
if n > mq and q ∗ := ∞ if n mq (so q ∗ is related to Sobolev embedding of
m
n
Wq (R )). The method of proof (going back to Davies [22]) is well known for operators in
divergence form. The new feature here is that it works (almost) quite as well for operators in
non-divergence form. The assumption (Hq ) that −Aq generates an analytic semigroup is here
a natural one (recall that the generation property always holds for divergence form operators
in L2 ). We give the details in Section 3. The reason that we do not need additional assumptions
2n
for q ∈ [ 2m+n
, 2] ∩ (1, ∞) is, of course, the equivalence of this condition to 2 ∈ [q, q ∗ ]. Thirdly,
we employ an extrapolation argument based on Sneiberg’s lemma to make good for the fact
that Theorem 1.2 does not give maximal regularity in Lq itself. For this we exploit the special
structure of non-divergence operators and assumption (Hq ). We give a little discussion on the
additional assumption in Theorem 1.1 in Section 4. In fact, the study of Brézis’ question led us
to a closer investigation of the Lq -theory of non-divergence operators on Rn . We present the
related results, which also comprise the relation of property (Hq ) to ellipticity assumptions, in
Section 5. In Section 6 we give some applications of the results of this paper, in particular we
relate our results to other approaches to maximal regularity. This demonstrates that our methods
yield completely new proofs for maximal regularity and some new results. In particular, we give
a new proof of maximal regularity for operators whose top order coefficients are in VMO, and
generalize a recent result due to Kim and Krylov on operators with bounded measurable coefficients depending on one coordinate only. Moreover, we illustrate our results by the discussion
of a famous class of counterexamples. Section 7 collects some remarks on variants of the results
presented here.
q∗
2. Maximal regularity and off-diagonal bounds
In this section we prove Theorem 1.2. We first recall the following characterization result
from [50].
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
Theorem 2.1. (See L. Weis [50,51].) Let q ∈ (1, ∞) and let −A be the generator of a bounded
analytic semigroup in Lq (Ω). Then A has maximal regularity if and only if, for some δ > 0, the
±iδ
set {e−zA : |arg z| δ} is R2 -bounded in Lq (Ω), which is the case if and only if {e−e tA : t 0}
is R2 -bounded in Lq (Ω).
We shall prove Theorem 1.2 as an application of the following improvement of [14, Theorem 1.1]. Here, let (Ω, d, | · |) be a (measured metric) space of homogeneous type, i.e. for some
cΩ , n > 0 we have
B(x, λρ) cΩ λn B(x, ρ), ρ > 0, λ 1, x ∈ Ω.
(4)
For a measurable function g : Ω → C and q ∈ [1, ∞], x ∈ Ω, ρ > 0 we define Nq,ρ g(x) :=
|B(x, ρ)|−1/q 1B(x,ρ) f q , Mq g(x) := supρ>0 Nq,ρ g(x). Observe that M1 g is the usual centered
maximal function Mg, and that Mq g = (M(|g|q ))1/q for q ∈ (1, ∞), M∞ g = g∞ . We shall
use the notation A(x, ρ, k) := B(x, (k + 1)ρ) \ B(x, kρ).
Theorem 2.2. Let (Ω, d, | · |) be a space of homogeneous type such that (4) holds. Let 1 q0 q1 ∞, and assume that (S(t))t∈τ is a family of linear operators on Lq0 ∩ Lq1 such that
1B(x,ρ(t)) S(t)1A(x,ρ(t),k) q0 →q1
−( 1 − 1 )
B x, ρ(t) q0 q1 h(k),
t ∈ τ, x ∈ Ω, k ∈ N0 , (5)
where ρ : τ → (0, ∞) is a function and the sequence (h(k))k∈N0 satisfies h(k) cδ (k + 1)−δ for
some δ > qn0 + q1 . Then we have
0
S(t): t ∈ τ is Rs -bounded in Lq (Ω)
for all (q, s) ∈ (q0 , q1 ) × [q0 , q1 ] ∪ {(q0 , q0 ), (q1 , q1 )}.
We start the preparations for the proof with a domination property.
Proposition 2.3. Let Ω be a space of homogeneous type that is (4) holds. Assume
1B(x,ρ(t)) S(t)1A(x,ρ(t),k) q0 →q1
−( 1 − 1 )
B x, ρ(t) q0 q1 h(k),
t ∈ τ, x ∈ Ω, k ∈ N0 ,
where the sequence (h(k))k∈N0 satisfies h(k) cδ (k + 1)−δ for some δ >
for a constant Cq0 ,
n
q0
+ q1 . Then we have,
0
∞
1/q0
n−1 −δ
q
Nq1 ,ρ(t) S(t)f (x) Cq0
k q0 Nq0 ,kρ(t) f (x) 0
,
t ∈ τ, x ∈ Ω, f
k=1
q0
∈ Lloc (Ω),
q0 ,
and in particular, for a constant C
q0 Mq0 f (x),
Nq1 ,ρ(t) S(t)f (x) C
(6)
q
0
t ∈ τ, x ∈ Ω, f ∈ Lloc
(Ω).
(7)
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2737
Proof. We study the case q1 < ∞. The modifications for q1 = ∞ are obvious. Let x ∈ Ω,
f ∈ Lq0 ∩ Lq1 , t ∈ τ and set ρ = ρ(t), T := S(t). We denote, for k ∈ N, vk := |B(x, kρ)|,
q
q
bk := 1B(x,kρ) f q00 , nk := Nq0 ,kρ f (x), and, for k ∈ N0 , ak := 1A(x,ρ,k) f q00 . We also let
v := v0 := v1 . Then we have
Nq1 ,ρ (Tf )(x) = v −1/q1 1B(x,ρ) Tf q1 v −1/q1
∞
1B(x,ρ) T 1A(x,ρ,k) f q1
k=0
∞
v
− q1
0
h(k)1A(x,ρ,k) f q0 cδ
k=0
∞
v −1/q0 (k + 1)−δ ak
1/q0
.
k=0
For q0 = 1 we use ak = bk+1 − bk and v −1 cΩ k n vk−1 and obtain
cδ
∞
∞
v −1 k −δ − (k + 1)−δ bk cδ
k n k −δ − (k − 1)−δ vk−1 bk C1
k n−δ−1 nk .
k=1
k=1
k=1
For q0 > 1 we choose α = qn−1
. Observe that αq0 − δ < −1 and n − αq0 − δ < 0. Let Cδ :=
0 q0
∞
αq0 −δ
. By Hölder’s inequality we have
k=0 (k + 1)
Nq1 ,ρ (Tf )(x) cδ
∞
(k + 1)αq0 −δ
1/q 0
k=0
1/q = c δ Cδ 0
1/q0
∞
(k + 1)−αq0 −δ v −1 (bk+1 − bk )
k=0
∞
−αq −δ
k 0 − (k + 1)−αq0 −δ v −1 bk
1/q0
k=1
1/q 1/q
= c δ Cδ 0 c Ω 0
∞
k k −αq0 −δ − (k + 1)−αq0 −δ vk−1 bk
1/q0
n
k=1
Cq0
∞
1/q0
q
k n−αq0 −δ−1 nk0
.
k=1
This is (6), and (7) follows by supk nk Mq0 f (x).
2
We shall combine this proposition with the next one that gives more information on the families Nq := {Nq,ρ : ρ > 0} of sublinear operators.
Proposition 2.4. Let q ∈ [1, ∞] and assume that (4) holds. For (s, r) ∈ [q, ∞] × (q, ∞) ∪
{(q, q), (∞, ∞)}, the family Nq is Rs -bounded in Lr (Ω). For (s, r) ∈ [1, q] × (1, q) ∪
{(1, 1), (q, q)}, the family Nq is Rs -bounded from below in Lr (Ω).
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
Here, we call a set T of sublinear operators Rs -bounded from below on Lr (Ω) if, for some
C > 0 and all n ∈ N, T1 , . . . , Tn ∈ T , and all f1 , . . . , fn ∈ Lr (Ω), we have
1/s 1/s s
s
.
|f
|
C
|T
f
|
j
j j
r
j
r
j
We call the infimum of all such C the lower Rs -bound of T in Lr (Ω).
Proof. The second assertion follows from the first one by a dualization argument (cf. [14]). The
first assertion is clear for q = ∞. For q < ∞, it may be reduced to the case q = 1 by considering
(s/q, r/q) in place of (s, r). By Fubini and (4) we have
N1,ρ f 1 =
−1 1B(y,ρ) (x)B(x, ρ) f (y) dx dy cΩ 2n f 1 ,
Ω Ω
1,ρ g(x) :=
i.e. N1 is uniformly bounded in L1 , which implies R1 -boundedness in L1 . Letting N
−1
y → 1B(x,ρ) |B(y, ρ)| g1 we have for s, r ∈ [1, ∞], families (fj ), (gj ) of functions and (ρj )
of positive numbers (with obvious modifications for ∞ ∈ {s, s })
gj N1,ρj fj dx j
Ω
|gj |N1,ρj |fj | dx =
j Ω
Ω
1/s s
cr |N1,ρj gj |
r
j
1,ρj gj · |fj | dx cΩ 2n
N
j
1/s s
.
|fj |
Ω
N1,ρj gj · |fj | dx
j
r
j
Hence Rs -boundedness of N1 in Lr implies Rs -boundedness of N1 in Lr . Moreover, letting
s = ∞, all gj = g and using Lr -boundedness of the uncentered maximal operator, the estimate
shows that N1 is R1 -bounded in all Lr , r ∈ (1, ∞). By the preceding argument, N1 is R∞ bounded in all Lr , r ∈ (1, ∞), and the first assertion follows by interpolation. 2
Putting together Propositions 2.3 and 2.4 we obtain the following.
the lower Rs -bound of Nq1 in Lq and by Cq,s the
Proof of Theorem 2.2. Denoting by Cq,s
q
upper Rs -bound of Nq0 in L we have
1/s 1/s C S(tj )fj s
Nq ,ρ(t ) S(tj )fj s
q,s j
1
j
q
q
j
∞
s/q0 q0 /s n−1
1/q0
−δ
Cq,s
Cq0 k q0 (Nq0 ,kρ(tj ) fj )q0 q/q0
j
k=1
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
Cq,s
Cq0
∞
k
n−1
q0 −δ
1/s q0 1/q0
s
|Nq0 ,kρ(tj ) fj |
Cq,s
Cq0
1/s q0 1/q0
∞ n−1
−δ
q
0 k q0 Cq,s
|fj |s
Cq,s
Cq0 Cq,s
q
j
k=1
∞
k=1
which proves the claim.
q
j
k=1
2739
k
n−1
q0 −δ
1/q0 1/s s
,
|fj |
j
q
2
We now can prove Theorem 1.2.
Proof of Theorem 1.2. We may assume ν = 0 in (3) where either (q0 , q1 ) = (q, 2) (if q < 2)
or (q0 , q1 ) = (2, q) (if q > 2). We choose θ ∈ (0, δ) (the δ from (3)) and apply Theorem 2.2 to
the operators S(t) = Sq (e±iθ t), t > 0 and the function ρ(t) = t 1/m . The assumption (5) holds
(just cover the annuli A(x, t 1/m , k) by balls B(yj , t 1/m ) where |x − yj | ∼ k, we refer to similar
arguments in [14,17] (cf. also Lemma 3.4 below)). We obtain that {S(t): t > 0} is R2 -bounded
in Lr (Ω) for any r ∈ (q, 2] or any r ∈ [2, q), respectively. From this the claim follows. 2
Actually, the same arguments prove the following more general statement.
Theorem 2.5. Let 1 q0 q q1 ∞. Let (Sq (z)) be a bounded analytic semigroup in Lq (Rn )
satisfying generalized Gaussian (q0 , q1 )-bounds (3) for some δ ∈ (0, π/2) and ν = 0. Then we
have:
(a) For all r ∈ [q0 , q1 ] \ {∞}, the operators Sq (z), z ∈ Σδ , extend to a bounded analytic semigroup (Sr (z)) in Lr (Rn ).
(b) For all (r, s) ∈ ((q0 , q1 ) × [q0 , q1 ]) ∪ {(q0 , q0 ), (q1 , q1 )} and all δ ∈ (0, δ) the set {Sr (z):
z ∈ Σδ } is Rs -bounded in Lr (Rn ).
(c) If q0 2 q1 then, for all r ∈ (q0 , q1 ) ∪ {2}, the negative generator Br of (Sr (t)) has
maximal regularity in Lr (Rn ).
We close this section with some remarks.
Remark 2.6. Theorem 2.2 improves over [14, Theorem 1.1]. On the one hand, we took up the
argument from [36] which allowed to include the limit cases q0 = 2 and q1 = 2. On the other
hand, [14, Theorem 1.1] would have required the condition δ > n. The given condition δ >
n
1
n
q0 + q also improves over the argument on [41, p. 184], which would have required δ > q0 + 1.
0
Remark 2.7. Part (a) of Theorem 2.5 is a generalization of the main result in [45] and has also
been proved in [43, Section 2]. In fact, the case r = q0 = 1 requires an additional argument to
obtain strong continuity. This shall be skipped here since it is very similar to what has been done
elsewhere. Part (b) is a new complement to these results. There is a recent result, namely [12,
Theorem 8.10], which, for q0 = 2 or q1 = 2, is related to part (c) of Theorem 2.5. The required
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
off-diagonal bounds are less restrictive there, but the proof is totally different, relying on abstract
Hardy spaces associated with the generator of the semigroup.
The following remark shows that, in the situation of Theorem 1.2, the above proof cannot give
maximal regularity of the negative generator Bq of Sq (·) in Lq (Ω).
Remark 2.8. We mention here that the range of (s, r) in the first assertion of Proposition 2.4 is
sharp since, for r = 1 in Ω = R, N1 is not Rs -bounded
in L1 for any s ∈ (1, ∞]. To see this, take
N
fj := f := 1[0,1] for j = 1, . . . , N . Then ( j =1 |fj |s )1/s 1 = N 1/s and N1,2j f (x) = 2−j −1
for x ∈ [0, 2j ]. Hence
j
N
1/s N 2
|N1,2j fj |s
N1,2j f (x) dx = N/4,
j =1
1
j =1 j −1
2
and N → ∞ yields the claim. By the duality argument in the proof we see that N1 is not Rs bounded in L∞ for any s ∈ [1, ∞). Of course, N∞ is Rs -bounded from below in Lr for all
r, s ∈ [1, ∞], since |f | N∞,ρ f a.e. for any ρ > 0.
3. Off-diagonal estimates for non-divergence operators
In this section we establish generalized Gaussian estimates for non-divergence operators under
the assumption (Hq ), and we give the proof of Theorem 1.1. We also provide Gaussian type
estimates for (higher) derivatives that shall be used in Section 5. As mentioned in the introduction
we use a modification of Davies’ method from [22]. To this end we let Em denote the set of all
real-valued C ∞ -functions φ with compact support on Rn satisfying ∂ α φ∞ 1 for all 1 |α| m. This restriction yields that
dm (x, y) := sup φ(x) − φ(y): φ ∈ Em
(8)
defines a distance which is equivalent to the Euclidean distance on Rn (cf. [22, Lemma 4]
where dm (x, y) |x − y| is shown). Davies’ method consists in considering the operators
Aλφ := e−λφ Aeλφ as perturbations of the operator A. For divergence form operators, perturbations act on the form domain. Since we are dealing with non-divergence operators we consider
perturbations of the operator A on its domain D(A). Observe that multiplication with eλφ , λ ∈ R,
nq
if
φ ∈ Em , takes D(A) = Wqm (Rn , C) into itself. As in the introduction, we let q ∗ := n−mq
n > mq and q ∗ := ∞ if n mq. If (Hq ) holds we denote by δ(Aq ) the angle of the semigroup
Tq (·) generated by −Aq , i.e.
δ(Aq ) := sup δ ∈ (0, π/2): Tq (·) has an analytic extension to Σδ .
(9)
Recall that (Hq ) implies the existence of ν 0 such that, for any δ ∈ (0, δq (A)), we have
−zA e
q→q
Mδ eν Re z ,
z ∈ Σδ .
(10)
As a last preparation we observe that (Hq ) implies in particular that Aq on Wqm (Rn ) is a closed
operator in Lq (Rn , C). However, Aq with domain Wqm (Rn ) is closed in Lq (Rn ) if and only if the
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2741
graph norm uq + Aq uq is equivalent to the norm uWqm of Wqm (Rn , C). By aα ∈ L∞ (Rn )
this holds if and only if, for some η > 0, the Gårding type inequality
uq + Aq uq η(−)m/2 uq ,
u ∈ Wqm Rn ,
(11)
holds.
Theorem 3.1. Let m ∈ 2N and q ∈ (1, ∞). Let A be an operator of the form (2) that
satisfies (Hq ), and let δ ∈ (0, δ(Aq )). Then there are constants C, ω only depending on
m, n, q, δ, δq (A), aα ∞ , the constants in (10), and the constant η in (11) such that
−λφ −zA λφ m
e
e
e q→q Ceω(1+λ ) Re z |arg z| δ, λ ∈ R, φ ∈ Em ,
m
(−)m/2 e−λφ e−zA eλφ C|z|−1 eω(1+λ ) Re z |arg z| δ, λ ∈ R, φ ∈ Em ,
q→q
(12)
(13)
and
−λφ −zA λφ e
e
e q→q ∗
n 1
−m
( q − q1∗ ) ω(1+λm ) Re z
C|z|
e
|arg z| δ, λ ∈ R, φ ∈ Em .
(14)
We call (12) a weighted (q, q)-estimate and (14) a weighted (q, q ∗ )-estimate.
Proof. Without loss of generality we may assume that the semigroup is bounded analytic, i.e.
ν = 0 in (10). We assume n = mq for simplicity and shall comment on the case n = mq later on.
Let θ := π/2 − δ, δ := (δ + δ(Aq ))/2, and θ := π/2 − δ . Denote Σ := Σπ−θ and Σ := Σπ−θ .
We let Bλφ := Aλφ − A. Then Bλφ is a differential operator of order m − 1 with L∞ -coefficients
and, recalling (11), we may see as in [22] (cf. also [41, p. 189]) that we have
Aλφ u − Auq γ Auq + Cγ 1 + λm uq ,
u ∈ Wqm Rn , λ ∈ R, φ ∈ Em ,
where we may choose γ > 0 so small that γ −1 > M + 1 where M := sup{μ(μ + A)−1 q→q :
μ ∈ Σ }. This yields
Bλφ (μ + A)−1 u γ A(μ + A)−1 + Cγ 1 + λm (μ + A)−1 u
q
q
q
m
−1
γ (M + 1) + Cγ 1 + λ M|μ| uq
for μ ∈ Σ where M is as above. Choosing ρ > MCγ /(1 − γ (M + 1)) we obtain for μ ∈ Σ with |μ| ρ(1 + λm ) that
Bλφ (μ + A)−1 q→q
which leads to
γ (M + 1) +
MCγ
=: d < 1,
ρ
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∞
−1 k
μ + (A + Bλφ ) = (μ + A)−1
−Bλφ (μ + A)−1 k=0
∞
M
(μ + A)−1 (1 − d)−1 .
dk |μ|
(15)
k=0
Thus Sλ := Σ \ {|μ| < ρ(1 + λm )} is a subset of ρ(−Aλφ ). Now, whenever r > 0 and s :=
r/ sin θ , then Σ + s ⊂ Σ \ {|μ| < r} (recall that the half opening angle of Σ is > π/2). Letting
λφ ) ⊃ Σ . Moreover, for μ ∈ Σ , we have
λφ := Aλφ + ρ (1 + λm ) we conclude that ρ(−A
A
sin θ
the following estimate
−1 M
μ(μ + A
λφ )−1 = μ μ + ρ 1 + λm + (A + Bλφ )
1 − d |μ +
sin θ
|μ|
.
ρ
m
sin θ (1 + λ )|
Now we observe that, for σ 0 and μ ∈ Σ , we have |μ + σ | sin θ |μ|. Indeed, we have
− Re μ |μ| cos θ for μ ∈ Σ , and
|μ + σ |2 = |μ|2 + σ 2 + 2σ Re μ |μ|2 + σ 2 − 2σ |μ| cos θ 2
2
= cos θ |μ| − σ + sin2 θ |μ|2 sin θ |μ| .
Hence, we have for all μ ∈ Σ and all λ, φ the bound
μ(μ + A
λφ )−1 M
.
(1 − d) sin θ (16)
λφ
By the generation theorem for bounded analytic semigroups we conclude that all operators −A
are generators and that
−zA λφ e
q→q
K
(17)
for all |arg z| δ and all λ, φ where K depends only on the bound in (16), i.e. only on M, θ ,
and d, but not on z, λ, or φ. This means in particular that (12) is proved. Letting δ := δ − δ we
have for z ∈ Σδ by Cauchy’s formula
f (ζ )
1
dζ,
f (z) =
2πi
(ζ − z)2
|ζ −z|=|z| sin δ where f (z) = e−zAλφ . Using (17), this yields
A
λφ e−zAλφ q→q
K
|z|−1
sin δ |arg z| δ
(18)
λφ and observe
for all λ, φ. Before we exploit this bound, we take a look back at the definition of A
ρ
m
m
that μλφ := sin θ (1 + λ ) belongs to the subset Sλ = Σ \ {|μ| < ρ(1 + λ )} of ρ(−Aλφ ). Hence
λφ = Aλφ + μλφ is boundedly invertible and, by the arguments used above,
A
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2743
A(A
λφ )−1 = A(μλφ + Aλφ )−1 = A μλφ − (A + Bλφ ) −1 ∞
M +1
A(μλφ − A)−1 dk 1−d
(19)
k=0
for all λ, φ. By (11), (19) and (15) we then have
M
(−)m/2 u η−1 Auq + uq η−1 M + 1 +
λφ uq
A
q
1−d
μλ,φ (1 − d)
M sin θ −1 M + 1
λφ uq .
+
A
η
1−d
ρ(1 − d)
We call the constant in the last line K2 . For fixed f ∈ Lq , z ∈ Σδ , λ ∈ R and φ ∈ Em we let
v(z) := e−zAλφ f and (18) yields
(−)m/2 v(z) K2 A
λφ v(z) = K2 A
λφ e−zAλφ f K2 K |z|−1 f q .
q
q
q
sin δ This means that we have proved (13). Now we distinguish the three cases n > mq, n < mq,
and n = mq. If n > mq then we use the Sobolev embedding uq ∗ L(−)m/2 uq . For fixed
f ∈ Lq , z, λ, φ, and v(z) as above we obtain
v(z)
q∗
K
L(−)m/2 v(z)q LK2
|z|−1 f q .
sin δ This means that we have proved the following weighted (q, q ∗ )-bound
−λφ −zA λφ m
e
e
e q→q ∗ C|z|−1 eω(1+λ ) Re z |arg z| δ, λ ∈ R, φ ∈ Em
where ω := ρ/ sin θ and C := LK2 K/ sin δ depend only on the constants M, γ , Cγ , η, θ , the
n
(1/q − 1/(q ∗ )) =
chosen ρ, and the Sobolev constant L, but not on z, λ, or φ. Observe that − m
−1. Hence (14) is proved in this case. In the case n < mq we use the estimate
n
1− n
u∞ L(−)m/2 uqmq uq mq
instead of the Sobolev embedding and obtain in a similar way
−λφ tA λφ n
ω(1+λm )t
e
− m (1/q−1/(q ∗ )) e
e e q→q ∗ Ct
|arg z| δ, λ ∈ R, φ ∈ Em
where we recall q ∗ := ∞. By an argument similar to [9, p. 59] the case n = mq may be reduced
to the other two cases as follows. We use [38, Theorem 8], an application of Sneiberg’s lemma
(cf., e.g., [9, p. 53]), to obtain an ε > 0 such that (Hq−ε ) and (Hq+ε ) hold where q + ε (q − ε)∗ .
Then we obtain weighted (q − ε, q − ε)-estimates and weighted (q − ε, (q − ε)∗ )-estimates, i.e.
(12) and (14) hold for q − ε in place of q. Since q + ε (q − ε)∗ , we obtain weighted (q −
ε, q + ε)-estimates by interpolation. Combining these via the semigroup property with weighted
(q + ε, ∞)-estimates (recall n < m(q + ε) which means (q + ε)∗ = ∞) yields weighted (qε , ∞)estimates. Interpolating those with the weighted (q + ε, ∞)-estimates, we get weighted (q, ∞)estimates as desired. 2
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Corollary 3.2. In the situation of Theorem 3.1 there also are constants C, ω only depending on
m, n, q, δ, δq (A), aα ∞ , the constants in (10), and the constant η in (11) such that
−λφ α −zA λφ m
e
(20)
∂ e
e q→q C|z|−1 eω(1+λ )|Re z| |arg z| δ, λ ∈ R, φ ∈ Em ,
for any multi-index α with |α| = m.
Proof. The arguments are similar to those used for weighted gradient estimates in [16]. We start
with the observation
∂j e−λφ g = e−λφ ∂j g − λ∂j φ · e−λφ g,
which implies
e−λφ ∂ α g =
n
(∂j + λ∂j φ)αj e−λφ g
j =1
for any multi-index α = (α1 , . . . , αn ). By induction we obtain
−λφ α −zA λφ e
∂ e
e f q C
βα;j +|β|m
|λ|j ∂ β e−λφ e−zA eλφ f q .
By interpolation, (12) and (13) yield
β −λφ −zA λφ m
∂ e
e
e f q C|z||β|/m eω(1+λ ) Re z f q
for any |β| m. Hence we have
−λφ α −zA λφ m
e
∂ e
e f q C|z|−1
|λ|j |z|(m−k)/m eω(1+λ ) Re z f q .
j +km
Finally, for j + k m, the inequality
|λ|j |z|(m−k)/m 1 + λm |z| cδ 1 + λm Re z
yields the assertion.
2
Corollary 3.3. Assume in the situation of Theorem 3.1 and Corollary 3.2 that the semigroup
e−zA satisfies generalized Gaussian (q0 , q)-estimates for |arg(z)| δ. Then we have, for some
C , ω > 0,
−λφ α −zA λφ e
∂ e
e q0 →q
n 1
−1− m
( q − q1 ) ω (1+λm )|Re z|
0
C |z|
for any multi-index α with |α| = m.
e
|arg z| δ, λ ∈ R, φ ∈ Em ,
(21)
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2745
Proof. We use the semigroup property. The only thing we have to do is get weighted
(q0 , q)-estimates from the assumed generalized Gaussian estimates. This can be done as, e.g.,
in [17]. 2
In order to obtain generalized Gaussian (p, q)-estimates from weighted (p, q)-estimates we
state the following lemma, which shall be used several times later on. Here we use again the
notation A(x, ρ, k) := B(x, (k + 1)ρ) \ B(x, kρ), where now the B(x, ρ) are open balls for the
max-norm in Rn , i.e. cubes with center x and side length 2ρ.
Lemma 3.4. Let G ⊂ C and let e−λφ Sz eλφ p→q Cz eωλ Re z for all λ ∈ R, φ ∈ Em , and
z ∈ G. Then we have, for some constants b, c > 0, two-ball estimates
m
m/(m−1)
1B(x,r) Sz 1B(y,r) p→q Cz e
−b |x−y| 1/(m−1)
(22)
(Re z)
√
for all r > 0, x, y ∈ Rn with |x − y| 2 nr, z ∈ G, and annular estimates
1B(x,r) Sz 1A(x,r,k) p→q Cz e−c((kr)
m / Re z)1/(m−1)
(23)
√
for all r > 0, x ∈ Rn , z ∈ G and k 2 n.
Proof. Recalling (8), we have
m
1B(x,r) Sz 1B(y,r) Cz inf e−λφ ∞,B(x,r) eλφ ∞,B(y,r) eωλ Re z
λ,φ
√
m
Cz inf e n|λ|r e−λ(φ(x)−φ(y)) eωλ Re z
λ,φ
√ Cz exp − sup λ dm (x, y) − nr − ωλm Re z
λ>0
√
1/(m−1) (dm (x, y) − nr)m
+
,
= Cz exp −cm
ω Re z
where cm = m−1/(m−1) − m−m/(m−1) . Since dm (x, y) |x − y|, we obtain (22). For the proof
of (23) we cover A(x, r, k) with 2n ((k + 1)n − k n ) balls B(y, r), use (22) and observe that
|x − y|∞ kr. 2
Corollary 3.5. Let m ∈ 2N and q ∈ (1, ∞). Let A be an operator of the form (2) that satisfies (Hq ). Then, for any r ∈ [q, q ∗ ] and any δ ∈ (0, δ(Aq )), the semigroup Tq (·) satisfies
generalized Gaussian and weighted (q, r)-estimates and generalized Gaussian and weighted
(r, q ∗ )-estimates for |arg(z)| δ.
Proof. We interpolate between (12) and (14) and apply Lemma 3.4.
2
2n
Proof of Theorem 1.1. If q ∈ [ 2m+n
, 2], then (Hq ) implies by Corollary 3.5 generalized Gaussian (q, 2)-estimates. If q < 2, we thus have generalized Gaussian (q, 2)-estimates, which we
translate into weighted (q, 2)-estimates. We use [38, Theorem 8] again and get (Hq−ε ) for some
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
ε > 0 such that 1 < q − ε < q (q − ε)∗ . By Corollary 3.5 we obtain weighted (q − ε, q)estimates. By the semigroup property we get weighted (q − ε, 2)-estimates, and an application
of Lemma 3.4 yields generalized Gaussian (q − ε, 2)-estimates. Now we can apply Theorem 1.2.
The proof for the case q > 2 is very similar. 2
Remark 3.6. By the cost of changing the constant Cz we could remove the restriction on |x − y|
in (22) and on k in (23). But the given estimates shall be sufficient for the applications in Section 5. We also refer to [14,17,22,39,43] for the kind of arguments used in the proof and for
converse statements.
Remark 3.7. Off-diagonal bounds of the type we derived in this section can be composed and
interpolated. For more on this technique we refer to [14,17,36,39,43,47–49], see also the recent
series of papers [5–8]. Here we just mention the following. As we see by Theorem 2.5, the operators T (t) := e−tA in Theorem 3.1 act boundedly in Lr (Rn ) for r ∈ [q, q ∗ ], and they induce
analytic semigroups (Tr (t)) in Lr (Rn ) for r ∈ [q, q ∗ ] \ {∞}. It is, however, not clear if the negative generator of (Tr (t)) equals the operator Ar . If we have equality then (Hr ) holds, and we may
use Theorem 3.1 again to obtain weighted (r, r ∗ )-bounds. By Corollary 3.5 we obtain weighted
(q, r)-bounds, which—by the semigroup property—can be composed with the weighted (r, r ∗ )bounds to obtain weighted (q, r ∗ )-bounds for the semigroup.
Remark 3.8. In [22], the bound corresponding to (14) was shown first for z = t > 0, i.e. for real
times. Estimates for complex times were obtained by a Phragmen–Lindelöf type argument. Here
we have shown bounds for complex times directly. We obtained the optimal angle for weighted
bounds, but those more sophisticated arguments are needed to obtain optimal values for some
constants in weighted estimates. We do not go into details here and refer to [13,21,22,43] for
further information.
4. On the additional assumption in Theorem 1.1
In this section we give a short discussion on the additional assumption in Theorem 1.1.
Proposition 4.1.
2n
(a) In Theorem 1.1, for q ∈
/ [ 2m+n
, 2] the following weaker additional assumption will do,
namely that Tq (·) satisfies generalized Gaussian (r, 2)-estimates for some q r < 2 (if q <
2n
2m+n ) or generalized Gaussian (2, r)-estimates for some 2 < r q (if q > 2).
(b) The additional assumption in Theorem 1.1 is satisfied if (H2 ) holds.
(c) The additional assumption in Theorem 1.1 holds if Tq (·) extends to a C0 -semigroup in
L2+ε (Rn ) (if q < 2) or in L2−ε (Rn ) (if q > 2).
Remark 4.2. It is not clear if the converse of the assertions in Proposition 4.1(b) or (c) holds. For
a further study of the situation in (b) and (c) we refer to the next section.
Proof of Proposition 4.1. We recall Remark 3.7 on composition and interpolation.
(a) Let q < 2. We obtain by Theorem 2.5 that the semigroup is analytic in Ls (Rn ) for any
s ∈ [q, q ∗ ] ∪ [r, 2]. By interpolation we get analytic semigroups in all Ls (Rn ), s ∈ [q, 2]. We
transfer the generalized Gaussian estimates to weighted estimates, use the semigroup property
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2747
and interpolation, and obtain weighted (q, 2)-estimates. Lemma 3.4 yields generalized Gaussian
(q, 2)-estimates. The proof for q > 2 is similar.
(b) If q > 2 then we obtain generalized Gaussian (2, r)-estimates for r = 2∗ by Theorem 2.5
and can use (a). If q < 2, we have to interpolate between the generalized Gaussian (q, q ∗ )- and
(2, 2∗ )-estimates to obtain generalized Gaussian (r, 2)-estimates for a suitable r < 2.
(c) If q > 2 then we can interpolate strong continuity and analyticity to any L2−ε , ε ∈ (0, ε).
Using Stein interpolation between weighted and unweighted estimates, and exploiting the semigroup property yields generalized Gaussian (2, q)-estimates. The proof for the case q < 2 is
similar. 2
5. Lq -theory for non-divergence operators
In this section we develop an Lq -theory for non-divergence operators satisfying (Hq ) for some
q ∈ (1, ∞). In Theorem 1.1, we did not require any specific form of ellipticity of the symbol
of Aq , but we recall that (Hq ) implied the Gårding type inequality (11) (cf. the paragraph before
Theorem 3.1). Combined with the sign condition that is part of (Hq ), the estimate (11) may
be viewed as a form of ellipticity. Of course, (11) does not change (except for the value of the
constant) if we replace (−)m/2 by the matrix ∇ m of all mth order derivatives. Below we shall
investigate how to get back from (11) to (Hq ) under suitable assumptions. The desire to enlarge
the scale of Lr -spaces via the arguments in Remark 3.7 leads to the following definition.
Definition 5.1. Let (aα )|α|m be a family of coefficients in L∞ (Rn ) and let A = |α|m aα(x)D α .
Suppose that (H
q ∈ (1, ∞) and denote the generated semigroup by T (·). Let
q ) holds for some IA be the largest interval I⊆ [1, ∞] that contains q and has the property that (3) holds for all
q0 , q1 ∈ I with q0 q1 , and let JA be the set of all q ∈ IA ∩ (1, ∞) such that (Hq ) holds. Let
KA be the set of all q ∈ [1, ∞) such that T (·) acts as a C0 -semigroup in Lq (Rn ). Let JA be
the largest interval J ⊆ KA that contains q and has the property that (11) holds for all q ∈ J.
Moreover, denote qmin,A := inf IA , qmax,A := sup IA , q−,A := inf JA , and q+,A := sup JA . For
r ∈ IA \ {∞} we denote by Br the negative generator of the semigroup T (·) acting in Lr (Rn ).
By interpolation, KA is an interval, and by Remark 3.7 we see that the interiors of KA and
IA \ {∞} coincide. As a further main result of the paper we now prove the following.
Theorem 5.2. Under the assumptions of Definition 5.1, the set JA is an interval and JA = JA .
We have qmin,A = q−,A and Br = Ar for all r ∈ JA . Moreover, we have qmax,A (q+,A )∗ , and
the closures of IA and JA coincide if and only if q+,A = ∞.
Convexity of JA is obtained by interpolation (since semigroups and resolvents are consistent),
and the inequality qmax,A (q+,A )∗ follows from Theorem 3.1. Equality of JA and JA means
that the question, whether (Hq ) holds for other q ∈ KA than q , becomes a question on elliptic
regularity. The identity qmin,A = q−,A relies on a modification of the main result in [16], which
gave conditions for boundedness of operators BA−α where α ∈ (0, 1). Here we need such a result
for α = 1. We shall give a direct proof, based on an amelioration (Theorem 5.4 below) of a result
from [15]. We view Theorem 5.2 as a first step in an investigation of non-divergence operators in
a way similar to what has been done by Pascal Auscher in [4] for operators in divergence form.
We start by proving the assertion on (11) in Theorem 5.2.
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Proposition 5.3. Under the assumptions of Theorem 5.2 we have JA = JA .
Proof. The inclusion JA ⊆ JA holds by the argument before Theorem 3.1. To prove the other
inclusion, we start with a preparation. Fix r ∈ KA and recall that (H
q ) holds (cf. Definition 5.1)
m
r
r
q
r
and that q ∈ IA ∩ (1, ∞). Let D := {u ∈ W
qu ∈ L ∩L
q ∩ L : Au ∈ L }. For u ∈ D and g := A
we have
t
T (t)u − u =
t > 0,
T (s)g ds,
0
q and in Lr . We conclude that D ⊆ D(B ) and B u = Au for
where the integral converges in L
r
r
u ∈ D. The set D is invariant under the semigroup T (·), and D is dense in Lr , since
m
m
W
q ∩ Wr ⊆ D.
(24)
m
m
m
Hence D is a core for Br in Lr . On W
q ∩ Wr , the norm · Wr is stronger than the graph
norm of A, hence stronger than the graph norm of Br . Taking closures in (24) we thus see that
Wrm ⊆ D(Br ) and Ar = Br on Wrm . If, in addition, r ∈ JA then (11) holds for r and we see that
Wrm is a closed subspace of D(Br ) for the graph norm of Br . If there existed an r ∈ JA \ JA then
either r inf JA or r sup JA . Let q0 := inf JA in the first case and q0 := sup JA in the second.
Since JA is open by [38, Theorem 8], q0 belongs to JA \ JA . Now we take λ > 0 large and obtain
for R := (λ + A)−1 by interpolation that ARq→q + Rq→q M for q ∈ JA ∩ V where V is
a neighborhood of q0 . By [32, Theorem 2.5] the set JA is open in KA and q0 has a neighborhood
U such that (11) holds for each q ∈ U with a uniform constant η > 0. Taking the limit q → q0
where q ∈ JA ∩ V ∩ U we obtain
m ∇ Rf = lim ∇ m Rf lim M f q = M f q
0
q0
q
q→q0
q→q0 η
η
by properties of the Lq -norms. By consistency, R = R(λ, Bq0 ), which means that D(Bq0 ) ⊆ Wqm0 .
Hence both spaces coincide, (Hq0 ) holds, and q0 ∈ JA , a contradiction. We thus have shown
JA = JA . 2
Proof of Theorem 5.2. It rests to prove qmin,A = q−,A . Since “” is clear we prove “”. To
this end we take q0 ∈ IA and q ∈ JA with q0 < q, and use the following result which is an
improvement of the weak type (q0 , q0 )-criterion in [15].
Theorem 5.4. Let (Ω, d, | · |) be a space of homogeneous type and dimension n. Let 1 q0 <
q < ∞ and let T : Lq (Ω) → Lq (Ω) be bounded. Suppose that there is an approximation of the
identity (Sr )r>0 satisfying
Sr f q Mf q ,
1(k+1)B\kB Sr 1B q0 →q |B|
1
1
q − q0
1
1(k+1)B\kB T (I − Sr )1B q0 →q |B| q
− q1
0
r > 0,
(25)
h(k),
k = 0, 1, 2, . . . ,
(26)
h(k),
k k0 ,
(27)
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2749
for some M > 0, k0 4 and any ball B of radius r where the sequence (h(k)) satisfies h(k) cδ (k + 1)−δ for some δ > n/q + 1/q. Then T is of weak type (q0 , q0 ) and bounded on Ls (Ω)
for s ∈ (q0 , q).
Remark 5.5. (a) In comparison to the result in [15] the bound on the sequence (h(k)) is relaxed.
This corresponds to the improved bound in Proposition 2.3. Moreover, Pascal Auscher observed
(cf. [4]) that it is sufficient to have the same q in (26) and (27). The proof of Theorem 5.4 can
be done similar to the proof in [4], if one takes into account the improved maximal estimates of
Proposition 2.3.
(b) Of course, we want to apply Theorem 5.4 to the operators T = CA−1 where C = ∂ β and
|β| = m. This will give boundedness of CA−1 in all Ls (Rn ), s ∈ (q0 , q), which means that
m ∇ u Ks As us , u ∈ D(As ).
s
The argument in the first part of the proof of Proposition 5.3 shows that we also have
Wsm → D(As ). Hence D(As ) and Wsm coincide with equivalent norms, and (Hs ) holds. Thus
(q0 , q) ⊂ JA . Since q0 ∈ IA was arbitrary we conclude qmin,A q−,A .
(c) For operators of the form T = CA−α with α ∈ (0, 1) we could have used the main result
in [16]. Since we deal with the case α = 1 here, the arguments have to be modified. Hence we
give full details in the sequel, for a general α > 0. The essential additional hypotheses on the
operator C is (29) below, which holds in our case if α = 1 (cf. below).
The approximation of identity (Sr(t) ) will be given by a linear combination of terms e−ktA ,
k = 0, . . . , N , namely via
I − Sr(t) =
N N
k=0
k
(−1)k e−ktA ,
(28)
where r(t) = t 1/m /2. The estimate (26) follows from Remark 3.7, and the
N that proved
argument
Lemma 3.4. We fix N ∈ N to be chosen later and define ψ(z) := N
k=0 k exp(−kz) for
Re z > 0. Notice that |ψ(z)| = O(|z|N ) for |z| → 0 in fixed sectors of half opening angle < π/2
and that |ψ(z)| = O(1) for |z| → ∞, so that we have, for any θ ∈ (0, π/2),
ψ(z) Cθ min |z|N , 1 , z ∈ Σθ .
In order to show (27) we need a representation of T (I − Sr(t) ), and we start by writing
T (I − Sr(t) ) = CA−α ψ(tA) = Ct α φ(tA),
where φ(z) := z−α ψ(z) and φ ∈ H ∞ (Σθ ) for any θ ∈ (0, π/2) if N > α with
φ(z) Cθ,α min |z|N −α , |z|−α , z ∈ Σθ .
By H0∞ -calculus (cf., e.g., [41, Section 9]), we can write
CA−α ψ(tA) = Ct α φ(tA) =
1
2πi
Ct α φ(tμ)R(μ, A) dμ
Γ
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
where Γ = ∂Σθ and θ ∈ (ω(A), π/2). We now represent R(μ, A) in terms of the semigroup. We
let Γ±θ = {re±iθ : r 0} and have, for μ ∈ Γ±θ ,
eμz e−zA dz,
R(μ, A) = −
μ ∈ Γ±
Γ±σ
where σ :=
π
2
−
Observe that, for j = ±1, μ ∈ Γj θ and z ∈ Γj σ , we have
ω+θ
2 .
Re(μz) = |μ||z| cos j (σ + θ ) = −c|μ||z|
where c := − cos(σ + θ ) > 0 since σ + θ ∈ (π/2, π). We thus have shown
−α
T (I − Sr(t) ) = CA
tα
ψ(tA) =
2πi
φ(tμ)eμz dμ Ce−zA dz.
+
Γθ Γσ
Γ−θ Γ−σ
Aiming at (27) we calculate, for ρ = r(t),
1A(x,ρ,k) T (I − Sr(t) )1B(x,ρ) q0 →q
φ(tμ)eμz |dμ| 1A(x,ρ,k) Be−zA 1B(x,ρ) α
Ct
+
q
0 →q
Γθ Γσ
|dz|.
Γ−θ Γ−σ
If we have a bound
−λφ −zA λφ e
Ce
e q0 →q
n 1
−α− m
( q − q1 ) ωλm Re z
0
C|z|
e
(this is the case for α = 1 by Corollary 3.3) and recall ρ = r(t) = t 1/m /2, then Lemma 3.4 leads
to
1A(x,ρ,k) Ce−zA 1B(x,ρ) q
0 →q
n 1
−α− m
( q − q1 ) −b(k m t/|z|)1/(m−1)
0
C|z|
e
√
for k k0 2 n. Writing u = |μ| and v = |z|, we hence have
1A(x,ρ,k) T (I − Sr(t) )1B(x,ρ) q
0 →q1
Ct α
∞∞
n 1
−α− m
( q − q1 ) −b(k m t/v)1/(m−1)
0
min (tu)N −α , (tu)−α e−cuv du v
e
dv
0 0
= C
∞∞
0 0
n 1
−α− m
( q − q1 ) −b(k m t/v)1/(m−1)
0
min (tu)N , 1 u−α e−cuv du v
e
dv.
(29)
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
We let γ :=
leads to
n 1
m ( q0
2751
− q1 ) temporarily, and estimate the min by (tu)N . The substitution v = k m tη
∞∞
1−α−γ
m
−1/(m−1)
C
(tu)N u−α e−cuk tη du η−α−γ e−bη
dη k m t
,
0 0
and the subsitution u = ξ/(k m tη) yields
=C
∞
ξ
N −α −cξ
e
0
∞
dξ
η−1−N −γ e−bη
−1/(m−1)
−γ −N
dη t −γ k m
.
0
Observe that the t-exponent is the one we are aiming at and that, choosing N large, we can make
the negative k-exponent as large as we wish. Hence (27) is proved, Theorem 5.4 applies, and
boundedness of CA−1 is proved. The arguments in Remark 5.5(b) above end the proof. 2
A combination of Theorems 3.1, 2.5 and 5.2 yields the following.
Theorem 5.6. Under the assumptions of Definition 5.1 the following holds: if 2 ∈ IA , then Br
has maximal regularity in Lr (Rn ) for all r in the interior of IA , in particular, Ar has maximal
regularity for all r in the interior of JA .
Thus, modulo the requirement 2 ∈ IA , this result is a positive answer to Brézis’ question for
non-divergence operators with complex-valued bounded measurable coefficients. We conclude
that, for operators of this kind, the main task is to prove generation of analytic semigroups.
Once this is done (on a sufficiently large scale of Lq -spaces), maximal Lp -regularity follows
“automatically.”
Open problem. We do not know if there is an operator A of form (2) satisfying q+,A < ∞.
6. Applications and examples
In this section we indicate some applications of our results. We discuss elliptic operators with
highest order coefficients in BUC and in VMO. This gives new insight in these operators, and
leads to new proofs for their maximal regularity. For elliptic operators whose coefficients are
only measurable in one variable, we also obtain a new proof of maximal regularity. Moreover,
our results allow to give a complement to some results in [34]. First we shortly discuss the relation of our approach to other approaches. Under the assumptions of Definition 5.1 the semigroup
T (·) has kernel bounds of Gaussian type if and only if IA = [1, ∞]. In this case maximal regularity of type Lp –Lq , p, q ∈ (1, ∞), has been shown in [19,30]. The usual approach to elliptic
non-divergence operators is the following: assuming ellipticity and suitable regularity properties
of the highest order coefficients, one can show, via localization and perturbation arguments, that
−Aq (with D(Aq ) = Wqm (Rn )) generates an analytic semigroup in Lq (Rn ), q ∈ (1, ∞). Having
done this, a refinement of the arguments usually leads to maximal Lp -regularity, p ∈ (1, ∞). We
refer to [2,23,26,29,37,40,41,46]. Here, we do not use localization, nor do we “freeze” coefficients. If the highest order coefficients are regular enough then one usually obtains an analytic
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
semigroup in the whole range of Lq (Rn )-spaces, i.e., one has JA = (1, ∞). This is the case
for bounded uniformly continuous coefficients [2,23,41] or, more generally, for coefficients in
L∞ ∩ VMO [3,26,29]. In all these cases, our method provides completely new proofs for the
property of maximal regularity of type Lp –Lq , p, q ∈ (1, ∞) (see Section 6.1 below). Our results allow to give a complement to recent results by Kim and Krylov [33,34] on elliptic second
order operators whose top order coefficients are measurable and depend only on one coordinate,
see Section 6.2. We draw attention to the fact that, in perturbation arguments, smallness conditions on the deviation of the coefficients of an operator with variable coefficients to an operator
with constant coefficients usually depend on q. If, e.g. the highest order coefficients are uniformly
continuous, then a suitable localization allows to make the deviation arbitrarily small, and one
obtains generation in all Lq , q ∈ (1, ∞). For a fixed deviation, however, one might only obtain
generation for a restricted scale of Lq -spaces. Nevertheless, our method applies to such slightly
more general operators. We illustrate this by discussing in detail a well-known (counter)example
in Section 6.3, usually attributed to Pucci–Talenti or Gilbarg–Serrin. For this example, we took
inspiration from [42,44].
6.1. Elliptic operators with coefficients in BUC and VMO
Assume that the operator A has the form (2) with aα ∈ BUC(Rn , C) for |α| = m and that the
following ellipticity condition holds:
aα (x)ξ α ∈ Σω ,
x, ξ ∈ Rn ,
|α|=m
α
aα (x)ξ η|ξ |m ,
x, ξ ∈ Rn ,
(30)
|α|=m
for some η > 0 and some ω ∈ (0, π/2). By, e.g., [2], those operators generate analytic semigroups
in all spaces Lq (Rn ), 1 < q < ∞. For such operators, maximal regularity of type Lp –Lq follows
via the Dore–Venni approach from the main result in [25]. However, the proof in [25] is quite
involved, making use of techniques related to the T1-theorem. Simpler proofs have been given
(cf. [37], [40], [41], and—with infinite-dimensional state space—[23]) that rely on L. Weis’
characterization (cf. [50]) mentioned in the introduction, which allowed to refine the perturbation
and localization technique that is usually used to show generation of an analytic semigroup (cf.,
e.g., [2]). Another way, which also uses L. Weis’ characterization, is via generation results in
Lp -spaces with Muckenhoupt weights [28]. Concerning our method we first remark that, as
mentioned above, the assumption (Hq ) holds for all q ∈ (1, ∞). Via Theorem 3.1 and Remark 3.7
we obtain that the estimate (3) holds for q1 = ∞ and every q0 > 1. Already this result seems to be
new. Recall that kernel estimates of Gaussian type, which would correspond to (q0 , q1 ) = (1, ∞)
in (3), do not hold in general for operators of this kind (cf. [10]). Applying Theorem 2.5 we thus
have proved
Theorem 6.1. Under the assumptions above, we have for all p, q, s ∈ (1, ∞) that the semigroup
(e−tAq ) satisfies (3) for (q0 , q1 ) = (s, ∞), that the semigroup is Rs -bounded in Lq (Rn ), and that
Aq has maximal Lp -regularity in Lq (Rn ).
Remark 6.2. The assertions of Theorem 6.1 also hold if the top order coefficients are in VMO.
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2753
For elliptic operators with top order coefficients in VMO, generation of an analytic semigroup
has been shown in [3], [26, Lemma 3.1], [29]. For the case m = 2 it has been shown in [26]
that suitable translates ν + Aq even have a bounded H ∞ -calculus, from which maximal Lp regularity for Aq follows. In [29] maximal Lp -regularity for Aq of arbitrary order m 2 has
been proved via generation results in weighted Lq -spaces with Muckenhoupt weights. In [35],
maximal regularity has been shown for non-autonomous situations. Again, our method allows
for a new proof which only uses the generation result from [3] or [26, Lemma 3.1] without
using the deeper results on boundedness of the H ∞ -calculus or weighted Lq -spaces. Moreover,
the assertions on off-diagonal bounds (3) and on Rs -boundedness are new and of independent
interest.
6.2. Elliptic operators with measurable coefficients
Recently, the VMO-regularity assumption for the top order coefficients in case m = 2 has been
further relaxed [33,34], and we cite the following (recall Definition 5.1 for the notation IA ).
Theorem 6.3. (See [33,34].) If A has the form (2) where the elliptic second order coefficients
aj k are measurable and depend only on one coordinate, then we have [2, ∞) ⊆ JA and Aq has
maximal Lp -regularity in Lq for q ∈ [2, ∞).
In fact, the results in [34] are more general, and maximal Lp -regularity has been proved even
in non-autonomous situations. Here, our results yield a new proof for maximal regularity, but
recalling [38, Theorem 8] we can even give the following extension of this result.
Theorem 6.4. Under the assumptions of Theorem 6.3 the set JA ⊇ [2, ∞) is an open subset
of (1, ∞) and Aq has maximal Lp -regularity in Lq for all q ∈ JA . Moreover, the semigroup is
Rs -bounded in Lq for q, s ∈ IA and satisfies (3) for q1 = ∞ and all q0 ∈ JA .
Here, the assertions for q 2 and the assertions on Rs -boundedness and on (3) are new.
Proof. By [38, Theorem 8], the set JA is open. Combining Theorem 6.3 with our main results,
the claims follow. 2
6.3. A well-known example
xj xk
) in Ω = Rn , the whole space, or Ω =
|x|2
1 (B), respectively.
n
2,q
n
B = B(0, 1) ⊂ R , the unit ball, with domain W (R ) and Wq2 (B) ∩ Wq,0
We let e(x) := (xj |x|−1 )j ∈ Rn . For the coefficient matrix a γ (x) of A(γ ) we then have a γ (x) =
I + γ e(x)e(x)t . This leads for ξ ∈ Rn to the estimate
For n 2 and γ ∈ R we let A(γ ) :=
j k (δj k
+γ
2
2
γ 0,
|ξ | ,
ξ t a γ (x)ξ = |ξ |2 + γ e(x)t ξ (1 + γ )|ξ |2 , γ < 0
and this is optimal. In particular, we conclude that A(γ ) is uniformly elliptic if and only if
γ > −1. It may be interesting that, for |γ | = 1, the inverse of the matrix a γ (x) = I + γ e(x)e(x)t
is given by (1 − γ 2 )−1 (I − γ e(x)e(x)t ) (cf. [42]).
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
Case Ω = B
B. Now we take Ω = B and let u(x) := |x|μ − 1 where μ ∈ R \ {0} is specified later.
Then u = 0 on the boundary. We calculate
∂j u = μxj |x|
μ−2
μ−2
xj xk
δj k + (μ − 2) 2 ,
|x|
δj k + (γ + μ − 2)δj k
xj2 xk2
xj xk
+
γ
(μ
−
2)
|x|2
|x|4
and ∂k ∂j u = μ|x|
which leads to
A(γ )u = μ|x|μ−2
jk
= μ|x|
μ−2
n + (γ + μ − 2) + γ (μ − 2) .
−n
Choosing μ = 2+γ
1+γ we see that u is in the kernel of A(γ ). Observe that μ = 0 corresponds
to u = 0, hence we have an eigenfunction of A(γ ) only if 2 + γ = n. Next we observe that
|x|μ ∈ Wq2 (B) if and only if |x|μ−2 ∈ Lq (B) which in turn is equivalent to μ − 2 > −n/q. For
μ=
2+γ −n
1+γ
we thus have
|x|μ ∈ Wq2 (B)
⇐⇒
−
n+γ
n
>−
1+γ
q
⇐⇒
q<
n(1 + γ )
=: fn (γ ).
n+γ
Discussion. For any n 2 the function fn is strictly increasing on (−1, ∞) and tends to n as
γ → ∞. Moreover fn (0) = 1. This means: whenever γ > 0 and 2 + γ = n then
1
(B) → Lq (B),
Aq (γ ) : Wq2 (B) ∩ Wq,0
n(1 + γ )
q ∈ 1,
,
n+γ
has non-trivial kernel. On the other hand, we find for fixed q ∈ (1, ∞) a small γq > 0 such
that, for each γ ∈ (0, γq ), the operator Aq (γ ) is a small perturbation of the Dirichlet Laplacian
1 (B), in the sense that (A (γ ) − )−1 q with domain D(q ) = Wq2 (B) ∩ Wq,0
q
q
q→q < 1.
q
By the usual perturbation theorem Aq (γ ) generates a bounded analytic semigroup (Tt (γ )) in
Lq (Ω) for γ ∈ (0, γq ). Resolvents of Aq (γ ) are compact in Lq (Ω). For any γ ∈ (0, γq ) we
denote by qmin (γ ) the infimum of all r ∈ [1, q] such that (Tt (γ )) acts as a C0 -semigroup in
Lr (Ω). Since the semigroup satisfies generalized Gaussian bounds, we conclude that Ar (γ )
generates an analytic semigroup in each space Lr (Ω), r ∈ (qmin (γ ), q), and resolvents of Ar (γ )
are compact. But then the spectrum of Ar (γ ) does not depend on r ∈ (qmin (γ ), q). Hence we
)
conclude that qmin (γ ) n(1+γ
n+γ > 1, which means that none of the semigroups (Tt (γ )) acts in
all Lr -spaces, r ∈ (1, ∞). It is remarkable that this holds also for n = 2.
Case Ω = Rn . We turn to the case of Ω = Rn . Here we take inspiration from [42].
−n
Again we set μ = 2+γ
1+γ with γ > −1 and let
u(t, x) := t −α |x|μ e−c
|x|2
t
,
t > 0, x ∈ Rn ,
(31)
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
where α and c > 0 are specified later. Observe u(t, ·)q = t
is our candidate for a non-trivial solution of the equation
∂t u − A(γ )u = 0,
n
−α+ μ2 + 2q
2755
u(1, ·)q . The function u
t > 0, x ∈ Rn .
We calculate partial derivatives:
|x|2
|x|2
∂j ∂k u = t −α ∂j ∂k |x|μ e−c t + t −α ∂j |x|μ ∂k e−c t
|x|2
|x|2
+ t −α ∂k |x|μ ∂j e−c t + t −α |x|μ ∂j ∂k e−c t
(1)
(2)
(3)
(4)
= Sj k + Sj k + Sj k + Sj k ,
where
∂k e−c
∂j (|x|μ ) = μxj |x|μ−2 ,
∂j ∂k e−c
|x|2
t
|x|2
t
=−
2cxk −c |x|2
e t ,
t
2cδj k 4c2 xj xk −c |x|2
e t .
+
= −
t
t2
For ∂t u we obtain
|x|2
∂t u = ct −α−2 |x|μ+2 − αt −α−1 |x|μ e−c t .
(32)
Applying A(γ ) to u we obtain
A(γ )u = S (1) + S (2) + S (3) + S (4)
γ (ν)
where S (ν) = j k aj k Sj k . By choice of μ we have S (1) = 0 (cf. above). By symmetry we have
S (2) = S (3) , and we calculate
S (2) = −
xj xk
2cμ −α μ−2 −c |x|2 δj k + γ
x
t |x|
e t
x
j k
t
|x|2
jk
|x|2
= −2cμ(1 + γ )t −α−1 |x|μ e−c t ,
|x|2
xj xk
S (4) = −2ct −α−1 |x|μ e−c t
δj k + γ δj k 2
|x|
jk
2 −α−2
+ 4c t
μ −c |x|t
|x| e
2
xj2 xk2
δj k x j x k + γ
|x|2
jk
= −2c(n + γ )t −α−1 |x|μ e−c
|x|2
t
+ 4c2 (1 + γ )t −α−2 |x|μ+2 e−c
|x|2
t
.
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P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
Sorting terms according to powers of t and |x| and comparing with (32) we see that u is a solution
to (31) if and only if
−4cμ(1 + γ ) − 2c(n + γ ) = −α
and 4c2 (1 + γ ) = c.
By c > 0 the only solution is
c=
1
4(1 + γ )
and α = 4cμ(1 + γ ) + 2c(n + γ ) = μ +
n+γ
4 + 3γ − n
=
.
2(1 + γ )
2(1 + γ )
)
The condition for u(t, ·) ∈ Wq2 (Rn ) is the same as for |x|μ ∈ Wq2 (B(0, 1)), i.e. q < n(1+γ
n+γ . For
those q, Aq (γ ) cannot be the generator of a semigroup. Again, for any given q there exists a
small γq such that, for γ ∈ (0, γq ), Aq (γ ) generates an analytic semigroup. But none of these
semigroups acts in the whole scale Lr , 1 < r < ∞.
7. Miscellaneous remarks
Remark 7.1. Our methods do not only apply to scalar equations but also to elliptic systems, i.e.
to operators of the form (2) where the functions aα take values in the complex matrices CN ×N
with N > 1. The usual ellipticity assumption in this case is the one from [2], i.e.
σ
aα (x)ξ
α
⊆ Σω \ {0},
|α|=m
−1 α
M|ξ |−m ,
a
(x)ξ
α
(33)
|α|=m
for all x ∈ Rn , ξ ∈ Rn \ {0}, where M > 0 and ω ∈ (0, π/2). With top order coefficients in BUC
or VMO, condition (Hq ) holds for q ∈ (1, ∞). We refer to Remark 7.2 for the case ω ∈ (0, π).
At the end we comment on the modifications to be made for the case ω ∈ (0, π) in (30) or (33).
Remark 7.2. So far, we have concentrated on operators A of the form (2) satisfying (Hq ), i.e.
generators of analytic semigroups, as this is the right class of operators for Brézis’ question. If
Aq satisfies (Hq ) and the semigroup generated by −Aq is bounded analytic in Lq then Aq is
sectorial of some angle ω ∈ (0, π/2), i.e. it has spectrum σ (Aq ) ⊆ Σω and, for any θ ∈ (ω, π),
the set of operators
(34)
λ(λ + A)−1 : λ ∈ Σθ
is bounded in norm · q→q . In some applications, operators A arise that are sectorial of type ω
for ω ∈ (0, π) without being generators of bounded analytic semigroups. Such an operator A is
called Rs -sectorial of type ω in Lq , where s ∈ [1, ∞], if for any θ ∈ (ω, π) the sets in (34) are Rs bounded in Lq in the sense of (1). For an operator Aq of the form (2) that is sectorial in Lq (Rn ),
an analog of Theorem 3.1 in Section 3 can be proved where the operators (1 + zA)−1 take the
role of the semigroup operators e−zA . In this case, one can argue in the proof directly with (16)
and (19) without passing via (17) and (18). Again, the bounds corresponding to (12) and (14) can
P.C. Kunstmann / Journal of Functional Analysis 255 (2008) 2732–2759
2757
be composed and interpolated in the sense of Remark 3.7, but by lack of the semigroup property,
this leads in general to estimates (3) where T (t) has to be replaced by operators (1 + tA)−l
for a suitable l ∈ N. The analog of Theorem 2.5 would state that the resolvent operators extend
consistently on the scale Lr , r ∈ [q0 , q1 ], and that they are resolvents of Rs -sectorial operators Br
for (r, s) ∈ (q0 , q1 )×[q0 , q1 ]∪{(q0 , q0 ), (q1 , q1 )}. One can also obtain an analog of Theorem 5.2.
In the proof, when applying Theorem 5.4, the semigroup operators e−tkA in (28) have to be
replaced by (1 + ktA)−l for a suitable l ∈ N.
Those modified statements then apply to, e.g., elliptic operators with BUC- or VMOcoefficients satisfying (30) only for some ω ∈ (0, π). For elliptic systems (cf. Remark 7.1) one
has to require (33) for some ω ∈ (0, π).
Acknowledgment
The author is indebted to the unknown referee for a number of very useful remarks that helped
a lot to improve the paper.
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