A MAXIMAL THEOREM FOR HOLOMORPHIC SEMIGROUPS
ON VECTOR-VALUED SPACES
GORDON BLOWER, IAN DOUST, AND ROBERT J. TAGGART
Abstract. Suppose that 1 < p ≤ ∞, (Ω, µ) is a σ-finite measure space and
E is a closed subspace of a Lebesgue–Bochner space Lp (Ω; X), consisting of
functions on Ω that take their values in some complex Banach space X. Suppose also that −A is injective and generates a bounded holomorphic semigroup
{Tz } on E. If 0 < α < 1 and f belongs to the domain of Aα then the maximal
function supz kTz f kX , where the supremum is taken over any given sector
contained in the sector of holomorphy, belongs to Lp . A similar result holds
for generators that are not injective. This extends earlier work of Blower and
Doust [BD].
1. Introduction
Suppose that {Tt }t≥0 is a C0 -semigroup of bounded linear operators on a Banach
space E. In the case that E is a space of functions f from a measurable set Ω to
a normed space X, an important tool in the analysis of such a semigroup is the
maximal function M f where
M f (ω) = ess-sup kTt f (ω)kX .
t≥0
The classical theorems of Stein [St] and Cowling [Co] apply to symmetric diffusion
semigroups on E, where E = Lp (Ω) and 1 < p < ∞, and show that in this case
kM f kp ≤ c kf kp for all f in Lp (Ω).
Taggart [Ta] extended this to the vector-valued context where E = Lp (Ω; X)
and X satisfies a geometric condition weak enough to include, for example, many
of the classical reflexive function spaces.
Under much weaker hypotheses, Blower and Doust [BD] showed that in the
scalar-valued case, if the semigroup {Tt }t>0 can be extended to a bounded holomorphic semigroup on a sector of the complex plane, then M f lies in Lp (Ω), at
least for f in a large submanifold of Lp (Ω).
In this paper we show that the result of [BD] may be extended to the vectorvalued case where E is a subspace of Lp (Ω; X) and X is any complex Banach
space. Moreover, the result also holds when the supremum used to define the
maximal function is taken over sectors contained in the sector of holomorphy of
the semigroup (c.f., the results of [Co] and [Ta]). We will deduce both of these
extensions by modifying the original argument of [BD].
The paper is organised as follows. In Section 2 we introduce some notation and
state the main theorem of the paper. As with the result of [BD], the theorem is
proved by representing the semigroup in terms of fractional powers of its generator
Date: Received 15 October 2009 / Accepted 7 December 2009.
2000 Mathematics Subject Classification. 47D06 (47A60, 34G10, 46E40).
Key words and phrases. Holomorphic semigroup, maximal theorem, sectorial operator, vectorvalued Lebesgue–Bochner spaces.
The third author was supported by the Australian Research Council and the Centre for Mathematics and its Applications, Canberra.
105
106
G. BLOWER, I. DOUST, AND R. TAGGART
and obtaining good Lp bounds for parts of this representation. Some of the arguments of [BD] which made use of Yosida approximants to the semigroup’s generator
have been replaced by direct appeals to the functional calculus for sectorial operators. Salient facts about the functional calculus are presented in Section 3, while
the representation of the semigroup and corresponding bounds are established in
Section 4. In Section 5 we complete the proof of the theorem.
As an application, we show how the main theorem can be used to deduce almost
everywhere pointwise convergence for the semigroup on a large submanifold of Lp .
The precise details are stated in Section 2 and proved at the end of the paper. For
examples of semigroups to which the theorem applies, see [BD, Sections 1 and 4].
2. Notation and theorem
We begin by introducing some notation. Given θ in [0, π), let Sθ0 and Sθ denote
the open and closed sectors of C given by
Sθ0 = ζ ∈ C \ {0} : | arg ζ| < θ
and
Sθ = ζ ∈ C \ {0} : | arg ζ| ≤ θ ∪ {0},
where arg ζ denotes the principal argument of a nonzero complex number ζ. Note
that S0 = [0, ∞).
Throughout, suppose that X is a complex Banach space and that (Ω, µ) is a positive σ-finite measure space. When 1 ≤ q ≤ ∞, let Lq (Ω; X) denote the Lebesgue–
Bochner space of strongly measurable functions f : Ω → X whose norm is given
by
Z
1/q
q
kf kLq (Ω;X) =
kf (ω)kX dµ(ω)
Ω
if q < ∞ and kf kL∞ (Ω;X) = ess-supω∈Ω kf (ω)kX . We write Lq (Ω) for Lq (Ω; C).
Definition 2.1. Suppose that E is a Banach space and 0 < θ < π/2. A family
{Tz : z ∈ Sθ0 } of bounded linear operators acting on E is said to be a bounded
holomorphic semigroup of angle θ on E if
i) Tz Tw = Tz+w whenever z, w ∈ Sθ0 ,
ii) for each θ0 in [0, θ) there exists a positive constant Kθ0 such that kTz k ≤ Kθ0
whenever z ∈ Sθ0 ,
iii) the mapping z 7→ Tz is a holomorphic function from Sθ0 into the space of
bounded linear operators on E, and
iv) for each f in E, Tz f → f in E as z → 0 with z in Sθ0 .
The setting for our main result is as follows. Suppose that E is a closed subspace
of Lp (Ω; X), where 1 < p ≤ ∞, and suppose that {Tz : z ∈ Sθ0 } is a bounded
holomorphic semigroup of angle θ on E, for some θ in (0, π/2). Let −A denote
the infinitesimal generator of this semigroup. (See [RS2, Section X.8] or [Da] for
definitions of these terms.) When 0 < α < 1 we can define the fractional power
Aα of A and, if A is injective, the power A−α (see Remark 3.3). Given f in
D(Aα ) ∩ D(A−α ), define kf kp,α by
kf kp,α = kAα f kLp (Ω;X) + A−α f Lp (Ω;X) .
Whenever 0 ≤ θ0 < θ and f ∈ E, define the maximal function Mθ0 f by
(Mθ0 f )(ω) = sup {k(Tz f )(ω)kX : z ∈ Sθ0 }
Sθ0 }
∀ω ∈ Ω.
Since {Tz : z ∈
is a holomorphic semigroup, the maximal function Mθ0 f is
well-defined and measurable (see [St, pp. 72–73] for further details).
HOLOMORPHIC SEMIGROUPS ON VECTOR-VALUED SPACES
107
The following is the main result of the paper.
Theorem 2.2. Suppose E, A and {Tz : z ∈ Sθ0 } are as above, that A is injective
and that 0 < α < 1. If f ∈ D(Aα ) ∩ D(A−α ) and 0 ≤ θ0 < θ then Mθ0 f ∈ Lp (Ω)
and there is a constant C(A, α, θ0 ) such that
kMθ0 f kLp (Ω) ≤ C(A, α, θ0 ) kf kp,α .
The proof of the theorem will be deferred to Section 5. We conclude the present
section with a remark and corollary.
Remark 2.3. The constant C(A, α, θ0 ) is bounded by
Cη sec(θ0 + η) sec(απ/2)
πα
for any η such that π/2 − θ < η < π/2 − θ0 , where Cη is the constant appearing in
the resolvent bound (3.1) for A on Lp (Ω; X). Note that if the semigroup acts on a
range of Lp spaces then these quantities may vary with p.
The following corollary shows that the injectivity hypothesis of Theorem 2.2
may be discarded provided that one modifies the maximal function appropriately.
Moreover, this modified maximal function can be used to deduce almost everywhere
pointwise convergence for the semigroup. To be precise, assume the setting introduced prior to the statement of Theorem 2.2. For each positive number s and each
function f in E, define the maximal function Mθs0 f by
∀ω ∈ Ω.
(Mθs0 f )(ω) = sup{e−zs (Tz f )(ω)X : z ∈ Sθ0 }
Corollary 2.4. Suppose E, A and {Tz : z ∈ Sθ0 } are as in the setting of Theorem
2.2 but that A is not injective. If 0 < α < 1, s > 0, f ∈ D(Aα ) and 0 ≤ θ0 < θ
then Mθs0 f ∈ Lp (Ω) and there is a constant C(A, α, θ0 , s) such that
kMθs0 f kLp (Ω)
≤ C(A, α, θ0 , s) k(sI + A)α f kLp (Ω;X) + (sI + A)−α f Lp (Ω;X) .
Moreover, for µ-almost all ω in Ω, Tz f (ω) → f (ω) as z → 0 in Sθ0 .
The proof of the corollary is given at the end of Section 5.
3. Functional calculus calculus for sectorial operators
In this section we summarise for use in Section 4 a few pertinent facts about the
holomorphic functional calculus for sectorial operators.
Definition 3.1. Suppose that 0 ≤ ϑ < π and that E is any Banach space. We say
that an operator A in E is sectorial of type ϑ if A is closed, σ(A) ⊆ Sϑ and for each
η in (ϑ, π) there exists a constant Cη such that
(ζI − A)−1 ≤ Cη |ζ|−1
∀ζ ∈ C \ Sη .
(3.1)
We recall the following important characterisation of generators of bounded holomorphic semigroups. Details may be found in [Da] or [RS2].
Theorem 3.2. Suppose that E is a Banach space and 0 < θ < π/2. If {Tz : z ∈ Sθ0 }
is a bounded holomorphic semigroup on E with generator −A, then A is a densely
defined sectorial operator in E of type π/2 − θ. Conversely, if A is a sectorial
operator in E of type π/2 − θ then −A is the generator of a bounded holomorphic
semigroup {Tz : z ∈ Sθ0 } on E.
108
G. BLOWER, I. DOUST, AND R. TAGGART
We now describe a holomorphic functional calculus for sectorial operators. Suppose that 0 < ϑ < ν < π. Let ψ denote the complex-valued function defined on C
by
ψ(ζ) = ζ/(1 + ζ)2
∀ζ ∈ C.
0
Denote by H(Sν ) the space of all holomorphic functions on Sν0 . Following the
notation of [CDMY], we define the following subspaces of H(Sν0 ):
H ∞ (Sν0 ) = f ∈ H(Sν0 ) : sup |f (ζ)| < ∞ ,
ζ∈Sν0
Ψ(Sν0 )
= f∈
H(Sν0 )
: f ψ −s ∈ H ∞ (Sν0 ) for some s > 0
and
F (Sν0 ) = f ∈ H(Sν0 ) : f ψ s ∈ H ∞ (Sν0 ) for some s > 0 .
Note that
Ψ(Sν0 ) ⊂ H ∞ (Sν0 ) ⊂ F (Sν0 ) ⊂ H(Sν0 ).
It is well known (see [CDMY, Section 2] and [Ha, Chapter 2]) that if A is an
injective sectorial operator of type ϑ on a Banach space E, then A has an F (Sν0 )
functional calculus. Moreover, if f ∈ Ψ(Sν0 ) then f (A), defined by the contour
integral
Z
1
f (A) =
(ζI − A)−1 f (ζ) dζ,
(3.2)
2πi γ
is a bounded operator on E. Here the integral converges absolutely in the uniform
topology and the contour γ is given by
(
−teiη if − ∞ < t ≤ 0
γ(t) =
te−iη
if 0 < t < ∞,
where η is any angle strictly between ϑ and ν. It can be shown that the definition
of f (A) is independent of the choice of angle η in this range.
Remark 3.3. The functional calculus defined above allows one to define fractional
powers for injective sectorial operators, and in particular for injective generators of
holomorphic semigroups. If 0 < |α| < 1 and ζ ∈ Sν , then we define the fractional
power ζ α by
ζ α = exp(α ln |ζ| + iα arg ζ).
Note that the function ζ 7→ ζ α belongs to F (Sν0 ). Thus if A has an F (Sν0 )
functional calculus then the operator Aα may be defined by Aα = g(A), where
g(ζ) = ζ α . The fractional powers of A (and of the sectorial operator sI + A for
positive s) have the following properties:
(1) D(A) ⊂ D(Aβ ) ⊂ D(Aα ) whenever 0 < α < β < 1,
(2) D(Aα ) = D((sI + A)α ) whenever 0 < α < 1 and s > 0, and
(3) if A is invertible and 0 < α < 1 then A−α is bounded.
See [Ha, Chapter 3] for further details.
4. A representation for the semigroup
Suppose that t ∈ R, 0 < α < 1, ϕ ∈ R, ζ ∈ C and | arg(eiϕ ζ)| < π/2. By Fourier
inversion,
Z
iϕ
1
eiϕ ζ
e−|t|e ζ =
eitu du
iϕ
π R (e ζ)2 + u2
!
Z
Z
eiϕ
itu
−α
itu
α
=
e Fu,ϕ (ζ) ζ du +
e Gu,ϕ (ζ) ζ du
π
|u|<1
|u|>1
HOLOMORPHIC SEMIGROUPS ON VECTOR-VALUED SPACES
109
where 0 < α < 1,
Fu,ϕ (ζ) =
ζ 1+α
(eiϕ ζ)2 + u2
and
Gu,ϕ (ζ) =
ζ 1−α
.
(eiϕ ζ)2 + u2
This observation and the F (Sν0 ) functional calculus leads to the following lemma.
Lemma 4.1. Suppose that A and θ are as in the hypothesis of Theorem 2.2. If
0 < α < 1, f ∈ D(Aα ) and z ∈ Sθ then
Z
Z
eiϕ
eiϕ
Tz f =
eitu Fu,ϕ (A)A−α f du +
eitu Gu,ϕ (A)Aα f du,
π |u|<1
π |u|>1
where t = |z| and ϕ = arg(z).
The proofs of both lemmata in this section make frequent use of the following
fact. If |φ| < π/2 then
t2 + u2
:
t
>
0,
u
>
0
= sec φ.
sup
|(teiφ )2 + u2 |
This may be deduced using planar trigonometry. We turn now to the proof of
Lemma 4.1.
Proof. Suppose that z = teiϕ ∈ Sθ0 and choose ν such that π/2 − θ < ν < π/2 − |ϕ|.
By the hypotheses on A and Theorem 3.2, A has an F (Sν0 ) functional calculus. If
ζ ∈ Sν0 then
|ζ|1+α
|Fu,ϕ (ζ)| ≤ sec(ϕ + ν) 2
|ζ| + u2
0
and hence Fu,ϕ ∈ Ψ(Sν ) for all nonzero u in R. In fact, if
Z
Z
itu
F̃z,ε (ζ) =
e Fu,ϕ (ζ) du and F̃z (ζ) =
eitu Fu,ϕ (ζ) du
|u|<1
ε<|u|<1
whenever 0 < ε < 1 then
Z
|F̃z,ε (ζ)| ≤ 2 sec(ϕ + ν)
1
|ζ|1+α
du
|ζ|2 + u2
ε
= 2 sec(ϕ + ν) |ζ|α arctan(1/|ζ|) − arctan(ε/|ζ|) .
Hence F̃z,ε ∈ Ψ(Sν0 ) and F̃z,ε converges to F̃z uniformly on compact subsets of Sν
as ε → 0+ . This convergence implies that F̃z is holomorphic in Sν and, combining
this with the bounds on F̃z, , one concludes that F̃z ∈ Ψ(Sν0 ).
One can similarly show that Gu,ϕ and G̃z , where
Z
G̃z (ζ) =
eitu Gu,ϕ (ζ) du,
|u|>1
Ψ(Sν0 ).
both belong to
Finally, since Tz f = e−zA f for all f in E and
eiϕ e−zζ =
F̃z (ζ) ζ −α + G̃z (ζ) ζ α
∀ζ ∈ Sν
π
by Fourier inversion, the lemma follows from the F (Sν0 ) functional calculus for A.
For the proof of Theorem 2.2 we require good bounds for Fu,ϕ (A) and Gu,ϕ (A),
which will be expressed in terms of two operators Γu,η and ∆u,η defined below.
Suppose that u is a nonzero real number, π/2 − θ < η < π and f ∈ Lp (Ω; X).
Recall that the map ζ 7→ (ζI − A)−1 is holomorphic from ρ(A)
of
into the space
bounded linear operators on E. Hence the map t 7→ Fu,0 (t) (teiη I − A)−1 f X is
110
G. BLOWER, I. DOUST, AND R. TAGGART
continuous from (0, ∞) to Lp (Ω). Therefore, for each finite interval [a, b] in (0, ∞),
the integral
Z b
Fu,0 (t) (teiη I − A)−1 f X dt
(4.1)
a
p
can be defined as an L (Ω) limit of Riemann sums. Moreover, the triangle inequality
and resolvent bounds (3.1) for A show that
Z b
Z b
tα
Fu,0 (t) (teiη I − A)−1 f X dt p
≤ Cη kf kLp (Ω;X)
dt
2
2
L (Ω)
a
a t +u
≤ Cη Ku,α kf kLp (Ω;X)
for some finite constant Ku,α independent of the interval [a, b] in (0, ∞). Hence the
integral
Z ∞
Fu,0 (t) (teiη I − A)−1 f X dt
0
p
can be defined as an L (Ω) limit of integrals of the form (4.1) and is itself in Lp (Ω).
In a similar manner, we define the scalar-valued functions Γu,η f and ∆u,η f by
the contour integrals
Z ∞
1
Fu,0 (t) (teiη I − A)−1 f X + (te−iη I − A)−1 f X dt
Γu,η f =
2π 0
and
1
∆u,η f =
2π
Z
0
∞
Gu,0 (t) (teiη I − A)−1 f X + (te−iη I − A)−1 f X dt.
By definition, both functions belong to Lp (Ω).
Lemma 4.2. Suppose that 0 < θ0 < π/2 − η < θ, f ∈ Lp (Ω; X) and u is a nonzero
real number. Then
k(Fu,ϕ (A)f )(ω)kX ≤ sec(θ0 + η)(Γu,η f )(ω)
and
k(Gu,ϕ (A)f )(ω)kX ≤ sec(θ0 + η)(∆u,η f )(ω)
for µ-almost every ω in Ω and for all ϕ in [−θ0 , θ0 ]. Moreover,
kΓu,η f kLp (Ω) ≤
Cη
sec(απ/2)|u|−1+α
2
k∆u,η f kLp (Ω) ≤
Cη
sec(απ/2)|u|−1−α ,
2
and
where Cη is the constant appearing in the resolvent bound (3.1) for A.
Proof. If ν is chosen such that π/2 − θ < η < ν < π/2 − θ0 then Fu,ϕ ∈ Ψ(Sν0 ).
Hence the function Fu,ϕ (A)f can be represented as a contour integral of the form
(3.2). Now if {gε } is a convergent net in Lp (Ω; X) then
lim gε = lim kgε kX ,
ε
X
ε
HOLOMORPHIC SEMIGROUPS ON VECTOR-VALUED SPACES
111
where convergence on the left is in Lp (Ω; X) while convergence on the right in is
Lp (Ω). So we may move the X-norm through the limit of Riemann sums representing Fu,ϕ (A)f to obtain
Z ∞
1
|Fu,ϕ (teiη )| (teiη I − A)−1 f X dt
kFu,ϕ (A)f kX ≤
2π 0
Z ∞
1
|Fu,ϕ (te−iη )| (te−iη I − A)−1 f X dt
+
2π 0
≤ sec(ϕ + η)Γu,η f
≤ sec(θ0 + η)Γu,η f.
By resolvent bounds for A, we also have
kΓu,η f kLp (Ω)
Z ∞ 1−α
Cη
dt
t
≤
kf kLp (Ω)
2 + u2 t
π
t
0
Cη
≤
sec(απ/2)|u|1−α kf kLp (Ω) .
2
The bounds for kGu,ϕ (A)f kX and k∆u,η f kLp (Ω) are verified in a similar fashion.
5. Proof of the maximal theorem and its corollary
In this final section we present a proof of Theorem 2.2 and Corollary 2.4.
Proof of Theorem 2.2. Assume the setting and hypotheses of Theorem 2.2. Suppose that f ∈ D(Aα ) and define v : Ω × Sθ0 → X by
v(ω, z) = Tz f (ω)
∀ (ω, z) ∈ Ω × Sθ0 .
Note that Mθ0 f ∈ Lp (Ω) if and only if v ∈ Lp (Ω; L∞ (Sθ0 ; X)), where
!1/p
Z
kvkLp (L∞ ) =
p
ess-sup kv(ω, z)kX dµ(ω)
Ω z∈Sθ0
and where we have written Lp (L∞ ) for Lp (Ω; L∞ (Sθ0 ; X)).
Our aim now is to embed Lp (L∞ ) inside the dual of a suitable Banach space Z
and to then show that
kvkLp (L∞ ) = sup |hg, vi| : kgkZ ≤ 1
(5.1)
is finite.
Each operator Tz of the semigroup acts on the closed subspace E of Lp (Ω; X).
Thus in particular v(ω, z) can be considered as an element of X ∗∗ for each ω and z.
Writing Y for X ∗ , we note that the standard duality theory for Lebesgue-Bochner
spaces (see [DU, Chapter IV]) says that L∞ (Sθ0 ; Y ∗ ) ⊆ L1 (Sθ0 ; Y )∗ isometrically,
and so
Lp (Ω; L∞ (Sθ0 ; Y ∗ )) ⊆ Lp (Ω; L1 (Sθ0 ; Y )∗ ).
But if
1
p
+
1
q
= 1, then
Lp (Ω; L1 (Sθ0 ; Y )∗ ) ⊆ Lq (Ω; L1 (Sθ0 ; Y ))∗
isometrically, and hence
Lp (Ω; L∞ (Sθ0 ; X)) ⊆ Lp (Ω; L∞ (Sθ0 ; X ∗∗ )) ⊆ Lq (Ω; L1 (Sθ0 ; Y ))∗ .
(5.2)
112
G. BLOWER, I. DOUST, AND R. TAGGART
As above we shall write Lq (L1 ) for Lq (Ω; L1 (Sθ0 ; Y )). From (5.1) and (5.2) it
follows that
kvkLp (L∞ )
Z
= sup hg(ω, ·), v(ω, ·)i dµ(ω) : kgkLq (L1 ) ≤ 1 .
)
( ZΩ Z
hg(ω, z), v(ω, z)ihY,Xi dz dµ(ω) : kgkLq (L1 ) ≤ 1 .
= sup Ω
Sθ 0
Suppose then that g ∈ Lq (Ω; L1 (Sθ0 ; Y )) and kgkLq (L1 ) ≤ 1. Writing z as teiϕ
and using Lemma 4.1, we find that
Z Z
hg, vi =
hg(ω, z), Tz f (ω)ihY,Xi dz dµ(ω)
Ω Sθ 0
Z
Z Z
1
=
hg(ω, z), eiϕ eitu Fu,ϕ (A)A−α f (ω)ihY,Xi dz dµ(ω) du
π |u|<1 Ω Sθ0
Z
Z Z
1
hg(ω, z), eiϕ eitu Gu,ϕ (A)Aα f (ω)ihY,Xi dz dµ(ω) du,
+
π |u|>1 Ω Sθ0
where the use of Fubini’s theorem is justified by estimates in Section 4. The modulus
of the first of these terms may be estimated using Hölder’s inequality and Lemma 4.2
as follows:
Z Z
Z
hg(ω, z), eiϕ eitu Fu,ϕ (A)A−α f (ω)ihY,Xi dz dµ(ω) du
|u|<1
Z
Ω Sθ0
1 Z Z
≤
−1
Ω
Sθ0
kg(ω, z)kY Fu,ϕ (A)A−α f (ω)X dz dµ(ω) du
Z
0
1
Z Z
≤ sec(θ + η)
≤ sec(θ0 + η)
≤ sec(θ0 + η)
−1
Z 1
−1
Z 1
−1
Ω
Z
Ω
Sθ0
kg(ω, z)kY (Γu,η A−α f )(ω) dz dµ(ω) du
kg(ω, · )kL1 (Sθ0 ;Y ) (Γu,η A−α f )(ω) dµ(ω) du
kgkLq (L1 ) Γu,η A−α f Lp (Ω;X) du
Cη
≤
sec(θ0 + η) sec(απ/2) A−α f Lp (Ω;X)
2
≤ Cη sec(θ0 + η) sec(απ/2) α−1 A−α f p
Z
1
|u|−1+α du
−1
L (Ω;X)
,
where θ0 < π/2 − η < θ. A similar calculation shows that
Z Z
Z
hg(ω, z), eiϕ eitu Gu,ϕ (A)Aα f (ω)ihY,Xi dz dµ(ω) du
|u|>1
Ω
Sθ0
0
≤ Cη sec(θ + η) sec(απ/2) α−1 kAα f kLp (Ω;X) .
It follows therefore that
|hg, vi| ≤
Cη sec(θ0 + η) sec(απ/2)
kf kp,α
πα
and hence
Cη sec(θ0 + η) sec(απ/2)
kf kp,α .
πα
This completes the proof of Theorem 2.2.
kMθ0 f kLp (Ω) ≤
HOLOMORPHIC SEMIGROUPS ON VECTOR-VALUED SPACES
113
Proof of Corollary 2.4. Adopt now the setting and hypotheses of Corollary 2.4.
Since −A is the generator of the holomorphic semigroup {Tz : z ∈ Sθ0 }, it is
sectorial of type π/2 − θ. Hence the operator sI + A is has a bounded inverse on E
and is also sectorial of type π/2 − θ. By Theorem 3.2, −(sI + A) is the generator
of a holomorphic semigroup, which we denote by {e−z(sI+A) : z ∈ Sθ0 }. Now apply
Theorem 2.2 to this semigroup, noting that e−z(sI+A) = e−zs Tz for each z in Sθ0
and that, by Remark 3.3, it is sufficient that f ∈ D(Aα ). This proves that the
maximal function Mθs0 f is in Lp (Ω) with the stated bounds.
We now prove almost everywhere pointwise convergence. For convenience, write
z → 0+ as short hand for z → 0 with z in Sθ0 . Fix positive s. Since the semigroup
is holomorphic, Tz+r f converges pointwise almost everywhere to Tr f as z → 0+
whenever r > 0 (see [St, p. 72]). Hence, for almost every ω in Ω,
lim sup k(Tz f )(ω) − f (ω)kX
z→0+
≤ lim sup |esz | (e−sz Tz (f − Tr f )(ω)X + k(Tr f )(ω) − f (ω)kX
z→0+
+ lim sup k(Tz+r f )(ω) − Tr f (ω)kX
z→0+
≤ es Mθs0 (f − Tr f )(ω) + kTr f (ω) − f (ω)kX .
Now, given positive , choose positive r such that kf − Tr f kLp (Ω;X) < and
k(sI + A)α f − Tr (sI + A)α f kLp (Ω;X) < . By the boundedness of the maximal
function and the operator (sI + A)−α , there exist constants C and c, depending
only on A, α, θ0 and s, such that
lim sup kTz f − f k X
z→0+
Lp (Ω)
α
≤ C k(sI + A) (f − Tr f )k p
+ (sI + A)−α (f − Tr f ) p
L (Ω;X)
L (Ω;X)
+ kTr f − f kLp (Ω;X)
< C k(sI + A)α f − Tr (sI + A)α f kLp (Ω;X) + c kf − Tr f kLp (Ω;X) + < (C + cC + 1).
Here we have used the fact that Tr and (sI + A)α commute on D(Aα ) by the
functional calculus (see [Ha, Theorem 1.3.2]). It follows that
lim sup k(Tz f )(ω) − f (ω)kX = 0
z→0+
for almost all ω and hence that Tz f converges pointwise almost everywhere to f as
z → 0+ . This completes the proof of Corollary 2.4.
References
[BD]
G. Blower and I. Doust, A maximal theorem for holomorphic semigroups, Q. J. Math.,
56 (2005), 21-30 .
[Co]
M.G. Cowling, Harmonic analysis on semigroups, Ann. Math., 117 (1983), 267-283.
[CDMY] M.G. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with a
bounded H ∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.
[Da]
E. B. Davies, One-parameter semigroups, Academic Press, London, 1980.
[DU]
J. Diestel and J.J. Uhl, Vector measures, Mathematical Surveys No. 15, American
Mathematical Society, Providence, 1977.
[Ha]
M. Haase, The functional calculus for sectorial operators, Operator Theory Advances
and Applications No. 169, Birkhäuser Verlag, Basel, 2006.
[RS2]
M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis
and self-adjointness, Academic Press, London, 1975.
114
[St]
[Ta]
G. BLOWER, I. DOUST, AND R. TAGGART
E.M. Stein, Topics in harmonic analysis related to the Littlewood–Paley theory, Princeton University Press, Princeton, N.J., 1970.
R.J. Taggart, Pointwise convergence for semigroups in vector-valued Lp spaces, Math.
Z., 261 (2009), 933–949.
Gordon Blower, Department of Mathematics and Statistics, Lancaster University,
Lancaster LA1 4YF, United Kingdom.
E-mail address: [email protected]
Ian Doust, School of Mathematics and Statistics, UNSW Sydney, NSW, 2052, Australia.
E-mail address: [email protected]
Robert Taggart, Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia
E-mail address: [email protected]