Criteria for R-boundedness of operator families^{∗}

MARIA GIRARDI^{†}

Department of Mathematics, University of South Carolina Columbia, SC 29208, U.S.A. e-mail: girardi@math.sc.edu

LUTZ WEIS^{‡}

Mathematisches Institut I, Universität Karlsruhe Englerstraße 2, 76128 Karlsruhe, Germany e-mail: Lutz.Weis@math.uni-karlsruhe.de

Abstract: The main results state that large classes of convolution and Fourier multiplier operators are R-bounded. R-boundedness is of importance in connection with maximal regularity and functional calculus. Also shown is that smooth operator-valued functions have a R-bounded range, where the degree of smoothness depends on the geometry of the Banach space.

1 INTRODUCTION

Recently the notion of R-boundedness has played an important role in the functional analytic approach to partial differential equations. It is shown in [36] (see also [2, 3, 7, 8, 14, 15, 23, 26, 35]) that, for a sectorial operator A, R-boundedness of the set {λ(λ − A)^{−1} : λ ∈ ℂ_{+}}, i. e. R-sectoriality, characterizes maximal regularity. Rboundedness is also a very useful notion in connection with the H^{∞}-calculus (see [23, 24]). Furthermore it was shown in [36] (see also [8, 19, 21, 33]) that R-boundedness provides a proper setting for boundedness theorems for operator-valued Fourier multipliers. Hence, workable criteria for R-boundedness are needed. It is already known that sectorial operators with Gaussian estimates ( [14, 35]), as well as large classes of partial differential operators ( [7, 15, 25, 26]), are R-sectorial.

∗ To appear: Recent Contributions to Evolution Equations, Lecture Notes in Math., Marcel Dekker.

† Girardi is supported in part by the Alexander von Humboldt Foundation.

‡ Weis is supported in part by Landesforschungsschwerpunkt Evolutionsgleichungen des Landes Baden-Württenberg.

This leads to the natural question of whether each sectorial operator from a Banach space into itself is R-sectorial. Kalton and Lancien [22] answered this question in the negative; however, their counterexample is rather abstract and no reasonable sectorial differential operator which is not R-sectorial is known. This note gives general criteria for R-boundedness, which may explain why the common operators of analysis are R-bounded and why it is so difficult to find counterexamples. This note’s R-boundedness criteria are based on the observation that the very boundedness theorems for Fourier multipliers, which have R-boundedness in their assumption, improve themselves by a trick to give R-boundedness of classes of operators that satisfy these assumptions in a uniform way. This is possible provided X has the UMD-property and Pisier’s property (α). As a rule of thumb, the reflexive Banach spaces commonly appearing in analysis (such as subspaces of Sobolev spaces or Hardy spaces) have these properties. This idea is related to the fact (see [23]) that for a sectorial operator A (on such spaces X) with an H^{∞}- calculus, the set {f (A) : ∥f∥_{H^{∞}} ≤ 1} is R-bounded. We think that our approach gives an unified way to prove R-boundedness for many families of operators relevant in applications and also can be applied to situations not considered here.

The main result of Section 3, Theorem 3.2, gives that the set of all Fourier multiplier operators that satisfy the Mihlin conditions in an uniform way form an R-bounded set. This is of particular interest for scalar-valued multiplier functions. Furthermore, Theorem 3.2 leads to (see Section 4) boundedness criteria for sets of convolution operators, which include the Gauß and Poisson semigroups. The results in Section 5 shows that the range of a sufficiently smooth function from ℝ^{N} into B (X) is R-bounded, where the Fourier type of X is taken into account in order to reduce the required degree of smoothness. Then further applications to semigroups and resolvents are given. Section 2 collects needed definitions and facts.

Theorem 3.2 was presented by the authors at the TULKA seminar in February 2001. Independently and with different methods, A. Venni [34] has shown a special case of Theorem 3.2.

2 DEFINITIONS AND NOTATION

Throughout this paper X, Y, and Z are complex Banach spaces. B (X, Y) is the space of bounded linear operators from X into Y; often B (X, X) is denoted by just B (X). The triple (Ω, Σ, μ) is a σ-finite complete measure space; corresponding to it is the usual Bochner-Lebesgue space L_{p} (Ω; X) of measurable functions from Ω into X with finite L_{p} (Ω; X)-norm where 1 ≤ p ≤ ∞; often L_{p} (Ω; X) is denoted by just L_{p} (X) or L_{p} if confusion seems unlikely. If Ω = [0, 1] or Ω = ℝ^{N} then (Ω, Σ, μ) is understood to be the usual Lebesgue measure space. The Schwartz class S (ℝ^{N}; X) is the space of rapidly decreasing smooth function from ℝ^{N} to X. The sequence {r_{j}}_{j=1}^{∞} denotes a sequence of independent, symmetric, {1, −1}– valued random variables on [0, 1], e.g. the Rademacher functions. ℕ is the set of natural numbers while ℕ_{0} = ℕ ∪ {0}.

R-boundedness is the central notion of this paper.

DEFINITION 2.1. Let τ be a subset of B (X, Y) and p ∈ [1, ∞). The number R_{p}(τ) is the smallest of the constants R ∈ [0, ∞] with the property that for each n ∈ ℕ and subset {T_{j}}_{j=1}^{n} of τ and subset {x_{j}}_{j=1}^{n} of X,

∥∑_{j=1}^{n} r_{j}(·)T_{j}(x_{j})∥_{L_{p}([0,1];Y)} ≤ R ∥∑_{j=1}^{n} r_{j}(·)x_{j}∥_{L_{p}([0,1];X)} .

The set τ is R-bounded provided R_{p}(τ) is finite for some (and thus then, by Kahane’s inequality, for each) p ∈ [1, ∞).

Thus a set τ is R-bounded provided Kahane’s contraction principle holds for operator coefficients from τ. Note that if X and Y are q-concave Banach lattices for some finite q (e.g. X = Y = L_{q} (Ω; ℂ) where 1 ≤ q < ∞) then R-boundedness is equivalent to the square function estimate

∥(∑_{j=1}^{m} |T_{j}x_{j}|^{2})^{1/2}∥_{Y} ≤ R ∥(∑_{j=1}^{n} |x_{j}|^{2})^{1/2}∥_{X}  (2.1)

known from harmonic analysis (cf. [28, Thm. 1.d.6]). For basic properties of Rbounded sets and further references, see [13, 36].

The space Rad(X) provides a convenient way to view R-boundedness.

DEFINITION 2.2. For a Banach space X, define

Rad(X) := {x̃ := {x_{j}}_{j∈ℕ} ∈ X^{ℕ} : ∑_{j=1}^{n} r_{j}(·)x_{j} converges in L_{2} ([0, 1] ; X)} .

Often Rad(X) is denoted by just X̃.

When equipped with one of the following equivalent norms, where 1 ≤ p < ∞:

∥{x_{j}}_{j∈ℕ}∥_{Rad_{p}(X)} := ∥∑_{j∈ℕ} r_{j} (·) x_{j}∥_{L_{p}([0,1];X)} ,

Rad_{p} (X) is a Banach space. When confusion seems unlikely, Rad_{p} (X) is denoted by just Rad(X). Much can be found about Rad(X) in the literature (see, e.g. [16]).

FACT 2.3. A sequence {T_{j}}_{j∈ℕ} from B (X, Y) is R-bounded if and only if the corresponding map

X̃ ∋ \{x_{j}\}_{j∈ℕ} \underrightarrow{T̃} \{T_{j}x_{j}\}_{j∈ℕ} ∈ Ỹ

defines an element in B (X̃, Ỹ); in which case, R_{p} ({T_{j}}_{j∈ℕ})=∥T̃∥_{B(Rad_{p}(X),Rad_{p}(Y))} for each p ∈ [1, ∞).

Fubini’s theorem yields the following useful fact.

FACT 2.4. Let p ∈ [1, ∞). The mapping

I_{X}: L_{p} (Ω; Rad_{p} (X)) → Rad_{p} (L_{p} (Ω; X))

given by

I_{X}f := {f_{j}}_{j∈ℕ} for f(·) = {f_{j} (·)}_{j∈ℕ} ∈ L_{p} (Ω; Rad_{p} (X))

is an isometry. Furthermore

f̂ = {f̂_{j}}_{j∈ℕ}  and f̌ = {f̌_{j}}_{j∈ℕ}  (2.2)

for each f(·) = {f_{j}(·)}_{j∈ℕ} ∈ L_{1} (ℝ^{N}; X̃).

Considering R-boundedness in B (X̃, Ỹ) leads to estimating double random series; thus, an additional property of the Banach spaces X and Y is often needed.

DEFINITION 2.5. ( [32, Def. 2.1]) Let {ε_{k}}_{k=1}^{∞} (resp. {ε′_{k}}_{k=1}^{∞}) be a sequence of independent, symmetric, {1, −1}-valued random variables on some probability space (Ω, Σ, μ) (resp. (Ω′, Σ′, μ′)) with {ε_{k}}_{k=1}^{∞} and {ε′_{k}}_{k=1}^{∞} independent. For a Banach space X and 1 ≤ p < ∞, the number α_{p}(X) is the smallest of the constants α ∈ [0, ∞] with the property that

∥ ∑_{j,k=1}^{n} α_{jk}ε_{j}ε′_{k}x_{jk}∥_{L_{p}(Ω×Ω′;X)} ≤ α ∥ ∑_{j,k=1}^{n} ε_{j}ε′_{k}x_{jk}∥_{L_{p}(Ω×Ω′;X)}  (2.3)

for each n ∈ ℕ and subset {x_{jk}}_{j,k=1}^{n} of X and choices {α_{jk}}_{j,k=1}^{n} of signs {1, −1}. The space X has property (α) provided α_{p}(X) is finite for some (and thus then, by Kahane’s inequality, for each) p ∈ [1, ∞).

Each subspace [32] of a Banach function space of finite cotype (e.g. a subspace of L_{p} (Ω; ℂ) with 1 ≤ p < ∞) has property (α). If X has property (α), then so does L_{q} (Ω; X) for 1 ≤ q < ∞. The estimate in (2.3) extends to R-bounded sets.

FACT 2.6. (cf. [13, Lemma 2.3.3]) Let X and Y be Banach spaces enjoying property (α) and τ be an R-bounded subset of B (X, Y). Then for p ∈ [1, ∞)

∥ ∑_{j,k=1}^{n} ε_{j}ε′_{k}T_{jk}x_{jk}∥_{L_{p}(Ω×Ω′;Y)} ≤ α_{p}(X) α_{p}(Y) R_{p}(τ) ∥ ∑_{j,k=1}^{n} ε_{j}ε′_{k}x_{jk}∥_{L_{p}(Ω×Ω′;X)}

for each n ∈ ℕ and subset {T_{jk}}_{j,k=1}^{n} of τ and subset {x_{jk}}_{j,k=1}^{n} of X. The UMD property is often needed for vector-valued multiplier theorems.

DEFINITION 2.7. A Banach space X is a UMD space provided the Hilbert transform

Hf(t) = P V − ∫ \frac{f(s)}{t − s} ds for f ∈ S(X)

extends to a bounded linear operator on L_{p} (ℝ; X) for some (and thus then for each) p ∈ (1, ∞).

Thus X is a UMD space if and only if m: ℝ \ {0} → B (X) given by

m(t) = sign(t) I_{X}

is a Fourier multiplier on L_{p} (ℝ; X) for some (and thus then for each) p ∈ (1, ∞). Closed subspaces of, the dual of, and quotients spaces of a UMD space are UMD spaces. Each Hilbert space is a UMD space. If X is a UMD space and q ∈ (1, ∞), then L_{q} (Ω; X) is also a UMD space.

For f ∈ L_{1} (ℝ^{N}; X) we denote by f̂ or Ff the Fourier transform of f, i.e.,

f̂(t) ≡ F(f)(t) := ∫_{ℝ^{N}} e^{−it·s}f(s) ds .

In the vector-valued setting the Hausdorff-Young inequality does not hold in general. Thus one considers the following class of Banach spaces.

DEFINITION 2.8. ( [30]) A Banach space X has Fourier type p, where p ∈ [1, 2], if the Fourier transform F defines a bounded linear operator from L_{p} (ℝ^{N}; X) to L_{p′} (ℝ^{N}; X) for some (and thus then for each) N ∈ ℕ. The Fourier type constant F_{p,N}(X) of X is then the norm of F ∈ B (L_{p} (ℝ^{N}; X) , L_{p′} (ℝ^{N}; X)).

The simple estimate ∥Ff(t)∥_{X} ≤ ∥f∥_{L_{1}(X)} shows that each Banach space X has Fourier type 1 with F_{1,N}(X) = 1. The notion becomes more restrictive as p increases to 2. A Banach space has Fourier type 2 if and only if X is isomorphic to a Hilbert space [27]. A space L_{q} (Ω; ℝ) has Fourier type p = min(q, q′) [30]. If X have Fourier type p ∈ [1, 2] and p ≤ q ≤ p′, then F_{p,N} (X) = F_{p,N} (X^{∗}) = F_{p,N} (L_{q} (ℝ^{N}; X)) for each N ∈ ℕ (cf. [18]). Each closed subspace (by definition) and quotient space (by duality) of a Banach space X has the same Fourier type as X.

REMARK 2.9. Recall the following relationships between property (α), UMD spaces, Fourier type, and Rad(X).

a) Property (α) and the UMD property are independent. An L_{1} (Ω; ℂ) space has property (α). However, infinite dimensional L_{1} (Ω; ℂ) spaces (as well as C(K)- spaces) are not UMD spaces. The Schatten classes S_{p} have UMD for p ∈ (1, ∞). However, an infinite dimensional S_{p} does not have property (α) when 1 ≤ p ≤ ∞ and p ≠ 2.

b) A UMD space has some non-trivial Fourier type p > 1. Indeed, a UMD space has a uniformly convex renorming [10]. A space with a uniformly convex renorming is reflexive and B-convex. A B-convex Banach space has some non-trivial Fourier type p > 1 [9, 12].

c) If X has property (α) (resp. the UMD property, Fourier type p), then so does Rad(X). This follows from viewing Rad(X) as a subspace of L_{2} ([0, 1]; X).

3 FOURIER MULTIPLIER OPERATORS

Let q ∈ (1, ∞). A bounded measurable function M: ℝ^{N} \ {0} → B (X, Y) is a Fourier multiplier on L_{q} (ℝ^{N}; X) provided the corresponding Fourier multiplier operator T_{M}, defined by

T_{M}f := [M (·) f̂(·)]^{∨} for f ∈ S (ℝ^{N}; X) ,

extends to a bounded operator T_{M} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )).

FACT 3.1. Let τ be a R-bounded subset of B (X, Y) and q ∈ (1, ∞). It is known that M is a Fourier multiplier on L_{q} (ℝ^{N}; X) if X and Y are UMD spaces with Fourier type p ∈ [1, 2] and M belongs to one of the following symbol classes:

\begin{matrix}M^{l} (τ) := \\ \{M: ℝ^{N} \ \{0\} → B (X, Y) : |t|^{|α|} M^{(α)} (t) ∈ τ for t ≠ 0 ,α ∈ ℕ_{0}^{N} , |α| ≤ l\} \end{matrix}

for some l ∈ ℕ where l > N/p (see [19, 21, 33, 36]);

\begin{matrix}H^{N} (τ) := \\ \{M: ℝ^{N} \ \{0\} → B (X, Y) : t^{α}M^{(α)} (t) ∈ τ for t ≠ 0 ,α ∈ℕ_{0}^{N} ,α ≤ (1, 1, …, 1)\} \end{matrix}

provided X and Y have property (α) (see [33]);

\begin{matrix}N^{r} (τ) := \\ \{M: ℝ \ \{0\}→B (X, Y) : M (t) ∈τ , |t|^{r} \frac{M (t + s) − M (t)}{|s|^{r}} ∈ τ for s, t ≠ 0\} \end{matrix}

for some r ∈ (1/p, 1) (see [19]). If X = Y = ℂ (i.e., scalar-valued multiplier functions) and τ is the unit ball of ℂ, then the above classes are denoted by just M^{l}, H^{N}, and N^{r}.

Fact 2.6 leads to our main criteria for R-boundedness of sets of Fourier multiplier operators.

THEOREM 3.2. Let X and Y be UMD Banach spaces having property (α) and Fourier type p. Let l = [N/p] + 1 and r ∈ (1/p, 1). Let τ be a R-bounded subset of B (X, Y). Then the sets of Fourier multiplier operators

M := {T_{M} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) : M ∈ M^{l}(τ)}
H := {T_{M} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) : M ∈ H^{N}(τ)}
N := {T_{M} ∈ B (L_{q} (ℝ^{1}; X) , L_{q} (ℝ^{1}; Y )) : M ∈ N^{r}(τ)}

are R-bounded for each q ∈ (1, ∞).

PROOF. Fix q ∈ (1, ∞) To simplify notation, denote L_{q} (ℝ^{N}; Z) by just L_{q}(Z) and Rad_{q} (Z) by just Z̃ for the assorted Banach spaces Z appearing.

Let {M_{j}}_{j=1}^{n} be a subset of M^{l}(τ) (resp. H^{N}(τ), N^{r}(τ)) and set M_{j} := 0 for j > n. Define

T̃ ∈ B ({ \atop L_{q} (X)}, { \atop L_{q} (Y)})

by

T̃ g̃ = \{T_{M_{j}}g_{j}\}_{j=1}^{∞} for g̃ = \{g_{j}\}_{j=1}^{∞} ∈ { \atop L_{q} (X)} .

Note that ∥T̃∥ = R_{q} ({T_{M_{j}}}_{j=1}^{n}) < ∞. By Fact 2.3, it suffices to show that there exists a constant C_{∗} that depends only on R_{q}(τ), X, Y, q, p, and N (but not on the particular chosen {M_{j}}_{j=1}^{n}) such that

∥T̃∥_{B({ \atop L_{q}(X)},{ \atop L_{q}(Y)})} ≤ C_{∗} .  (3.1)

For then R_{q} (M) (resp. R_{q} (H), R_{q} (N)) is bounded above by C_{∗}.

Towards this, define

M̃: ℝ^{N} \ {0} → B (X̃, Ỹ)

by

[M̃(·)] ({x_{j}}_{j∈ℕ}) := {M_{j} (·) x_{j}}_{j∈ℕ}  for  {x_{j}}_{j∈ℕ} ∈ X̃ .

Note that ∥M̃(t) ∥ = R_{q} ({M_{j} (t)}_{j=1}^{n}) ≤ R_{q} (τ) for each t ∈ ℝ^{N} \ {0}. To

show (3.1), it suffices to show that M̃ is a Fourier multiplier on L_{q}(X̃) with

∥TM̃∥_{B(L_{q} \binom{}{X},L_{q} \binom{}{Y})} ≤ C_{∗} .

Indeed, for then the diagram

Rad_{q} (L_{q} (X)) \underrightarrow{T̃} Rad_{q} (L_{q} (Y))
I_{X} ↑  ↑^{I_{Y}}
L_{q} (Rad_{q} (X)) ⟶^{TM̃} L_{q} (Rad_{q} (Y))

commutes since if f (·) = {f_{j} (·)}_{j∈ℕ} ∈ S (ℝ^{N}; X̃) then TM̃f = [M̃(·) f̂(·)]^{∨} and so, with the help of (2.2),

I_{Y} TM̃f = {T_{M_{j}}f_{j}}_{j∈ℕ} = T̃I_{X}f ; hence, ∥TM̃∥ = ∥T̃∥ .

First consider the set M. Note that the set

{|t|^{|α|} (D^{α}M̃) (t) ∈ B (X̃, Ỹ) : t ∈ ℝ^{N} \ {0} ,α ∈ ℕ_{0}^{N} , |α| ≤ l}

is R-bounded; indeed, for any fixed subsets

{t_{j}}_{j=1}^{m} ⊂ ℝ^{N} \ {0} and {x̃_{j} := {x_{k}^{j}}_{k∈ℕ}}_{j=1}^{m} ⊂ X̃

and α ∈ ℕ_{0} with |α| ≤ l, note that D^{α}M̃ = {D^{α}M_{k}}_{k∈ℕ} and so for

N_{k} (t) := |t|^{|α|} D^{α}M_{k} (t) and Ñ (t) := {N_{k} (t)}_{k∈ℕ} ,

by Fact 2.6,

\begin{matrix}∥∑_{j=1}^{m} r_{j} (·) Ñ (t_{j}) x̃_{j}∥ & {q \atop L_{q}([0,1];Ỹ)} \\ = ∫_{[0,1]} ∫_{[0,1]′} ∥∑_{k=1}^{n} ∑_{j=1}^{m} r′_{k} (u′) r_{j} (u) N_{k} (t_{j}) x_{k}^{j}∥_{Y}^{q} du′du \\ ≤ C_{0}^{q} ∫_{[0,1]} ∫_{[0,1]′} ∥∑_{k=1}^{n} ∑_{j=1}^{m} r′_{k} (u′) r_{j} (u) x_{k}^{j}∥_{X}^{q} du′du \\ ≤ C_{0}^{q} ∥∑_{j=1}^{m} r_{j} (·) x̃_{j}∥_{L_{q}([0,1];X̃)}^{q} , \end{matrix} (3.2)

where C_{0} = α_{q}(X) α_{q}(Y) R_{q} (τ). Thus by Remark 2.9c) and Fact 3.1, M̃ is a Fourier multiplier on L_{q} \binom{}{X} with TM̃ bounded by some constant C_{∗} that depends only on the items claimed.

The proofs for H and N are similar. The only difference is that for the estimate in (3.2) one considers α ∈ ℕ_{0} with α ≤ (1, 1, …, 1) and uses N_{k} (t) := t^{α} (D^{α}M_{k}) (t) in the case of H while one considers {s_{j}}_{j=1}^{m} ⊂ ℝ^{N} \ {0} also and uses N_{k} (s, t) := |t/s|^{r} [M_{k} (t + s) − M_{k} (t)] or N_{k} (t) = M_{k} (t) in the case of N.

Since X = Y = ℂ have UMD, property (α), and Fourier type 2, an interesting R-boundedness result follows for scalar-valued Fourier multipliers on L_{q} (ℝ^{N}; ℂ).

COROLLARY 3.3. Let l = [N/2] + 1 and choose r ∈ (1/2, 1). Then the sets of Fourier multiplier operators {T_{M}: M ∈ M^{l}}, {T_{M}: M ∈ H^{N}}, and {T_{M}: M ∈ N^{r}} are R-bounded on L_{q} (ℝ^{N}; ℂ) for each 1 < q < ∞ (of course, N = 1 in the latter case).

4 CONVOLUTION OPERATORS

The theorem in the previous section gives that certain families of Fourier multiplier operators are R-bounded. The results of this section give that certain families of convolution operators are R-bounded. Recall that for a kernel function k ∈ L_{1} (ℝ^{N}; B (X, Y)) and q ∈ [1, ∞], one can define the corresponding convolution operator K_{k} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) by

K_{k}f(t) := ∫_{ℝ^{N}} k(t − s)f(s) ds  for a.e. t ∈ ℝ^{N} .  (4.1)

Note that it follows from Lemma 4.1 that {k̂(t) : t ∈ ℝ^{N}} is R-bounded in B (X, Y).

LEMMA 4.1. Let X and Y be arbitrary Banach spaces.

a) Let N : Ω → B (X, Y) be strongly integrable on the measure space (Ω, Σ, μ).

Thus there is a constant A so that

∫_{Ω} ∥N (ω) x∥_{Y} dμ (ω) ≤ A ∥x∥_{X} for each x ∈ X . (4.2)

Then

τ := {X ∋ x → ∫_{Ω} h (ω) N (ω) x dμ (ω) ∈ Y : h ∈ L_{∞} (Ω; ℂ) , ∥h∥_{∞} ≤ 1}

is an R-bounded subset of B (X, Y) with R_{1}(τ) ≤ 2A.

b) If N ∈ L_{1} (Ω; B (X, Y)), then

R_{1} ({∫_{Ω} h (ω) N (ω) dμ (ω) ∈ B (X, Y) : h ∈ L_{∞} (Ω; ℂ) , ∥h∥_{∞} ≤ 1}) ≤

2 ∥N∥_{L_{1}(Ω;B(X,Y))} .

PROOF. Clearly it suffices to show just part a). For h ∈ L_{∞} (Ω; ℂ), let

T_{h}x = ∫_{Ω} h (ω) N (ω) xdμ (ω)

so that T_{h} ∈ B (X, Y) with ∥T_{h}∥_{B(X,Y)} ≤ A ∥h∥_{∞}. For h_{1}, …, h_{n} ∈ L_{∞} (Ω; ℂ) with ∥h_{i}∥_{∞} ≤ 1 and x_{1}, …, x_{n} ∈ X

∫_{[0,1]} ∥ ∑_{j=1}^{n} r_{j}(t)T_{h_{j}}(x_{j})∥_{Y} dt

= ∫_{[0,1]} ∥∫_{Ω} (∑_{j=1}^{n} r_{j} (t) h_{j} (ω) N (ω) x_{j}) dμ (ω)∥_{Y} dt

≤ ∫_{Ω} ∫_{[0,1]} ∥∑_{j=1}^{n} r_{j} (t) h_{j} (ω) N (ω) x_{j}∥_{Y} dtdμ (ω)

≤ 2 ∫_{Ω} ∫_{[0,1]} ∥∑_{j=1}^{n} r_{j} (t) N (ω) x_{j}∥_{Y} dt dμ (ω)

= 2∫_{[0,1]}(∫_{Ω} ∥N (ω) [∑_{j=1}^{n} r_{j} (t) x_{j}]∥_{Y} dμ (ω)) dt

≤ 2 A ∫_{[0,1]} ∥∑_{j=1}^{n} r_{j} (t) x_{j}∥_{X} dt

by Kahane’s contraction principle and (4.2). Lemma 4.1 leads to an R-boundedness criterion for certain families of kernels.

PROPOSITION 4.2. Let X and Y be UMD spaces with property (α). Let the kernel function k: ℝ^{N} → B (X, Y) satisfy, for each α ∈ ℕ_{0}^{ℕ} with α ≤ (1, 1, …, 1),

∫_{ℝ^{N}} |t^{α}| ∥D^{α}k (t)∥_{B(X,Y)} dt < ∞ .  (4.3)

Then the family {K_{γ} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) : γ ∈ ℝ}, defined by

(K_{γ}f) (t) := ∫ γ^{N}k (γ (t − s)) f (s) ds for f ∈ S (ℝ^{N}; X) ,

is R-bounded for each q ∈ (1, ∞).

REMARK 4.3. For X = Y = ℂ, Proposition 4.2 reduces to: if k ∈ ℝ^{N} → ℂ satisfies

∫_{ℝ^{N}} |t^{α}D^{α}k (t)| dt < ∞ for each α ∈ ℕ_{0}^{ℕ} with α ≤ (1, 1, …, 1)

then the family {K_{γ} ∈ B (L_{q} (ℝ^{N}; ℂ)) : γ ∈ ℝ} is R-bounded for each q ∈ (1, ∞).

REMARK 4.4. The following variant of Proposition 4.2 is also true. First replace (4.3) by

∫_{ℝ^{N}} |t^{α}| ∥D^{α}k (t) x∥_{Y} dt ≤ A ∥x∥_{X} for each x ∈ X .  (4.3′)

Note that then k is strongly integrable and so the function M: ℝ^{N} → B (X, Y), given by

M (t) x := [k (·) x] (t) for t ∈ ℝ^{N} and x ∈ X ,  (4.4)

is well-defined. Assume further that the distributional derivatives D^{α}M are represented by functions for each α ∈ ℕ_{0}^{ℕ} with α ≤ (1, 1, …, 1). Then the conclusion of Proposition 4.2 holds.

Proof of Proposition 4.2 and Remark 4.4. Let X and Y be UMD spaces with property (α). Let k satisfy either (4.3) or (4.3′) and M be as in (4.4). Define k_{x}, M_{x}: ℝ^{N} → Y by

k_{x} (·) := k (·) x M_{x} (·) := M (·) x = k̂_{x} (·)

for each x ∈ X.

Fix α ∈ ℕ_{0}^{ℕ} with α ≤ (1, 1, …, 1). Then for each x ∈ X

D^{α} (s^{α} k_{x} (s)) = ∑_{γ≤α} \binom{α}{γ} D^{α−γ} (s^{α}) D^{γ} (k_{x} (s)) = ∑_{γ≤α} C_{α,γ} s^{γ} D^{γ} (k_{x} (s)) (4.5)

and

(−1)^{|α|} [D^{α} (s^{α}k_{x} (s))] (t) = (−i)^{|α|} t^{α} [s^{α}k_{x} (s)] (t) = t^{α}D^{α}M_{x} (t) . (4.6)

Thus if k satisfies (4.3), then D^{α} (s^{α}k (s)) ∈ L_{1} (B (X, Y)) and so the distributional derivative D^{α}M is represented as a function. Thus it suffices to show just Remark 4.4; so, assume the setting of Remark 4.4.

By (4.6) and (4.5)

t^{α}D^{α}M_{x} (t) = (−1)^{|α|} ∑_{γ≤α} C_{α,γ} ∫_{ℝ^{N}} h_{t} (s) [s^{γ}D^{γ}k_{x} (s)] ds

where h_{t} (s) := e^{−it·s}. Thus by (4.3′) and Lemma 4.1, the set

τ_{α} := {t^{α}D^{α}M (t) : t ∈ ℝ^{N}}

is R-bounded.

Fix γ ∈ ℝ^{N} \ {0}. Then M_{γ}: ℝ^{N} → B (X, Y), defined below, satisfies

M_{γ} (t) (x) := [γ^{N}k_{x} (γ·)] (t) = (sign γ)^{N} k̂_{x} (γ^{−1}t) = (sign γ)^{N} M (γ^{−1}t) x

and so

t^{α}D^{α}M_{γ} (t) = (sign γ)^{N} (γ^{−1}t)^{α} D^{α}M (γ^{−1}t) ∈ ±τ_{α}

for each t ∈ ℝ^{N}. Hence M_{γ} ∈ H^{N}(∪_{α≤(1,...,1)}±τ_{α}) and so M_{γ} is a Fourier multiplier on L_{q} (ℝ^{N}; X) by Fact 3.1. Furthermore, Theorem 3.2 gives that {K_{γ}: γ ∈ ℝ} is R-bounded.

R-boundedness of the Gauß and Poisson semigroups follows from Proposition 4.2.

EXAMPLE 4.5. Keeping with Proposition 4.2’s notation, fix γ > 0. If

k(s) := (4π)^{−1/2} exp (\frac{−|s|^{2}}{4} ) ,

then

(K_{γ^{−1/2}}f) (t) = (4πγ)^{−N/2} ∫_{ℝ^{N}} exp (\frac{−|s|^{2}}{4γ} ) f(t − s)ds = (e^{γΔ}f) (t) .

If k(s) := C_{N} (1 + |s|^{2})^{−(N+1)/2} where C_{N} := Γ (\frac{N+1}{2}) π^{−(N+1)/2} then

(K_{γ^{−1}}f) (t) = C_{N} ∫_{ℝ^{N}} \frac{γ}{(γ^{2} + |s|^{2})^{(N+1)/2}} f (t − s) ds = (e^{−γ\sqrt{−Δ}}f) (t) .

Since both functions k satisfy (4.3), Remark 4.3 gives that the Gauß semigroup \{e^{γΔ} : γ > 0\} and the Poisson semigroup \{e^{−γ\sqrt{−Δ}} : γ > 0\} are R-bounded in B (L_{q} (ℝ^{N}; ℂ)) for each q ∈ (1, ∞).

A more general criterion for R-boundedness of a family of convolution operators is now formulated. For this, let τ be a R-bounded subset of B (X, Y) and τ_{0} be the closure (in the strong operator topology) of the complex absolute convex hull of τ. Note that τ_{0} is also R-bounded (indeed, R_{p}(τ_{0}) ≤ 2R_{p}(τ)). Let ∥·∥_{τ} : B (X, Y) → [0, ∞] be the Minkowski functional of τ_{0}, i.e.,

∥S∥_{τ} := inf{λ ≥ 0: S ∈ λτ_{0}} .

Define

K^{N}(τ):={k: ℝ^{N} →B (X, Y) :∫_{ℝ^{N}}∥ t^{α}D^{α}k(t) ∥_{τ} dt ≤ 1 for each α ≤ (1, …, 1)} .

Note that if k ∈ K^{n} (τ), then k satisfies (4.3); in particular, the convolution operator K_{k}, as defined in (4.1), is in B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) for each q ∈ (1, ∞).

THEOREM 4.6. Let X and Y be UMD spaces with property (α) and τ be an R-bounded subset of B (X, Y). Then the family of convolution operators

{K_{k} ∈ B (L_{q} (ℝ^{N}; X) , L_{q} (ℝ^{N}; Y )) : k ∈ K^{N}(τ)}

is R-bounded for each q ∈ (1, ∞).

REMARK 4.7. Let X = Y = ℂ (i.e. scalar-valued kernels) and τ be the unit ball of ℂ (so ∥·∥_{τ} = |·|). Then Theorem 4.6 implies that

{K_{k} ∈ B (L_{q} (ℝ^{N}; ℂ)) : ∫_{ℝ^{N}} | t^{α}D^{α}k(t) | dt ≤ 1 for each α ≤ (1, …, 1)}

is R-bounded for each q ∈ (1, ∞).

Proof of Theorem 4.6 Fix k ∈ K^{N}(τ). Then, keeping with the notation of Proposition 4.2,

t^{α}D^{α}M (t) = (−1)^{|α|} ∑_{γ≤α} C_{α,γ} ∫_{ℝ^{N}} e^{−it·s} [s^{γ}D^{γ}k (s)] ds .

Note that

S (t) := ∫_{ℝ^{N}} e^{−it·s} [s^{γ}D^{γ}k (s)] ds = ∫ e^{−it·s}h (s) k̃ (s) ds

where h (·) := ∥(·)^{γ} D^{γ}k (·)∥_{τ} is in the unit ball of L_{1} (ℝ^{N}; ℝ) and

k̃ (s) := [h (s)]^{−1} s^{γ}D^{γ}k (s) ∈ τ_{0} .

Therefore S (t) ∈ τ_{0}. So there is a constant C_{α} so that t^{α}D^{α}M (t) ∈ C_{α}τ_{0} for each t ∈ ℝ^{N} \ {0}. Now Theorem 3.2 gives the desired result.

5 SMOOTH OPERATOR-VALUED FUNCTIONS

The theorem in this section gives that the range of a sufficiently smooth function from ℝ^{N} into B (X, Y) is R-bounded. Smoothness is measured in terms of the modulus of smoothness s of a Besov space B_{q,r}^{s}. Also, the Fourier type of Y is taken into account in order to reduce the required degree of smoothness. The theorem’s corollaries measure smoothness in more classical ways (e.g. in terms of derivatives or a Lipschitz-like condition) rather than in terms of Besov spaces.

There are several equivalent well-known definitions of Besov spaces B_{q,r}^{s} (ℝ^{N}; Z); e.g. Besov spaces can be defined via a Paley-Littlewood decomposition (c.f., e.g., [18, Def. 2.4]), via real interpolation (c.f., e.g., [1]), or as in [31, Prop. 3.1].

THEOREM 5.1. Let X be an arbitrary Banach space and Y have Fourier type p ∈ [1, 2]. Let M ∈ B_{p,1}^{N/p} (ℝ^{N}; B (X, Y)) have norm A. Then the set

τ := {M (t) ∈ B (X, Y) : t ∈ ℝ^{N}}

is R-bounded and R_{1}(τ) ≤ CA for some constant C that depends only on F_{p,N} (Y).

PROOF. First assume in addition that M ∈ S (B (X, Y)). Fix x ∈ X. Note that { \atop [M (·) x]} (t) = M̂(t) x since M ∈ L_{1}. Thus by [18, Cor. 3.2], for each x ∈ X,

∫_{ℝ^{N}} ∥M̂(t) x∥_{Y} dt ≤ C_{1} ∥M (·) x∥_{B_{p,1}^{N/p}(ℝ^{N};Y)} ≤ C_{1} A ∥x∥_{X}

for some constant C_{1} depending only on F_{p,N} (Y). By the Fourier inversion formula

M (t) = (2π)^{−N} ∫_{ℝ^{N}} e^{it·s}M̂(s) ds for each t ∈ ℝ^{N} .

Applying Lemma 4.1 with h_{t} (s) = (2π)^{−N} e^{it·s} gives that

R_{1} ({M (t) : t ∈ ℝ^{N}}) ≤ 2C_{1}A .

This proves the claim if M ∈ S (B (X, Y)).

For an arbitrary function M ∈ B_{p,1}^{N/p} (ℝ^{N}; B (X, Y)), choose a sequence {M_{k}}_{k∈ℕ} from S (B (X, Y)) so that ∑_{k=1}^{n} M_{k} converges to M in B_{p,1}^{N/p} (ℝ^{N}; B (X, Y)) and

∑_{k∈ℕ} ∥M_{k}∥_{B_{p,1}^{N/p}} ≤ 2 ∥M∥_{B_{p,1}^{N/p}} .

Note that ∑_{k=1}^{n} M_{k} converges to M also in L_{∞} (ℝ^{N}; B (X, Y)) since the formal identity mapping from B_{p,1}^{N/p} (ℝ^{N}; Z) into L_{∞} (ℝ^{N}; Z) is continuous by the Sobolev embedding theorem. Thus

\begin{aligned}R_{1} (\{M (t) : t ∈ ℝ^{N}\}) & ≤ ∑_{k∈ℕ} R_{1} (\{M_{k} (t) : t ∈ ℝ^{N}\}) \\ & ≤ ∑_{k∈ℕ} 2 C_{1} ∥M_{k}∥_{B_{p,1}^{N/p}} ≤ 4 C_{1} ∥M∥_{B_{p,1}^{N/p}} \end{aligned}

by [36, Lemma 2.4] and the above argument.

REMARK 5.2. In Theorem 5.1, the assumption that M ∈ B_{p,1}^{N/p} (ℝ^{N}; B (X, Y)) with norm A can be replaced by a pointwise estimate; more precisely, it can be replaced by: M is strongly integrable and

∥M (·) x∥_{B_{p,1}^{N/p}(ℝ^{N};Y)} ≤ A ∥x∥_{X}  for each x ∈ X .  (5.1)

PROOF. For x ∈ X, define M_{x}: ℝ^{N} → Y by M_{x} (·) := M (·) x. Since M is strongly integrable, the function L: ℝ^{N} → B (X, Y), given by

[L (t)] (x) := (FM_{x}) (t) ,

is well-defined. By (5.1) and [18, Cor. 3.2], FM_{x} ∈ L_{1} (Y) with

∥M̂_{x}∥_{L_{1}(Y)} ≤ CA ∥x∥_{X}

for some constant C that depends only on F_{p,N} (Y). Thus by the Fourier inversion formula

M_{x} (t) = (2π)^{−N} ∫_{ℝ^{N}} e^{it·s}M̂_{x} (s) ds = ∫_{ℝ^{N}} h_{t} (s) [L (s)] (x) ds

where h_{t} (s) := (2π)^{−N} e^{it·s}. As in the proof of Theorem 5.1, Lemma 4.1 now gives the desired result.

The following corollary compares to [33, Thm. 4.1].

COROLLARY 5.3. Let X be an arbitrary Banach space and Y have Fourier type p ∈ [1, 2]. Let l = [N/p] + 1. Assume that t ∈ ℝ^{N} → M (t) ∈ B (X, Y) satisfies

(∫_{ℝ^{N}} ∥D^{α}M (t)∥_{B(X,Y)}^{p} dt)^{1/p} ≤ A

for each α ∈ ℕ_{0} with |α| ≤ l. Then {M (t) ∈ B (X, Y) : t ∈ ℝ^{N}} is R-bounded.

PROOF. Since N/p < l ∈ ℕ, the formal identity map from a Sobolev space W_{p}^{l} (ℝ^{N}; Z) into B_{p,1}^{N/p} (ℝ^{N}; Z) is continuous. Thus there are constants K_{i} so that

\begin{aligned}∥M∥_{B_{p,1}^{N/p}(ℝ^{N};B(X,Y))} & ≤ K_{1} ∥M∥_{W_{p}^{l}(ℝ^{N};B(X,Y))} \\ & := K_{1} ∑_{0≤|α|≤l} ∥D^{α}M∥_{L_{p}(ℝ^{N};B(X,Y))} ≤ K_{1} K_{2} A . \end{aligned}

Now apply Theorem 5.1.

Recall that for a function ℝ ⊃ (a, b) ∋ t → M (t) ∈ B (X, Y) with integrable derivative [36] (or more generally, for a function of bounded variation [33]) the set {M (t) ∈ B (X, Y) : t ∈ (a, b)} is R-bounded. A short proof of this in the integrable derivative case follows easily from Lemma 4.1 since

M(t) = M(a) + ∫_{a}^{b} χ_{[a,t]} (s) M′ (s) ds .

The next corollary is a variant of these results. Its proof uses the following (equivalent) definition [31, Prop. 3.1] of Besov spaces B_{p,1}^{1/p} (ℝ; Z) for p ∈ (1, 2]:

B_{p,1}^{1/p} (ℝ; Z) = {f ∈ L_{p} (ℝ; Z) : B_{p,1}^{1/p} (f) < ∞} ,  (5.2)

equipped with the norm ∥f∥′_{B_{p,1}^{1/p}} = ∥f∥_{L_{p}} + B_{p,1}^{1/p} (f), where

B_{p,1}^{1/p} (f) = ∫_{0}^{∞} sup_{|h|≤s} ∥f (· + h) − f (·)∥_{L_{p}(Z)} \frac{ds}{s^{1+ \frac{1}{p}}} .

COROLLARY 5.4. Let X be an arbitrary Banach space and Y have Fourier type p ∈ (1, 2]. Fix α ∈ (1/p, 1) and ε ∈ (α − 1/p, α]. If M ∈ L_{p} (ℝ; B (X, Y)) satisfies

∥M (t + s) − M (t)∥_{B(X,Y)} ≤ A \frac{|s|^{α}}{1 + |s|^{ε}} (1 + |t|)^{−α} for each s, t ∈ ℝ

for some constant A, then {M (t) ∈ B (X, Y) : t ∈ ℝ} is R-bounded.

PROOF. Note that \frac{|s|^{α}}{1+|s|^{ε}} is increasing in |s| since ε ≤ α. Thus for s ≥ 0

sup_{|h|≤s} [∫_{R} ∥M (t + s) − M (t)∥_{B(X,Y)}^{p} dt]^{1/p} ≤ A \frac{s^{α}}{1 + s^{ε}} [∫_{R} (1 + |t|)^{−αp} dt]^{1/p}

= A [\frac{2p}{αp − 1}]^{1/p} \frac{s^{α}}{1 + s^{ε}}

since 1/p < α. Note that

∫_{0}^{∞} \frac{s^{α}}{1 + s^{ε}} \frac{ds}{s^{1+ \frac{1}{p}}} ≤ ∫_{0}^{17}s^{α−1− \frac{1}{p}} ds + ∫_{17}^{∞}s^{α−1− \frac{1}{p} −ε} ds .  (5.3)

On the right-hand side of inequality (5.3), the first integral is finite since 1/p < α while the second integral is finite since α − 1/p < ε. Thus B_{p,1}^{1/p}(M) is finite and so M ∈ B_{p,1}^{1/p} (ℝ; B (X, Y)). Now apply Theorem 5.1.

Now for some applications to semigroup theory.

EXAMPLE 5.5. Let (T_{t})_{t≥0} be a c_{0}-semigroup on a Banach space X with generator A. Define

ω (T_{t}) := inf {w ∈ ℝ: there exists a constant C_{w} with∥T_{t}∥ ≤ C_{w}e^{wt} for t > 0}
s_{R} (A) := inf {w ∈ ℝ: {R (λ, A) : ℜλ > w} is R-bounded}
s_{N} (A) := inf {w ∈ ℝ: {R (λ, A) : ℜλ > w} is norm bounded}
s (A) := sup {ℜλ: λ ∈ σ (A)} .

a) Then s_{R} (A) ≤ ω (T_{t}); however, equality need not hold in general.

b) If furthermore (T_{t})_{t≥0} is positive and X is a q-concave Banach lattice for some q ∈ [1, ∞), then s (A) = s_{R} (A).

QUESTION 5.6. If A generates a c_{0}-semigroup (T_{t})_{t≥0} on a Banach space X, then

s (A) ≤ s_{N} (A) ≤ s_{R} (A) ≤ ω (T_{t}) .  (5.4)

The first (resp. last) inequality in (5.4) need not be an equality in general, as illustrated by several examples already in the literature (resp. by Example 5.5 above). It is not know whether the middle inequality in (5.4) need not be an equality in general.

Proof of Example 5.5 Towards part a), let ℜλ > w_{1} > ω (T_{t}). Then for each x ∈ X

R (λ, A) x = ∫_{0}^{∞}e^{−(λ−w_{1})t} [e^{−w_{1}t} T_{t}x] dt

(cf., eg., [17, Thm. II.1.10]) and so it follows from Lemma 4.1 that

{R (λ, A) : ℜλ > w_{1}}

is R-bounded. Hence s_{R}(A) ≤ ω(T_{t}).

Now assume the setting of part b). Let ℜλ ≥ λ_{0} > s(A). Then

|R (λ, A) x| ≤ R (λ_{0}, A) |x|

for each x ∈ X (cf., eg., [4, C-III: Thm. 1.2, Cor. 1.3]); hence,

{R (λ, A) : ℜλ > λ_{0}}

is R-bounded (just consider inequality (2.1) and [28, Prop. 1.d.9]) so s (A) = s_{R} (A).

However, there are positive c_{0}-semigroups on q-concave Banach lattices (q finite) for which ω (T_{t}) > s (A) (cf., eg., [29, Ch. 4 Ex. 4.2]).

EXAMPLE 5.7. Let −A be the generator of a c_{0}-semigroup (T_{t})_{t≥0} on a Banach space X having Fourier type p ∈ (1, 2] and ω(T_{t}) < 0. Denote by i_{α} the embedding of D(A^{α}) with its graph norm into X. Then

{T_{t}i_{α}: t ≥ 0} ⊂ B (D (A^{α}) , X)

is R-bounded for 1/p < α < 1.

PROOF. Let X_{θ} be the real interpolation space (X, D (A))_{θ,1}. By [5, Thm. 6.7.3], the norm ∥x∥_{θ} on X_{θ} is equivalent to

∥x∥_{X} + ∫_{0}^{∞}(t^{−θ} w (x, t)) \frac{dt}{t} where w (x, t) := sup_{s≤t} ∥T_{s}x − x∥ .

Let h_{x} (t) := T_{|t|} (x). Note that h_{x} ∈ L_{1} (ℝ; X) since ω(T_{t}) < 0. Furthermore, using the notation from (5.2),

B_{p,1}^{1/p} (h_{x}) ≤ (2 ∫_{0}^{∞}∥T_{u}∥ du) (∫_{0}^{∞}t^{−1/p} w (x, t) \frac{dt}{t})

and since D (A^{α}) ⊂ X_{1/p} for α > 1/p

∥h_{x}∥_{B_{p,1}^{1/p}(X)} ≤ C ∥x∥_{X_{1/p}} ≤ C_{1} ∥x∥_{D(A^{α})} .

Now just apply Remark 5.2 with M(t) = T_{|t|}.

Bibliography

1. Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Birkhäuser Boston Inc., Boston, MA, 1995, Abstract linear theory.

2. W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, (preprint).

3. , Tools for maximal regularity, (preprint).

4. W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Springer-Verlag, Berlin, 1986.

5. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.

6. Earl Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math. 112 (1994), no. 1, 13–49.

7. S. Blunck and P. C. Kunstmann, Weighted norm estimates and maximal regularity, (submitted).

8. Sönke Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math. 146 (2001), no. 2, 157–176.

9. J. Bourgain, A Hausdorff-Young inequality for B-convex Banach spaces, Pacific J. Math. 101 (1982), no. 2, 255–262.

10. , Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168.

11. , Vector-valued singular integrals and the H^{1}-BMO duality, Probability theory and harmonic analysis (Cleveland, Ohio, 1983), Dekker, New York, 1986, pp. 1–19.

12. , Vector-valued Hausdorff-Young inequalities and applications, Geometric aspects of functional analysis (1986/87), Springer, Berlin, 1988, pp. 239–249.

13. P. Clément, B. de Pagter, F. A. Sukochev, and H. Witvliet, Schauder decomposition and multiplier theorems, Studia Math. 138 (2000), no. 2, 135–163.

14. Philippe Clément and Jan Prüss, An operator-valued transference principle and maximal regularity on vector-valued L_{p}-spaces, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Dekker, New York, 2001, pp. 67–87.

15. R. Denk, M. Hieber, and J. Prüß, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, (submitted).

16. Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge University Press, Cambridge, 1995.

17. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

18. Maria Girardi and Lutz Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Mathematische Nachrichten, (to appear).

19. , Operator-valued Fourier multiplier theorems on L_{p}(X) and geometry of Banach spaces, (submitted).

20. Jerome A. Goldstein, Semigroups of linear operators and applications, The Clarendon Press Oxford University Press, New York, 1985.

21. R. Haller, H. Heck, and A. Noll, Mikhlin’s theorem for operator-valued Fourier multipliers on n-dimensional domains, (preprint).

22. N. J. Kalton and G. Lancien, A solution to the problem of L^{p}-maximal regularity, Math. Z. 235 (2000), no. 3, 559–568.

23. N. J. Kalton and L. Weis, The H^{∞}-calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345.

24. N. J. Kalton and Lutz Weis, Comparison and perturbation theorems for the H^{∞}-calculus, (in preparation).

25. P. C. Kunstmann, Maximal L_{p}-regularity for second order elliptic operators with uniformly continuous coefficients on domains, (submitted).

26. Peer Christian Kunstmann and Lutz Weis, Perturbation theorems for maximal L_{p}-regularity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 2, 415– 435.

27. S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI.

28. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Springer-Verlag, Berlin, 1979, Function spaces.

29. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

30. Jaak Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26.

31. A. Pełczyński and M. Wojciechowski, Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math. 107 (1993), no. 1, 61–100.

32. Gilles Pisier, Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), no. 1, 3–19.

33. Ž. Štrkalj and Lutz Weis, On operator-valued Fourier multiplier theorems, (submitted).

34. Alberto Venni, Marcinkiewicz and Mihlin multiplier theorems, and Rboundedness, (preprint).

35. Lutz Weis, A new approach to maximal L_{p}-regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Dekker, New York, 2001, pp. 195–214.

36. , Operator-valued Fourier multiplier theorems and maximal L_{p}- regularity, Math. Ann. 319 (2001), no. 4, 735–758.