OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) AND GEOMETRY OF BANACH SPACES

MARIA GIRARDI AND LUTZ WEIS to appear: Journal of Functional Analysis

Abstract. This paper gives the optimal order l of smoothness in the Mihlin and Hörmander conditions for operator-valued Fourier multiplier theorems. This optimal order l is determined by the geometry of the underlying Banach spaces (e.g. Fourier type). This requires a new approach to such multiplier theorems, which in turn leads to rather weak assumptions formulated in terms of Besov norms.

1. INTRODUCTION

In recent years, operator-valued multiplier theorems have had many applications in the theory of evolutions equations, in particular in connection with: maximal regularity of parabolic equations [4, 7, 11, 23, 37, 38], stability theory [17, 26, 36], elliptic operators on infinite dimensional state spaces [3], and pseudo differential operators on manifolds with singularities [31]. In these applications one often has a multiplier function m, from ℝ^{N} into the space B (X) of bounded operators on a Banach space X, such that Mihlin’s condition holds, that is, the set

τ_{l} := { |t|^{|α|} D^{α}m (t) : t ∈ ℝ^{N} \ {0} , |α| ≤ l}  (1.1)

is norm bounded in B (X) and then one wants to conclude that the operator

T_{m}: S (ℝ^{N}, X) → S′ (ℝ^{N}, X) given by T_{m}f := [mf̂]^{∨} ,

defined on the Schwartz class, extends to a bounded operator on L_{q} (ℝ^{N}, X) for 1 < q < ∞.

It is a classical result of J. Schwartz (cf. [6, Section 6.1]) that such an extension exists if X is a Hilbert space and l = [N/2] + 1. Furthermore, Pisier showed that Hilbert spaces are the only Banach spaces for which the boundedness of τ_{l} is sufficient for the Mihlin theorem to hold; he showed that such spaces must have type 2 and cotype 2. Bourgain [10] showed that, for a scalarvalued multiplier function m and N = 1, Mihlin’s theorem holds if and only if X is a UMD Banach space. This result was extended to higher dimensions in [27, 40] with l = N. For operator-valued

Date: 30 July 2002.

2000 Mathematics Subject Classification. Primary 42B15, 46E40, 46B09.

Key words and phrases. Operator-valued Fourier multiplier theorems, R-boundedness, UMD spaces, Fourier type of Banach spaces.

Girardi is supported in part by the Alexander von Humboldt Foundation. Weis is supported in part by Landesforschungsschwerpunkt Evolutionsgleichungen des Landes Baden-Württenberg.

2 MARIA GIRARDI AND LUTZ WEIS

multiplier functions, it was first shown in [35, 38] that for a UMD Banach space X and l = N, R-boundedness of τ_{l} is sufficient for Mihlin’s theorem to hold. Recall that τ_{l} is R-bounded if there is a constant C so that for each n ∈ ℕ, subset {T_{j}}_{j=1}^{n} of τ_{l}, and subset {x_{j}}_{j=1}^{n} of X one has that

𝔼 ∥∑_{j=1}^{n} r_{j} (·) T_{j}(x_{j})∥ ≤ C 𝔼 ∥∑_{j=1}^{n} r_{j} (·) x_{j}∥

where {r_{j}}_{j} are the Rademacher functions. For variants of these multiplier theorems, see [4, 11, 16].

This paper presents a new method of proof which allows for the determination of the optimal smoothness of the multiplier function; indeed, the best exponent l in (1.1) for a given Banach space X depends on the geometry of X, specifically, on its Fourier type. Recall that a Banach space X has Fourier type p ∈ [1, 2] provided the Fourier transform defines a bounded operator from L_{p} (X) into L_{p′} (X), i.e. the Hausdorff Young inequality holds for the exponent p.

Corollary 4.4 shows that Mihlin’s theorem holds with l = [N/p] + 1 if X is a UMD space with Fourier type p. Since a Hilbert space has Fourier type 2, one recovers Schwartz’s result. Since each UMD space has Fourier type p for some p > 1, one obtains the results in [35, 38] with l = N. If X is a subspace of an L_{q}(Ω) space, then X has Fourier type p = min(q, q′) and so one may use l = [N/p] + 1; hence, the l in (1.1) improves (i.e. decreases) as q tends towards 2. Furthermore, the exponent l = [N/p] + 1 is best possible for L_{p} spaces.

The main result of this paper, Theorem 4.1, is a general multiplier theorem. This theorem’s assumption, which uses vector-valued Besov spaces, may not look very attractive at first sight; indeed, the assumption is stated in a rather general form. However, this formulation allows as fairly easy corollaries (see Corollaries 4.4, 4.10, 4.11) vector-valued generalizations of several classical multiplier theorems conditions (à la Mihlin, Hörmander, or Lipschitz estimates); the latter corollary gives, for scalar-valued multiplier functions, an improvement of Bourgain’s original result in [10]. Theorem 4.1 also gives multiplier theorems in Sobolev spaces, H^{1} (X), and BMO (X); see Corollaries 4.6 and 4.9.

Section 3 prepares for the proof of the main result of this paper. Bourgain’s work [10] leads to a variant of the Littlewood-Paley decomposition (Corollary 3.3) and some precise norm estimates for scalar multiplier functions (Proposition 3.6). Also needed is a sharping, Corollaries 3.10 and 3.11, of results in [14] on Fourier estimates on Besov spaces. Section 2 collects necessary definitions (such as UMD and Fourier type) and some basic properties of these classes of Banach spaces.

Analogous results for multiplier theorems on Besov spaces B_{q,r}^{s} (ℝ^{N}, X) are contained in [14]. In this setting, the proofs are much more elementary and the UMD assumption and R-boundedness are not necessary. The new technique developed in this paper can also be used to prove boundedness results for singular integral operators (see [19]).

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 3

2. DEFINITIONS and NOTATION

Notation is standard. Throughout this paper X, Y, Z are Banach spaces over the field ℂ and X^{∗} is the (topological) dual space of X. The space B (X, Y) of bounded linear operators from X to Y is endowed with the usual uniform operator topology, unless otherwise stated. The Bochner-Lebesgue space L_{p} (ℝ^{N}, X), where 1 ≤ p ≤ ∞, is endowed with its usual norm topology. The space C^{l} (ℝ^{N} \ {0} , X) of functions f: ℝ^{N} \ {0} → X so that D^{α}f is continuous and bounded for |α| ≤ l ∈ ℕ_{0} is endowed with the usual supremum norm of f. If convenient and confusion seems unlikely, the various function spaces E (ℝ^{N}, X) in this paper are denoted simply by just E (X) or E, with the exception of the Schwartz class function space. ℕ = {1, 2, …} is the set of natural numbers while ℕ_{0} = {0} ∪ ℕ. If s is a positive real number, then [s] := max {n ∈ ℕ_{0} : n ≤ s}. The conjugate exponent p′ of p ∈ [1, ∞] is given by \frac{1}{p} + \frac{1}{p′} = 1. Non-numerical subscripts on constants indicate dependency.

The Schwartz class S (ℝ^{N}, X), or simply S (X), is the space of X-valued rapidly decreasing smooth functions φ on ℝ^{N}, equipped with its usual topology generated by seminorms. As customary, S (ℝ^{N}, ℂ) is often denoted by just S. Recall S (X) is norm dense in L_{q} (X) when 1 ≤ q < ∞. The space of X-valued tempered distributions S′ (ℝ^{N}, X) is the space of continuous linear operators L : S → X, equipped with the bounded convergence topology. Each m ∈ L_{q} (ℝ^{N}, X), where 1 ≤ q ≤ ∞, defines an L_{m} ∈ S′ (ℝ^{N}, X) by L_{m}(φ) := ∫_{ℝ^{N}} φ(t)m(t) dt; when convenient and confusion seems unlikely, such a function m is identified with L_{m}.

It is well-known that the Fourier transform F: S (X) → S (X) defined by

(Fφ) (t) ≡ φ̂ (t) := ∫_{ℝ^{N}} e^{−it·s}φ (s) ds

is an isomorphism whose inverse is given by

(F^{−1}φ) (t) ≡ φ̌ (t) = (2π)^{−N} ∫_{ℝ^{N}} e^{it·s}φ (s) ds ,

where φ ∈ S (X) and t ∈ ℝ^{N}. Also, the Fourier transform F: S′ (X) → S′ (X) defined by

(FL) (φ) ≡ L̂ (φ) := L \binom{}{φ} where L ∈ S′ (X) , φ ∈ S

is an isomorphism whose inverse is given by (F^{−1}L) (φ) ≡ Ľ (φ) := L \binom{}{φ̌}. The set

S_{o} (X) := {φ ∈ S (X) : supp φ̂ is compact , 0 ∉ supp φ̂}

is norm dense in L_{p} (X) for each 1 < p < ∞ (cf. [40, Lemma 2.3]).

For completeness a proof of the following fact is included.

Fact 2.1. Let f ∈ L_{1} (ℝ^{N}, X) with N > 1. Define F ∈ L_{1} (ℝ, L_{1} (ℝ^{N−1}, X)) by

F (t) (u) = f (t, u) for u ∈ ℝ^{N−1} and a.e. t ∈ ℝ .

4 MARIA GIRARDI AND LUTZ WEIS

If the (N-dimensional) Fourier transform f̂ of f has support in

K_{N} (r) := {(v_{1}, …, v_{N}) ∈ ℝ^{N}: |v_{j}| ≤ r} ,

then the (1-dimensional) Fourier transform F̂ of F has support in [−r, r].

Proof. It suffices to show Fact 2.1 for the case X = ℂ; indeed, just consider functions of the form x^{∗} ∘ f for x^{∗} ∈ X^{∗}.

If f̂: ℝ × ℝ^{N−1} → ℂ has the special form

\begin{matrix}g (t, u) & = ∑_{j=1}^{m} h_{j} (t) k_{j} (u) \\ h_{j} ≡ h_{j} |_{K_{1}(r)} ∈ S (ℝ, ℂ) & and k_{j} ≡ k_{j} |_{K_{N−1}(r)} ∈ S (ℝ^{N−1}, ℂ) , \end{matrix}  (2.1)

then f = ∑_{j=1}^{m} ȟ_{j} ǩ_{j} and F̂ (t) (u) = ∑_{j=1}^{m} h_{j} (t) ǩ_{j} (u); so, supp F̂ ⊂ [−r, r].

For a general f ∈ L_{1} (ℝ^{N}, ℂ) with supp f̂ ⊂ K_{N} (r), approximate f̂ by functions g_{n} of the special form in (2.1) in the L_{2} (ℝ^{N}, ℂ)-norm. It then follows from Plancherel’s Theorem, first applied in L_{2} (ℝ^{N}, ℂ) and then applied in L_{2} (ℝ, L_{2} (ℝ^{N−1}, ℂ)), that F̂ can be approximated in the L_{2} (ℝ, L_{2} (ℝ^{N−1}, ℂ))-norm by functions with support in [−r, r]. Hence supp F̂ ⊂ [−r, r]. ■

The derivative, translation, and dilation properties of F and F^{−1} that hold in the scalar-valued case also hold in the vector-valued case. However, the Hausdorff-Young inequality need not hold; thus, one has to consider the following class of Banach spaces that was introduced by Peetre [28].

Definition 2.2. Let 1 ≤ p ≤ 2. A Banach space X has Fourier type p provided 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, by [22], 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)).

Remark 2.3. 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 it is isomorphic to a Hilbert space [24]. A space L_{q} ((Ω, Σ, μ) , ℝ) has Fourier type p = min(q, q′) [28]. If X has 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. [14]). Each closed subspace (by definition) and each quotient space (by duality) of a Banach space X has the same Fourier type as X. Each uniformly convex Banach space has some non-trivial Fourier type p > 1.

Definition 2.4. Let 1 < q < ∞ and m : ℝ^{N} \ {0} → B (X, Y) be a bounded measurable function. Consider the L_{∞} (ℝ^{N}, Y ) functions

T_{m}f := [m f̂]^{∨} ∈ L_{∞} (ℝ^{N}, Y ) for f ∈ S (ℝ^{N}, X) .  (2.2)

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 5

Then m is a Fourier multiplier from L_{q} (ℝ^{N}, X) to L_{q} (ℝ^{N}, Y ) provided there is a constant C so that

∥T_{m}f∥_{L_{q}(ℝ^{N} ,Y)} ≤ C ∥f∥_{L_{q}(ℝ^{N} ,X)} for each f ∈ S (ℝ^{N}, X) ;

in which case, the operator T_{m} ∈ B (L_{q} (ℝ^{N}, X) , L_{q} (ℝ^{N}, Y )) uniquely determined by (2.2) is the Fourier multiplier operator induced by m.

Since multiplier theorems need not, in general, extend from the scalar case to the Banach space case, the class of UMD Banach spaces is considered. There are several (equivalent) formulations of this geometric property of a Banach space; below is one which is pertinent to the setting here.

Definition 2.5. A Banach space X is a UMD space if and only if the Hilbert transform

Hf(t) = PV − ∫ \frac{f(s)}{t − s} ds , f ∈ S (X)

extends to a bounded 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, 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, ∞).

Remark 2.6. a) L_{q} ((Ω, Σ, μ) , ℝ) spaces, where 1 < q < ∞, are examples of UMD spaces. Closed subspaces of, the dual of, and quotient spaces of a UMD space are UMD spaces. If X has UMD and 1 < q < ∞, then L_{q}(ℝ^{N}, X) has UMD. (cf., e.g., [1, Thm. 4.5.2]).

b) A UMD space has a uniformly convex renorming. A space with a uniformly convex renorming is reflexive and B-convex. Bourgain [8, 10] has shown that each B-convex Banach space has some non-trivial Fourier type p > 1.

The notion (cf. [38]) of R-boundedness, which provides a vector-valued substitute for Kahane’s contraction principle, is needed to extend scalar-valued multiplier theorems to operator-valued multiplier theorems.

Notation 2.7. Let {ε_{k}}_{k∈ℤ} be a sequence of {−1, +1}–valued, independent, symmetric random variables on some probability space (Ω, Σ, μ). For example, one can take an enumeration of the sequence {r_{n}}_{n∈ℕ} of Rademacher functions where r_{n}(u) = sign sin(2^{n}πu) on [0, 1]. Let {ε_{k}} and {ε′_{k}} be independent copies of {−1, +1, }–valued, independent, symmetric random variables.

Definition 2.8. Let τ ⊂ 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} ε_{j} (·) T_{j}(x_{j})∥_{L_{p}(Ω,Y)} ≤ R ∥∑_{j=1}^{n} ε_{j} (·) x_{j}∥_{L_{p}(Ω,X)} .

6 MARIA GIRARDI AND LUTZ WEIS

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, ∞). In this case, R_{p}(τ) is called the R_{p}-bound of τ.

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

Definition 2.9. Let X be a Banach space. Then Rad (X) := { {x_{j}}_{j∈ℤ} ∈ X^{ℤ}: ∑_{j=−n}^{n} ε_{j} (·) x_{j}: Ω → X is convergent in L_{2}(Ω, X) }.

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

∥{x_{j}}_{j∈ℤ}∥_{Rad_{p}(X)} := ∥∑_{j∈ℤ} ε_{j} (·) x_{j}∥_{L_{p}(Ω,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. [12]).

Remark 2.10. a) A sequence {T_{j}}_{j∈ℤ} from B (X, Y) is R-bounded if and only if the mapping

Rad_{p} (X) ∋ \{x_{j}\}_{j∈ℤ} \underrightarrow{T̃} \{T_{j}x_{j}\}_{j∈ℤ} ∈ Rad_{p} (Y)

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

b) If X has Fourier type p then so does Rad (X). If X has UMD then so does Rad (X). This follows from Remark 2.3, Remark 2.6, and the fact that Rad (X) is a subspace of L_{2} (X).

c) Note that {x_{j}}_{j∈ℤ} ∈ Rad (X) if and only if the series ∑_{j=−n}^{j=n} ε_{j} (·) x_{j} converges almost surely. If X does not (isomorphically) contain c_{0} and sup_{n} ∥∑_{j=−n}^{j=n} ε_{j} (·) x_{j}∥_{L_{p}(X)} is finite for some p ∈ [1, ∞), then {x_{j}}_{j∈ℤ} ∈ Rad (X).

Besov spaces serve as a tool in this paper (see [15] for further details). Among the many equivalent descriptions of Besov spaces, the most useful one in this context is given in terms of the so-called Littlewood-Paley decomposition. Roughly speaking this means that one considers f ∈ S′ (X) as a distributional sum f = ∑_{k} f_{k} of analytic functions f_{k} := φ̌_{k} ∗ f whose Fourier transforms have support in dyadic-like intervals {supp φ_{k}}_{k} and then defines the Besov norm in terms of the f_{k}’s. Here, {φ_{k}}_{k∈ℕ_{0}} is a partition of unity chosen as follows.

Notation 2.11. Take a nonnegative function ψ ∈ S (ℝ, ℝ) that has support in [2^{−1}, 2] and satisfies ∑_{k∈ℤ} ψ(2^{−k}s) = 1 for each s ∈ ℝ\{0}. Let φ_{k}(t) := ψ (2^{−k}|t|) if k ∈ ℕ and φ_{0}(t) := 1−∑_{k∈ℕ} φ_{k}(t). Note that φ_{k} ∈ S (ℝ^{N}, ℝ) for each k ∈ ℕ_{0}.

Definition 2.12. Let 1 ≤ q, r ≤ ∞ and the smoothness index s ∈ ℝ. The Besov space B_{q,r}^{s} (ℝ^{N}, X) is the space of all f ∈ S′ (ℝ^{N}, X) for which

∥f∥_{B_{q,r}^{s}(ℝ^{N} ,X)} := ∥{2^{ks} (φ̌_{k} ∗ f) }_{k=0}^{∞} ∥_{ℓ_{r}(L_{q}(X))}  (2.3)

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 7

is finite. B_{q,r}^{s} (ℝ^{N}, X), together with the norm in (2.3), is a Banach space.

Different choices of {φ_{k}}’s lead to equivalent norms on B_{q,r}^{s} (ℝ^{N}, X) (cf. [29, Lemma 3.2]). Weighted Besov spaces will give a precise estimate for the Fourier transform on Besov spaces.

Definition 2.13. Let 1 ≤ q ≤ ∞ and A = {a_{k}}_{k∈ℕ_{0}} be a sequence of non-negative real numbers. Then B_{q}^{A} (ℝ^{N}, X) is the space of all f ∈ S′ (ℝ^{N}, X) for which φ̌_{k} ∗ f ∈ L_{q} (ℝ^{N}, X) for each k ∈ ℕ_{0} and

∥f∥_{B_{q}^{A}(ℝ^{N} ,X)} := ∑_{k∈ℕ_{0}} a_{k} ∥φ̌_{k} ∗ f∥_{L_{q}(ℝ^{N} ,X)}  (2.4)

is finite. B_{q}^{A} (ℝ^{N}, X), endowed with the norm in (2.4), is normed linear space.

Other function spaces are also considered in this paper.

Definition 2.14. Let 1 ≤ q ≤ ∞ and m ∈ ℕ_{0}. The Sobolev space W_{q}^{m} (ℝ^{N}, X) is

W_{q}^{m} (ℝ^{N}, X) := {f ∈ S′ (ℝ^{N}, X) : D^{α}f ∈ L_{q} (ℝ^{N}, X) for each α ∈ ℕ_{0}^{N} with |α| ≤ m} , equipped with the norm

∥f∥_{W_{q}^{m}(ℝ^{N} ,X)} := ∑_{0≤|α|≤m} ∥D^{α}f∥_{L_{q}(ℝ^{N} ,X)} .

It is well-known that the Sobolev spaces are Banach spaces. For more information regarding Besov and Sobolev spaces, see [2, 14, 29, 30].

Definition 2.15. Let 1 < q < ∞ and s ∈ ℝ. The Bessel potential J^{s}: S (ℝ^{N}, X) → S (ℝ^{N}, X) is given by J^{s}f := [(1 + |·|^{2})^{s/2} f̂(·) ]^{∨}. The fractional Sobolev space H_{q}^{s} (ℝ^{N}, X) is the completion of S (ℝ^{N}, X) with respect to the norm ∥f∥_{H_{q}^{s}(ℝ^{N} ,X)} := ∥J^{s}f∥_{L_{q}(ℝ^{N} ,X)}.

The Banach spaces H_{q}^{s} (ℝ^{N}, X) are also called Liouville spaces and Bessel potential spaces. If X has UMD then H_{q}^{n} (ℝ^{N}, X) = W_{q}^{n} (ℝ^{N}, X) for q ∈ (1, ∞) and n ∈ ℕ (see [34, (15.55)]).

Definition 2.16. Let X be a Banach space.

a) For a locally integrable function f: ℝ^{N} → X let

∥f∥_{BMO(X)} := sup \frac{1}{|Q|} ∫_{Q} ∥f(t) − f_{Q}∥_{X} dt  (2.5)

where the sup is taken over all cubes of ℝ^{N} and f_{Q} = |Q|^{−1} ∫_{Q} f (t) dt. BMO (X) is the space of such functions f for which ∥f∥_{BMO(X)} is finite, endowed with the seminorm given by (2.5).

b) A function a ∈ L_{∞} (ℝ^{N}, X) is an atom if a is supported in a cube Q and ∫ a (t) dt = 0 and ∥a∥_{L_{∞}} ≤ |Q|^{−1}. The Hardy space H_{1} (ℝ^{N}, X) is the space of all f ∈ L_{1} (ℝ^{N}, X) which can be represented as f = ∑_{n} λ_{n}a_{n} where {λ_{n}}_{n} ∈ ℓ_{1} and each a_{n} is an atom. The norm ∥f∥_{H_{1}} is the

8 MARIA GIRARDI AND LUTZ WEIS

infimum of ∑_{n} |λ_{n}| over all such representations.

c) The weak-L_{1} space L_{1}^{wk} (ℝ^{N}, X) consists of all measurable functions f: ℝ^{N} → X that satisfy

∥f∥_{L_{1}^{wk}(ℝ^{N} ,X)} := sup_{λ>0} λ μ ({t ∈ ℝ^{N}: ∥f (t)∥_{X} > λ}) < ∞ .  (2.6)

It is well-known that the expression ∥·∥_{L_{1}^{wk}} in (2.6) is a quasi-norm on L_{1}^{wk} (ℝ^{N}, X) with

∥f + g∥_{L_{1}^{wk}(ℝ^{N} ,X)} ≤ 2 [∥f∥_{L_{1}^{wk}(ℝ^{N} ,X)} + ∥g∥_{L_{1}^{wk}(ℝ^{N} ,X)}] .

The balls with respect to ∥·∥_{L_{1}^{wk}} define a linear topology on L_{1}^{wk} (ℝ^{N}, X) and L_{1}^{wk} (ℝ^{N}, X), endowed with this topology, is a quasi-Banach space.

3. STEPS TOWARDS MULTIPLIER THEOREMS

Bourgain [9, N=1], and in the higher dimensional case McConnell [27] and Zimmermann [40], showed a generalization of the Mihlin’s multiplier theorem: Theorem 3.2. The following notation simplifies the statements of their result and results to follow.

Notation 3.1. Let M_{l}^{N} (X) be the set of all measurable functions m: ℝ^{N} \ {0} → X whose distributional derivatives D^{α}m are represented by measurable functions for each α ∈ ℕ_{0}^{N} with |α| ≤ l and

∥m∥_{M_{l}^{N} (X)} := sup {|t|^{|α|} ∥D^{α}m (t)∥_{X} : t ∈ ℝ^{N} \ {0} , α ∈ ℕ_{0}^{N} , |α| ≤ l} < ∞

where l ∈ ℕ_{0}.

Theorem 3.2 ([9, 27, 40]). Let X be a UMD space and 1 < q < ∞. If m ∈ M_{N}^{N} (ℂ) then m(·) I_{X} is a Fourier multiplier from L_{q}(ℝ^{N}, X) to L_{q}(ℝ^{N}, X) with ∥T_{m}∥ ≤ C_{X,N,q} ∥m∥_{M_{N}^{N} (ℂ)}.

Multiplier theorems such as Theorem 3.2 imply Littlewood-Paley decompositions such as Corollary 3.3. (Recall that {ε_{k}}_{k∈ℤ} was defined in Notation 2.7).

Corollary 3.3. Let X be a UMD space and 1 < q < ∞. Let ϑ_{0} ∈ S (ℝ^{N}, ℝ) be a nonnegative function that satisfies, for some n ∈ ℕ,

supp ϑ_{0} ⊂ {t ∈ ℝ^{N}: 2^{l} ≤ |t| ≤ 2^{L}} for some l, L ∈ ℤ
∑_{k∈ℤ} ϑ_{k} (t) = 1 for each t ∈ ℝ^{N} \ {0} where ϑ_{k} (t) := ϑ_{0} (2^{−nk}t) .

Then there is a constant C = C_{X,N,q,ϑ_{0}} so that

\frac{1}{C} ∥f∥_{L_{q}(ℝ^{N} ,X)} ≤ 𝔼_{ε} ∥∑_{k∈ℤ}ε_{k} (ϑ̌_{k} ∗ f)∥_{L_{q}(ℝ^{N} ,X)} ≤ C ∥f∥_{L_{q}(ℝ^{N} ,X)}  (3.1)

for each f ∈ L_{q} (ℝ^{N}, X).

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 9

Proof. For K ∈ ℕ and u ∈ Ω, let

m_{u,K} (·) := ∑_{k=−K}^{K} ε_{k}(u)ϑ_{k} (·) : ℝ^{N} \ {0} → ℂ .  (3.2)

Note that, for a fixed · ∈ ℝ^{N} \ \{0\}, there are at most \frac{L−l}{n} + 1 non-zero summands in (3.2). Thus m_{u,K} ∈ M_{N}^{N} (ℂ) with

∥m_{u,K}∥_{M_{N}^{N} (ℂ)} ≤ (\frac{L − l}{n} + 1) ∥ϑ_{0}∥_{M_{N}^{N} (ℂ)} .

So by Theorem 3.2, there is a constant C_{X,N,q} so that

sup_{\begin{matrix}u∈Ω \\ K∈ℕ \end{matrix}} ∥ ∑_{k=−K}^{K} ε_{k} (u) (ϑ̌_{k} ∗ f)∥_{L_{q}(X)} ≤ C_{X,N,q} (\frac{L − l}{n} + 1) ∥ϑ_{0}∥_{M_{N}^{N} (ℂ)} ∥f∥_{L_{q}(X)}  (3.3)

for f ∈ S (X). This gives the desired upper estimate in (3.1).

As usual one obtains the lower estimate from (3.3) and the corresponding inequality for X^{∗}. Since supp ϑ_{k} and supp ϑ_{k+j} can overlap only for |j| < \frac{L−l}{n}, for each f ∈ S_{o} (X) and g ∈ S_{o} (X^{∗})

⟨ g, f ⟩_{L_{q}(X)} = ∑_{|j|< \frac{L−l}{n}} ∑_{k∈ℤ}⟨ ϑ̌_{k+j} ∗ g , ϑ̌_{k} ∗ f ⟩_{L_{q}(X)}

= ∑_{|j|< \frac{L−l}{n}} 𝔼 ( ⟨ ∑_{m∈ℤ}ε_{m} (ϑ̌_{m+j} ∗ g) , ∑_{k∈ℤ}ε_{k} (ϑ̌_{k} ∗ f) ⟩_{L_{q}(X)} )

≤ ∑_{|j|< \frac{L−l}{n}} (𝔼 ∥∑_{m∈ℤ}ε_{m} (ϑ̌_{m+j} ∗ g)∥_{L_{q′}(X^{∗})}^{2})^{1/2} (𝔼 ∥∑_{k∈ℤ}ε_{k} (ϑ̌_{k} ∗ f)∥_{L_{q}(X)}^{2})^{1/2}

≤ C̃_{X,N,q,ϑ_{0}} (\frac{2 (L − l)}{n} + 1) ∥g∥_{L_{q′}(X^{∗})} 𝔼 ∥∑_{k∈ℤ}ε_{k} (ϑ̌_{k} ∗ f)∥_{L_{q}(X)}

by Kahane’s inequality and (3.3) applied to g ∈ L_{q′} (X^{∗}). Supping over all g ∈ S_{o} (X^{∗}) with ∥g∥_{L_{q′}(X^{∗})} ≤ 1 gives the lower estimate in (3.1). ■

Notation 3.4. There is (cf., eg. [6, Lemma 6.1.7]) a nonnegative function ϕ_{0} ∈ S (ℝ^{N}, ℝ) satisfying

supp ϕ_{0} ⊂ {t ∈ ℝ^{N}: 2^{−1} ≤ |t| ≤ 2^{1}}
∑_{k∈ℤ} ϕ_{k} (t) = 1 for each t ∈ ℝ^{N} \ {0} where ϕ_{k} (t) := ϕ_{0} (2^{−k}t) .

Thus one can take ϑ_{0} ≡ ϕ_{0} in Corollary 3.3.

Note that

supp ϕ_{k} ⊂ {t ∈ ℝ^{N}: 2^{k−1} ≤ |t| ≤ 2^{k+1}} ;

10 MARIA GIRARDI AND LUTZ WEIS

thus, supp ϕ_{k} and supp ϕ_{k+j} can overlap only for j ∈ {−1, 0, 1}. So fix j ∈ {−1, 0, 1} and let

ψ_{0} := ϕ_{j−1} + ϕ_{j} + ϕ_{j+1}
ψ_{k} (·) := ψ_{0} (2^{−3k} · ) for each k ∈ ℤ .

Then

supp ψ_{0} ⊂ {t ∈ ℝ^{N}: 2^{j−2} ≤ |t| ≤ 2^{j+2}}
ψ_{k} = ϕ_{j+3k−1} + ϕ_{j+3k} + ϕ_{j+3k+1} for each k ∈ ℤ .

Thus one can also take ϑ_{0} ≡ ψ_{0} in Corollary 3.3. Note that if

supp h ⊂ {t ∈ ℝ^{N}: 2^{j+3n−1} ≤ |t| ≤ 2^{j+3n+1}} ,

then

h ψ_{k} = \{ \begin{matrix}h & if k = n \\ 0 & if k ≠ n \end{matrix}

for each k ∈ ℤ.

The next lemma transfers [9, Lemma 10] from 𝕋 to ℝ^{N}.

Lemma 3.5. Let X be a UMD space and 1 < p < ∞. Consider translations {τ_{θ_{j}s}}_{j=1}^{m},

L_{p} (ℝ^{N}, X) ∋ f ⟶^{τ_{θ_{j} s}} (τ_{θ_{j}s}f) (·) := f (· + θ_{j}s) ∈ L_{p} (ℝ^{N}, X) ,

for θ_{j} ∈ ℝ and a fixed unit (in Euclidean norm) vector s ∈ ℝ^{N}. Let f_{j} ∈ L_{p} (ℝ^{N}, X) satisfy

supp f̂_{j} ⊂ B_{N} (r_{j}) := {(v_{1}, …, v_{N}) ∈ ℝ^{N}: [∑_{j=1}^{N} |v_{j}|^{2}]^{1/2} ≤ r_{j}} .

Assume that |θ_{j}| < \frac{K}{r_{j}} for a fixed constant K > 2 and \frac{r_{j+1}}{r_{j}} ≥ 2. Then

𝔼 ∥∑_{j=1}^{m} ε_{j}τ_{θ_{j}s}f_{j}∥_{L_{p}(ℝ^{N} ,X)} ≤ C_{X,N,p} (ln K) 𝔼 ∥∑_{j=1}^{m} ε_{j}f_{j}∥_{L_{p}(ℝ^{N} ,X)} .

Proof. Without loss of generality, {f_{j}}_{j=1}^{m} ⊂ S (ℝ^{N}, X). Indeed, for each f ∈ L_{p} (ℝ^{N}, X) with supp f̂ ⊂ B_{N} (r) there is a sequence g_{n} ∈ S (ℝ^{N}, X), with g_{n} → f in L_{p}-norm, along with φ ∈ S (ℝ^{N}, ℂ) such that supp φ ⊂ B_{N} (r + ε) and φ ≡ 1 on B_{N} (r); thus, φ̌ ∗ g_{n} → f in L_{p}- norm and supp (φ̌ ∗ g_{n}) ⊂ B_{N} (r + ε). \underline{Step 1}: reduction to the one-dimensional case. Since the L_{p} (ℝ^{N}, X)-norm is invariant under rotations, it suffices to take s = e_{1} := (1, 0, …, 0) ∈ ℝ^{N}. If N > 1, then define F_{j} ∈ L_{p} (ℝ, L_{p} (ℝ^{N−1}, X)) ∩ L_{1} (ℝ, L_{1} (ℝ^{N−1}, X)) by

F_{j} (t) (u) = f_{j} (t, u) for u ∈ ℝ^{N−1} and a.e. t ∈ ℝ .

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 11

Then supp F̂_{j} ⊂ [−r_{j}, r_{j}] by Fact 2.1. If Lemma 3.5 holds in the one-dimensional case, then, with the help of Remark 2.6a,

𝔼 ∥∑_{j} ε_{j}τ_{θ_{j}e_{1}}f_{j}∥_{L_{p}(ℝ^{N} ,X)} = 𝔼 ∥∑_{j} ε_{j}τ_{θ_{j}}F_{j}∥_{L_{p}(ℝ,L_{p}(ℝ^{N−1},X))}
≤ C_{X,N,p} (ln K) 𝔼 ∥∑_{j} ε_{j}F_{j}∥_{L_{p}(ℝ,L_{p}(ℝ^{N−1},X))}
= C_{X,N,p} (ln K) 𝔼 ∥∑_{j} ε_{j}f_{j}∥_{L_{p}(ℝ^{N} ,X)} .

Thus, without loss of generality, N = 1 and s = e_{1} ∈ ℝ^{1}. \underline{Step 2}: the transference. For f ∈ S (ℝ, X) and ε > 0, define the 2π-periodic function

F_{ε} (s) := ∑_{n∈ℤ} \frac{1}{ε} f (\frac{s + 2πn}{ε}) .  (3.4)

By the Poisson summation formula (cf. eg. [20])

F_{ε} (s) = \frac{1}{2π} ∑_{n∈ℤ} f̂(εn) e^{ins} ,  (3.5)

i.e., the n^{th} Fourier coefficient F̂_{ε} (n) of F_{ε} with respect to the discrete Fourier transform on L_{1} (−π, π) is \frac{1}{2π} f̂(εn).

For a fixed θ ∈ ℝ

∫_{−π}^{π} ∥ε^{1− \frac{1}{p}} F_{ε} (s + εθ)∥_{X}^{p} ds ⟶^{ε ↘ 0} ∫_{−∞}^{∞} ∥f (t + θ)∥_{X}^{p} dt.  (3.6)

To see (3.6), first note that, by (3.4),

∫_{−π}^{π} ∥ε^{1− \frac{1}{p}} F_{ε} (s + εθ)∥_{X}^{p} ds =  (3.7)

∫_{−π}^{π} ∥ε^{− \frac{1}{p}} f (\frac{s + εθ}{ε} ) + ∑_{0≠n∈ℤ} ε^{− \frac{1}{p}} f (\frac{s + εθ + 2πn}{ε})∥_{X}^{p} ds .

For the n = 0 term in (3.7)

∫_{−π}^{π} ∥ε^{− \frac{1}{p}} f (\frac{s + εθ}{ε} )∥_{X}^{p} ds = ∫_{− \frac{π}{ε}}^{+ \frac{π}{ε}} ∥f (t + θ)∥_{X}^{p} dt ⟶^{ε ↘ 0} ∫_{−∞}^{∞} ∥f (t + θ)∥_{X}^{p} dt.

Next estimate the remaining terms in (3.7) for ε |θ| < π/2 using the fact that, since f ∈ S (X), there is a constant C so that ∥f (t)∥_{X} ≤ C|t|^{−2} :

[∫_{−π}^{π} ∥ε^{−1/p} ∑_{0≠n∈ℤ}f (\frac{s + 2πn + εθ}{ε} )∥_{X}^{p} ds]^{1/p} ≤ C ε^{2− \frac{1}{p}} ∑_{0≠n∈ℤ}[∫_{−π}^{π} |s + 2πn + εθ|^{−2p} ds]^{1/p}

≤ C ε^{2− \frac{1}{p}} (2π)^{1/p} \frac{2}{π^{2}} ∑_{n=1}^{∞} (2n − \frac{3}{2})^{−2}

⟶^{ε ↘ 0} 0 .

Thus (3.6) holds. Furthermore (3.6) clearly extends to linear combinations:

∥∑_{j=1}^{m} ε_{j} ε^{1− \frac{1}{p}} (F_{j})_{ε} (· + εθ_{j})∥_{L_{p}(𝕋,X)} ⟶^{ε ↘ 0} ∥∑_{j=1}^{m} ε_{j} f_{j} (· + θ_{j})∥_{L_{p}(ℝ,X)}

for each choice {ε_{j}}_{j=1}^{m} of signs ±1. Thus

𝔼 ∥∑_{j=1}^{m} ε_{j}ε^{1− \frac{1}{p}} (F_{j})_{ε} (· + εθ_{j})∥_{L_{p}(𝕋,X)} ⟶^{ε ↘ 0} 𝔼 ∥∑_{j=1}^{m} ε_{j}f_{j} (· + θ_{j})∥_{L_{p}(ℝ,X)} .  (3.8)

If supp f̂_{j} ⊂ [−r_{j}, r_{j}], then by (3.5), supp ((F_{j})_{ε}) ⊂ [−\frac{r_{j}}{ε} , \frac{r_{j}}{ε} ]. If |θ_{j}| < \frac{K}{r_{j}} and \frac{r_{j+1}}{r_{j}} ≥ 2, then, for ε small enough, there is a strictly increasing sequence {n_{j}}_{j=1}^{m} of positive integers so that 2^{n_{j}−1} < \frac{r_{j}}{ε} ≤ 2^{n_{j}} and thus |εθ_{j}| < \frac{K}{r_{j}/ε} ≤ \frac{2K}{2^{n_{j}}}. Hence by [9, Lemma 10] applied to \{(F_{j})_{ε}\} ∈ L_{p} (𝕋, X) and {εθ_{j}},

𝔼 ∥∑_{j=1}^{m} ε_{j}ε^{1− \frac{1}{p}} (F_{j})_{ε} (· + εθ_{j})∥_{L_{p}(𝕋,X)} ≤ C (ln K) 𝔼 ∥∑_{j=1}^{m} ε_{j}ε^{1− \frac{1}{p}} (F_{j})_{ε}∥_{L_{p}(𝕋,X)}  (3.9)

for ε small enough. Letting ε ↘ 0 in (3.8) and (3.9) finishes the proof. ■

Lemma 3.5 leads to the following corollary to Theorem 3.2.

Proposition 3.6. Let X be a UMD space, {a_{k}}_{k∈ℤ} be a sequence from ℂ with |a_{k}| ≤ 1, and 1 < q < ∞. Fix h ∈ M_{N}^{N} (ℂ).

(a) Assume that supp h ⊂ {t ∈ ℝ^{N}: b^{−d} ≤ |t| ≤ b^{d}} with some b > 1 and d ∈ ℕ. Then

n (t) := ∑_{k∈ℤ}a_{k}h (b^{−k}t)

is a Fourier multiplier on L_{q} (ℝ^{N}, X) with ∥T_{n}∥ ≤ C_{X,N,q} d ∥h∥_{M_{N}^{N} (ℂ)}.

(b) Assume that supp h ⊂ {t ∈ ℝ^{N}: 2^{−1} ≤ |t| ≤ 2} and s ∈ ℝ^{N}. Then

m_{s} (t) := ∑_{k∈ℤ}a_{k} exp (is · 2^{−k} t) h (2^{−k}t)

is a Fourier multiplier on L_{q} (ℝ^{N}, X) with ∥T_{m_{s}}∥ ≤ C_{X,N,q} ln (2 + |s|) ∥h∥_{M_{N}^{N} (ℂ)}.

Proof. Throughout this proof, the C_{i}’s are constants that depend on at most: X, N, and q. To simplify notation, let M := M_{N}^{N} (ℂ). Note that h (a ·) ∈ M and ∥h (a ·)∥_{M} = ∥h∥_{M} for each a > 0.

Part (a) follows easily from Theorem 3.2: indeed, the support of h (b^{−k}·) overlaps with the support of h (b^{−m}·) only if |m − k| < 2d; hence, n ∈ M and ∥n∥_{M} ≤ 4d ∥h∥_{M}.

Now to show part (b). Note that the function h_{s} (∘) := exp (is · ∘) h (∘) is in M with ∥h_{s}∥_{M} ≤ C_{1} [max {1, |s|^{N}}] ∥h∥_{M} for some constant C_{1} that is independent of s. So the desired conclusion for |s| ≤ 1 follows from part (a). Thus assume that |s| > 1.

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 13

Fix f ∈ S_{o} (ℝ^{N}, X). Then

T_{m_{s}}f = ∑_{k∈ℤ}a_{k} τ_{k} (ȟ_{k} ∗ f)

where

L_{q} (ℝ^{N}, X) ∋ g ⟶^{τ_{k}} (τ_{k}g) (·) := g (· + 2^{−k} s) ∈ L_{q} (ℝ^{N}, X) ,

and

h_{k} (·) := h (2^{−k} · ) ,

which is a Fourier multiplier on L_{q} (ℝ^{N}, X) by part (a). Thus, by the Littlewood-Paley decomposition Corollary 3.3, for an appropriate choice of ϑ_{0} (eg., ϑ_{0} = ψ_{0} of Notation 3.4 so that for then ϑ_{k}h_{3n+j} = δ_{k,n}h_{3n+j}),

∥T_{m_{s}}f∥_{L_{q}(ℝ^{N} ,X)} ≤ ∑_{j=−1}^{1} ∥∑_{n∈ℤ}a_{3n+j} τ_{3n+j} (ȟ_{3n+j} ∗ f)∥_{L_{q}(ℝ^{N} ,X)}

≤ C_{2} ∑_{j=−1}^{1} 𝔼 ∥∑_{k∈ℤ} ε_{k} a_{3k+j} τ_{3k+j} (ȟ_{3k+j} ∗ f)∥_{L_{q}(ℝ^{N} ,X)} .

Thus, by Kahane’s contraction principle, Lemma 3.5, and then part (a),

∥T_{m_{s}}f∥_{L_{q}(ℝ^{N} ,X)} ≤ 6C_{2} 𝔼 ∥∑_{k∈ℤ}ε_{k} τ_{k} (ȟ_{k} ∗ f)∥_{L_{q}(ℝ^{N} ,X)}

≤ C_{3} ln (2 |s|) 𝔼 ∥∑_{k∈ℤ}ε_{k} (ȟ_{k} ∗ f)∥_{L_{q}(ℝ^{N} ,X)}

≤ C_{4} ln (2 + |s|) ∥h∥_{M} ∥f∥_{L_{q}(ℝ^{N} ,X)} .

This completes the proof of Proposition 3.6. ■

The logarithmic estimate in Proposition 3.6b enters the proof of the next result, which is a centerpiece of the proof of Theorem 4.1.

Proposition 3.7. Let X have UMD and 1 < q < ∞. Let k : ℝ^{N} → B (X, Y) be a strongly integrable function such that

supp k̂ ⊂ supp ϕ_{0}  (3.10)

and

∫_{ℝ^{N}} ∥[k(s)]^{∗} g∥_{L_{q′}(ℝ^{N} ,X^{∗})} w (s) ds ≤ A ∥g∥_{L_{q′}(ℝ^{N} ,Y^{∗})}  for each g ∈ L_{q′} (ℝ^{N}, Y^{∗}) (3.11)

where w (·) := ln (2 + |·|). Set k_{j} (·) := 2^{Nj}k (2^{j} ·).

(a) For each finitely supported scalar sequence {a_{j}}_{j∈ℤ} with |a_{j}| ≤ 1 and

L (·) := ∑_{j∈ℤ}a_{j}k_{j} (·) : ℝ^{N} → B (X, Y)  (3.12)

14 MARIA GIRARDI AND LUTZ WEIS

the operator

T f := L ∗ f for f ∈ S (ℝ^{N}, X)

extends to a bounded operator T : L_{q} (ℝ^{N}, X) → L_{q} (ℝ^{N}, Y ) with ∥T ∥ ≤ C_{X,N,q} A.

(b) For each f ∈ S (ℝ^{N}, X) and n ∈ ℕ

𝔼 ∥ ∑_{j=−n}^{n} ε_{j}k_{j} ∗ f∥_{L_{q}(ℝ^{N} ,Y)} ≤ C_{X,N,q} A ∥f∥_{L_{q}(ℝ^{N} ,X)} .  (3.13)

Also, if Y does not contain c_{0}, then the summand in (3.13) can be taken over j ∈ Z.

Proof. Recall that the ϕ_{k}’s were defined in Notation 3.4.

Since k is strongly integrable, f → k_{j} ∗ f is a bounded operator from L_{1} (ℝ^{N}, X) to L_{1} (ℝ^{N}, Y ). Fix q ∈ (1, ∞).

Fix f = ∑_{i=1}^{m} x_{i}g_{i} with x_{i} ∈ X and g_{i} ∈ S (ℝ^{N}, ℂ) and m ∈ ℕ. Let h_{j} := ϕ_{j−1} + ϕ_{j} + ϕ_{j+1}; note that h_{j} = 1 on supp ϕ_{j}. Put f_{j} := ȟ_{j} ∗ f. Note that [k_{j} ∗ f_{j}] = k̂_{j} h_{j} f̂ = k̂_{j} f̂ = [k_{j} ∗ f] by (3.10). Thus, for a.e. t ∈ ℝ^{N},

(L ∗ f) (t) = ∑_{j∈ℤ}a_{j} (k_{j} ∗ f) (t)  = ∑_{j∈ℤ}a_{j} ∫_{ℝ^{N}} 2^{Nj}k(2^{j}s)f_{j}(t − s)ds
= ∫_{ℝ^{N}} k(s) ∑_{j∈ℤ}a_{j} f_{j}(t − 2^{−j}s)ds = ∫_{ℝ^{N}} k(s) ∑_{j∈ℤ} a_{j} (τ_{2^{−j}s}f_{j})(t)ds  (3.14)
= ∫_{ℝ^{N}} k(s)(m̌_{s} ∗ f)(t)ds

where τ_{a}g (·) := g (· − a) and m_{s}(t) := ∑_{j} a_{j} exp (−is · 2^{−j}t) h_{0}(2^{−j}t). By Proposition 3.6

∥m̌_{s} ∗ f∥_{L_{q}(ℝ^{N} ,X)} ≤ Cw (s) ∥f∥_{L_{q}(ℝ^{N} ,X)}  (3.15)

for some constant C = C_{X,N,q} ∥ϕ_{0}∥_{M_{N}^{N} (ℂ)}. If g ∈ L_{q′} (ℝ^{N}, Y^{∗}), then by (3.14), (3.15), and (3.11)

\begin{aligned}|⟨g, L ∗ f⟩_{L_{q}(Y)}| & ≤ ∫_{ℝ^{N}} ∫_{ℝ^{N}} | ⟨ g (t) , k(s) (m̌_{s} ∗ f) (t) ⟩_{Y} | ds dt \\ & ≤ ∫ ∥[k(s)]^{∗} g∥_{L_{q′}(X^{∗})} ∥m̌_{s} ∗ f∥_{L_{q}(X)} ds ≤ C A ∥g∥_{L_{q′}(Y^{∗})} ∥f∥_{L_{q}(X)} . \end{aligned} (3.16)

Thus ∥L ∗ f∥_{L_{q}(ℝ^{N} ,Y)} ≤ CA ∥f∥_{L_{q}(ℝ^{N} ,X)}. Hence part (a) holds.

Part (b) follows formally from part (a). Indeed, for a fixed n ∈ ℕ and u ∈ Ω, define L_{u,n} (·) := ∑_{j=−n}^{n} ε_{j} (u) k_{j} (·). Then by part (a)

𝔼 ∥ ∑_{j=−n}^{n} ε_{j}k_{j} ∗ f∥_{L_{q}(ℝ^{N} ,Y)} = ∫_{ℝ^{N}} ∥L_{u,n} ∗ f∥_{L_{q}(ℝ^{N} ,Y)} du ≤ C_{X,N,q} A ∥f∥_{L_{q}(ℝ^{N} ,X)} . (3.17)

If Y does not contain c_{0}, then L_{q} (ℝ^{N}, Y ) does not contain c_{0} [25] and so, by Remark 2.10c, the sum in (3.17) can be taken over j ∈ ℤ. Thus part (b) holds. ■

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 15

Remark 3.8. a) In Proposition 3.7, if (3.11) is replaced by ∫_{ℝ^{N}} ∥k(s)∥_{B(X,Y)} w(s)ds ≤ A, then this modified version of Proposition 3.7 remains true.

b) In Proposition 3.7, if:

(1) in addition, Y also has UMD,

(2) in addition, the function k^{∗} (·) := [k (·)]^{∗} : ℝ^{N} → B (Y^{∗}, X^{∗}) is strongly integrable,

(3) the condition (3.11) is replaced by:

(i) ∫_{ℝ^{N}} ∥k(s)f∥_{L_{r}(Y)} w(s) ds ≤ A ∥f∥_{L_{r}(X)} for each f ∈ L_{r} (ℝ^{N}, X) and 1 < r ≤ 2 (ii) ∫_{ℝ^{N}} ∥k^{∗}(s)g∥_{L_{r}(X^{∗})} w(s) ds ≤ A ∥g∥_{L_{r}(Y^{∗})} for each g ∈ L_{r} (ℝ^{N}, Y^{∗}) and 1 < r ≤ 2 ,

(4) C_{X,N,q} is replaced by C_{X,Y,N,q}, then this modified version of Proposition 3.7 remains true (for each q ∈ (1, ∞)).

Proof of Remark 3.8. a) Indeed, the calculation in (3.16) can then be replaced by

∥L ∗ f∥_{L_{q}(Y)} = ∥∫_{ℝ^{N}} k (s) (m̌_{s} ∗ f) (·) ds∥_{L_{q}(Y)}  ≤ ∫_{ℝ^{N}} ∥k (s)∥_{B(X,Y)} ∥m̌_{s} ∗ f∥_{L_{q}(X)} ds
≤ C ∥f∥_{L_{q}(X)} ∫_{ℝ^{N}} ∥k (s)∥_{B(X,Y)} w (s) ds ≤ CA ∥f∥_{L_{q}(X)}

with the help of (3.14) and (3.15).

b) If q ∈ [2, ∞), then the modified Proposition 3.7 follows from the original Proposition 3.7 since condition (ii) implies (3.11). So fix q ∈ (1, 2] and consider a function L of the form (3.12). Since X and Y are UMD spaces, they are reflexive. Applying the original Proposition 3.7 to k^{∗} gives, by condition (i), that

Kf := [L (·)]^{∗} ∗ [f ( − ·)] for f ∈ S (ℝ^{N}, Y^{∗})

extends to a bounded operator K: L_{q′} (ℝ^{N}, Y^{∗}) → L_{q′} (ℝ^{N}, X^{∗}) with ∥K∥ ≤ C_{Y^{∗},N,q′} A. Since X and Y are reflexive, K^{∗} is in B (L_{q} (X) , L_{q} (Y)) and K^{∗} restricted to S (ℝ^{N}, X) is L (·)∗f ( − ·). ■

In the proof of Theorem 4.1, the assumptions Remark 3.8b will be checked by applying estimates of the operator norm of the Fourier transform on Besov spaces. The Hausdorff-Young inequality holding in spaces with Fourier type gives a starting point for such norm estimates.

Corollary 3.9 ([14]). Let X have Fourier type p ∈ [1, 2]. Then the Fourier transform defines a bounded operator

F : B_{p,1}^{N/p} (ℝ^{N}, X) ⟶ L_{1} (ℝ^{N}, X) .  (3.18)

Furthermore, the norm of F is bounded above by a constant depending only on F_{p,N}(X).

A sharping of this result, in the spirit of the logarithmic estimate in Proposition 3.6, is needed in this paper. Recall that B_{p}^{A} was given in Definition 2.13 and {φ_{k}}_{k∈ℕ_{0}} was given in Notation 2.11.

16 MARIA GIRARDI AND LUTZ WEIS

Corollary 3.10. Let w: ℝ^{N} → [0, ∞) be a measurable function so that

a_{k} := 2^{Nk/p} ∥wχ_{supp φ_{k}}∥_{L_{∞}(ℝ^{N} ,ℝ)} < ∞ for each k ∈ ℕ_{0} .

Set A := {a_{k}}_{k∈ℕ_{0}}. If X has Fourier type p ∈ [1, 2] then the Fourier transform defines a bounded operator

F : B_{p}^{A} (ℝ^{N}, X) ⟶ L_{1} ((ℝ^{N}, w (t) dt) , X) .  (3.19)

Furthermore, the norm of F is bounded above by a constant depending only on F_{p,N}(X).

Note that if w ≡ 1 in Corollary 3.10, then (3.19) reduces to (3.18). Corollary 3.9 and Corollary 3.10 remain valid if F is replaced with F^{−1}.

Proof of Corollary 3.10. Let {J_{k}}_{k∈ℕ_{0}} be the partitioning of ℝ^{N} given by

J_{k} := {t ∈ ℝ^{N} : 2^{k−1} < |t| ≤ 2^{k}} for k ∈ ℕ and J_{0} := {t ∈ ℝ^{N} : |t| ≤ 1}

and set φ_{−1} ≡ 0 and J_{−1} = ∅ to simplify notation.

Fix f ∈ B_{p}^{A} (ℝ^{N}, X). For each k ∈ ℕ_{0}, since φ̌_{k} ∗ f ∈ L_{p} (ℝ^{N}, X) and X has Fourier type p, one has that φ_{k} · f̂ = F (φ̌_{k} ∗ f) ∈ L_{p′} (ℝ^{N}, X); in particular, the distributional Fourier transform of f is represented as a measurable function. Thus for each m ∈ ℕ_{0}

f̂· χ_{J_{m}} = ∑_{k=m−1}^{m} φ_{k} f̂χ_{J_{m}}

and so (with a_{−1} := 0)

∥f̂· χ_{J_{m}} · w∥_{L_{1}(X)} ≤ ∑_{k=m−1}^{m} ∥f̂φ_{k} [\frac{1 + |·|}{4}]^{N/p} χ_{J_{m}} ∥_{L_{p′}(X)} ∥[\frac{1 + |·|}{4}]^{−N/p} χ_{J_{m}}wχ_{supp φ_{k}} ∥_{L_{p}(ℝ)}

≤ ∑_{k=m−1}^{m} ∥[\frac{1 + |·|}{4}]^{N/p} χ_{J_{m}} ∥_{L_{∞}(ℝ)} ∥f̂φ_{k}∥_{L_{p′}(X)} ∥wχ_{supp φ_{k}}∥_{L_{∞}(ℝ)} ∥[\frac{1 + | · |}{4}]^{−N/p} χ_{J_{m}} ∥_{L_{p}(ℝ)}

≤ ∑_{k=m−1}^{m} (2^{k})^{N/p} ∥f̂φ_{k}∥_{L_{p′}(X)} a_{k} 2^{−Nk/p} [∫_{J_{m}} (\frac{1 + |t|}{4})^{−N} dt]^{1/p}

≤ C ∑_{k=m−1}^{m} a_{k} ∥f̂φ_{k}∥_{L_{p′}(X)} ≤ C F_{p,N}(X) ∑_{k=m−1}^{m} a_{k} ∥φ̌_{k} ∗ f∥_{L_{p}(X)}

for some universal constant C. Thus

∫_{ℝ^{N}} ∥f̂(t)∥_{X} w (t) dt ≤ 2 C F_{p,N}(X) ∥f∥_{B_{p}^{A}(ℝ^{N} ,X)} ,

as needed. ■

Let X and Y have Fourier type p and m ∈ B_{p,1}^{N/p} (ℝ^{N}, B (X, Y)). Then Corollary 3.9 implies that m̂ ∈ L_{1} (ℝ^{N}, B (X, Y)) only if p = 1 since if p > 1 then B (X, Y) usually does not also have Fourier

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 17

type p. However, for p > 1, one can at least conclude from Corollary 3.9 that for each x ∈ X

∫_{ℝ^{N}} ∥ [m (·) x] (t) ∥_{Y} dt ≤ ∥F∥_{B_{p,1}^{N/p}(ℝ^{N} ,Y)→L_{1}(ℝ^{N} ,Y)} ∥m∥_{B_{p,1}^{N/p}(ℝ^{N} ,B(X,Y))} ∥x∥_{X} .

A somewhat stronger statement than this, which is made possible by Corollary 3.10, is needed.

Corollary 3.11. Let X and Y have Fourier type p ∈ (1, 2] and 1 < q ≤ p. Let w: ℝ^{N} → [0, ∞) be a measurable function so that

a_{k} := 2^{Nk/q} ∥wχ_{supp φ_{k}}∥_{L_{∞}(ℝ^{N} ,ℝ)} < ∞ for each k ∈ ℕ_{0} .

Set A := {a_{k}}_{k∈ℕ_{0}} and let m ∈ B_{q}^{A} (ℝ^{N}, B (X, Y)) ∩ L_{1} (ℝ^{N}, B (X, Y)) have norm M in B_{q}^{A}. Then then there is a constant C, depending only on F_{q,N} (X) and F_{q,N} (Y), so that

∫_{ℝ^{N}} ∥ m̂ (t) [f (·)] ∥_{L_{r}(ℝ^{N} ,Y)} w (t) dt ≤ C M ∥f∥_{L_{r}(ℝ^{N} ,X)} for each f ∈ L_{r} (ℝ^{N}, X) (3.20)
∫_{ℝ^{N}} ∥ [m̂ (t)]^{∗} [g (·)] ∥_{L_{r}(ℝ^{N} ,X^{∗})} w (t) dt ≤ C M ∥g∥_{L_{r}(ℝ^{N} ,Y^{∗})} for each g ∈ L_{r} (ℝ^{N}, Y^{∗}) (3.21)

for each 1 ≤ r ≤ q′.

Note that if w ≡ 1, then B_{q}^{A} (B (X, Y)) = B_{q,1}^{N/q} (B (X, Y)). Also, Corollary 3.11 remains valid if one replaces m̂ with m̌ in (3.20) and (3.21).

Proof. The first step is to show that the operator

L_{r} (ℝ^{N}, X) ∋ f ⟶^{T_{r}} m̂ (·) fw (·) ∈ L_{1} (ℝ^{N}, L_{r} (ℝ^{N}, Y ))

is bounded for r = 1 and r = q′.

Fix x ∈ X. Note that m (·) x ∈ B_{q}^{A} (Y) has norm at most M ∥x∥_{X} since m ∈ B_{q}^{A} (B (X, Y)) has norm M and that [m (·) x] (t) = [m̂ (t)] (x) since m ∈ L_{1} (B (X, Y)). Thus by Corollary 3.10 applied to m (·) x, there is a constant C_{1}, depending only on F_{q,N} (Y), so that

∫_{ℝ^{N}} ∥ [m̂ (t)] x ∥_{Y} w(t) dt ≤ C_{1} M ∥x∥_{X} .

Thus, for simple functions f ∈ L_{1} (X),

∫_{ℝ^{N}} ∥ [ m̂ (t)] f (·) ∥_{L_{1}(Y)} w(t) dt = ∫_{ℝ^{N}} ∫_{ℝ^{N}} ∥ [m̂ (t)] f (s) ∥_{Y} w(t) dt ds ≤ C_{1} M ∥f∥_{L_{1}(X)} .

Thus ∥T_{1}∥ ≤ C_{1}M.

Let \overline{X} := L_{q′} (ℝ^{N}, X) and \overline{Y} := L_{q′} (ℝ^{N}, Y ). Then F_{q,N} (X) = F_{q,N} (X^{∗}) = F_{q,N} (\overline{X}), and likewise for Y, by Remark 2.3. Extend m to an operator ℝ^{N} ∋ t → \overline{m}(t) ∈ B (\overline{X}, \overline{Y} ) by

[ \overline{m} (t) f ] (s) := [m (t)] (f (s)) for f ∈ \overline{X} , s ∈ ℝ^{N} .

Note that \overline{m} ∈ B_{q}^{A} (B (\overline{X}, \overline{Y} )) ∩ L_{1} (B (\overline{X}, \overline{Y} )) has norm M in B_{q}^{A} since for each t ∈ ℝ^{N}

∥\overline{m}(t)∥_{B(\overline{X},\overline{Y})} = ∥m(t)∥_{B(X,Y)} and ∥(φ̌_{k} ∗ \overline{m}) (t)∥_{B(\overline{X},\overline{Y})} = ∥(φ̌_{k} ∗ m) (t)∥_{B(X,Y)} .

18 MARIA GIRARDI AND LUTZ WEIS

Fix a simple function f ∈ \overline{X}. Note \overline{m} (·) f ∈ B_{q}^{A} (\overline{Y} ) has norm at most M ∥f∥\overline{X} and

[\overline{m} (·) f] (t) = m̂ (t) [f (·)] .

Thus by Corollary 3.10 applied to \overline{m} (·) f, there is a constant C_{2}, depending only on F_{q,N} (\overline{Y} ), so that

∫_{ℝ^{N}} ∥m̂ (t) [f (·)]∥\overline{Y} w (t) dt ≤ C_{2} M ∥f∥\overline{X} .

Thus ∥T_{q′} ∥ ≤ C_{2}M.

Thus (3.20) holds for 1 ≤ r ≤ q′ by interpolation in the scale of Bochner-spaces (see [6, Thm. 5.1.2]). Claim (3.21) is obtained by applying the same argument to [m(t)]^{∗} ∈ B (Y^{∗}, X^{∗}). ■

One more preparatory result is needed for the proof of Theorem 4.1. The proof of the following lemma is due to N.J. Kalton and replaces our original (more complicated) proof. Recall that {ε_{k}}_{k∈ℤ} and {ε′_{k}}_{k∈ℤ} were given in Notation 2.7.

Lemma 3.12. For any Banach space X and any array {x_{kl}}_{k,l∈ℤ} from X

𝔼_{ε} ∥∑_{k∈ℤ}ε_{k}x_{kk}∥_{X} ≤ 𝔼_{ε,ε′} ∥ ∑_{k,l∈ℤ}ε_{k}ε′_{l}x_{kl}∥_{X} .

Proof. For α_{k} ∈ {−1, 1}, the sequences of random variables {ε_{k}}, {ε′_{k}}, {α_{k} ε_{k}}, and {α_{k} ε′_{k}} have the same distribution. Hence by Fubini’s theorem

𝔼_{ε,ε′} ∥ ∑_{k,l∈ℤ}ε_{k}ε′_{l}x_{kl}∥_{X} = 𝔼_{ε,ε′} ∥ ∑_{k,l∈ℤ}α_{k} α_{l} ε_{k}ε′_{l}x_{kl}∥_{X} .  (3.22)

Let {η_{k}} be a sequence of {−1, +1}–valued, independent, symmetric random variables that is independent of {ε_{k}} and {ε′_{k}}. Integrating (3.22) gives

𝔼_{ε,ε′} ∥ ∑_{k,l∈ℤ}ε_{k}ε′_{l}x_{kl}∥_{X} = 𝔼_{ε,ε′} 𝔼_{η} ∥ ∑_{k,l∈ℤ}η_{k}η_{l}ε_{k}ε′_{l}x_{kl}∥_{X} ≥ 𝔼_{ε,ε′} ∥𝔼_{η} ( ∑_{k,l∈ℤ}η_{k}η_{l}ε_{k}ε′_{l}x_{kl})∥_{X}

= 𝔼_{ε,ε′} ∥∑_{k∈ℤ}ε_{k}ε′_{k}x_{kk}∥_{X} = 𝔼_{ε} ∥∑_{k∈ℤ}ε_{k}x_{kk}∥_{X} ,

again since u → ε_{k} (u) · ε′_{k} (v), for each fixed v, is a sequence of independent random variables with the same distribution as {ε_{k}}. ■

4. FOURIER MULTIPLIER THEOREMS

This section explores operator-valued Fourier multiplier operators. The assumptions of the main result, Theorem 4.1, may be somewhat awkward looking. However, with respect to the smoothness of the multiplier function, it is a rather weak assumption that will make it easy to derive, in a

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 19

unified way, several results modeled after classical conditions, such as of Mihlin and Hörmander, as shown by the corollaries to follow.

Theorem 4.1. Let X and Y be UMD spaces with Fourier type p ∈ (1, 2] and 1 < q < ∞. Set X̃ := Rad_{q} (X) and Ỹ := Rad_{q} (Y). Let m : ℝ^{N} \ {0} → B (X, Y) be a measurable function that induces a mapping M: ℝ^{N} → B (X̃, Ỹ),

M(s) := {ϕ_{0}(s)m(2^{k}s)}_{k∈ℤ}  for s ≠ 0  and  M(0) := 0 ,

that satisfies M ∈ B_{p}^{A} (ℝ^{N}, B (X̃, Ỹ)) with norm D where A = {(k ∨ 1) 2^{Nk/p}}_{k∈ℕ_{0}}. Then m is a Fourier multiplier from L_{q}(ℝ^{N}, X) to L_{q}(ℝ^{N}, Y). Furthermore, ∥T_{m}∥_{L_{q}→L_{q}} ≤ C D for some constant C depending on: X, Y, N, p, q, and ϕ_{0}.

Remark 4.2. a) Recall that each UMD space has Fourier type p for some p > 1.

b) Remark 2.10 provides a convenient way to compute the norm of M; indeed, ∥M (s)∥_{B(X̃,Ỹ)} = ϕ_{0}(s)R_{q} ({m(2^{k}s): k ∈ ℤ}) for each s ∈ ℝ^{N}.

c) For the A in Theorem 4.1, there are continuous embeddings

W_{p}^{l} (ℝ^{N}, Z) ⊂ B_{p,r}^{s} (ℝ^{N}, Z) ⊂ B_{p}^{A} (ℝ^{N}, Z) ⊂ B_{p,1}^{N/p} (ℝ^{N}, Z)

where N/p < s < l ∈ ℕ and r ∈ [1, ∞].

d) The exponent N/p is best possible in the following sense. Let A_{p} := {(k ∨ 1) 2^{Nk/p}}_{k∈ℕ_{0}}. Note that the spaces B_{p}^{A_{p}} form a scale since, for 1 ≤ p < q < ∞,

B_{p}^{A_{p}} (ℝ^{N}, Z) ⊂ B_{q}^{A_{q}} (ℝ^{N}, Z) .

In general it is not possible to replace B_{p}^{A_{p}} in Theorem 4.1 by a larger space B_{q}^{A_{q}} for some q > p. This follows as in [14, Remark 4.9].

Proof of Theorem 4.1. Throughout this proof, the C_{i}’s denotes constants which depend on at most: X, Y, N, p, q, and ϕ_{0}. Note that X̃ and Ỹ are UMD spaces with Fourier type p by Remark 2.10.

Since M ∈ L_{∞} (ℝ^{N}, B (X̃, Ỹ)) has bounded support, M ∈ L_{1} (ℝ^{N}, B (X̃, Ỹ)). Define the function K by:

ℝ^{N} ∋ s → K(s) := (F^{−1}M) (s) ∈ B (X̃, Ỹ) .

Corollary 3.11 applied to M ∈ B_{p}^{A} (B (X̃, Ỹ)) with w(t) = ln (2 + |t|) gives that for 1 ≤ r ≤ 2

∫_{ℝ^{N}} ∥K(t)F (·)∥_{L_{r}\binom{}{Y}} w (t) dt ≤ C_{1}D ∥F ∥_{L_{r}\binom{}{X}}  for each F ∈ L_{r} \binom{}{X}
∫_{ℝ^{N}} ∥[K(t)]^{∗} G (·)∥_{L_{r}\binom{}{X^{∗}}} w (t) dt ≤ C_{1}D ∥G∥_{L_{r}\binom{}{Y^{∗}}}  for each G ∈ L_{r} \binom{}{Y^{∗}} .

20 MARIA GIRARDI AND LUTZ WEIS

Also, K and K^{∗} are strongly integrable by Corollary 3.9 and supp K̂ ⊂ supp ϕ_{0}. Thus Remark 3.8b can be applied to K ∈ L_{∞} (B (X̃, Ỹ)). Hence for K_{j} (·) := 2^{Nj}K (2^{j} · ), condition (3.13) gives that for each F ∈ S (ℝ^{N}, X̃)

𝔼 ∥∑_{j∈ℤ}ε_{j}K_{j} ∗ F ∥_{L_{q}\binom{}{Y}} ≤ C_{2} C_{1} D ∥F ∥_{L_{q}\binom{}{X}} .  (4.1)

To see that inequality (4.1) implies the boundedness of T_{m}, fix f ∈ S_{o} (X) and let h_{k} := ϕ_{k−1} + ϕ_{k} + ϕ_{k+1}. Then F := \{ȟ_{k} ∗ f\}_{k∈ℤ} ∈ S \binom{}{X} and by Corollary 3.3

∥F ∥_{L_{q}\binom{}{X}} ≤ ∑_{j=−1}^{1} ∥\{ϕ̌_{k+j} ∗ f\}_{k∈ℤ}∥_{L_{q}(X̃)} = 3 ∥\{ϕ̌_{k} ∗ f\}_{k∈ℤ}∥_{L_{q}(X̃)} ≤ C_{3} ∥f∥_{L_{q}(X)} . (4.2)

Note that M ∈ L_{∞} implies that m ∈ L_{∞}; thus it follows from Remark 3.8b that

K_{j} ∗ F = \{ϕ̌_{j} ∗ ȟ_{k} ∗ T_{m(2^{k−j} ·)}f\}_{k∈ℤ} ∈ L_{q} \binom{}{Y} = Rad_{q} (L_{q} (Y))

since, for a.e. s,

(K_{j} ∗ F) (s) = M (2^{−j}s) F̂ (s) = {ϕ_{0} (2^{−j}s) m (2^{k−j}s) h_{k} (s) f̂(s)}_{k∈ℤ}
= {ϕ_{j} (s) h_{k} (s) (T_{m(2^{k−j} ·)}f) (s)}_{k∈ℤ} .

Since h_{k} is 1 on the support of ϕ_{k}, if j = k then ϕ̌_{j} ∗ ȟ_{k} ∗ T_{m(2^{k−j} ·)} (f) = ϕ̌_{j} ∗ (T_{m}f). Hence, by Corollary 3.3 and Lemma 3.12,

∥T_{m}f∥_{L_{q}(Y)} ≤ C_{4} 𝔼_{ε} ∥∑_{j∈ℤ}ε_{j} ϕ̌_{j} ∗ T_{m}f∥_{L_{q}(Y)}

≤ C_{4} 𝔼_{ε,ε′} ∥∑_{k∈ℤ} ε′_{k} [∑_{j∈ℤ}ε_{j}ϕ̌_{j} ∗ ȟ_{k} ∗ T_{m(2^{k−j} ·)}f]∥_{L_{q}(Y)}  (4.3)

= C_{4} 𝔼_{ε} ∥{∑_{j∈ℤ}ε_{j}ϕ̌_{j} ∗ ȟ_{k} ∗ T_{m(2^{k−j} ·)}f}_{k∈ℤ}∥_{Rad_{1}(L_{q}(Y))}

≤ C_{5} 𝔼_{ε} ∥∑_{j∈ℤ}ε_{j} \{ϕ̌_{j} ∗ ȟ_{k} ∗ T_{m(2^{k−j} ·)}f\}_{k∈ℤ}∥_{L_{q}\binom{}{Y}} = C_{5} 𝔼_{ε} ∥∑_{j∈ℤ}ε_{j}K_{j} ∗ F ∥_{L_{q}\binom{}{Y}} .

Combining (4.1), (4.2), and (4.3) gives that ∥T_{m}f∥_{L_{q}(Y)} ≤ C_{6}D ∥f∥_{L_{q}(X)}. ■

The following notation simplifies the statements of corollaries to come.

Notation 4.3. Let RM_{l}^{N} (B (X, Y)) be the set of all measurable functions m: ℝ^{N} \ {0} → B (X, Y) whose distributional derivatives D^{α}m are represented by measurable functions for each α ∈ ℕ_{0}^{N}

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 21

with |α| ≤ l and

∥m∥_{RM_{l}^{N} (B(X,Y))} := max_{\begin{matrix}α∈ℕ_{0}^{N} \\ |α|≤l \end{matrix}} R_{2} (\{|t|^{|α|} D^{α}m (t) : t ∈ ℝ^{N} \ \{0\}\}) < ∞  (4.4)

where l ∈ ℕ_{0}.

Note that if X and Y are Hilbert spaces, then ∥m∥_{M_{l}^{N} (B(X,Y))} = ∥m∥_{RM_{l}^{N} (B(X,Y))}.

The following Mihlin-type multiplier theorem is an immediate consequence of Theorem 4.1.

Corollary 4.4. Let X and Y be UMD spaces with Fourier type p ∈ (1, 2] and l := [\frac{N}{p} ] + 1. If

m ∈ RM_{l}^{N} (B (X, Y)) with ∥m∥_{RM_{l}^{N} (B(X,Y))} := A

then m is a Fourier multiplier from L_{q}(ℝ^{N}, X) to L_{q}(ℝ^{N}, Y) for each q ∈ (1, ∞). Furthermore, ∥T_{m}∥_{L_{q}→L_{q}} ≤ C A for some constant C independent of m.

Remark 4.5. a) Corollary 4.4 applies to Hilbert spaces X and Y with l = [N/2] + 1, which recovers Schwartz’s classical result. For arbitrary UMD spaces, Corollary 4.4, but with l = N, was first shown in [35], for a second proof see [16]. Furthermore, [36, Remark 3.7] shows that the exponent l := [N/p] + 1 is best possible for L_{p}-spaces.

b) The proof will show that the assumption (4.4) on m can be replaced by the following formally weaker assumption:

max_{\begin{matrix}α∈ℕ_{0}^{N} \\ |α|≤l \end{matrix}} sup_{\begin{matrix}t∈ℝ^{N} \\ 2^{−1}≤|t|≤2^{1} \end{matrix}} R_{2} (\{|2^{k}t|^{|α|} D^{α}m(2^{k}t): k ∈ ℤ\}) := A < ∞ .  (4.5)

Proof of Corollary 4.4. Let B := B (Rad_{q} (X) , Rad_{q} (Y)). Without loss of generality, via an approximation argument, supp m ⊂ {t ∈ ℝ^{N}: 2^{−n} ≤ |t| ≤ 2^{n}} for some n ∈ ℕ. Thus, if s ∈ ℝ^{N} then M (s) := {ϕ_{0} (s) m(2^{k}s)}_{k∈ℤ} ∈ B with the k^{th} coordinate of M (s) being zero for |k| > n. By Leibniz’s rule and Remark 2.10a,

∥M∥_{W_{p}^{l}(ℝ^{N} ,B)} ≤ ∑_{|α|≤l} 2^{|α|} ∥|t|^{|α|} D^{α}M (t) χ_{supp ϕ_{0}} (t)∥_{L_{p}(dt,B)}

≤ ∑_{|α|≤l} ∑_{β≤α} 2^{|α|}\binom{α}{β} ∥|t|^{|α−β|} D^{α−β}ϕ_{0} (t) \{|2^{k}t|^{|β|} D^{β}m (2^{k}t)\}_{k∈ℤ}∥_{L_{p}(dt,B)}

= ∑_{|α|≤l} ∑_{β≤α} 2^{|α|}\binom{α}{β} ∥|t|^{|α−β|} D^{α−β}ϕ_{0} (t) R_{q} (\{|2^{k}t|^{|β|} D^{β}m (2^{k}t) : k ∈ ℤ\}) χ_{[2^{−1}≤|t|≤2]} ∥_{L_{p}(dt,ℝ)} .

By (4.5) and Hölder’s inequality, M ∈ W_{p}^{l} (ℝ^{N}, B). By Remark 4.2c, Theorem 4.1 applies. ■

Corollary 4.6. Let X and Y be UMD spaces with Fourier type p ∈ (1, 2] and l := [\frac{N}{p} ] + 1. Let m ∈ RM_{l}^{N} (B (X, Y)) and set ∥m∥_{RM_{l}^{N} (B(X,Y))} := A.

(a) Then m is a Fourier multiplier from H_{1} (ℝ^{N}, X) to L_{1} (ℝ^{N}, Y ) and ∥T_{m}∥_{H_{1}→L_{1}} ≤ C A for

22 MARIA GIRARDI AND LUTZ WEIS

some constant C independent of m.

(b) Then m is a Fourier multiplier from L_{1} (ℝ^{N}, X) to L_{1}^{wk} (ℝ^{N}, Y ) and ∥T_{m}∥_{L_{1}→L_{1}^{wk}} ≤ C A for some constant C independent of m.

(c) Then m is a Fourier multiplier from L_{∞} (ℝ^{N}, X) to BMO (ℝ^{N}, Y ) and ∥T_{m}∥_{L_{∞}→BMO} ≤ C A for some constant C independent of m.

Remark 4.7. A related result, which also covers the spaces H_{p} for p < 1 was found independently by T. Hytönen [18].

A word about what is meant by a Fourier multiplier for UMD spaces X and Y for the function spaces in Corollary 4.6 is in order. Since S_{H} (ℝ^{N}, X) := {f ∈ S (ℝ^{N}, X) : ∫_{ℝ^{N}} f (t) dt = 0} is norm-dense in H_{1} (ℝ^{N}, X), it is clear what is meant by a Fourier multiplier from H_{1} to L_{1}: in Definition 2.4, just replace S by S_{H} and L_{q} by either H_{1} or L_{1}, accordingly. A linear mapping T: L_{1} (ℝ^{N}, X) → L_{1}^{wk} (ℝ^{N}, Y ) is (uniformly) continuous if and only if

∥T ∥_{L_{1}→L_{1}^{wk}} := sup {∥T f∥_{L_{1}^{wk}} : ∥f∥_{L_{1}} ≤ 1} < ∞ .

If a linear mapping T: S (ℝ^{N}, X) → L_{1}^{wk} (ℝ^{N}, Y ) satisfies

∥T f∥_{L_{1}^{wk}(ℝ^{N} ,Y)} ≤ C ∥f∥_{L_{1}(ℝ^{N} ,X)} for each f ∈ S (ℝ^{N}, X) ,

then, since S (ℝ^{N}, X) is a norm-dense subspace of L_{1} (ℝ^{N}, X), there is a unique continuous linear extension T̃: L_{1} (ℝ^{N}, X) → L_{1}^{wk} (ℝ^{N}, Y ) of T and ∥T̃∥_{L_{1}→L_{1}^{wk}} ≤ 2C. Thus it is clear what is meant by a Fourier multiplier from L_{1} to L_{1}^{wk}: in Definition 2.4, just replace L_{q} by either L_{1} or L_{1}^{wk}, accordingly. The Schwartz class is not norm-dense in L_{∞}; however, it is weak^{∗}-dense in L_{∞}. Since X and Y are UMD spaces (in particular, reflexive)

[H_{1} (ℝ^{N}, Y^{∗})]^{∗} = BMO (ℝ^{N}, Y ) and [L_{1} (ℝ^{N}, Y^{∗})]^{∗} = L_{∞} (ℝ^{N}, Y ) ,

(cf. [9] for H_{1}–BMO duality). Thus m is a Fourier multiplier from L_{∞} (ℝ^{N}, X) to BMO (ℝ^{N}, Y ) provided there is an operator T_{m} ∈ B (L_{∞} (ℝ^{N}, X) , BMO (ℝ^{N}, Y )) satisfying:

T_{m}f := [m f̂]^{∨} for each f ∈ S (ℝ^{N}, X)
T_{m} is weak^{∗}-to-weak^{∗} continuous .

(4.6)

Note that (4.6) guarantees the uniqueness of a norm-to-norm continuous operator, if it exists.

Proof of Corollary 4.6. Throughout this proof, the C_{i}’s are constants that are independent of m and the fixed n ∈ ℕ.

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 23

For each j ∈ ℤ and a fixed n ∈ ℕ, let

m_{j} := m ϕ_{j}  M_{n} := ∑_{j=−∞}^{n} m_{j}
k_{j} := m̌_{j}  K_{n} := M̌_{n} = ∑_{j=−∞}^{n} k_{j} .

Note that M_{n} ∈ RM_{l}^{N} (B (X, Y)) with

∥M_{n}∥_{RM_{l}^{N} (B(X,Y))} ≤ C_{0} A .

So T_{M_{n}} ∈ B (L_{2} (ℝ^{N}, X) , L_{2} (ℝ^{N}, Y )) by Corollary 4.4. That T_{M_{n}} satisfies parts (a) and (b), for some constant C independent of m and the fixed n ∈ ℕ, follows from the Benedek-Calderon-Panzone theorem for convolutions on vector-valued spaces (see [13, Ch. V, Thm 3.4 and Remark (3.3′) on p. 494]) if one can show that

∫_{|t|>2|s|} ∥K_{n} (t − s) x − K_{n} (t) x∥_{Y} dt ≤ A C_{1} ∥x∥_{X}  (4.7)

for each s ∈ ℝ^{N} and x ∈ X. Inequality (4.7) is now shown via an adaption of the classical argument in [32, Ch. VI, Sect. 4.4].

Fix x ∈ X. For α ∈ ℕ_{0}^{N} with |α| ≤ l, by Leibniz’s rule,

∥D^{α}m_{j} (t)∥_{B(X,Y)} ≤ A C_{3}2^{−j|α|} ;

thus, since (D^{α}m_{j})^{∨} (t) = (−it)^{α} k_{j} (t) and Y has Fourier type p,

[∫_{ℝ^{N}} ∥(−it)^{α} k_{j} (t) x∥_{Y}^{p′} dt]^{1/p′} ≤ C_{4} [∫_{supp ϕ_{j}} ∥D^{α}m_{j} (t) x∥_{Y}^{p} dt]^{1/p} ≤ A C_{5} 2^{−j|α|} 2^{\frac{jN}{p}} ∥x∥_{X} .

Thus for each l_{0} ∈ {0, 1, …, l}

[∫_{ℝ^{N}} ∥|t|^{l_{0}} k_{j} (t) x∥_{Y}^{p′} dt]^{1/p′} ≤ A C_{6} 2^{\frac{jN}{p}} 2^{−jl_{0}} ∥x∥_{X} .  (4.8)

Using (4.8) with l = l_{0}, Hölder’s inequality gives, for any a > 0, since N/p < l,

\begin{aligned}∫_{|t|≥a} ∥k_{j} (t) x∥_{Y} dt & ≤ [∫_{ℝ^{N}} ∥|t|^{l} k_{j} (t) x∥_{Y}^{p′} dt]^{1/p′} [∫_{|t|≥a} |t|^{−lp} dt]^{1/p} \\ & ≤ A C_{6} 2^{\frac{jN}{p}} 2^{−jl} ∥x∥_{X} a^{\frac{N}{p}} a^{−l} [∫_{|t|≥1} |t|^{−lp} dt]^{1/p} \\ & ≤ A C_{7} (2^{j}a)^{\frac{N}{p} −l} ∥x∥_{X} . \end{aligned}  (4.9)

Similarly, taking l_{0} = 0 in (4.8) gives

∫_{|t|≤a} ∥k_{j} (t) x∥_{Y} dt ≤ [∫_{ℝ^{N}} ∥k_{j} (t) x∥_{Y}^{p′} dt]^{1/p′} [∫_{|t|≤a} 1 dt]^{1/p} ≤ A C_{8} (2^{j}a)^{\frac{N}{p}} ∥x∥_{X} . (4.10)

Choosing a = 2^{−j} in (4.9) and (4.10) gives

∫_{ℝ^{N}} ∥k_{j} (t) x∥_{Y} dt ≤ A C_{9} ∥x∥_{X} .

Let n_{j} (t) := [D^{α}k_{j}] (t) = (i)^{|α|} t^{α}m_{j} (t) for each j ∈ ℤ. For β ∈ ℕ_{0}^{N} with |β| ≤ l

∥D^{β}n_{j} (t)∥_{B(X,Y)} ≤ A C_{10} 2^{j|α|} 2^{−j|β|}

by Leibniz’s rule; thus, since [D^{β}n_{j}]^{∨} (t) = (−i)^{|β|} t^{β} D^{α}k_{j} (t) and Y has Fourier type p,

[∫_{ℝ^{N}} ∥t^{β}D^{α}k_{j} (t) x∥_{Y}^{p′} dt]^{1/p′} ≤ C_{11} [∫_{ℝ^{N}} ∥D^{β}n_{j} (t) x∥_{Y}^{p} dt]^{1/p}
≤ A C_{12} 2^{j|α|} 2^{−j|β|}2^{jN/p} .

Arguing as above gives that

∫_{ℝ^{N}} ∥D^{α}k_{j} (t) x∥_{Y} dt ≤ A C_{13}2^{j|α|} ∥x∥_{X} .  (4.11)

Thus, for each h = (h_{1}, …, h_{N}) ∈ ℝ^{N},

\begin{aligned}∫_{ℝ^{N}} ∥k_{j} (t + h) x − k_{j} (t) x∥_{Y} dt & ≤ ∫_{0}^{1} (∑_{i=1}^{N} |h_{i}| ∫_{ℝ^{N}} ∥\frac{∂}{∂t_{i}} k_{j} (t + sh) x∥_{Y} dt) ds \\ & ≤ A C_{14} 2^{j} |h| ∥x∥_{X} ; \end{aligned}  (4.12)

indeed, just apply the Fundamental Theorem of Calculus to ℝ^{n} ∋ s → k_{j} (t + sh) x ∈ Y and then use (4.11). Fix s ∈ ℝ^{N}. Note that the inequality

∑_{j∈ℤ} ∫_{|t|>2|s|} ∥k_{j} (t − s) x − k_{j} (t) x∥_{Y} dt ≤ A C_{15} ∥x∥_{X}  (4.13)

implies (4.7). To see (4.13), let Γ_{1} := {j ∈ ℕ: 2^{j} ≤ |s|^{−1}} and Γ_{2} := {j ∈ ℕ: |s|^{−1} < 2^{j}}. By (4.12),

∑_{j∈Γ_{1}} ∫_{|t|>2|s|} ∥k_{j} (t − s) x − k_{j} (t) x∥_{Y} dt ≤ A C_{14} ∥x∥_{X} [|s| ∑_{j∈Γ_{1}} 2^{j}] ≤ A C_{16} ∥x∥_{X} .

By (4.9)

∑_{j∈Γ_{2}} ∫_{|t|>2|s|} ∥k_{j} (t − s) x − k_{j} (t) x∥_{Y} dt ≤ 2 ∑_{j∈Γ_{2}} ∫_{|t|>1 |s|} ∥k_{j} (t) x∥_{Y} dt

≤ 2 A C_{7} ∥x∥_{X} [|s|^{\frac{N}{p} −l} ∑_{j∈Γ_{2}} (2^{\frac{N}{p} −l})^{j}] ≤ A C_{17} ∥x∥_{X}

since N/p < l. This completes the proof of (4.7). Thus T_{M_{n}} satisfies parts (a) and (b) for some constant C_{18} independent of m and n ∈ ℕ.

Towards showing part (a), fix f ∈ S_{H} (ℝ^{N}, X) and let

G := {g ∈ S (ℝ^{N}, Y^{∗}) : ∥g∥_{L_{∞}(Y^{∗})} ≤ 1} .

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 25

Then

∥T_{m}f∥_{L_{1}(ℝ^{N} ,Y)} = sup_{g∈G} ⟨g, T_{m}f⟩ = sup_{g∈G} lim_{n→∞} ⟨g, T_{M_{n}}f⟩ ≤ sup_{g∈G} \underline{lim}_{n→∞} ∥g∥_{L_{∞}(Y^{∗})} ∥T_{M_{n}}f∥_{L_{1}(Y)}

≤ \underline{lim}_{n→∞} ∥T_{M_{n}}∥_{H_{1}(X)→L_{1}(Y)} ∥f∥_{H_{1}(X)} ≤ A C_{18} ∥f∥_{H_{1}(X)} .

Thus part (a) holds.

If f ∈ S_{∗} (ℝ^{N}, X) := {g ∈ S (ℝ^{N}, X) : supp ĝ is compact}, then T_{m}f = T_{M_{n}}f for n sufficiently large. Since S_{∗} (ℝ^{N}, X) is norm-dense in L_{1} (ℝ^{N}, X), part (b) holds.

Part (c) follows from part (a) and a duality argument. Since X and Y have nontrivial (Fourier) type, m^{∗} ∈ RM_{l}^{N} (B (Y^{∗}, X^{∗})) and ∥m^{∗}∥_{RM_{l}^{N} (B(Y^{∗},X^{∗}))} ≤ C_{19} ∥m∥_{RM_{l}^{N} (B(X,Y))} (cf., eg., [39, Thm. 2.2.14]). Thus T_{m^{∗}} ∈ B (H_{1} (ℝ^{N}, Y^{∗}) , L_{1} (ℝ^{N}, X^{∗})) with ∥T_{m^{∗}}∥_{H_{1}→L_{1}} ≤ A C_{20} by part (a). Hence (T_{m^{∗}})^{∗} ∈ B (L_{∞} (ℝ^{N}, X) , BMO (ℝ^{N}, Y )) is weak^{∗}-to-weak^{∗} continuous and is of norm at most A C_{20}. Furthermore, (T_{m^{∗}})^{∗} |_{S(ℝ^{N} ,X)} = T_{m(−·)} |_{S(ℝ^{N} ,X)} . Thus part (c) holds. ■

Remark 4.8. One can use the estimates in the proof of Corollary 4.6 to give a more direct proof of Corollary 4.4 avoiding Besov spaces. We indicate this for p = 1, l = [N] + 1: By assumption M(t) = (ϕ_{0}(t)m(2^{k}t)) and M^{(α)}, where |α| ≤ N +1, have norm less than C in B (Rad (X) , Rad (Y)). Estimates (4.9) and (4.10) applied to K(t) = M(t)^{∨} with p = 1 give ∫_{ℝ^{N}} ∥k(t)∥ dt < ∞ for the operator norm on B (Rad (X) , Rad (Y)). Now apply Remark 3.8a and finish the proof as in the last part of the proof of Theorem 4.1.

Our results are easily adapted to Sobolev spaces. Corollary 4.9 gives a flavor of this for fractional Sobolev spaces (see Definition 2.15).

Corollary 4.9. Let X and Y have UMD and Fourier type p ∈ (1, 2] and l := [\frac{N}{p} ] + 1 and s ∈ ℝ. Assume that m ∈ C^{l} (ℝ^{N} \ {0} , B (X, Y)) satisfies that

max_{\begin{matrix}α∈ℕ_{0}^{N} \\ |α| ≤ l \end{matrix}} R_{2} (\{ (1 + |t|^{2})^{s/2} |t|^{|α|} (D^{α}m) (t) : t ∈ ℝ^{N} \ \{0\}\} ) := A ≤ ∞ .  (4.14)

Then m is a Fourier multiplier from H_{q}^{r} (ℝ^{N}, X) to H_{q}^{r+s} (ℝ^{N}, Y ) for each r ∈ ℝ and q ∈ (1, ∞). Also, ∥T_{m}∥_{H_{q}^{r}→H_{q}^{r+s}} ≤ C A for some constant C independent of m.

Proof. It suffices to show that n (·) := (1 + |·|^{2})^{s/2} m (·) is a Fourier multiplier from L_{q} (ℝ^{N}, X) to L_{q} (ℝ^{N}, Y ); for then, the following diagram

H_{q}^{r} (ℝ^{N}, X) ⟶^{T_{m}} H_{q}^{r+s} (ℝ^{N}, Y )
J^{r}↓ ↓_{J^{r+s}}
L_{q} (ℝ^{N}, X) ⟶^{T_{n}} L_{q} (ℝ^{N}, Y )

commutes, where J^{u}: H_{q}^{u} (ℝ^{N}, Z) → L_{q} (ℝ^{N}, Z) is the isometry defined by

J^{u} (f) := [(1 + |·|^{2})^{u/2} f̂(·)]^{∨} for f ∈ S (ℝ^{N}, Z) .

But

|t|^{|α|} D^{α}n (t) = |t|^{|α|} D^{α} [(1 + |·|^{2})^{s/2} m (·)] (t)

= ∑_{β≤α} \binom{α}{β} \{|t|^{|β|} \frac{D^{β} [(1 + |·|^{2})^{s/2}] (t)}{(1 + |t|^{2})^{s/2}} \} [(1 + |t|^{2})^{s/2} |t|^{|α−β|} D^{α−β}m (t)] (4.15)

and the terms {⋯ } in (4.15) are uniformly bounded on ℝ^{N} \ {0}. Thus the assumption (4.14) gives that n ∈ RM_{l}^{N} (B (X, Y)) and so Corollary 4.4 finishes off the proof. ■

Hörmander’s condition takes here the following form.

Corollary 4.10. Let X and Y be UMD spaces with Fourier type p ∈ (1, 2] and l := [\frac{N}{p} ] + 1. Let m: ℝ^{N} \ {0} → B (X, Y) be a measurable function whose distributional derivatives D^{α}m are represented by measurable functions for each α ∈ ℕ_{0}^{N} with |α| ≤ l and

max_{\begin{matrix}α∈ℕ_{0}^{N} \\ |α| ≤ l \end{matrix}} ∫_{2^{−1}<|t|<2} [ R_{2} ( \{ |2^{k}t|^{|α|} D^{α}m (2^{k}t) : k ∈ ℤ \} ) ]^{p} dt := A < ∞ .  (4.16)

Then m is a Fourier multiplier from L_{q}(X) to L_{q}(Y) for each 1 < q < ∞. Also, ∥T_{m}∥_{L_{q}→L_{q}} ≤ CA for some constant C independent of m.

Proof. In the proof of Corollary 4.4, just replace (4.5) by (4.16) to see that Theorem 4.1 applies in this setting also. ■

The following corollary shows that our result improves on the multiplier theorem of Bourgain even in the case of scalar-valued multiplier functions m. Its proof uses the following equivalent norm [29, Prop. 3.1] on the Besov spaces B_{p,1}^{s} (ℝ, Z) for 0 < s < 1 and 1 ≤ p < ∞:

∥f∥′_{B_{p,1}^{s}(ℝ,Z)} := ∥f∥_{L_{p}(ℝ,Z)} + B_{p,1}^{s} (f)

where f_{h} (u) := f (u + h),

B_{p,1}^{s} (f) := ∫_{0}^{∞} t^{−s} w_{p} (f, t) \frac{dt}{t}
w_{p} (f, t) := sup_{|h|≤t} ∥f_{h} − f∥_{L_{p}(ℝ,Z)} .

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON L_{p}(X) 27

Corollary 4.11. Let X and Y be UMD spaces with Fourier type p ∈ (1, 2]. Assume that for the

function m : ℝ \ \{0\} → B (X, Y) there is some l ∈ (\frac{1}{p}, 1) and h_{0} > 0 so that, for some constant A,

\begin{matrix}R_{2} (\{m (u) : u ∈ ℝ \ \{0\}\}) ≤ A \\ R_{2} (\{|u|^{l} \frac{m (u + h) − m (u)}{|h|^{l}} : u, u + h ∈ ℝ \ \{0\} and 0 < |h| ≤ |u| h_{0}\}) ≤ A . \end{matrix} (4.17)

Then m is a Fourier multiplier from L_{q} (ℝ, X) to L_{q} (ℝ, Y) for each q ∈ (1, ∞). Furthermore,

∥T_{m}∥_{L_{q}→L_{q}} ≤ C A for some constant C independent of m.

Proof. Throughout this proof, the C_{j}’s are constants independent of m. Without loss of general-

ity, h_{0} < \frac{1}{4}. Fix s so that 0 < \frac{1}{p} < s < l < 1.

Set m (0) := 0. For t ∈ ℝ, let m_{k}(t) := ϕ_{0}(t)m(2^{k}t) and M (t) := {m_{k} (t)}_{k∈ℤ}. Note that

|\frac{u}{h}|^{l} [m_{k}(u + h) − m_{k}(u)] =

ϕ_{0}(u) |\frac{u}{h}|^{l} [m (2^{k} (u + h)) − m (2^{k}u)] + m (2^{k} (u + h)) |\frac{u}{h}|^{l} [ϕ_{0} (u + h) − ϕ_{0} (u)] .

So, since ϕ_{0} ∈ S and l < 1, the assumption (4.17) yields, for each u, h ∈ ℝ\{0} with 0 < |h| ≤ |u| h_{0},

R_{2} ({m_{k} (u + h) − m_{k} (u) : k ∈ ℤ}) ≡

|\frac{h}{u}|^{l} R_{2} (\{|\frac{u}{h}|^{l} [m_{k} (u + h) − m_{k} (u)] : k ∈ ℤ\}) ≤ |\frac{h}{u}|^{l} A C_{1} .

Thus, if |h| ≤ \frac{h_{0}}{4} then, with B := B (Rad_{q} (X) , Rad_{q} (Y)),

∥M_{h} − M∥_{L_{p}(B)} = [∫ 3_{\frac{1}{4}} ∥M (u + h) − M (u)∥_{B}^{p} du]^{\frac{1}{p}} ≤ A C_{2} [∫ 3_{\frac{1}{4}} |\frac{h}{u}|^{lp} du]^{\frac{1}{p}} ≤ A C_{3} |h|^{l}

since \frac{1}{p} < l. Also, ∥M_{h} − M∥_{L_{p}(B)} ≤ 2 ∥M∥_{L_{p}(B)} ≤ A C_{4} for any h ∈ ℝ. Thus

w_{p} (M, t) ≤ \begin{cases}A C_{5} t^{l} & if t ≤ \frac{h_{0}}{4} \\ A C_{6} & if t > \frac{h_{0}}{4} . \end{cases}

Thus B_{p,1}^{s} (M) ≤ A C_{8} since 0 < s < l and so M ∈ B_{p,1}^{s} (ℝ, B) with norm at most A C_{9}. By

Remark 4.2c, since \frac{1}{p} < s, we can apply Theorem 4.1, one last time. ■

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Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

Current address: Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany

   E-mail address: girardi@math.sc.edu

Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany

   E-mail address: Lutz.Weis@math.uni-karlsruhe.de