Math. Nachr. [to appear] (2002), 0 – 0

Operator-valued Fourier multiplier theorems on Besov spaces

By Maria Girardi of Columbia, South Carolina, Lutz Weis of Karlsruhe, Germany

(Received 2 July 2001; accepted 4 January 2002)

Abstract. Presented is a general Fourier multiplier theorem for operator-valued multiplier functions on vector-valued Besov spaces where the required smoothness of the multiplier functions depends on the geometry of the underlying Banach space (specifically, its Fourier type). The main result covers many classical multiplier conditions, such as Mihlin and Hörmander conditions.

1. INTRODUCTION

In recent years, Fourier multiplier theorems with operator-valued multiplier functions have found many applications in the theory of evolution equations, in particular, in connection with: maximal regularity of parabolic equations [2, 28, 27, 6, 18, 4], stability theory [29, 15, 20], elliptic operators on infinite dimensional state spaces [1], and pseudo differential operators on manifolds with singularities [25]. Of particular interest are versions of Mihlin’s multiplier theorem for functions m, from ℝ^{N} into the space B (X) of bounded operators on a Banach space X, which satisfy that

(1.1) the set  { (1 + |t|)^{|α|} D^{α}m (t) : t ∈ ℝ^{N} , |α| ≤ l}  is norm bounded ;

specifically, one would like to know for which X-valued function spaces E (ℝ^{N}, X) can one extend the operator

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

defined on the Schwartz class, to a bounded operator from E (ℝ^{N}, X) into itself.

For the Bochner spaces L_{p} (ℝ^{N}, X), additional hypotheses are needed, specifically, extension is possible only when the underlying Banach space X has the UMD property, 1 < p < ∞, and the set in (1.1) is R-bounded (see [9, 30, 21] for scalar-valued m and [28, 26] for operator-valued m; see also [10, 14, 4]).

1991 Mathematics Subject Classification. Primary 42B15, 46E40, 46E35; Secondary 46B20.

Keywords and phrases. vector-valued Besov spaces, Mihlin multiplier theorem, Fourier type \begin{matrix}c \\ ◯ \end{matrix}WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0025-584X/02/[to appear]00-0000 $ 17.50+.50/0

In sharp contrast to these results for L_{p} (ℝ^{N}, X), it was discovered independently by H. Amann and L. Weis that for Besov spaces B_{q,r}^{s} (ℝ^{N}, X), additional restrictions on X and m are not needed and all indices s ∈ ℝ and q, r ∈ [1, ∞] are allowed (for a detailed proof of the Mihlin case with l = N + 1 see [2], in [29] a more general result was announced).

This paper presents a more elementary proof of these multiplier theorems (avoiding the theory of vector-valued distributions), which has rather general assumptions on m (again expressed in terms of Besov spaces) and which also allows for the determination of the optimal order l of derivatives in (1.1) in terms of the geometry of the underlying Banach space X. Indeed, Corollary 4.11 gives that for a Banach space X of Fourier type p ∈ [1, 2] one may choose l = [N/p]+1. 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. Since a Hilbert space has Fourier type 2, one recovers the classical result of J. Schwartz (cf. [5, Section 6.1]). Since each Banach space has Fourier type 1, one obtains the main result in [2]. It follows from a result of Bourgain [9] that each uniformly convex space has Fourier type p for some p > 1; thus, for uniformly convex spaces one may choose l = N. If X is a subspace of an L_{q}(Ω) space, then X has Fourier type p = min(q, q′); thus, the l in (1.1) improves (i.e. decreases) as q tends towards 2. Furthermore, the order l = [N/p] + 1 is optimal with respect to the scale of Besov spaces. In the case of Bochner spaces L_{p} (ℝ^{N}, X) on can obtain analogous results to the ones described above (see [13]); however, the method of proof is quite different.

The paper is organized as follows. Section 2 collects definitions and basic properties of vector-valued function spaces, in particular, Besov spaces. Section 3 contains basic estimates for the Fourier transform on Besov spaces B_{q,r}^{s} (ℝ^{N}, X) for a Banach space X with Fourier type p. Section 4 presents the multiplier theorems. Theorem 4.3 gives a multiplier theorem on L_{q} (ℝ^{N}, X); its assumption is much stronger than (1.1). Applying this result to the blocks of the Paley Littlewood decomposition gives a multiplier theorem for Besov spaces B_{q,r}^{s} (ℝ^{N}, X), Theorem 4.8, whose proof is almost trivial if q, r ∈ [1, ∞). The case q = ∞ or r = ∞ requires a bit more care. The assumptions in Theorem 4.8 are stated in a very general form. From this general formulation follows as immediate corollaries (see Corollaries 4.11, 4.13, 4.14) vector-valued generalizations of several classical multiplier theorems conditions (Mihlin’s condition, Hörmanders condition, Lipschitz condition).

Theorem 4.8 was announced in [29]; the present expanded paper would have never seen the light of day without the collaboration of the first named author.

2. BASICS

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 conjugate exponent p′ of p ∈ [1, ∞] is given by \frac{1}{p} + \frac{1}{p′} = 1. Also, ℕ = \{1, 2, …\} is the set of natural numbers while ℕ_{0} = {0} ∪ ℕ. 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.

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 the locally convex topology generated by the seminorms

∥φ∥_{k,α} := sup_{t∈ℝ^{N}} (1 + |t|^{2})^{k} ∥(D^{α}φ) (t)∥_{X}

for all k ∈ ℕ_{0} and α ∈ ℕ_{0}^{N}. As customary, we often denote S (ℝ^{N}, ℂ) by just S. Note that S (X) is norm (resp. σ (L_{p} (X) , L_{p′} (X^{∗}))) dense in L_{p} (X) when 1 ≤ p < ∞ (resp. 1 ≤ p ≤ ∞).

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 sufficiently bounded function m : ℝ^{N} → X (e.g. a measurable function which grows at most polynominally as |x| → ∞ or an L_{p} (X) function for 1 ≤ p ≤ ∞) 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, we will identify such a function m with L_{m} ∈ S′ (X).

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

(2.1)  (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 derivative, translation, and dilation properties of F and F^{−1} which hold in the scalar-valued case also hold in the vector-valued case. However, the Hausdorff-Young inequality need not hold. Thus we have to consider the following class of Banach spaces that was introduced by Peetre [22].

Definition 2.1. Let X be a Banach space and 1 ≤ p ≤ 2. Let F_{p,N}(X) be the smallest C ∈ [0, ∞] so that

(2.2)  ∥Ff∥_{L_{p′} (ℝ^{N} ,X)} ≤ C ∥f∥_{L_{p}(ℝ^{N} ,X)}  for each f ∈ S (ℝ^{N}, X) .

X has Fourier type p provided the Fourier type constant F_{p,N}(X) is finite for some (and thus then, by [17], for each) N ∈ ℕ.

Remark 2.2. 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 [19]. A space L_{q} ((Ω, Σ, μ) , ℝ) has Fourier type p = min(q, q′) [22]. Each closed subspace (by definition) and each quotient space (by duality) of a Banach space X has the same Fourier type as X. Bourgain [7, 9] has shown that each B– convex Banach space (thus, in particular, each uniformly convex Banach space) has some non-trivial Fourier type p > 1.

For completeness we include the proof the the following well-known proposition.

Proposition 2.3. Let X have Fourier type p ∈ [1, 2] and p ≤ q ≤ p′. Then X^{∗} and L_{q}(ℝ^{N}, X) also have Fourier type p. Specifically, F_{p,M} (X^{∗}) = F_{p,M} (X) and F_{p,M} (L_{q} (ℝ^{N}, X)) = F_{p,M} (X) for each M ∈ ℕ.

Proof. If X is an isometric subspace of Z then F_{p,M} (X) ≤ F_{p,M} (Z); thus, it suffices to show just ≤ in the two claimed equalities. The claim for X^{∗} follows from the fact that the adjoint of F: L_{p} (X) → L_{p′} (X), restricted to L_{p} (X^{∗}), is the Fourier transform map from L_{p} (X^{∗}) to L_{p′} (X^{∗}). Let h be a simple function with finite support in L_{p} (ℝ^{M}, L_{q} (ℝ^{N}, X)); thus, [ĥ (t)] (u) = [h (·) (u)] (t). By the general Minkowski-Jessen inequality, since q ≤ p′ and p ≤ q (and wlog p ≠ 1),

[ ∫_{ℝ^{M}} ( ∫_{ℝ^{N}} ∥[ĥ (t)] (u)∥^{q} du)^{p′/q} dt]^{1/p′}

≤ [∫_{ℝ^{N}} (∫_{ℝ^{M}} ∥[h (·) (u)] (t)∥^{p′} dt)^{q/p′} du]^{1/q}

≤ C [∫_{ℝ^{N}} (∫_{ℝ^{M}} ∥h (s) (u)∥^{p} ds)^{q/p} du]^{1/q}

≤ C [∫_{ℝ^{M}} (∫_{ℝ^{N}} ∥h (s) (u)∥^{q} du)^{p/q} ds]^{1/p} ,

where C = F_{p,M} (X). This shows the claim for L_{q}(ℝ^{N}, X). □

In order to define Besov spaces we consider the dyadic-like subsets, {J_{k}}_{k=0}^{∞} and {I_{k}}_{k=0}^{∞}, of ℝ^{N}. Let {J_{k}}_{k=0}^{∞} be the partitioning of ℝ^{N} given by

(2.3) J_{0} = {t ∈ ℝ^{N} : |t| ≤ 1} , J_{k} = {t ∈ ℝ^{N} : 2^{k−1} < |t| ≤ 2^{k}} for k ∈ ℕ .

Enlarge each J_{k} to form a sequence {I_{k}}_{k=0}^{∞} of overlapping subsets defined by

(2.4) I_{0} = {t ∈ ℝ^{N}: |t| ≤ 2} , I_{k} = {t ∈ ℝ^{N}: 2^{k−1} < |t| ≤ 2^{k+1}} for k ∈ ℕ .

To simplify notation later, let J_{k} = ∅ for k < 0. Next define a partition of unity {φ_{k}}_{k∈ℕ_{0}} of functions from S (ℝ^{N}, ℝ) as follows. Take a nonnegative function ψ ∈ S (ℝ, ℝ) with support in [2^{−1}, 2] which satisfies

∑_{k=−∞}^{∞} ψ(2^{−k}s) = 1 for s ∈ ℝ \ {0}

and let, for t ∈ ℝ^{N},

φ_{k}(t) = ψ (2^{−k}|t|) for k ∈ ℕ  and  φ_{0}(t) = 1 − ∑_{k=1}^{∞} φ_{k}(t) .

To simplify notation later on, let φ_{k} ≡ 0 if k < 0. Note the following useful properties:

supp φ_{k} ⊂ \overline{I_{k}} for each k ∈ ℕ_{0}
∑_{k=0}^{∞} φ_{k}(s) = 1 for each s ∈ ℝ^{N}
J_{m} ∩ supp φ_{k} = ∅ if |m − k| > 1
φ_{k−1}(s) + φ_{k}(s) + φ_{k+1}(s) = 1 for each s ∈ supp φ_{k} , k ∈ ℕ_{0} .

Among the many equivalent descriptions of Besov spaces, the most useful one for us is given in terms of the so called Littlewood–Paley decomposition. Roughly speaking this means that we consider f ∈ S′ (X) as a distributional sum f = ∑_{k} f_{k} of analytic functions f_{k} whose Fourier transforms have support in dyadic-like I_{k} and then define the Besov norm in terms of the f_{k}’s.

Definition 2.4. Let 1 ≤ q, r ≤ ∞ and s ∈ ℝ. The Besov space B_{q,r}^{s} (ℝ^{N}, X) is the space of all f ∈ S′ (ℝ^{N}, X) for which

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

≡ \{ \begin{matrix}[ ∑_{k=0}^{∞} 2^{ksr} ∥φ̌_{k} ∗ f∥_{L_{q}(X)}^{r} ]^{1/r} & if r ≠ ∞ \\ sup_{k∈ℕ_{0}} [ 2^{ks} ∥φ̌_{k} ∗ f∥_{L_{q}(X)} ] & if r = ∞ \end{matrix}

is finite; q is the main index while s is the smoothness index. B_{q,r}^{s} (ℝ^{N}, X), together

with the norm in (2.5), is a Banach space. B^{∘} _{q,r}^{s} (ℝ^{N}, X) is the closure of S (ℝ^{N}, X) in B_{q,r}^{s} (ℝ^{N}, X), with the induced norm.

Different choices of {φ_{k}} lead to equivalent norms on B_{q,r}^{s} (ℝ^{N}, X) [23, Lem. 3.2]. A well-known [23, Prop. 3.1] equivalent norm on the Besov spaces B_{q,r}^{s} (ℝ, X), for 0 < s < 1 and 1 ≤ q, r < ∞, is

(2.6)  ∥f∥B_{q,r}^{s}(X)^{′} = ∥f∥_{L_{q}(X)} + B_{q,r}^{s} (f)

where f_{h} (s) = f (s + h),

\begin{aligned}B_{q,r}^{s} (f) & = (∫_{0}^{∞} [t^{−s}w_{q} (f, t)]^{r} \frac{dt}{t})^{1/r} \\ w_{q} (f, t) & = sup_{|h|≤t} ∥f_{h} − f∥_{L_{q}(X)} ; \end{aligned} (2.7)

there are similar expressions for other values of s and N > 1. Besov spaces also can be introduced via the real interpolation method (cf. [2]): for m ∈ ℕ_{0}, s ∈ (0, m), q ∈ [1, ∞), and r ∈ [1, ∞]

B_{q,r}^{s}(ℝ^{N}, X) = (L_{q} (ℝ^{N}, X) , W_{q}^{m} (ℝ^{N}, X))_{\frac{s}{m} ,r}

where W_{q}^{m} (ℝ^{N}, X) are usual Sobolev spaces, which are defined below.

Definition 2.5. 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| ≤ 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-know that the Sobolev spaces are Banach spaces.

We collect some known facts about Besov and Sobolev spaces (cf. [2]).

Fact 2.6. Let s ∈ ℝ and 1 ≤ q, r ≤ ∞ and m ∈ ℕ_{0}. Here, ↪ denotes that the natural injection is a continuous linear operator. Then:

S (X) ↪ W_{q}^{m} (X) ↪ S′ (X)

S (X) ↪ B_{q,r}^{s} (X) ↪ S′ (X) (2.8)

(2.9) W_{q}^{m+1}(X) ↪ B_{q,r}^{s}(X) ↪ W_{q}^{m}(X) ↪ L_{q}(X)  if m < s < m + 1
B_{q,r}^{s}(X) ↪ B_{q,r+ε}^{s}(X)  if ε > 0
B_{q,∞}^{s}(X) ↪ B_{q,1}^{s−ε}(X)  if ε > 0
B^{∘} _{q,r}^{s}(X) = B_{q,r}^{s}(X)  if q, r < ∞ .

Also B_{q_{1},1}^{N/q_{1}} (ℝ^{N}, X) ↪ B_{q_{2},1}^{N/q_{2}} (ℝ^{N}, X) ↪ L_{∞} (ℝ^{N}, X) provided 1 ≤ q_{1} ≤ q_{2} < ∞.

Let (E (Z) , E^{∗} (Z^{∗})) be one of the pairs, where 1 ≤ q, r ≤ ∞ and s ∈ ℝ:

(L_{q} (Z) , L_{q′} (Z^{∗})) or (B_{q,r}^{s} (Z) , B_{q′,r′}^{−s} (Z^{∗})) .

There is a natural embedding of E^{∗} (Z^{∗}) into [E (Z)]^{∗} as a norming subspace for E (Z). This embedding is given by the duality map

⟨ · , · ⟩_{E(Z)} : E^{∗} (Z^{∗}) × E (Z) → ℂ

where

⟨ g , f ⟩_{L_{q}(Z)} := ∫_{ℝ^{N}} ⟨ g (t) , f (t) ⟩_{Z} dt = ∫_{ℝ^{N}} g (t) [f (t)] dt

in the Lebesgue space setting with E = L_{q} and

(2.10)  ⟨ g , f ⟩_{E(Z)} := ∑_{n,m∈ℕ_{0}} ⟨ φ̌_{n} ∗ g , φ̌_{m} ∗ f ⟩_{L_{q}(Z)}

in the Besov space setting with E = B_{q,r}^{s}. One can check that this definition of duality is independent of the choice of the φ_{k}’s. Furthermore, as seen by [2] and the fact that

B^{∘} _{q,r}^{s}(ℝ^{N}, X) norms B_{q′ ,r′}^{−s} (ℝ^{N}, X^{∗}),

(2.11)  B^{∘} _{q,r}^{s}(X) is σ(B_{q,r}^{s}(X), B_{q′,r′}^{−s} (X^{∗}))–dense in B_{q,r}^{s}(X) .

Often E^{∗} (Z^{∗}) = [E (Z)]^{∗}, for example, provided Z^{∗} has the Radon-Nikodým property and either E = L_{q} (cf. [11]) or E = B_{q,r}^{s}(cf. [2]) where q, r ∈ [1, ∞) and s ∈ ℝ. Recall that if Z is reflexive or if Z^{∗} is separable, then Z^{∗} has the Radon-Nikodým property.

For more information regarding Besov spaces, see [2, 23, 24].

3. THE FOURIER TRANSFORM ON BESOV SPACES

As a consequence of the Hausdorff-Young inequality we get the following estimates for the Fourier transform on Besov spaces.

Theorem 3.1. Let X be a Banach space with Fourier type p ∈ [1, 2]. Let 1 ≤ q ≤ p′ and s ≥ N (\frac{1}{q} − \frac{1}{p′} ) and 1 ≤ r ≤ ∞. Then there exists a constant C, depending only on F_{p,N}(X), so that if

f ∈ B_{p,r}^{s} (ℝ^{N}, X)

then

(3.1)  ∥ {f̂· χ_{J_{m}}}_{m=0}^{∞}∥_{ℓ_{r}(L_{q}(ℝ^{N} ,X))} ≤ C ∥f∥_{B_{p,r}^{s}(ℝ^{N} ,X)} .

A variant of Theorem 3.1 for Besov spaces on the multidimensional torus and Lorentz sequence spaces is already contained in [17, Thm. 4]; the proof is based on interpolation theory and does not give the statements we need here. An immediate corollary of Theorem 3.1 follows by choosing q = r = 1 (for (3.2)) or r = q = p′ (for (3.3)).

Corollary 3.2. Let X have Fourier type p ∈ [1, 2]. Then the Fourier transform F defines bounded operators

F : B_{p,1}^{N/p} (ℝ^{N}, X) ⟶ L_{1} (ℝ^{N}, X) (3.2)

F : B_{p,p′}^{0} (ℝ^{N}, X) ⟶ L_{p′} (ℝ^{N}, X) . (3.3)

The norms of the above maps F are bounded above by a constant depending only on F_{p,N}(X).

Theorem 3.1 and Corollary 3.2 remain valid if F is replaced with F^{−1}.

Proof.[Proof of Theorem 3.1] Fix f ∈ B_{p,r}^{s} (ℝ^{N}, X). Then, for each k ∈ ℕ_{0}, since φ̌_{k} ∗ f ∈ L_{p}(X) and X has Fourier type p,

φ_{k} · f̂ = F (φ̌_{k} ∗ f) ∈ L_{p′}(X) .

Thus

f̂· χ_{J_{m}} = ( ∑_{k=m−1}^{m+1} φ_{k}f̂) · χ_{J_{m}} ∈ L_{q}(X)

for each m ∈ ℕ_{0}.

If there exists a constant C_{1} so that

(3.4)  ∥f̂· χ_{J_{m}} ∥_{L_{q}(X)} ≤ C_{1} ∑_{k=m−1}^{m+1} 2^{ks} ∥f̂· φ_{k}∥_{L_{p′} (X)} for each m ∈ ℕ_{0}

then

∥f̂· χ_{J_{m}} ∥_{L_{q}(X)} ≤ C_{1} ∑_{k=m−1}^{m+1} 2^{ks} ∥F (φ̌_{k} ∗ f)∥_{L_{p′} (X)}

≤ C_{1} F_{p,N}(X) ∑_{k=m−1}^{m+1} 2^{ks} ∥φ̌_{k} ∗ f∥_{L_{p}(X)}

and so

∥{f̂· χ_{J_{m}}}_{m=0}^{∞}∥_{ℓ_{r}(L_{q}(X))} ≤ 9 C_{1} F_{p,N}(X) ∥f∥_{B_{p,r}^{s}(X)} .

Thus it suffices to show that there exists C_{1} so that (3.4) holds.

First consider the case where q ≠ p′. Choose 1 ≤ u < ∞ so that \frac{1}{q} = \frac{1}{p′} + \frac{1}{u}; thus, \frac{N}{u} ≤ s. By the generalized Hölder’s inequality, for each m ∈ ℕ_{0},

∥f̂· χ_{J_{m}} ∥_{L_{q}(X)} ≤ ∑_{k=m−1}^{m+1} ∥f̂· φ_{k} · χ_{J_{m}} ∥_{L_{q}(X)}

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

≤ ∑_{k=m−1}^{m+1} ∥[\frac{1 + |·|}{4}]^{N/u} χ_{J_{m}} ∥_{L_{∞}(ℝ)} ∥f̂φ_{k}∥_{L_{p′} (X)} [∫_{J_{m}} [\frac{1 + |t|}{4}]^{−N} dt]^{1/u} (3.5)

≤ ∑_{k=m−1}^{m+1} (2^{m−1})^{N/u} ∥f̂φ_{k}∥_{L_{p′} (X)} [C_{2}]^{\frac{1}{u}} ≤ C_{2} ∑_{k=m−1}^{m+1} 2^{ks} ∥f̂φ_{k}∥_{L_{p′} (X)}

for some universal constant C_{2} > 1.

If q = p′, then

∥f̂· χ_{J_{m}} ∥_{L_{q}(X)} ≤ ∑_{k=m−1}^{m+1} ∥f̂· φ_{k} · χ_{J_{m}} ∥_{L_{p′} (X)} ≤ ∑_{k=m−1}^{m+1} 2^{ks} ∥f̂φ_{k}∥_{L_{p′} (X)}

for each m ∈ ℕ_{0}. □

Remark 3.3. The statement of Theorem 3.1 remains valid if B_{p,r}^{s} is replaced by W_{p}^{j} where s < j ∈ ℕ (with a new constant C which is just the product of the original constant C from Theorem 3.1 and the embedding constant from W_{p}^{j} (ℝ^{N}, X) ↪ B_{p,r}^{s} (ℝ^{N}, X)). Also, it follows from Corollary 3.2 that if X has Fourier type p ∈ [1, 2] and N/p < j ∈ N, then the Fourier transform F defines a bounded operator

(3.2’)  F : W_{p}^{j} (ℝ^{N}, X) ⟶ L_{1} (ℝ^{N}, X) .

Note that (3.2’) is the appropriate vector-valued version of the well-known Bernstein theorem. Furthermore, if X has Fourier type p ∈ [1, 2] and N/p < j ∈ N then there is a constant C so that

(3.6)  ∥f̂∥_{L_{1}(ℝ^{N} ,X)} ≤ C ∥f∥_{L_{p}(ℝ^{N} ,X)}^{1− \frac{N}{jp}} ( ∑_{|α|=j} ∥D^{α}f∥_{L_{p}(ℝ^{N} ,X)})^{\frac{N}{jp}}

for each f ∈ W_{p}^{j} (ℝ^{N}, X). A short proof of (3.6) is obtained from the classical argument of Bernstein’s theorem (see, e.g. [3, Lemma 8.2.1]) by replacing the Cauchy-Schwarz inequality by Hölder’s inequality and using the fact that the appropriate Hausdorff-Young inequality holds in a space with Fourier type p.

4. FOURIER MULTIPLIERS ON BESOV SPACES

For a bounded measurable function m : ℝ^{N} → B (X, Y), its corresponding Fourier multiplier operator T_{0} is defined by the following (commutative) diagram

S (ℝ^{N}, X) ⟶^{T_{0}} S′ (ℝ^{N}, Y )
F↓  ↑_{F^{−1}}
S (ℝ^{N}, X) ⟶^{M_{m}} S′ (ℝ^{N}, Y )

where M_{m} is the multiplication operator induced by m; thus,

(4.1)  T_{0}(f) = F^{−1}[m(·) (Ff)(·)] .

Note that since m is bounded

(4.2)  (T_{0}f) (·) = m (·) [f̂(·)] ∈ L_{1} (Y)

and T_{0} maps S (X) into L_{∞} (Y).

In this section we identify conditions on m, generalizing the classical Mihlin condition, so that

(4.3)  ∥T_{0}f∥_{B_{q,r}^{s}(Y)} ≤ C ∥f∥_{B_{q,r}^{s}(X)}  for each f ∈ S (X) .

Note that (4.3) implies that T_{0} extends uniquely to a bounded operator

T̃_{0} : B^{∘} _{q,r}^{s} (X) → B_{p,r}^{s} (Y) .

If we can find a further extention

(4.4)  T_{m}: B_{q,r}^{s} (X) → B_{p,r}^{s} (Y)

of T̃_{0} which is σ (B_{q,r}^{s}(X), B_{q′,r′}^{−s} (X^{∗})) –to–σ (B_{q,r}^{s}(Y), B_{q′,r′}^{−s} (Y^{∗})) continuous then, by (2.11), T_{m} is uniquely determined by (4.1). This leads to the following definition of a Fourier multiplier.

Definition 4.1. Let (E (ℝ^{N}, Z) , E^{∗} (ℝ^{N}, Z^{∗})) be one of the following dual systems, where 1 ≤ q, r ≤ ∞ and s ∈ ℝ:

(L_{q} (Z) , L_{q′} (Z^{∗}))  or  (B_{q,r}^{s} (Z) , B_{q′,r′}^{−s} (Z^{∗})) .

A bounded measurable function m : ℝ^{N} → B (X, Y) is called a Fourier multiplier from E(X) to E(Y) if there is a bounded linear operator

T_{m} : E(X) → E(Y)

such that

T_{m}(f) = F^{−1}[m(·) (Ff)(·)] for each f ∈ S (X) (4.5) (4.6) T_{m} is σ(E(X), E^{∗}(X^{∗}))–to–σ(E(Y), E^{∗}(Y^{∗})) continuous .

The (uniquely determined) operator T_{m} is the Fourier multiplier operator induced by m.

Note that

(4.7)  (T_{m}f) (·) = m (·) [f̂(·)] ∈ L_{1} (Y) for each f ∈ S (X)

and T_{m} maps S (X) into L_{∞} (Y).

Remark 4.2. If T_{m} ∈ B (E (X) , E (Y)) and T_{m}^{∗} maps E^{∗} (Y^{∗}) into E^{∗} (X^{∗}) then T_{m} satisfies the continuity condition (4.6).

We will first give rather general criteria for Fourier multipliers in terms of the Besov norm of the multiplier function; later we derive from these results analogues of the classical Mihlin and Hörmander conditions. To simplify the statements of our results, we let

M_{p}(m) := inf {∥m (a·)∥_{B_{p,1}^{N/p}(ℝ^{N} ,B(X,Y))} : a > 0} .

First we give a multiplier result on L_{q}(X) in the spirit of Steklin’s theorem which, in spite of its strong hypothesis, is still useful in many circumstances.

Theorem 4.3. Let X and Y have Fourier type p ∈ [1, 2]. Then there is a constant C, depending only on F_{p,N}(X) and F_{p,N}(Y), so that if

(4.8)  m ∈ B_{p,1}^{N/p} (ℝ^{N}, B (X, Y))

then m is a Fourier multiplier from L_{q}(ℝ^{N}, X) to L_{q}(ℝ^{N}, Y) with

∥T_{m}∥_{L_{q}(X)→L_{q}(Y)} ≤ C M_{p} (m)

for each q ∈ [1, ∞].

Remark 4.4. a) Since W_{p}^{m}(ℝ^{N}, Z) ⊂ B_{p,1}^{N/p}(ℝ^{N}, Z) for m > N/p, one can estimate with Sobolev-norms in order to verify the assumption (4.8) (e.g. see [16, Prop. 6.4] or Lemma 4.10).

b) If m(s) = n(s)A, where n is a scalar-valued function and A ∈ B (X, Y), then the result holds if n ∈ B_{2,1}^{N/2}(ℝ^{N}, ℂ) without any Fourier type requirement on X and Y. See the proof of Corollary 4.15.

c) We can replace m ∈ B_{p,1}^{N/p}(ℝ^{N}, B (X, Y)) by the pointwise conditions

∥m(·)x∥_{B_{p,1}^{N/p}(Y)} ≤ M ∥x∥_{X} for x ∈ X
∥m(·)^{∗}y^{∗}∥_{B_{p,1}^{N/p}(X^{∗})} ≤ M ∥y^{∗}∥_{Y^{∗}} for y^{∗} ∈ Y^{∗} .

Combine the proof below with [12].

The proof of Theorem 4.3 uses the following lemma.

Lemma 4.5. Let k ∈ L_{1} (ℝ^{N}, B (X, Y)) and 1 ≤ q ≤ ∞. Assume that there exists constants C_{i} so that for each x ∈ X and y^{∗} ∈ Y^{∗}

(4.9) ∫_{ℝ^{N}} ∥k (s) x∥_{Y} ds ≤ C_{0} ∥x∥_{X} and ∫_{ℝ^{N}} ∥k (s)^{∗} y^{∗}∥_{X^{∗}} ds ≤ C_{1} ∥y^{∗}∥_{Y^{∗}} .

Then the convolution operator K: L_{q} (ℝ^{N}, X) → L_{q} (ℝ^{N}, Y ) defined by

(4.10)  (Kf) (t) = ∫_{ℝ^{N}} k(t − s)f(s) ds  for t ∈ ℝ^{N}

satisfies that ∥K∥_{L_{q}→L_{q}} ≤ C0^{\frac{1}{q}} C_{1}^{1− \frac{1}{q}}.

Proof.[Proof of Lemma 4.5] Since k ∈ L_{1} (ℝ^{N}, B (X, Y)) it is well-known that (4.10) defines a bounded operator on L_{q} (X). Indeed, for f ∈ L_{q} (X)∩L_{∞} (X) we have, where f_{s}(t) := f(t − s),

∫_{ℝ^{N}} ∥k (t − s) f (s)∥_{Y} ds = ∫_{ℝ^{N}} ∥k(s)f_{s}(t)∥_{Y} ds ≤ ∥k∥_{L_{1}(B(X,Y))} ∥f∥_{L_{∞}(X)}

for each t ∈ ℝ^{N} and

∥Kf (·)∥_{L_{q}(Y)} ≤ ∫_{ℝ^{N}} ∥k (s) f_{s} (·)∥_{L_{q}(Y)} ds

≤ ∫_{ℝ^{N}} ∥k (s)∥_{B(X,Y)} ∥f_{s} (·)∥_{L_{q}(X)} ds = ∥k∥_{L_{1}(B(X,Y))} ∥f∥_{L_{q}(X)} .

But what is important for us is that, by using (4.9), we can get a better norm estimate on ∥K∥ than ∥k∥_{L_{1}}.

For q = 1 we have from (4.9)

∥Kf∥_{L_{1}(Y)} ≤ ∫ ∫ ∥k(t − s)f(s)∥_{Y} dt ds
≤ C_{0} ∫ ∥f(s)∥_{X} ds = C_{0} ∥f∥_{L_{1}(X)} .

Thus ∥K∥_{L_{1}→L_{1}} ≤ C_{0}. If q = ∞, then for each f ∈ L_{∞} (X) and y^{∗} ∈ Y^{∗} and t ∈ ℝ^{N}

|⟨y^{∗}, (Kf) (t)⟩_{Y} | ≤ ∫ |⟨k(t − s)^{∗}y^{∗}, f(s)⟩_{X}| ds

≤ ∫ ∥k(t − s)^{∗}y^{∗}∥_{X^{∗}} ∥f(s)∥_{X} ds ≤ C_{1} ∥y^{∗}∥_{Y^{∗}} ∥f∥_{L_{∞}(X)} .

Hence ∥K∥_{L_{∞}→L_{∞}} ≤ C_{1}. Let L_{∞}^{0}(X) denotes the closure, in the L_{∞}(X) norm, of the simple functions ∑_{k=1}^{m} x_{k}χ_{A_{k}}, where x_{k} ∈ X and vol(A_{k}) < ∞ and m ∈ ℕ. Then one can check that K maps L_{∞}^{0}(X) into L_{∞}^{0}(Y). Indeed, for f = x · χ_{A}, we have Kf(t) = ∫_{t−A} k(s)xds → 0 for |t| → ∞ and Kf is a continuous function from ℝ^{N} to Y. Now the Riesz-Thorin Theorem (cf. [5, Thm 5.1.2]) yields the claim for 1 < p < ∞.

□

Remark 4.6. The assumption in Lemma 4.5 that k ∈ L_{1} (ℝ^{N}, B (X, Y)) can be replaced by much weaker measurability conditions. This is explored in [12].

Proof.[Proof of Theorem 4.3] First assume in addition that m ∈ S (B (X, Y)).

Thus m̌ ∈ S (B (X, Y)). Fix x ∈ X. For an appropriate choice of a > 0, we can apply Corollary 3.2 to the function

t → m (at) x in B_{p,1}^{N/p} (Y)

and use that [m (a·) x]^{∨} (s) = a^{−N} m̌ (\frac{s}{a}) x to get

\begin{aligned}∥m̌ (·) x∥_{L_{1}(Y)} & = ∥[m (a·) x]^{∨} ∥_{L_{1}(Y)} \\ & ≤ C_{1} ∥m (a·)∥_{B_{p,1}^{N/p}} ∥x∥_{X} ≤ 2 C_{1} M_{p}(m) ∥x∥_{X} \end{aligned} (4.11)

for some constant C_{1} which depends on F_{p,N} (Y).

By the additional assumption on m

[m (·)]^{∗} ∈ S (B (Y^{∗}, X^{∗})) and [m (·)^{∗}]^{∨} = [m̌ (·)]^{∗} ∈ S (B (Y^{∗}, X^{∗})) .

Let y^{∗} ∈ Y^{∗}. Similarly, by applying Corollary 3.2 to an appropriate function

t → [m (at)]^{∗} y^{∗} in B_{p,1}^{N/p} (X^{∗})

and using the fact that M_{p}(m) = M_{p}(m^{∗}), one has

(4.12)  ∥ [m̌ (·)]^{∗} y^{∗} ∥_{L_{1}(X^{∗})} ≤ 2C_{2} M_{p}(m) ∥y^{∗}∥_{Y^{∗}}

for some constant C_{2} which depends F_{p,N} (X^{∗}).

By Lemma 4.5, the convolution operator

(T_{m}f) (t) := ∫_{ℝ^{N}} m̌(t − s)f(s) ds

satisfies, where C = 2 max{C_{1}, C_{2}},

∥T_{m}∥_{B(L_{q}(X),L_{q}(Y))} ≤ C M_{p}(m) .

Furthermore T_{m} satisfies (4.5) since m ∈ L_{1} (B (X, Y)). Also, T_{m} satisfies (4.6); indeed, T_{m}^{∗} restricted to L_{q′} (Y^{∗}) is just T_{m(−·)^{∗}} and so Remark 4.2 holds.

For the general case where m ∈ B_{p,1}^{N/p} (B (X, Y)), choose a sequence {m_{n}}_{n} from S (B (X, Y)) that converges to m in the B_{p,1}^{N/p}–norm and obtain operators T_{m_{n}} ∈ B (L_{q}(X), L_{q}(Y)) with ∥T_{m_{n}} − T_{m_{l}}∥ at most C ∥m_{n} − m_{l}∥_{B_{p,1}^{N/p}}. Then

T := ∥·∥_{B(L_{q}(X),L_{q}(Y))} − lim_{n→∞} T_{m_{n}}

has the desired properties. Indeed, conditions (4.5) and (4.6) pass from T_{m_{n}} to T_{m}. One also has that ∥T_{m}∥ ≤ C ∥m∥_{B_{p,1}^{N/p}}. To get the desired bound, fix a > 0 such that

m(a·) ∈ B_{p,1}^{N/p} (B (X, Y)). Then I_{Y} ∘ T_{m(a·)} = T_{m} ∘ I_{X} where I_{Z}: L_{q} (Z) → L_{q} (Z) is the isometry (T f) (t) = a^{N/q} f (at). Thus

∥T_{m}∥ = ∥T_{m(a·)}∥ ≤ C ∥m(a·)∥_{B_{p,1}^{N/p}(B(X,Y))}

and so ∥T_{m}∥ ≤ CM_{p} (m). □

The following remark collects some basic facts about the Fourier multiplier operators T_{m} given in Theorem 4.3 that will be used in the proof of Theorem 4.8. Recall that for a distribution g ∈ S′ (ℝ^{N}, X) and a closed subset Ω of ℝ^{N}

supp g ⊂ Ω if and only if g (φ) = 0 for each φ ∈ S with Ω ∩ supp φ = ∅ .

Remark 4.7. Assume that we are in the setting of Theorem 4.3. Let f ∈ L_{q} (X) and Ω be a closed subset of ℝ^{N}.

a) Viewing f and T_{m}f as distributions, if supp f̂ ⊂ Ω then supp T̂_{m}f ⊂ Ω.

b) T_{m_{1}} + T_{m_{2}} = T_{m_{1}+m_{2}}. If φ ∈ S, then φ̌ ∗ (T_{m}f) = T_{m} (φ̌ ∗ f) = T_{φm} (f).

c) If φ ∈ S is 1 on supp f̂, then T_{φm}f = T_{m}f. d) T_{m}^{∗} restricted to L_{q′} (Y^{∗}) is T_{m(−·)^{∗}} .

Proof. a) If f ∈ S (X) then the claim follows from the fact that T_{m} has the form (4.7). In general, choose a sequence {f_{n}} from S (X) that converges to f in σ (L_{q} (X) , L_{q′} (X^{∗})) and with supp f̂_{n} ⊂ Ω. Then {T_{m}f_{n}} converges to T_{m}f in σ (L_{q} (Y) , L_{q′} (Y^{∗})). If φ ∈ S with Ω ∩ supp φ = ∅ and y^{∗} ∈ Y^{∗} then

y^{∗} [(T_{m}f) (φ)] = (y^{∗} ⊗ φ̂) (T_{m}f)

= lim_{n→∞} (y^{∗} ⊗ φ̂) (T_{m}f_{n}) = lim_{n→∞} y^{∗} [(T_{m}f_{n}) (φ)] .

Thus (T_{m}f) (φ) = 0.

b) The claim is trivial if T_{m} is a convolution operator with kernel m̌ ∈ S (B (X, Y)). But according to the proof of Theorem 4.3, T_{m} can be approximated by such operators in operator norm.

c) Follows directly from b).

d) The proof is contained in the proof of Theorem 4.3. □

By applying this Theorem 4.3 to the blocks of the Littlewood–Paley decomposition of B_{q,r}^{s} we will now get the main result of this section.

Theorem 4.8. Let X and Y be Banach spaces with Fourier type p ∈ [1, 2]. Then there is a constant C, depending only on F_{p,N} (X) and F_{p,N} (Y), so that if m : ℝ^{N} → B (X, Y) satisfy (4.13)

φ_{k} · m ∈ B_{p,1}^{N/p} (ℝ^{N}, B (X, Y))  and  M_{p}(φ_{k} · m) ≤ A  for each k ∈ ℕ_{0}

then m is a Fourier multiplier from B_{q,r}^{s}(ℝ^{N}, X) to B_{q,r}^{s}(ℝ^{N}, Y) and ∥T_{m}∥ ≤ CA for each s ∈ ℝ and q, r ∈ [1, ∞].

Remark 4.9. a) Note that each Banach space has Fourier type 1 and each uniformly convex Banach space has Fourier type p for some p > 1. Our result shows that the required smoothness of the multiplier function m depends not only on the dimension of ℝ^{N} but also on the geometry of the Banach spaces X and Y.

b) Our results are sharp in the following sense. Note that the spaces B_{p,1}^{N/p}(ℝ^{N}, Z) form a scale since

B_{p,1}^{N/p}(ℝ^{N}, Z) ⊂ B_{q,1}^{N/q}(ℝ^{N}, Z)

if p < q. In general it is not possible to replace B_{p,1}^{N/p} in Theorem 4.8 by a larger space B_{q,1}^{N/q} for some q > p. Indeed, it was shown in [29, Remark 3.7] that counterexamples for such a stronger statement can be found in the context of stability theory for semigroups on L_{p}-spaces.

c) We can replace (4.13) by a weaker pointwise condition:

∥φ_{k}m (·) x∥_{B_{p,1}^{N/p}(Y)} ≤ A ∥x∥ for x ∈ X
∥φ_{k}m (·)^{∗} y^{∗}∥_{B_{p,1}^{N/p}(X^{∗})} ≤ A ∥y^{∗}∥ for y^{∗} ∈ Y^{∗} .

See the proof below and Remark 4.4c.

The heuristic idea behind the proof of Theorem 4.8 is to formally decompose the desired Fourier multiplier operator T_{m} as

T_{m}f = [mf̂ ]^{∨} = ∑_{k∈ℕ_{0}} [(φ_{k−1} + φ_{k} + φ_{k+1}) m (φ̌_{k} ∗ f) ]^{∨}

= ∑_{k∈ℕ_{0}} T_{(φ_{k−1}+φ_{k}+φ_{k+1})m} (φ̌_{k} ∗ f)

where T_{(φ_{k−1}+φ_{k}+φ_{k+1})m} is the Fourier multiplier operator on L_{q} (X) given by Theorem 4.3. It then follows easily that such a decomposition yields a bounded operator from B^{∘} _{q,r}^{s} (X) to B_{q,r}^{s} (Y). The case when B^{∘} _{q,r}^{s} (X) ≠ B_{q,r}^{s} (X) (i.e. q = ∞ or r = ∞) requires further consideration.

Proof.[Proof of Theorem 4.8] Theorem 4.3 gives that φ_{k} · m induces a Fourier multiplier operator T_{mφ_{k}} with

∥T_{mφ_{k}}∥_{B(L_{q}(X),L_{q}(Y))} ≤ CM_{p}(φ_{k} · m) ≤ CA

for some constant C depending only on F_{p,N} (X) and F_{p,N} (Y). Let ψ_{k} = φ_{k−1} + φ_{k} + φ_{k+1}; note that ψ_{k} is 1 on supp φ_{k}. Then ψ_{k} · m induces the Fourier multiplier operator T_{mψ_{k}} with

(4.14)  T_{mψ_{k}} = T_{mφ_{k−1}} + T_{mφ_{k}} + T_{mφ_{k+1}} ∈ B (L_{q} (X) , L_{q} (Y))

and ∥T_{mψ_{k}}∥ ≤ 3CA.

As in the introduction of Section 4, define T_{0} : S(X) → S′(Y) by (4.1); namely,

T_{0}(f) = F^{−1}[m(·) (Ff)(·)] .

If f ∈ S (X), then

φ̌_{k} ∗ T_{0}f = T_{mψ_{k}} (φ̌_{k} ∗ f)

for each k ∈ ℕ_{0} since

[T_{mψ_{k}} (φ̌_{k} ∗ f)] (·) = ψ_{k} (·) m (·) [(φ̌_{k} ∗ f) (·)] = ψ_{k} (·) m (·) [φ_{k} (·) f̂(·)]

= φ_{k} (·) [m (·) f̂(·)] = φ_{k} (·) (T_{0}f) (·) = (φ̌_{k} ∗ T_{0}f) (·) ;

so, by the definition of the Besov norm

∥T_{0}f∥_{B_{q,r}^{s}(Y)} ≤ 3CA ∥f∥_{B_{q,r}^{s}(X)} .

Thus T_{0} extends to a bounded linear operator from B^{∘} _{q,r}^{s} (X) to B_{q,r}^{s} (Y). If q, r < ∞,

then B^{∘} _{q,r}^{s} (X) = B_{q,r}^{s} (X) and so all that would remain is to verify the weak continuity condition (4.6). However, we continue with the proof in order to also cover the case q = ∞ or r = ∞.

We shall show that

T_{m}: B_{q,r}^{s} (X) → B_{q,r}^{s} (Y)

defined by

(4.15)  T_{m}f := ∑_{k=0}^{∞} f_{k}  where f_{k} := T_{mψ_{k}} (φ̌_{k} ∗ f) ∈ L_{q} (Y)

is indeed a (norm) continuous operator. Towards this, fix f ∈ B_{q,r}^{s} (X).

First we show that the formal series T_{m}f in (4.15) defines an element in S′ (Y). Towards this, fix φ ∈ S. Remark 4.7 gives that supp f̂_{k} ⊂ \overline{I_{k}}. Thus

f_{k} (φ) = f̂_{k} \binom{}{φ̌} = f̂_{k} (ψ_{k} (−·) φ̌) = f_{k} ((2π)^{−N} ψ_{k} (−·) ∗ φ) = f_{k} (ψ̌_{k} ∗ φ)

and so

∑_{k=0}^{∞} ∥f_{k} (φ)∥_{Y} ≤ ∑_{k=0}^{∞} ∥f_{k}∥_{L_{q}(Y)} ∥ψ̌_{k} ∗ φ∥_{L_{q′} (ℂ)}

≤ 3AC ∑_{k=0}^{∞} (2^{ks} ∥φ̌_{k} ∗ f∥_{L_{q}(X)}) (2^{−ks} ∥ψ̌_{k} ∗ φ∥_{L_{q′} (ℂ)})

≤ 27AC 2^{|s|} ∥f∥_{B_{q,r}^{s}(X)} ∥φ∥_{B_{q′,r′}^{−s} (ℂ)} .

Thus

(4.16)  ∥·∥_{Y} − lim_{n→∞} ∑_{k=0}^{n} f_{k} (φ) := (T_{m}f) (φ) for φ ∈ S

defines a linear map T_{m}f from S into Y which is continuous by (2.8).

By Remark 4.7, for each j, k ∈ ℕ_{0}

φ̌_{j} ∗ T_{mψ_{k}} (φ̌_{k} ∗ f) = T_{mψ_{k}} (φ̌_{j} ∗ φ̌_{k} ∗ f) = φ̌_{k} ∗ T_{mψ_{k}} (φ̌_{j} ∗ f) .

Thus, since the support of φ_{k} intersects the support of φ_{j} only for |k − j| ≤ 1, applying Remark 4.7 further gives

\begin{aligned}φ̌_{k} ∗ T_{m}f & = ∑_{j=k−1}^{k+1} φ̌_{k} ∗ T_{mψ_{j}} (φ̌_{j} ∗ f) = ∑_{j=k−1}^{k+1} φ̌_{j} ∗ T_{mψ_{j}} (φ̌_{k} ∗ f) \\ & = ∑_{j=k−1}^{k+1} T_{φ_{j}ψ_{j}m} (φ̌_{k} ∗ f) = T_{mψ_{k}} (φ̌_{k} ∗ f) . \end{aligned} (4.17)

Hence φ̌_{k} ∗ T_{m}f ∈ L_{q} (Y) and

∥φ̌_{k} ∗ T_{m}f∥_{L_{q}(Y)} ≤ 3CA ∥φ̌_{k} ∗ f∥_{L_{q}(X)} ,

from which it follows that range of T_{m} is contained in B_{q,r}^{s}(X) and that norm of T_{m} as an operator from B_{q,r}^{s}(X) to B_{q,r}^{s}(Y) is bounded by a constant depending on the items claimed.

Furthermore, T_{m} extends T_{0}; indeed, if f ∈ S (X) then (viewing equality as pointwise in S′ (Y))

(T_{m}f) = ∑_{k=0}^{∞} [T_{mψ_{k}} (φ̌_{k} ∗ f)] = ∑_{k=0}^{∞} (ψ_{k} m φ_{k} f̂)

= ∑_{k=0}^{∞} (φ_{k} m f̂)  = (∑_{k=0}^{∞} φ_{k}) m f̂ = (T_{0}f) ,

where the first equality follows from (2.1) and (4.16) while the fourth equality follows from viewing each φ_{k}mf̂ as an element of L_{1}(Y).

It remains to show only that T_{m} satisfies (4.6). Since [m (−·)]^{∗} : ℝ^{N} → B (Y^{∗}, X^{∗}) also satisfies condition (4.13), the Fourier multiplier operator T_{m(−·)^{∗}}, defined by (4.5), extends to T_{m(−·)^{∗}} ∈ B (E^{∗} (Y^{∗}) , E^{∗} (X^{∗})) where E = B_{q,r}^{s}. By Remark 4.2, it suffices to show that T_{m}^{∗} restricted to E^{∗} (Y^{∗}) is T_{m(−·)^{∗}}. Towards this, fix g ∈ E^{∗} (Y^{∗}) and f ∈ E (X). By (4.17) and (2.10)

\begin{aligned}⟨ T_{m}^{∗}g, f ⟩_{E(X)} & = ∑_{n,k∈ℕ_{0}} ⟨ φ̌_{n} ∗ g, φ̌_{k} ∗ T_{m}f ⟩_{L_{q}(Y)} \\ & = ∑_{n,k∈ℕ_{0}} ⟨ φ̌_{n} ∗ g, T_{mψ_{k}} (φ̌_{k} ∗ f) ⟩_{L_{q}(Y)} \end{aligned} (4.18)

and

\begin{aligned}⟨ T_{m(−·)^{∗}}g, f ⟩_{E(X)} & = ∑_{n,k∈ℕ_{0}} ⟨ φ̌_{n} ∗ T_{m(−·)^{∗}}g, φ̌_{k} ∗ f ⟩_{L_{q}(X)} \\ & = ∑_{n,k∈ℕ_{0}} ⟨ T_{m(−·)^{∗}ψ_{n}(·)} (φ̌_{n} ∗ g) , φ̌_{k} ∗ f ⟩_{L_{q}(X)} . \end{aligned} (4.19)

Fix K_{0} ∈ ℕ_{0} and choose a radial ψ ∈ S with compact support such that ψ is 1 on ∪_{k=1}^{K_{0}+1}supp φ_{k}. If n, k ∈ {0, 1, …, K_{0}} then ψ_{k} = ψψ_{k} and ψ_{n} = ψψ_{n} and so, by Remark 4.7,

T_{mψ_{k}} (φ̌_{k} ∗ f) = T_{mψψ_{k}} (φ̌_{k} ∗ f) = T_{mψ} (φ̌_{k} ∗ f) (4.20)

and

T_{m(−·)^{∗}ψ_{n}(·)} (φ̌_{n} ∗ g) = T_{m(−·)^{∗}ψ(·)ψ_{n}(·)} (φ̌_{n} ∗ g) = T_{m(−·)^{∗}ψ(·)} (φ̌_{n} ∗ g) (4.21)

since ψm and ψ(·)m(−·)^{∗} satisfy the assumptions of Theorem 4.3 (indeed, ψm = ∑_{k=0}^{l} ψφ_{k}m for some l large enough, likewise for ψ(·)m(−·)^{∗}). Thus ⟨ T_{m}^{∗}g, f ⟩ = ⟨ T_{m(−·)^{∗}}g, f ⟩ by (4.18),(4.19), (4.20), (4.21), and Remark 4.7. □

The next lemma gives a convenient way to verify the assumption of Theorem 4.8 in terms of derivatives.

Lemma 4.10. Let \frac{N}{p} < l ∈ ℕ and u ∈ [p, ∞]. If m ∈ C^{l}(ℝ^{N}, B (X, Y)) satisfies, for some A,

(4.22)  ∥D^{α}m |_{I_{0}}∥_{L_{u}(B(X,Y))} ≤ A

and for each k ∈ ℕ, with m_{k}(·) := m (2^{k−1}·),

(4.23)  ∥D^{α}m_{k} |_{I_{1}}∥_{L_{u}(B(X,Y))} ≤ A

for each α ∈ ℕ_{0}^{N} with |α| ≤ l, then m satisfies condition (4.13) of Theorem 4.8.

Proof. Since l > \frac{N}{p} one has that W_{p}^{l} (ℝ^{N}, B (X, Y)) ⊂ B_{p,1}^{N/p} (ℝ^{N}, B (X, Y)), say with embedding constant K (see (2.9)). Let ũ ∈ [p, ∞] be so that 1/u + 1/ũ = 1/p. Note that, for i ∈ ℕ_{0},

Cl ; ũ (φ_{i}) := ∑_{|α|≤l} ∑_{β≤α} (\begin{matrix}α \\ β \end{matrix}) ∥D^{β}φ_{i}∥_{Lũ}

is finite. Thus, by (4.22),

M_{p} (φ_{0}m) ≤ ∥φ_{0}m∥_{B_{p,1}^{N/p}} ≤ K ∥φ_{0}m∥_{W_{p}^{l}}

= K ∑_{|α|≤l} ∥∑_{β≤α} (\begin{matrix}α \\ β \end{matrix}) D^{β}φ_{0} D^{α−β}m |_{I_{0}} ∥_{L_{p}} ≤ K A Cl ; ũ (φ_{0}) .

For each k ∈ ℕ,

(4.24)  M_{p} (φ_{k}m) ≤ ∥φ_{k} (2^{k−1}·) m (2^{k−1}·)∥_{B_{p,1}^{N/p}} = ∥φ_{1}m_{k}∥_{B_{p,1}^{N/p}} ;

thus by (4.23),

M_{p} (φ_{k}m) ≤ K ∥φ_{1}m_{k}∥_{W_{p}^{l}} ≤ K A Cl ; ũ (φ_{1}) .

Thus for each k ∈ ℕ_{0}

M_{p} (φ_{k}m) ≤ K A [Cl ; ũ (φ_{0}) ∨ Cl ; ũ (φ_{1})] ;

so m does indeed satisfy condition (4.13) of Theorem 4.8. □

Now we show that the classical Mihlin conditions imply assumption (4.13) of Theorem 4.8.

Corollary 4.11. Let q, r ∈ [1, ∞] and s ∈ ℝ. If m ∈ C^{l}(ℝ^{N}, B (X, Y)) satisfies, for some constant A,

(4.25)  sup_{t∈ℝ^{N}} ∥(1 + |t|)^{|α|} D^{α}m (t)∥_{B(X,Y)} ≤ A

for each multi-index α with |α| ≤ l, then m is a Fourier multiplier from B_{q,r}^{s}(ℝ^{N}, X) to B_{q,r}^{s}(ℝ^{N}, Y) provided one of the following conditions hold.

a) X and Y are arbitrary Banach spaces and l = N + 1.

b) X and Y are uniformly convex Banach spaces and l = N.

c) X and Y have Fourier type p and l = [\frac{N}{p}] + 1.

Remark 4.12. Part a) was shown by H. Amann in [2]. By Remark 2.2, part c) can be applied to Banach spaces X and Y that are subspaces of L_{q} ((Ω, Σ, μ) , ℝ) where p = min(q′, q).

Proof. In case c), X and Y are assumed to have Fourier type p. In case b), since X and Y are uniformly convex, they have Fourier type p for some p > 1. In case a), let p = 1. Thus, in each of the three cases, X and Y have Fourier type p and, by design, N/p < l ∈ ℕ.

Keeping with the notation from Lemma 4.10, let u = ∞. Fix α ∈ ℕ_{0}^{n} with |α| ≤ l. Clearly (4.25) implies (4.22) since

∥D^{α}m |_{I_{0}}∥_{L_{∞}} ≤ ∥(1 + |·|)^{|α|} D^{α}m (·)∥_{L_{∞}} ≤ A .

For each k ∈ ℕ and t ∈ I_{1} (4.26) ∥D^{α}m_{k} (t)∥_{B(X,Y)} ≤ ∥|t|^{|α|} D^{α}m_{k} (t)∥_{B(X,Y)} = ∥|2^{k−1}t|^{|α|} D^{α}m (2^{k−1}t)∥_{B(X,Y)} ;

thus (4.25) implies (4.23) since

∥D^{α}m_{k} |_{I_{1}}∥_{L_{∞}} ≤ ∥(1 + |·|)^{|α|} D^{α}m (·)∥_{L_{∞}} ≤ A .

So by Lemma 4.10, m satisfies condition (4.13) of Theorem 4.8. □

The next results generalizes Hörmanders condition.

Corollary 4.13. Let X and Y have Fourier type p and l = [\frac{N}{p} ] + 1. Let m ∈

C^{l}(ℝ^{N}, B (X, Y)) satisfy, for some constant A,

(4.27)  [ ∫_{|t|≤2} ∥D^{α}m(t)∥^{p} dt]^{1/p} ≤ A

and, for 1 ≤ R < ∞,

(4.28)  [R^{−N} ∫_{R<|t|<4R} ∥D^{α}m(t)∥^{p} dt]^{1/p} ≤ AR^{−|α|}

for each multi-index α ∈ ℕ_{0}^{N} with |α| ≤ l. Then m is a Fourier multiplier from B_{q,r}^{s} (ℝ^{N}, X) to B_{q,r}^{s} (ℝ^{N}, Y ) for each s ∈ ℝ and q, r ∈ [1, ∞].

Proof. Keeping with the notation from Lemma 4.10, let u = p. Fix α ∈ ℕ_{0}^{N} with |α| ≤ l. Then (4.27) is just (4.22). Also, (4.28) and (4.26) give that

∥D^{α}m_{k} |_{I_{1}}∥_{L_{p}} ≤ 4^{l}A

for each k ∈ ℕ. So by Lemma 4.10, m satisfies condition (4.13) of Theorem 4.8. □

Variants of Corollary 4.11, where the assumption (4.25) of bounds on derivatives is replaced by Lipschitz conditions, follow from the formal statement of Theorem 4.8. The proof of the next corollary shows how to obtain such variants.

Corollary 4.14. Let X and Y have Fourier type p ∈ (1, 2]. Assume that m: ℝ → B (X, Y) satisfies, for some constant A and l ∈ (1/p, 1),

∥m (t)∥ ≤ A for t ∈ ℝ (4.29)

(1 + |t|)^{l} ∥\frac{m (t + u) − m (t)}{|u|^{l}} ∥ ≤ A for u, t ∈ ℝ , u ≠ 0 . (4.30)

Then m is a Fourier multiplier from B_{q,r}^{s} (X) to B_{q,r}^{s} (Y) for each s ∈ ℝ and 1 ≤ q, r ≤ ∞.

Proof. It suffices to show that there is a constant C so that, following the notation from (2.7) and with m_{k} (·) := m (2^{k−1}·),

(4.31)  B_{p,1}^{1/p} (φ_{0}m) ≤ C and B_{p,1}^{1/p} (φ_{1}m_{k}) ≤ C for each k ∈ ℕ ;

indeed, for then m satisfies (4.13) from Theorem 4.8 by (4.24), (2.6), and (4.29). Let u_{0} = 9; thus, if |h| ≥ u_{0} then I_{j} ∩ (I_{j} − h) = ∅ for j = 0, 1. Fix k ∈ ℕ. Note that if t ∈ I_{1} and h ∈ ℝ \ {0} then by (4.30)

(4.32)  ∥\frac{m_{k} (t + h) − m_{k} (t)}{|h|^{l}} ∥ ≤ |t|^{l} ∥\frac{m_{k} (t + h) − m_{k} (t)}{|h|^{l}} ∥ ≤ A .

If |h| ≤ u_{0}, then by (4.32) and (4.29),

∥φ_{1}m_{k} (· + h) − φ_{1}m_{k} (·) ∥_{L_{p}} = ∥φ_{1} (·) |h|^{l} \frac{m_{k} (· + h) − m_{k} (·)}{|h|^{l}}χ_{I_{1}} (·) +

m_{k} (· + h) |h| \frac{φ_{1} (· + h) − φ_{1} (·)}{|h|}χ_{(I_{1}−h)∪I_{1}} (·)∥_{L_{p}}

≤ ∥φ_{1}∥_{L_{p}} |h|^{l} A + A |h| ∥φ′_{1}∥_{L_{∞}} ∥χ_{(I_{1}−h)∪I_{1}} ∥_{L_{p}} (4.33)

≤ A [∥φ_{1}∥_{L_{p}} + u_{0}^{1−l} ∥φ′_{1}∥_{L_{∞}} (2 |I_{1}|)^{1/p}] |h|^{l} .

If |h| ≥ u_{0}, then by (4.29)

(4.34)  ∥φ_{1}m_{k} (· + h) − φ_{1}m_{k} (·)∥_{L_{p}} = 2^{1/p} ∥φ_{1}m_{k}∥_{L_{p}} ≤ 2^{1/p} ∥φ_{1}∥_{L_{p}} A .

Thus, by (4.33) and (4.34), there is a constant C_{1}, dependent on φ_{1} but independent of k, so that (following the notation from (2.7))

\begin{aligned}w_{p} (φ_{1}m_{k}, u) & := sup_{|h|≤u} ∥φ_{1}m_{k} (· + h) − φ_{1}m_{k} (·)∥_{L_{p}} \\ & ≤ \begin{cases}C_{1}u^{l} & if 0 ≤ u ≤ u_{0} \\ C_{1} & if u ≥ u_{0} . \end{cases} \end{aligned} (4.35)

Similar calculations show that (4.35) holds for φ_{0}m with some constant C_{0} depending on φ_{0}. Thus (4.31) holds. □

The proof of Theorem 4.8 also gives a result for essentially scalar valued multipliers which has the same smoothness requirement as in the classical theorems but without any Fourier type assumptions on the Banach spaces.

Corollary 4.15. Let {X_{j}}_{j=1}^{n_{1}} and {Y_{i}}_{i=1}^{n_{2}} be Banach spaces and S_{ij} ∈ B (X_{j}, Y_{i}) for i = 1, …, n_{2} and j = 1, …, n_{1}. Put X = X_{1} × … × X_{n_{1}} and Y = Y_{1} × … × Y_{n_{2}} and S = {S_{ij}}_{i,j}. Assume that n: ℝ^{N} → ℂ^{n_{2}} satisfies for each k ∈ ℕ_{0}

(4.36)  φ_{k}n ∈ B_{2,1}^{N/2} (ℝ^{N}, ℂ^{n_{2}})  and  M_{2} (φ_{k}n) ≤ A .

Then t → m (t) := n (t) S ∈ B (X, Y) is a Fourier multiplier from B_{q,r}^{s} (ℝ^{N}, X) to B_{q,r}^{s} (ℝ^{N}, Y ) for each s ∈ ℝ and q, r ∈ [1, ∞].

Remark 4.16. Condition (4.36) follows from either condition (4.25) of Corollary 4.11 or conditions (4.27) and (4.28) of Corollary 4.13 or conditions (4.29) and (4.30) of Corollary 4.14 with p = 2 adjusted to this special setting.

Proof. Looking back at the proof of Theorem 4.8, one sees that one only needs that the φ_{k}m’s induce Fourier multiplier operators T_{mφ_{k}}’s on L_{q} (ℝ^{N}, X) which satisfy the conditions in Remark 4.7 and

∥T_{mφ_{k}}∥_{L_{q}(X)→L_{q}(Y)} ≤ CM_{2} (φ_{k}n) .

But this follows from (4.36); indeed, just make minor modifications to the proof of Theorem 4.3, applying Corollary 3.2 to φ_{k}n ∈ B_{2,1}^{N/2} (ℝ^{N}, ℂ^{n_{2}}), which is valid since ℂ^{n_{2}} has Fourier type 2 (Plancherel’s Theorem). Towards this, note that there exists a constant C_{1}, dependent on the choice of product topologies, so that

∥αy∥_{Y} ≤ C_{1} ∥α∥_{ℂ^{n_{2}}} ∥y∥_{Y}

for each α ∈ ℂ^{n_{2}} and y ∈ Y. If φ_{k}n ∈ S (ℂ^{n_{2}}), then (4.11) takes the form

∥(φ_{k}n)^{∨} Sx∥_{L_{1}(Y)} ≤ 2 C_{1} ∥S∥ M_{2} (φ_{k}n) ∥x∥_{X}

for each x ∈ X while (4.12) takes the form

∥(φ_{k}n)^{∨} S^{∗}y^{∗}∥_{L_{1}(X^{∗})} ≤ 2 C_{1} ∥S∥ M_{2} (φ_{k}n) ∥y^{∗}∥_{Y^{∗}}

for each y^{∗} ∈ Y^{∗}. Thus by Lemma 4.5, φ_{k}m induces the Fourier multiplier operator

(T_{φ_{k}m}f) (t) := ∫_{ℝ^{N}} [φ_{k}n]^{∨} (t − s)Sf(s) ds

with ∥T_{φ_{k}m}∥_{L_{q}→L_{q}} ≤ 2 C_{1} ∥S∥ M_{2} (φ_{k}n). The general case where φ_{k}n ∈ B_{2,1}^{N/2} follows, as in the proof of Theorem 4.3, by approximation. □

Acknowledgements

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

22  Math. Nachr. [to appear] (2002)

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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: girardi@math.sc.edu and girardi@math.uni-karlsruhe.de

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