VECTOR-VALUED EXTENSIONS OF SOME CLASSICAL THEOREMS IN HARMONIC ANALYSIS ∗

Maria Girardi ^{†}

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

girardi@math.sc.edu

Lutz Weis ^{‡}

Mathematisches Institut I Universität Karlsruhe Englerstraße 2 76128 Karlsruhe Germany

Lutz.Weis@math.uni-karlsruhe.de

Abstract This paper surveys some recent results on vector-valued Fourier multiplier theorems and pseudo differential operators, which have found important application in the theory of evolution equations. The approach used combines methods from Fourier analysis and the geometry of Banach spaces, such as R-boundedness. Keywords R-boundedness, Mihlin multiplier theorem, pseudo differential operators, Fourier type, Littlewood-Paley decomposition, vectorvalued Besov spaces Subject Classification Primary 42B15, 46E40, 46B09. Secondary 46B20, 46E35.

∗To appear: Analysis and Applications - ISAAC 2001 (H. G. W. Begehr, R. P. Gilbert, and M. W. Wong, eds.), Kluwer, Dordrecht. †Girardi is supported in part by the Alexander von Humboldt Foundation. ‡Weis is supported in part by Landesforschungsschwerpunkt Evolutionsgleichungen des Landes Baden-Württenberg.

1. Introduction

Boundedness theorems for Fourier multiplier operators, singular integral operators, and pseudo differential operators play an important role in analysis. In recent years it has become apparent that one needs not only the classical theorems but also vector-valued extensions with operator-valued multiplier functions or symbols. These extensions allow one to treat certain problems for evolution equations with partial differential operators in an elegant and efficient manner in analogy to ordinary differential equations. For example, such theorems are used: in studying maximal regularity of parabolic equations (see, e.g., [1, 4, 5, 9, 17, 18, 19, 27, 28]), in stability theory (see, e.g., [21, 26]), in the theory of pseudo differential operators on manifolds with singularities (see, e.g., [23]),and for elliptic operators on infinite dimensional state spaces (see, e.g., [3, 10]).

This paper surveys some recent results in harmonic analysis of Banach space valued functions and tries to elucidate the interesting interplay with the geometry of the underlying Banach space, which in the end leads to significant applications to evolution equations.

The first Fourier multiplier theorem for operator-valued multiplier functions was J. Schwartz’s version of Mihlin’s theorem.

Theorem 1 (J. Schwartz) Let H be a Hilbert space and

m: ℝ^{N} \ {0} → B (H)

be such that the sets

{|t|^{|α|} D^{α}m (t) : t ∈ ℝ^{N} \ {0}}  (1)

are norm bounded for each multi-index α ∈ ℕ_{0}^{N} with |α| ≤ [N/2] + 1. Then

T_{m} (f) := F^{−1} [m (Ff)] for f ∈ S (ℝ^{N}, H)

defines a bounded operator on L_{q} (ℝ^{N}, H) for each q ∈ (1, ∞).

Here F is the Fourier transform and S (ℝ^{N}, H) is the Schwartz class of rapidly decreasing functions from ℝ^{N} to H. Does Theorem 1 formally generalize by replacing the Hilbert space H by an arbitrary Banach space X? No; G. Pisier showed that (isomorphic images of) Hilbert spaces are the only Banach spaces for which Theorem 1 holds in the above form. In recent years, two approaches were found to circumvent this difficulty.

Vector-valued extensions of some classical theorems in harmonic analysis 3

(i) Replace Bochner spaces L_{q} (ℝ^{N}, X) by Besov spaces B_{q,r}^{s} (ℝ^{N}, X). Using the characterization of Besov spaces in terms of the Paley-Littlewood decomposition, one can prove a Mihlin-type theorem as well as boundedness results for pseudo differential operators for any Banach space X. Section 4 elaborates on this approach. Note that in this setting one needs only norm boundedness, and not R-boundedness, of the sets in (1).

(ii) In the case of the Bochner spaces L_{q} (ℝ^{N}, X), it makes sense to consider only those Banach spaces X for which the simplest multiplier function, i.e. M (·) = sign (·) χ_{X}, is a Fourier multiplier (or equivalently, for which the Hilbert transform is bounded) on L_{q} (ℝ^{N}, X). Such Banach spaces are called UMD spaces. Subspaces of L_{q} (Ω, ℂ)-spaces, for 1 < q < ∞, are examples of UMD spaces. There are many results showing the UMD spaces form the proper class of Banach spaces for vector-valued harmonic analysis (see, e.g., [6, 7, 29]). For starters, if X is a UMD space, then there is a Paley-Littlewood decompositition for L_{q} (ℝ^{N}, X). But this decomposition is more delicate than the corresponding decomposition for Besov spaces. Therefore one has to replace the norm bounded condition in (1) by an R-bounded condition. This leads to boundedness results for Fourier multipliers and pseudo differential operators. Section 5 elaborates on this approach. Since large classes of classical operators are R-bounded (cf. [12, and references therein]), the assumptions in this approach are not too restrictive for applications.

2. Definitions and Notation

Notation is standard; consult [14, 15] for the needed definitions and notations. Here some basics are recalled.

Schwartz used Plancherel’s identity for L_{2} (ℝ^{N}, H) in his proof of Theorem 1. Since Plancherel’s identity holds only for Hilbert space valued Bochner spaces L_{2} (ℝ^{N}, X), the following concept from Banach space theory is needed.

Definition 2 ([22]) 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 for each) N ∈ ℕ.

The simple estimate ∥Ff(t)∥_{X} ≤ ∥f∥_{L_{1}(X)} shows that each Banach space X has Fourier type 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 [20]. A space L_{q} (Ω, ℝ) has Fourier type p = min(q, q′) [22]. Each closed subspace, the dual, and quotient space of a Banach space X has the same Fourier type as X; the first fact holds by definition, the last by duality.

The Fourier type is connected with the minimal smoothness assumptions on the multiplier function. For example, the condition (1) holding for |α| ≤ [N/2] + 1, instead of |α| ≤ N for the Bochner case and |α| ≤ N + 1 for the Besov case, expresses the fact that a Hilbert space has Fourier type 2.

To define Besov spaces, first consider a partition of unity {φ_{k}}_{k∈ℕ_{0}} of functions from S (ℝ^{N}, ℝ) as follows. Take a nonnegative function ψ in S (ℝ, ℝ) with support in [2^{−1}, 2] that 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, let φ_{k} ≡ 0 if k < 0. Note that φ_{k} and φ_{j} have overlapping support if and only if |k − j| ≤ 1.

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=0}^{∞} [ φ_{k} f̂]^{∨} = ∑_{k=0}^{∞} φ̌_{k} ∗ f = ∑_{k=0}^{∞} f_{k} , where f_{k} := φ̌_{k} ∗ f ,

of analytic functions f_{k} whose Fourier transforms have support in the (slightly overlapping dyadic-like) intervals {supp φ_{k}}_{k∈ℕ_{0}} and then one defines the Besov norm in terms of the blocks {f_{k}}_{k∈ℕ_{0}} of the Littlewood-Paley decomposition of f.

Definition 3 The Besov space B_{q,r}^{s} (ℝ^{N}, X), where 1 ≤ q, r ≤ ∞ and s ∈ ℝ, 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∥_{L_{q}(X)} }_{k=0}^{∞} ∥_{ℓ_{r}}  (2)

≡ \{ \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. The space B_{q,r}^{s} (ℝ^{N}, X), together with the norm in (2), is a Banach space.

3. A weak Fourier multiplier theorem for Bochner spaces

The following theorem is a weak Fourier multiplier theorem in the sense that its assumption (3) is quite strong; indeed, (3) implies that the Fourier multiplier function m is in L_{p} (ℝ^{N}, B (X, Y)).

Theorem 4 ([14]) Let X and Y have Fourier type p ∈ [1, 2] and

m ∈ B_{p,1}^{N/p} (ℝ^{N}, B (X, Y)) .  (3)

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}(X)→L_{q}(Y)} ≤ C M_{p} (m) ,  (4)

where C is a constant independent of m and

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

Theorem 4 leads to vector-valued extentions of classical multiplier theorems (such as Mihlin-, Hörmanders-, and Lipschitz-type theorems) for Besov spaces (see Section 4) and Bochner spaces (see Section 5) by considering a Littlewood-Paley decomposition of these spaces and then applying Theorem 4 to the blocks of the decomposition with the multiplier function m restricted to the support of the blocks. This is the charm behind using Littlewood-Paley decompositions: the classical (weak) assumptions on m imply the (strong) assumption of Theorem 4 when restricted to the support of the blocks.

The first step in proving Theorem 4 is to extend two other classical results to the vector-valued setting. The first extension shows that, for spaces with Fourier type p, the Sobolev Embedding (SE) factors through L_{1} via the Fourier transform F.

[figure: B_{p,1}^{N/p} (ℝ^{N}, X); L_{∞} (ℝ^{N}, X); L_{1} (ℝ^{N}, X); i; SE; F; F^{−1}]

Lemma 5 ([14]) Let X have Fourier type p ∈ [1, 2]. Then the Fourier transform defines bounded operator from B_{p,1}^{N/p} (ℝ^{N}, X) to L_{1} (ℝ^{N}, X).

The next lemma extends a well-known boundedness result for classical integral operators.

Lemma 6 ([13]) Let F ⊆ Y^{∗} be a subspace that norms Y. Let

k: ℝ^{N} → B (X, Y)

be such that k and k^{∗} are srongly measurable and satisfy

∫_{ℝ^{N}} ∥k (s) x∥_{Y} ds ≤ C_{0} ∥x∥_{X}  for each x ∈ X
∫_{ℝ^{N}} ∥k (s)^{∗} y^{∗}∥_{X^{∗}} ds ≤ C_{1} ∥y^{∗}∥_{Y^{∗}}  for each y^{∗} ∈ F

(5)

for some constants C_{i}. Then the convolution operator K, defined for finitely-valued functions f: ℝ^{N} → X with finite support by

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

extends to a bounded operator K: L_{q} (ℝ^{N}, X) → L_{q} (ℝ^{N}, Y ) for

each q ∈ [1, ∞); furthermore, ∥K∥_{L_{q}→L_{q}} ≤ C0^{\frac{1}{q}} C_{1}^{1− \frac{1}{q}}. If, in addition, Y does not contain c_{0}, then the same holds true for q = ∞.

To prove Theorem 4, first consider a function m in the Schwartz class. Applying Lemma 5 to the functions

t → m (t) x for x ∈ X  and  t → m^{∗} (t) y^{∗} for y^{∗} ∈ Y^{∗}

gives that k := m̌ satisfies the assumptions in (5) and so one has that the corresponding operator T_{m} satisfies (4) (even if Y contains c_{0}). Now, thanks to the bound in (4) on the norm of the T_{m}’s, a density argument finishes the job.

4. Fourier multiplier theorems for Besov spaces

One can think of a Besov space as a direct sum

B_{q,r}^{s} (ℝ^{N}, X) = ∑_{k∈ℕ_{0}} Z_{k} where Z_{k} := {φ̌_{k} ∗ f: f ∈ B_{q,r}^{s} (ℝ^{N}, X)} .

To see how a Fourier multiplier operator T_{m}, for a multiplier function m: ℝ^{N} → B (X, Y), formally behaves on a blocks Z_{k} of the above Littlewood-Paley decomposition, fix an f ∈ S (ℝ^{N}, X). Since the function ψ_{k} := φ_{k−1} + φ_{k} + φ_{k+1} is 1 on supp φ_{k},

T_{m} (φ̌_{k} ∗ f) =[ψ_{k} m φ_{k} f̂ ]^{∨}= φ̌_{k} ∗ [m ψ_{k} f̂ ]^{∨}= φ̌_{k} ∗ T_{m} (ψ̌_{k} ∗ f) ∈ Z_{k}.

Thus T_{m} leaves the blocks Z_{k} invariant. Furthermore,

[φ̌_{k} ∗ T_{m}f] = ψ_{k} φ_{k} m f̂ = m ψ_{k} (φ̌_{k} ∗ f) = [T_{mψ_{k}} (φ̌_{k} ∗ f)] , (6)

and so

f = ∑_{k∈ℕ_{0}} φ̌_{k} ∗ f and T_{m}f = ∑_{k∈ℕ_{0}} φ̌_{k} ∗ T_{m}f = ∑_{k∈ℕ_{0}} T_{mψ_{k}} (φ̌_{k} ∗ f) .

Thus T_{m} behaves as a Fourier multiplier operator T_{ψ_{k} m} on each block Z_{k} of the Littlewood-Paley decomposition; furthermore, the operator T_{ψ_{k} m} depends only on the values of m on the supports of ψ_{k}. This suggests the following approach to boundedness results for T_{m}.

(1^{st}) Estimate ∥T_{mψ_{k}}∥_{L_{q}→L_{q}} on each block of the Littlewood-Paley decomposition separably. For this, apply Theorem 4 to the multiplier function m ψ_{k}.

(2^{nd}) Sum over the blocks; with the help of (6):

∥T_{m}f∥_{B_{p,r}^{s}(X)} = [ ∑_{k=0}^{∞} 2^{ksr} ∥T_{mψ_{k}} (φ̌_{k} ∗ f)∥_{L_{q}(X)}^{r} ]^{1/r}

≤ sup_{k} ∥T_{mψ_{k}}∥_{L_{q}→L_{q}} ∥f∥_{B_{p,r}^{s}(X)} .

This gives the heuristic idea behind the proof of the main result of this section (additional considerations are necessary if q or r is ∞):

Theorem 7 [14] Let X and Y be Banach spaces with Fourier type p. Let m : ℝ^{N} → B (X, Y) satisfy, for each k ∈ ℕ_{0},

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

Then m is a Fourier multiplier from B_{q,r}^{s}(ℝ^{N}, X) to B_{q,r}^{s}(ℝ^{N}, Y) for each q, r ∈ [1, ∞] and s ∈ ℝ. Furthermore, ∥T_{m}∥_{B_{q,r}^{s}→B_{q,r}^{s}} ≤ CA for some constant C that is independent of m.

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. It follows from results in [26] that the smoothness N/p is sharp for the Besov scale.

An advantage of the rather general formulation of the assumptions in Theorem 7 is that one can deduce from them, by simple estimates, several multiplier theorems with classical assumptions. For example, the Mihlin-type multiplier theorem below follows easily; it was the first multiplier theorem of this kind and its parts i) and iii) are due independently to H. Amann [2] and L. Weis [26], respectively. Corollaries of Theorem 7. [14] Let q, r ∈ [1, ∞] and s ∈ ℝ. \underline{Mihlin condition} If m: ℝ^{N} → B (X, Y) satisfies, for some constant A, the estimate

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

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) provided one of the following conditions hold:

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

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

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

\underline{Hörmander condition} Let X and Y have Fourier type p and l = [\frac{N}{p} ]+1.

Let m: ℝ^{N} → B (X, Y) satisfy, for some constant A and each R ∈ [1, ∞), the estimates:

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

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

for each 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 ).

\underline{Lipschitz condition} Let X and Y have Fourier type p and l ∈ (1/p, 1). Assume that m: ℝ → B (X, Y) satisfies, for some constant A, the estimates:

∥m (t)∥ ≤ A for t ∈ ℝ

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

Then m is a Fourier multiplier from B_{q,r}^{s} (X) to B_{q,r}^{s} (Y).

It is also useful to consider pseudo differential operators with operatorvalued symbols. A pseudo differential operators Ψ_{a} with symbol a is formally defined by

Ψ_{a}f (t) := (2π)^{−N} ∫_{ℝ^{N}} e^{it·s}a (t, s) f̂(s) ds , f ∈ S (ℝ^{N}, X) .

In analogy to classical symbol classes, for δ ∈ [0, 1) and an r > 0, let S_{1,δ}^{0} (r, X) be the the class of symbols a: ℝ^{N} × ℝ^{N} → B (X) so that for all multi-indices α there is a constant C_{α} with

∥(1 + |s|)^{|α|} ∂_{s}^{α}a (t, s)∥ ≤ C_{α} for each t, s ∈ ℝ^{N}  (7)
∥∂_{s}^{α}a (·, s)∥_{B_{∞,∞}^{r}} ≤ C_{α} (1 + |s|)^{δr−|α|} .

By extending the Coifman-Meyer decomposition of symbols to the operator-valued case, Z. Štrkalj showed the following theorem.

Theorem 8 [24] Let X be a separable Banach space. Let q, r ∈ [1, ∞] and − (1 − δ) r < s < r. If a ∈ S_{1,δ}^{0} (r, X) then Ψ_{a} is bounded on B_{q,r}^{s} (ℝ^{N}, X).

5. Fourier multiplier theorems for Bochner spaces

This section presents Fourier multiplier theorems on Bochner spaces. The methods are similar to those in the Besov case: one uses Theorem 4 and a Littlewood-Paley decomposition for Bochner spaces. For this decomposition, one needs to decompose ℝ^{N} not only for |t| → ∞ but also for |t| → 0.

So consider a partition of unity {ϕ_{k}}_{k∈ℤ} of functions from S (ℝ^{N}, ℝ) defined as follows. Take a nonnegative function ϕ_{0} ∈ C^{∞}(ℝ^{N}, ℝ) that has support in {t : 2^{−1} ≤ |t| ≤ 2} and satisfies, for ϕ_{k} (t) := ϕ_{0}(2^{−k}t) for each k ∈ ℤ, that ∑_{k∈ℤ} ϕ_{k}(t) = 1 for each t ≠ 0. Note that

\left∥{ \atop φ̌_{k}}\right∥_{L_{1}} = \left∥{ \atop φ̌_{0}}\right∥_{L_{1}} and supp ϕ_{k} ⊂ \{t : 2^{k−1} ≤ |t| ≤ 2^{k+1}\}

for each k ∈ ℤ.

Bourgain [7, N = 1] and Zimmermann [29, N > 1] proved that if a scalar-valued function m: ℝ^{N} \ {0} → ℂ satisfied a certain Mihlin-type smoothness condition and X is a UMD space, then m (·) I_{X} is a Fourier multiplier from L_{q}(ℝ^{N}, X) to L_{q}(ℝ^{N}, X) for each q ∈ (1, ∞). Their result leads to a Littlewood-Paley decomposition for Bochner spaces. Henceforth, {r_{k}}_{k∈ℤ} is just any enumeration of the Rademacher functions.

Corollary 9 ([15]) Let X be a UMD space and 1 < q < ∞. There is a constant C so that

\frac{1}{C} ∥f∥_{L_{q}(ℝ^{N} ,X)} ≤ ∫_{[0,1]} ∥∑_{k∈ℤ}r_{k} (t) (ϕ̌_{k} ∗ f)∥_{L_{q}(ℝ^{N} ,X)} dt ≤ C ∥f∥_{L_{q}(ℝ^{N} ,X)} (8)

for each f ∈ L_{q} (ℝ^{N}, X).

To see the fundamental difference between the Besov- and Bochnerspace case (for 1 < q < ∞), let’s compare the norms. If f ∈ B_{q,2}^{0} (X) (which is closest to L_{q} (X)), then

∥f∥_{B_{q,2}^{0}(X)} = [ ∑_{k=0}^{∞} ∥φ̌_{k} ∗ f∥_{L_{q}(X)}^{2} ]^{1/2} ;  (9)

thus, \{φ̌_{k} ∗ f\}_{k∈ℕ_{0}} is \underline{absolutely 2-summable} in L_{q} (X). If f ∈ L_{q} (X), then Corollary 9 gives not only that \{ϕ̌_{k} ∗ f\}_{k∈ℤ} is \underline{almost uncondi-} \underline{tionally summable} in L_{q} (X) but also (in the scalar case) that

∥f∥_{L_{q}(ℂ)} ∼ ∥[∑_{k∈ℤ} |ϕ̌_{k} ∗ f|^{2}]^{1/2}∥_{L_{q}(ℂ)} ,  (10)

with the help of Kahane’s and Khintchine’s inequalities. Compare (9) and (10)! In the Besov case, one can estimate the Bochner norm of each block φ̌_{k} ∗ f of f separately (via Theorem 4) and then sum over the blocks. But this approach is not possible in the Bochner case since one sums over the blocks inside the Bochner norm. Therefore one needs tools to estimate the blocks simultaneously as an unconditionally summable sequence. Definition 10 is the first tool; Definition 12 is the second tool. Definition 10 Let X be a Banach space. Then the space Rad (X), or simply X̃, is

Rad (X) =^{or} X̃ := { {x_{k}}_{k∈ℤ} ∈ X^{ℤ}:

∑_{k=−n}^{n} r_{k} (·) x_{k}: [0, 1] → X is convergent in L_{1}([0, 1], X) }.

For 1 ≤ p < ∞, when equipped with one of the following norms, which are equivalent by Kahane’s inequality,

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

Rad_{p} (X) becomes a Banach space. Much can be found about Rad (X) in the literature (see, e.g. [11]).

Condition (8) can thus be reformulated as follows:

∥f∥_{L_{q}(ℝ^{N} ,X)} ∼ ∥{ϕ̌_{k} ∗ f}_{k∈ℤ}∥_{L_{q}(ℝ^{N} ,X̃)} .

It is now possible to estimate the blocks simultaneously in the following way: for a given function m: ℝ^{N} \ {0} → B (X, Y), simultaneously roll in all the blocks to the 0^{th} block by defining the corresponding mapping M: ℝ^{N} \ {0} → B (X̃, Ỹ) by

M(s) := {ϕ_{0}(s)m(2^{k}s)}_{k∈ℤ} .  (11)

Next (if possible), apply Theorem 4 to the function M to get a Fourier multiplier operator T_{M}: L_{q} \binom{}{X} → L_{q} \binom{}{Y}, which then can be rolled back out to a Fourier multiplier operator T_{m}: L_{q} (X) → L_{q} (Y). This approach leads to the following theorem (note that if X has Fourier type p (resp. UMD), then so does X̃).

Theorem 11 [15] Let X and Y be UMD Banach spaces with Fourier type p ∈ (1, 2] and 1 < q < ∞. Let m : ℝ^{N} \ {0} → B (X, Y) be a measurable function so that the corresponding mapping M, as defined in (11), satisfies that M ∈ B_{p,1}^{s} (ℝ^{N}, B (X̃, Ỹ)) for some s > N/p.

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

The assumption in Theorem 11 may look awkward; however, it is general enough to yield classical multiplier theorems, with the help of our second tool.

Definition 12 A subset τ of B (X, Y) is R-bounded provided there is a constant C_{p} so that for each n ∈ ℕ and subset {T_{j}}_{j=1}^{n} of τ and subset {x_{j}}_{j=1}^{n} of X

∥∑_{j=1}^{n} r_{j} (·) T_{j}(x_{j})∥_{L_{p}(Ω,Y)} ≤ C_{p} ∥∑_{j=1}^{n} r_{j} (·) x_{j}∥_{L_{p}(Ω,X)}  (12)

for some (and thus then, by Kahane’s inequality, for each) p ∈ [1, ∞). The R-bound of τ, R (τ), is smallest constant C_{1} for which (12) holds.

Note the following connection between our two tools: Definition 10 and Definition 12.

Remark 13 A sequence {T_{j}}_{j∈ℤ} from B (X, Y) is R-bounded if and only if the mapping

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

defines an element in B (X̃, Ỹ) for some (or equiv., for each) p ∈ [1, ∞).

The statements, and proofs, of the following corollaries to Theorem 11 are similar to the corresponding corollaries to Theorem 7. Corollaries of Theorem 11. [15] Let X and Y be UMD spaces with Fourier type p and 1 < q < ∞. \underline{Mihlin condition} If for m: ℝ^{N} \ \{0\} → B (X, Y) the set

{|t|^{|α|} D^{α}m (t) : t ∈ ℝ^{N} \ {0}}

is R-bounded for each multi-index α ∈ ℕ_{0}^{N} with |α| ≤ [\frac{N}{p}] + 1, then m is a Fourier multiplier from L_{q} (ℝ^{N}, X) to L_{q} (ℝ^{N}, Y ). \underline{Hörmander condition} If for m: ℝ^{N} \ \{0\} → B (X, Y) the term

[∫_{\frac{1}{2} <|t|<2} R (\{|2^{k}t|^{|α|} D^{α}m (2^{k}t)\}_{k∈ℤ})^{p} dt]^{1/p}

is finite for each multi-index α ∈ ℕ_{0}^{N} with |α| ≤ [\frac{N}{p}] + 1, then m is a Fourier multiplier from L_{q} (ℝ^{N}, X) to L_{q} (ℝ^{N}, Y ). \underline{Lipschitz condition} If for m: ℝ → B (X, Y) the set

\{m (t) , |t|^{l} \frac{m (t + s) − m (t)}{|s|^{l}} : t, s ∈ ℝ \ \{0\}\}

is R-bounded for some l ∈ (1/p, 1), then m is a Fourier multiplier from L_{q} (ℝ^{N}, X) to L_{q} (ℝ^{N}, Y ).

For the connection between R-boundedness and multiplier theorems with scalar multiplier functions, see [8]. The operator-valued Mihlin multiplier theorem was first proved for N = 1 in [28] and for higher dimensions in [25]. For variants of the proof of the Mihlin-type result, see [4, 5, 9, 10, 16].

Now returning to pseudo differential operators, define the symbol class ℜS_{1,δ}^{0} (r, X) similarly to the symbol class S_{1,δ}^{0} (r, X): just replace (7) by the condition that

R ({(1 + |s|)^{|α|} ∂_{s}^{α}a (t, s) : s ∈ ℝ^{N}}) ≤ C_{α} for each t ∈ ℝ^{N} .

In this context, using the same tools and the Coifman-Meyer decomposition of symbols, Z. Štrkalj has shown, among other things, the following boundedness result.

Theorem 14 [24] Let X be a separable UMD Banach space, δ ∈ [0, 1), and r > 0. If a ∈ ℜS_{1,δ}^{0} (r, X) then Ψ_{a} is bounded on L_{q} (ℝ^{N}, X) for each q ∈ (1, ∞).

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