Marcinkiewicz and Mihlin multiplier

theorems, and R-boundedness

Alberto Venni

Dipartimento di Matematica, Università di Bologna Piazza di Porta S.Donato, 5 I-40127 Bologna, Italy e-mail: venni@dm.unibo.it

Abstract

We prove the R-boundedness of a family of operators in L^{p}(𝕋, X) (where X is a UMD Banach space and 1 < p < ∞) that are defined by scalar valued Fourier multipliers, under the assumption that the multipliers satisfy a uniform condition of Marcinkiewicz type. An analogous result is obtained for the Mihlin multiplier theorem in L^{p}(ℝ, X).

A.M.S. Subject Classification: 47B38, 46E30, 46E40, 42A45. Keywords: Marcinkiewicz multiplier theorem, Mihlin multiplier theorem.

1 Introduction

Let T be a set of bounded linear operators on a Banach space X to a Banach space Y. T is said to be R-bounded if ∃ C > 0 such that for any positive integer N and for arbitrary choices of T_{1}, …, T_{N} ∈ T and x_{1}, …, x_{N} ∈ X one has

∑_{ε∈{−1,1}^{N}} ∥ ∑_{k=1}^{N} ε_{k} T_{k} x_{k}∥_{Y} ≤ C ∑_{ε∈{−1,1}^{N}} ∥ ∑_{k=1}^{N} ε_{k} x_{k}∥_{X}.  (1)

If j_{1}, …, j_{N} are positive integers such that j_{h} ≠ j_{k} for h ≠ k, then the Rademacher functions r_{j_{1}}, …, r_{j_{N}} are equidistributed, so that for any function f on {−1, 1}^{N} we have

∫_{0}^{1} f(r_{j_{1}}(t), …, r_{j_{N}} (t)) dt = 2^{−N} ∑_{ε∈{−1,1}^{N}} f(ε). Therefore (1) becomes

∫_{0}^{1} ∥ ∑_{k=1}^{N} r_{j_{k}}(t) T_{k} x_{k}∥ dt ≤ C ∫_{0}^{1} ∥ ∑_{k=1}^{N} r_{j_{k}}(t) x_{k}∥ dt,

and now, because of the Khintchine-Kahane inequality (see [3, 11.1]), formula (1) can be written in an equivalent form as

( ∑_{ε∈{−1,1}^{N}} ∥ ∑_{k=1}^{N} ε_{k} T_{k} x_{k}∥_{Y}^{q} )^{1/q} ≤ C_{q} ( ∑_{ε∈{−1,1}^{N}} ∥ ∑_{k=1}^{N} ε_{k} x_{k}∥_{X}^{q})^{1/q}

where q ∈ ℝ^{+}. The best C_{q} will be called the R_{q}-bound of T .

In the last years the notion of R-boundedness has proven to be very useful in connection with such problems as the maximal regularity for linear differential equations of parabolic type and the boundedness of the H^{∞} functional calculus for sectorial operators. We refer to [2, 5] for further information on the subject.

In this note it is proved that if a family of Fourier multipliers on the real line satisfies a uniform bound of Marcinkiewicz type, then the corresponding set of bounded operators on L^{p}(𝕋, X) to itself (where X is a UMD Banach space and 1 < p < ∞) is R-bounded. An analogous result for a family of Mihlin multipliers in L^{p}(ℝ, X) is proved.

2 The discrete case

Let X be a complex Banach space, and let 𝕋 = ℝ/ℤ be the one-dimensional torus. Let (λ_{k})_{k∈ℤ} be a family of complex numbers, and let Λ be the operator defined on the space of the X-valued trigonometric polynomials in the following way: (Λf) (k) = λ_{k} f̂(k),

where for an arbitrary function g ∈ L^{1}(𝕋, X) ĝ(k) := ∫_{0}^{1} e^{−2πikt}g(t) dt is the k-th Fourier

coefficient of g. One says that (λ_{k})_{k∈ℤ} is a family of multipliers in L^{p}(𝕋, X) (where p ∈ [1, ∞[ ) if the operator Λ has a continuous extension on L^{p}(𝕋, X) to L^{p}(𝕋, X). In this case, as the operator g ↦ ĝ(k) is continuous on L^{p}(𝕋, X) to X, the equality (Λf) (k) = λ_{k} f̂(k) also holds when Λ denotes the continuous extension to L^{p}(𝕋, X) of the homonymous operator originally defined on the space of the trigonometric polynomials.

In theorem 2 we state an extension of the Marcinkiewicz multiplier theorem to the case of vector valued functions that was proved in [1, Theorem 4]. We agree that (Δ_{k})_{k∈ℤ} is the usual dyadic decomposition of ℤ: precisely we set Δ_{0} = {0}, and for any positive integer k, Δ_{k} = {n ∈ ℤ; 2^{k−1} ≤ n < 2^{k}}, Δ_{−k} = −Δ_{k}.

Theorem 2 Let (λ_{k})_{k∈ℤ} be a family of complex numbers. Assume that

max { sup_{r∈ℤ}|λ_{r}|, sup_{k∈ℤ} ∑_{r,r−1∈Δ_{k}} |λ_{r} − λ_{r−1}|} = γ ∈ ℝ^{+}.

Then for each Banach space X of type UMD and ∀p ∈]1, ∞[ (λ_{k})_{k∈ℤ} is a family of multipliers in L^{p}(𝕋, X), and the norm of the corresponding operator Λ ∈ L(L^{p}(𝕋, X)) satisfies the inequality ∥Λ∥_{L(L^{p}(𝕋,X))} ≤ C(p, X) γ.

Our first aim is to prove the following R-version of theorem 2.

Theorem 3 Let Ξ be any set of indices, and ∀ξ ∈ Ξ let (λ_{k,ξ})_{k∈ℤ} be a family of complex numbers. Assume that

sup_{ξ∈Ξ} ( max { sup_{r∈ℤ}|λ_{r,ξ}|, sup_{k∈ℤ} ∑_{r,r−1∈Δ_{k}} |λ_{r,ξ} − λ_{r−1,ξ}|}) = γ ∈ ℝ^{+}.

Let X be a UMD Banach space, p ∈]1, ∞[, and let Λ_{ξ} be the operator in L^{p}(𝕋, X) associated to the family of multipliers (λ_{k,ξ})_{k∈ℤ}. Then {Λ_{ξ}; ξ ∈ Ξ} is a R-bounded set in L(L^{p}(𝕋, X)), and its R_{p}-bound is ≤ C(p, X) γ.

Let us establish some notations and recall some results.

If X is a UMD Banach space, then (−i sgn k)_{k∈ℤ} is a family of multipliers in L^{p}(𝕋, X) for every p ∈]1, ∞[ : in fact that could be taken as a definition of the class UMD. The corresponding operator is the discrete Hilbert transform H. The Riesz projection operator R = R_{0} is the operator defined by Rf = \frac{1}{2}(f +iHf)+ \frac{1}{2} f̂(0). This is obviously a bounded operator on L^{p}(𝕋, X) to itself (p ∈]1, ∞[ ), whose norm only depends on the norm of H in L(L^{p}(𝕋, X)), and it is the operator associated to the family (λ_{k})_{k∈ℤ} of multipliers, where λ_{k} = 1 if k ≥ 0 and λ_{k} = 0 if k < 0.

Let m ∈ ℤ. The shifted Riesz operator R_{m} is defined by R_{m}f = φ_{m} R(φ_{−m}f), where, ∀k ∈ ℤ, φ_{k}(t) = e^{2πikt}. This is obviously a bounded operator on L^{p}(𝕋, X) to itself (always, as it is understood from now on, for p ∈]1, ∞[ ), with ∥R_{m}∥_{L(L^{p}(𝕋,X))} = ∥R∥_{L(L^{p}(𝕋,X))}, and it is in fact the operator associated to the family of multipliers (λ_{k})_{k∈ℤ}, where λ_{k} = 1 if k ≥ m and λ_{k} = 0 if k < m. Therefore, if Δ is a “segment” of ℤ, say Δ = {k ∈ ℤ; m ≤ k ≤ n}, then the characteristic function of Δ is a family of multipliers, and the associated operator is S_{Δ} = R_{m} − R_{n+1}, so that the operator norm of S_{Δ} is ≤ 2 ∥R∥_{L(L^{p}(𝕋,X))}.

In order to prove theorem 3, we need three lemmas.

Lemma 4 Let X be a UMD Banach space, 1 < p < ∞. Suppose that T_{1}, …, T_{n} belong to the linear subspace of L(L^{p}(𝕋, X)) generated by {R_{k}; k ∈ ℤ}, T_{k} = ∑_{j∈ℤ}λ_{kj}R_{j}. If

μ = sup_{1≤k≤n} ∑_{j∈ℤ}|λ_{kj}|, then

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{k=1}^{n} ε_{k} T_{k} f_{k}∥_{L^{p}(𝕋,X)}^{p})^{1/p} ≤ μ C(p, X) ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{k=1}^{n} ε_{k} f_{k}∥_{L^{p}(𝕋,X)}^{p})^{1/p}.

Lemma 5 Let X be a UMD Banach space, 1 < p < ∞. Then for f ∈ L^{p}(𝕋, X) and m, n ∈ ℤ with m ≤ n,

(2^{−(n−m+1)} ∑_{ε} ∥ ∑_{k=m}^{n} ε_{k} S_{Δ_{k}} f∥_{L^{p}(𝕋,X)}^{p})^{1/p} ≤ C(p, X)∥f∥_{L^{p}(𝕋,X)}.

Lemma 6 Let X be a UMD Banach space, 1 < p < ∞, m, n ∈ ℤ with m ≤ n, f_{m}, …, f_{n} ∈ L^{p}(𝕋, X). Then

∥ ∑_{k=m}^{n} S_{Δ_{k}} f_{k}∥_{L^{p}(𝕋,X)} ≤ C(p, X)(2^{−(n−m+1)} ∑_{ε} ∥ ∑_{k=m}^{n} ε_{k} S_{Δ_{k}} f_{k}∥_{L^{p}(𝕋,X)}^{p})^{1/p}.

A proof of lemma 4 can be found in [1, Lemma 7] (see also [6, Lemma 1.2]) and relies on the so-called “contraction principle” (that is the R-boundedness of {λI_{X}; |λ| ≤ M}, see [3, 12.2]) plus a convexity argument. The Littlewood-Paley type lemma 5 above is

Theorem 3 of [1]. In order to obtain lemma 6 it is enough to apply to f := ∑_{k=m}^{n} S_{Δ_{k}}f_{k} the

inequality

∥f∥_{L^{p}(𝕋,X)} ≤ C(p, X)(2^{−(n−m+1)} ∑_{ε} ∥ ∑_{j=m}^{n} ε_{j} S_{Δ_{j}} f∥_{L^{p}(𝕋,X)}^{p})^{1/p}

that can be obtained from lemma 5 by means of a duality argument (see [6, (1.9)]).

Proof of theorem 6. We have to prove that for a positive integer n, (ξ_{1}, …, ξ_{n}) ∈ Ξ^{n} and f_{1}, …, f_{n} ∈ L^{p}(𝕋, X) the following inequality holds:

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{i=1}^{n} ε_{i} Λ_{ξ_{i}}f_{i}∥_{L^{p}(𝕋,X)}^{p})^{1/p} ≤ C(p, X) γ ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{i=1}^{n} ε_{i}f_{i}∥_{L^{p}(𝕋,X)}^{p})^{1/p}.

It is enough to prove this inequality when f_{1}, …, f_{n} are X-valued trigonometric polynomials, since the space of the trigonometric polynomials is dense in L^{p}(𝕋, X). In this case there exist q, r ∈ ℤ, with q < r, such that the support of every f_{i} is con-

tained in ⋃_{k=q}^{r} Δ_{k}. Now we call Λ_{i,k} the operator associated to the family of multipliers

(χ_{Δ_{k}}(m) λ_{m,ξ_{i}})_{m∈ℤ} which vanishes outside of Δ_{k} and equals λ_{m,ξ_{i}} if m ∈ Δ_{k}. We have obviously Λ_{i,k} = Λ_{ξ_{i}} S_{Δ_{k}} = S_{Δ_{k}} Λ_{ξ_{i}}, and we have also Λ_{i,k} = Λ_{i,k} S_{Δ_{k}} = S_{Δ_{k}} Λ_{i,k}. Moreover

the support of Λ_{ξ_{i}}f_{i} is contained in ⋃_{k=q}^{r} Δ_{k}, as well as the support of f_{i}; therefore

Λ_{ξ_{i}}f_{i} = ∑_{k=q}^{r} S_{Δ_{k}} Λ_{ξ_{i}} f_{i} = ∑_{k=q}^{r} Λ_{i,k} f_{i} = ∑_{k=q}^{r} S_{Δ_{k}} Λ_{i,k} f_{i}.

Hence we have

∑_{ε∈{−1,1}^{n}} ∥ ∑_{i=1}^{n} ε_{i} Λ_{ξ_{i}} f_{i}∥_{L^{p}(𝕋,X)}^{p} = ∑_{ε∈{−1,1}^{n}} ∥ ∑_{k=q}^{r} S_{Δ_{k}} ∑_{i=1}^{n} ε_{i} Λ_{i,k} f_{i}∥_{L^{p}(𝕋,X)}^{p} ≤

(by lemma 6)

≤ \frac{C(p, X)^{p}}{2^{r−q+1}} ∑_{ε} ∑_{ε′} ∥ ∑_{k=q}^{r} ε′_{k} S_{Δ_{k}} ∑_{i=1}^{n} ε_{i} Λ_{i,k}f_{i}∥_{L^{p}(𝕋,X)}^{p} =

\frac{C(p, X)^{p}}{2^{r−q+1}} ∑_{(ε, ε′)} ∥ ∑_{i,k} ε_{i} ε′_{k} Λ_{i,k} S_{Δ_{k}}f_{i}∥_{L^{p}(𝕋,X)}^{p}.

Now we remark that the operators Λ_{i,k} satisfy the assumptions of lemma 4, with μ ≤ 3γ. Indeed, if we set Δ_{k} = {α_{k}, …, β_{k}}, then

Λ_{i,k} = ∑_{j=α_{k}}^{β_{k}} λ_{i,j} (R_{j} − R_{j+1}) = λ_{i,α_{k}}R_{α_{k}} − λ_{i,β_{k}}R_{β_{k}+1} + ∑_{j,j−1∈Δ_{k}} (λ_{i,j} − λ_{i,j−1})R_{j}.

Therefore, by lemma 4 we obtain

∑_{ε∈\{−1,1\}^{n}} ∥ ∑_{i=1}^{n} ε_{i} Λ_{ξ_{i}} f_{i}∥_{L^{p}(𝕋,X)}^{p} ≤ \frac{(3 γ C(p, X))^{p}}{2^{r−q+1}} ∑_{(ε, ε′)} ∥ ∑_{i,k} ε_{i} ε′_{k} S_{Δ_{k}} f_{i}∥_{L^{p}(𝕋,X)}^{p} =

\frac{(3 γ C(p, X))^{p}}{2^{r−q+1}} ∑_{ε} ∑_{ε′} ∥ ∑_{k} ε′_{k} S_{Δ_{k}}( ∑_{i} ε_{i} f_{i})∥_{L^{p}(𝕋,X)}^{p} ≤

(by lemma 5)

≤ (3 γ C(p, X))^{p} ∑_{ε} ∥ ∑_{i} ε_{i} f_{i}∥_{L^{p}(𝕋,X)}^{p}.

3 The continuous case

Now we want to obtain a result analogous to theorem 3 for multipliers of Mihlin type in L^{p}(ℝ, X). This can be made by a transference trick based on lemmas 7 and 8 (see [6]).

∀f ∈ L^{1}(ℝ, X), the Fourier transform of f is the function Ff : ℝ → X defined by

(Ff)(τ) = ∫_{ℝ} e^{−2πitτ}f(t) dt. We denote by J the operator acting on functions on ℝ in the

following way: (Jf)(t) = f(−t): thus (JFf)(τ) = ∫_{ℝ} e^{2πitτ}f(t) dt = (FJf)(τ).

The following results hold.

Lemma 7 (Poisson summation formula) Let X be a Banach space and let f : ℝ → X be a continuous function. Suppose that for some ε > 0 we have, as |t| → +∞, ∥f(t)∥ = O(|t|^{−1−ε}) and ∥(Ff)(t)∥ = O(|t|^{−1−ε}). Then ∀t ∈ ℝ and ∀a > 0

∑_{n∈ℤ}f(t + na) = \frac{1}{a} ∑_{n∈ℤ}e^{2πint/a}(Ff)(n/a).

Lemma 8 (see [6, Lemma 2.1]). Let X be a Banach space, f ∈ S(ℝ, X), p ∈]1, ∞[.

∀k ∈ ℕ we define g_{k} : ℝ → X, g_{k}(t) = 2^{−k/p′} ∑_{n∈ℤ}e^{2πint}(Ff)(2^{−k}n), where p′ is the

exponent conjugate to p. Then g_{k} ∈ L^{p}(𝕋, X) and ∥f∥_{L^{p}(ℝ,X)} = lim_{k→+∞} ∥g_{k}∥_{L^{p}(𝕋,X)}.

A proof of lemma 7 can be given by comparing the Fourier coefficients of the 1-periodic functions t ↦ ∑_{n∈ℤ}f(t + n) and t ↦ ∑_{n∈ℤ}e^{2πint}(Ff)(n). A sketch of proof of lemma 8 works

as follows: g_{k} is a C^{∞} 1-periodic function on ℝ, and therefore can be considered as an element of L^{p}(𝕋, X). Taking into account the Poisson summation formula, one obtains

∥g_{k}∥_{L^{p}(𝕋,X)} = ( ∫_{−2^{k−1}}^{2^{k−1}} ∥ ∑_{n∈ℤ}f(s + 2^{k} n)∥_{X}^{p} ds)^{1/p} ≤

( ∫_{−2^{k−1}}^{2^{k−1}} ∥f(s)∥_{X}^{p} ds)^{1/p} + ∑_{0≠n∈ℤ} 2^{k/p} sup_{|s|≥2^{k}(|n|−(1/2))} ∥f(s)∥_{X}.

Here, as k → +∞, the first addendum tends to ∥f∥_{L^{p}(ℝ,X)}, while the following sum tends to 0 since f is rapidly decreasing at infinity.

The R-version of the Mihlin multipliers theorem for Banach valued functions is the following.

Theorem 9 Let (φ_{i})_{i∈I} be a family of functions that belong to L^{∞}(ℝ) ∩ Wloc^{1,1}(ℝ \ {0}), and assume that the functions t ↦ t φ′_{i}(t) (shortly denoted t φ′_{i}) belong to L^{∞}(ℝ). Assume moreover that

sup_{i∈I} max {∥φ_{i}∥_{L^{∞}(ℝ)}, ∥t φ′_{i}∥_{L^{∞}(ℝ)}} =: η < +∞.

Let X be a UMD Banach space, and p ∈]1, ∞[ . Then ∃ C(p, X) ∈ ℝ^{+} such that for any positive integer n, (i_{1}, …, i_{n}) ∈ I^{n} and (f_{1}, …, f_{n}) ∈ (S(ℝ, X))^{n} we have

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r})∥_{L^{p}(ℝ,X)}^{p})^{1/p} ≤ C(p, X) η ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p}.

Proof. We perform the proof in several steps.

(I) Assume that ∀i ∈ I there is a sequence (ψ_{i,k})_{k∈ℕ} of functions satisfying the following conditions:

(a) φ_{i}(t) = lim_{k→+∞} ψ_{i,k}(t) almost everywhere on ℝ

(b) ∀(i, k) ∈ I × ℕ ψ_{i,k} ∈ L^{∞}(ℝ) and ∀i ∈ I ∃ M_{i} ∈ ℝ^{+} such that sup_{k∈ℕ} ∥ψ_{i,k}∥_{L^{∞}(ℝ)}

≤ M_{i}

(c) ∃ M̃ ∈ ℝ^{+} such that for any positive integer n, ∀ k ∈ ℕ and for arbitrary (i_{1}, …, i_{n}) ∈ I^{n}, (f_{1}, …, f_{n}) ∈ (S(ℝ, X))^{n} it holds

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} F(ψ_{i_{r},k} Ff_{r})∥_{L^{p}(ℝ,X)}^{p})^{1/p} ≤

M̃ ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p}.

If we now choose n, (i_{1}, …, i_{n}) and (f_{1}, …, f_{n}) as above, since Ff_{r} ∈ S(ℝ, X) ⊆ L^{1}(ℝ, X), from (a), (b) and the dominated convergence theorem we get

∥ψ_{i_{r},k} Ff_{r} − φ_{i_{r}} Ff_{r}∥_{L^{1}(ℝ,X)} →_{k→+∞} 0,

so that

∥F(ψ_{i_{r},k} Ff_{r}) − F(φ_{i_{r}} Ff_{r})∥_{L^{∞}(ℝ,X)} →_{k→+∞} 0

and also

∥ ∑_{r=1}^{n} ε_{r} F(ψ_{i_{r},k} Ff_{r}) − ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r})∥_{L^{∞}(ℝ,X)} →_{k→+∞} 0.

Therefore, by Fatou’s lemma and assumption (c) we get

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r})∥_{L^{p}(ℝ,X)}^{p})^{1/p} ≤ M̃ ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p}.

(II) Assume that ∀i ∈ I φ_{i} ∈ C^{∞}(ℝ). Then ∀f ∈ S(ℝ, X) we have φ_{i} Ff ∈ S(ℝ, X), and so φ_{i} Ff = FJF(φ_{i} Ff). Now let us fix n, (i_{1}, …, i_{n}) and (f_{1}, …, f_{n}) as above. If ε ∈ {−1, 1}^{n}, then ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r}) ∈ S(ℝ, X), and hence, by lemma 8,

∥ ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r})∥_{L^{p}(ℝ,X)} = ∥ ∑_{r=1}^{n} ε_{r} JF(φ_{i_{r}} Ff_{r})∥_{L^{p}(ℝ,X)} = lim_{k→+∞} ∥h_{k,ε}∥_{L^{p}(𝕋,X)}

where h_{k,ε}(t) = 2^{−k/p′} ∑_{m∈ℤ}e^{2πimt} ∑_{r=1}^{n} ε_{r} φ_{i_{r}}(2^{−k}m) (Ff_{r})(2^{−k}m), so that h_{k,ε} is the 1-

periodic function whose m-th Fourier coefficient is

2^{−k/p′} ∑_{r=1}^{n} ε_{r} φ_{i_{r}}(2^{−k}m) (Ff_{r})(2^{−k}m).

Likewise

∥ ∑_{r=1}^{n} ε_{r}f_{r}∥_{L^{p}(ℝ,X)} = lim_{k→+∞} ∥g_{k,ε}∥_{L^{p}(𝕋,X)}

where g_{k,ε}(t) = 2^{−k/p′} ∑_{m∈ℤ}e^{2πimt} ∑_{r=1}^{n} ε_{r} (Ff_{r})(2^{−k}m), so that g_{k,ε} is the 1-periodic

function whose m-th Fourier coefficient is 2^{−k/p′} ∑_{r=1}^{n} ε_{r} (Ff_{r})(2^{−k}m). Now we remark that sup_{m∈ℤ}|φ_{i}(2^{−k}m)| ≤ ∥φ_{i}∥_{L^{∞}(ℝ)} ≤ η. Moreover, if the dyadic interval Δ_{j} contains two different elements, then |j| ≥ 2; if j ≥ 2, then

∑_{m,m−1∈Δ_{j}} |φ_{i}(2^{−k}m) − φ_{i}(2^{−k}(m − 1))| = ∑_{m=2^{j−1}+1}^{2^{j}−1} | ∫_{2^{−k}(m−1)}^{2^{−k} m} φ′_{i}(s) ds| ≤

∑_{m=2^{j−1}+1}^{2^{j}−1} ∫_{m−1}^{m} |φ′_{i}(2^{−k}σ)| 2^{−k} dσ = ∫_{2^{j−1}}^{2^{j}−1} |φ′_{i}(t)| dt ≤

∥tφ′_{i}∥_{L^{∞}(ℝ)} ∫_{2^{j−1}}^{2^{j}−1} t^{−1} dt ≤ η log 2.

The same estimate is obtained when j ≤ −2, by means of a similar argument. Therefore if we apply theorem 3, we obtain

( ∑_{ε∈{−1,1}^{n}} ∥h_{k,ε}∥_{L^{p}(𝕋,X)}^{p})^{1/p} ≤ C(p, X) (log 2) η ( ∑_{ε∈{−1,1}^{n}} ∥g_{k,ε}∥_{L^{p}(𝕋,X)}^{p})^{1/p},

and as k → +∞ we get

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} F(φ_{i_{r}} Ff_{r})∥_{L^{p}(ℝ,X)}^{p})^{1/p} ≤

C(p, X) (log 2) η ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p}.

(III) Finally we drop the additional assumption that φ_{i} ∈ C^{∞} ∀i. We fix ω ∈ C_{0}^{∞}(ℝ),

with ω ≥ 0, ∫_{ℝ} ω = 1 and supp ω ⊆ [−1, 1]. ∀ρ ∈ ℝ^{+} we set ω_{ρ}(t) = ρ^{−1}ω(ρ^{−1}t)

and ψ_{i,ρ} = ω_{ρ} ∗ φ_{i}. Then ψ_{i,ρ} ∈ C^{∞}(ℝ) ∩ L^{∞}(ℝ) and (by Young’s inequality) ∥ψ_{i,ρ}∥_{L^{∞}(ℝ)} ≤ ∥φ_{i}∥_{L^{∞}(ℝ)}. Moreover

t ψ′_{i,ρ}(t) = t (ω′_{ρ} ∗ φ_{i})(t) = t ∫_{t−ρ}^{t+ρ} ω′_{ρ}(t − s)φ_{i}(s) ds,

and if 0 < ρ < |t| partial integration also gives

t ψ′_{i,ρ}(t) = t ∫_{t−ρ}^{t+ρ} ω_{ρ}(t − s)φ′_{i}(s) ds.

Let t ≠ 0 be fixed. If ρ ≥ \frac{|t|}{2}, then

|t ψ′_{i,ρ}(t)| = |t ∫_{ℝ} ω′_{ρ}(t − s)φ_{i}(s) ds| ≤
2ρ ∥φ_{i}∥_{∞} ∫_{ℝ} ρ^{−2} |ω′(ρ^{−1}(t − s))| ds = 2∥ω′∥_{L^{1}(ℝ)} ∥φ_{i}∥_{L^{∞}(ℝ)}.

If, on the contrary, ρ ≤ \frac{|t|}{2}, then

|t ψ′_{i,ρ}(t)| = |t ∫_{t−ρ}^{t+ρ} ω_{ρ}(t − s)φ′_{i}(s) ds| ≤

|t| ∫_{ℝ} |ω_{ρ}(σ)| dσ sup_{|t−s|≤ρ} |φ′_{i}(s)| ≤ ∥s φ′_{i}∥_{∞} sup_{|t−s|≤|t|/2} \frac{|t|}{|s|} = 2 ∥s φ′_{i}∥_{L^{∞}(ℝ)}.

Therefore

max{∥ψ_{i,ρ}∥_{L^{∞}(ℝ)}, ∥t ψ′_{i,ρ}∥_{L^{∞}(ℝ)}} ≤
max{∥φ_{i}∥_{L^{∞}(ℝ)}, 2∥ω′∥_{L^{1}(ℝ)} ∥φ_{i}∥_{L^{∞}(ℝ)}, 2∥t φ′_{i}∥_{L^{∞}(ℝ)}} ≤ 2η(1 + ∥ω′∥_{L^{1}(ℝ)}).

Next we remark that on ℝ \ {0} φ_{i} is almost everywhere equal to a continuous function. Therefore for almost all t ≠ 0 we have

|ψ_{i,ρ}(t) − φ_{i}(t)| = | ∫_{ℝ} ω_{ρ}(s) (φ_{i}(t − s) − φ_{i}(t)) ds| ≤

sup_{|s|≤ρ} |φ_{i}(t − s) − φ_{i}(t)| →_{ρ→0} 0.

Thus, combining steps (I) and (II), we get the desired inequality.

Corollary 10 Let (ψ_{i})_{i∈I} be a family of elements of L^{1}(ℝ) and assume that ∀i ∈ I Ff_{i} ∈ Wloc^{1,1}(ℝ \ {0}), with t (Ff_{i})′ ∈ L^{∞}(ℝ). Assume moreover that

sup_{i∈I} max {∥Ff_{i}∥_{L^{∞}(ℝ)}, ∥t (Ff_{i})′∥_{L^{∞}(ℝ)}} =: η < +∞.

Let p ∈]1, ∞[ and let X be a UMD Banach space. ∀i ∈ I let T_{i} : L^{p}(ℝ, X) → L^{p}(ℝ, X) be the bounded linear operator defined by T_{i}f = ψ_{i} ∗ f. Then {T_{i}; i ∈ I} is a R-bounded set of operators. More precisely, for any positive integer n and for arbitrary (i_{1}, …, i_{n}) ∈ I^{n}, (f_{1}, …, f_{n}) ∈ (L^{p}(ℝ, X))^{n} the following inequality holds:

( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} T_{i_{r}}f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p} ≤ C(p, X) η ( ∑_{ε∈{−1,1}^{n}} ∥ ∑_{r=1}^{n} ε_{r} f_{r}∥_{L^{p}(ℝ,X)}^{p})^{1/p}.

Proof. It is sufficient to deal with the particular case of f_{1}, …, f_{n} ∈ S(ℝ, X). In this case ψ_{i_{r}} ∗ f_{r} ∈ L^{1}(ℝ, X) and F(ψ_{i_{r}} ∗ f_{r}) = (Fψ_{i_{r}}) (Ff_{r}) ∈ L^{1}(ℝ, X) (since Fψ_{i_{r}} ∈ L^{∞}(ℝ) and Ff_{r} ∈ S(ℝ, X)). Therefore ψ_{i_{r}} ∗f_{r} = J F((Fψ_{i_{r}}) (Ff_{r})), and so the result follows at once from theorem 9.

Corollary 10 has been used in [4].

References

[1] J. Bourgain: Vector-valued singular integrals and the H_{1}-BMO duality; in: Probability Theory and Harmonic Analysis; Dekker, New York 1986, pp. 1-19.

[2] P. Clément, B. de Pagter, F.A. Sukochev, H. Witvliet: Schauder decompositions and multiplier theorems; Studia Math. 138 (2000), 135-163.

[3] J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators; Cambridge Studies in Advanced Mathematics vol. 43, Cambridge University Press, Cambridge 1995.

[4] G. Dore, A. Venni: H^{∞} functional calculus for an elliptic operator on a half-space with general boundary conditions; preprint.

[5] L. Weis: Operator-valued Fourier multiplier theorems and maximal L^{p}-regularity; preprint.

[6] F. Zimmermann: On vector-valued Fourier multiplier theorems; Studia Math. 93 (1989), 201-222.