Fourier Multiplier Theorems Involving Type and Cotype
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Abstract
In this paper we develop the theory of Fourier multiplier operators \(T_{m}:L^{p}({\mathbb R}^{d};X)\rightarrow L^{q}({\mathbb R}^{d};Y)\), for Banach spaces X and Y, \(1\le p\le q\le \infty \) and \(m:{\mathbb R}^d\rightarrow \mathcal {L}(X,Y)\) an operatorvalued symbol. The case \(p=q\) has been studied extensively since the 1980s, but far less is known for \(p<q\). In the scalar setting one can deduce results for \(p<q\) from the case \(p=q\). However, in the vectorvalued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for \(p<q\) other geometric conditions on X and Y, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for \(T_m\) without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as \(\tfrac{1}{p}\tfrac{1}{q}\) remains constant.
Keywords
Operatorvalued Fourier multipliers Type and cotype Fourier type Hörmander condition \(\gamma \)boundednessMathematics Subject Classification
Primary: 42B15 Secondary: 42B35 46B20 46E40 47B381 Introduction
The simplest class of examples of Fourier multipliers can be obtained by taking \(p=q=2\). Then \(T_m\) is bounded if and only if \(m\in L^{\infty }({\mathbb {R}^{d}})\), and \(\Vert T_m\Vert _{{\mathcal {L}}(L^{2}({\mathbb {R}^{d}}))} = \Vert m\Vert _{L^{\infty }({\mathbb {R}^{d}})}\). For \(p=q=1\) and \(p=q=\infty \) one obtains only trivial multipliers, namely Fourier transforms of bounded measures. The case where \(p=q\in (1, \infty )\setminus \{2\}\) is highly nontrivial. In general only sufficient conditions on m are known that guarantee that \(T_m\) is bounded, although also here it is necessary that \(m\in L^{\infty }({\mathbb {R}^{d}})\).
Hörmander also introduced an integral/smoothness condition on the kernel K which allows one to extrapolate the boundedness of \(T_{m}\) from \(L^{p_0}({\mathbb {R}^{d}})\) to \(L^{q_0}({\mathbb {R}^{d}})\) for some \(1<p_{0}\le q_{0}<\infty \) to boundedness of \(T_{m}\) from \(L^{p}({\mathbb {R}^{d}})\) to \(L^{q}({\mathbb {R}^{d}})\) for all \(1<p\le q<\infty \) satisfying \(\tfrac{1}{p}  \tfrac{1}{q} = \tfrac{1}{p_0}  \tfrac{1}{q_0}\). This led to extensions of the theory of Calderón and Zygmund in [13]. In the case \(p_0 = q_0\) it was shown that the smoothness condition on the kernel K can be translated to a smoothness condition on the multiplier m which is strong enough to deduce the classical Mihlin multiplier theorem. From here the field of harmonic analysis has quickly developed itself and this development is still ongoing. We refer to [23, 24, 35, 53] and references therein for a treatment and the history of the subject.

even for \(p=q=2\) one does not have \(T_m\in {\mathcal {L}}(L^{2}({\mathbb {R}^{d}};X))\) for general \(m\in L^{\infty }({\mathbb {R}^{d}})\) unless X is a Hilbert space.
In this article we complement the theory of operatorvalued Fourier multipliers by studying the boundedness of \(T_{m}\) from \(L^{p}({\mathbb {R}^{d}};X)\) to \(L^{q}({\mathbb {R}^{d}};Y)\) for \(p<q\). One of our main results is formulated under \(\gamma \)boundedness assumptions on \(\{\xi ^{\frac{d}{r}} m(\xi )\mid \xi \in {\mathbb {R}^{d}}\setminus \{0\}\}\). We note that Rboundedness implies \(\gamma \)boundedness (see Subsection 2.4). The result is as follows (see Theorem 3.18 for the proof):
Theorem 1.1
The condition \(p\le 2\le q\) cannot be avoided in such results (see below (1.1)). Note that no smoothness on m is required. Theorem 1.1 should be compared to the sufficient condition in (1.1) due to Hörmander in the case where \(X=Y={\mathbb C}\). We will give an example which shows that the \(\gamma \)boundedness condition (1.2) cannot be avoided in general. Moreover, we obtain several converse results stating that type and cotype are necessary.
We note that, in case m is scalarvalued and \(X = Y\), the \(\gamma \)boundedness assumption in Theorem 1.1 reduces to the uniform boundedness of (1.2). Even in this setting of scalar multipliers our results appear to be new.
In Theorem 3.21 we obtain a variant of Theorem 1.1 for pconvex and qconcave Banach lattices, where one can take \(p = p_0\) and \(q=q_0\). In [49] we will deduce multiplier results similar to Theorem 1.1 in the Besov scale, where one can let \(p = p_0\) and \(q=q_0\) for Banach spaces X and Y with type p and cotype q.
The exponents p and q in Theorem 1.1 are fixed by the geometry of the underlying Banach spaces. However, Corollary 4.2 shows that under smoothness conditions on the multiplier, one can extend the boundedness result to all pairs \((\tilde{p},\tilde{q})\) satisfying \(1<\tilde{p}\le \tilde{q}<\infty \) and \(\frac{1}{\tilde{p}}  \frac{1}{\tilde{q}} = \frac{1}{p}  \frac{1}{q} = \frac{1}{r}\). Here the required smoothness depends on the Fourier type of X and Y and on the number \(r\in (1,\infty ]\). We note that even in the case where \(X=Y={\mathbb C}\), for \(p<q\) we require less smoothness for the extrapolation than in the classical results (see Remark 4.4).
We will mainly consider multiplier theorems on \({\mathbb R}^d\). There are two exceptions. In Remark 3.11 we deduce a result for more general locally compact groups. Moreover, in Proposition 3.4 we show how to transfer our results from \({\mathbb R}^d\) to the torus \({\mathbb T}^d\). This result appears to be new even in the scalar setting. As an application of the latter we show that certain irregular Schur multipliers with sufficient decay are bounded on the Schatten class \({\mathscr {C}}^p\) for \(p\in (1,\infty )\).
We have pointed out that questions about operatorvalued Fourier multiplier theorems were originally motivated by stability and regularity theory. We have already successfully applied our result to stability theory of \(C_0\)semigroups, as will be presented in a forthcoming paper [50]. In [48] the firstnamed author has also applied the Fourier multiplier theorems in this article to study the \(\mathcal {H}^{\infty }\)calculus for generators of \(C_{0}\)groups.
Other potential applications could be given to the theory of dispersive equations. For instance the classical Strichartz estimates can be viewed as operatorvalued \(L^{p}\)\(L^{q}\)multiplier theorems. Here the multipliers are often not smooth, as is the case in our theory. More involved applications probably require extensions of our work to oscillatory integral operators, which would be a natural next step in the research on vectorvalued singular integrals from \(L^{p}\) to \(L^{q}\).
This article is organized as follows. In Sect. 2 we discuss some preliminaries on the geometry of Banach spaces and on function space theory. In Sect. 3 we introduce Fourier multipliers and prove our main results on \(L^{p}\)\(L^{q}\)multipliers in the vectorvalued setting. In Sect. 4 we present an extension of the extrapolation result under Hörmander–Mihlin conditions to the case \(p\le q\).
1.1 Notation and Terminology
We write \({{\mathbb N}}:=\left\{ 1,2,3,\ldots \right\} \) for the natural numbers and \({{\mathbb N}}_{0}:={{\mathbb N}}\cup \left\{ 0\right\} \).
We denote nonzero Banach spaces over the complex numbers by X and Y. The space of bounded linear operators from X to Y is \({\mathcal {L}}(X,Y)\), and \({\mathcal {L}}(X):={\mathcal {L}}(X,X)\). The identity operator on X is denoted by \(\mathrm {I}_{X}\).
The Hölder conjugate of p is denoted by \(p'\) and is defined by \(1=\frac{1}{p}+\frac{1}{p'}\). We write \(\ell ^{p}\) for the space of psummable sequences \((x_{k})_{k\in {{\mathbb N}}_{0}}\subseteq {\mathbb C}\), and denote by \(\ell ^{p}({{\mathbb Z}})\) the space of psummable sequences \((x_{k})_{k\in {{\mathbb Z}}}\subseteq {\mathbb C}\).
We say that a function \(m:\Omega \rightarrow {\mathcal {L}}(X,Y)\) is Xstrongly measurable if \(\omega \mapsto m(\omega )x\) is a strongly measurable Yvalued map for all \(x\in X\). We often identify a scalar function \(m:{\mathbb {R}^{d}}\rightarrow {\mathbb C}\) with the operatorvalued function \(\widetilde{m}:{\mathbb {R}^{d}}\rightarrow {\mathcal {L}}(X)\) given by \(\widetilde{m}(\xi ):=m(\xi )\mathrm {I}_{X}\) for \(\xi \in {\mathbb {R}^{d}}\).
We will use the convention that a constant C which appears multiple times in a chain of inequalities may vary from one occurrence to the next.
2 Preliminaries
2.1 Fourier Type
We recall some background on the Fourier type of a Banach space. For these facts and for more on Fourier type see [19, 28, 45].
A Banach space X has Fourier type \(p\in [1,2]\) if the Fourier transform \({\mathcal {F}}\) is bounded from \(L^{p}({\mathbb {R}^{d}};X)\) to \(L^{p'}({\mathbb {R}^{d}};X)\) for some (in which case it holds for all) \(d\in {{\mathbb N}}\). We then write \({\mathcal {F}}_{p,X,d}:=\Vert {\mathcal {F}}\Vert _{{\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{p'}({\mathbb {R}^{d}};X))}\).
Each Banach space X has Fourier type 1 with \({\mathcal {F}}_{1,X,d}=1\) for all \(d\in {{\mathbb N}}\). If X has Fourier type \(p\in [1,2]\) then X has Fourier type r with \({\mathcal {F}}_{r,X,d}\le {\mathcal {F}}_{p,X,d}\) for all \(r\in [1,p]\) and \(d\in {{\mathbb N}}\). We say that X has nontrivial Fourier type if X has Fourier type p for some \(p\in (1,2]\). In order to make our main results more transparent we will say that X has Fourier cotype \(p'\) whenever X has Fourier type p.
Let X be a Banach space, \(r\in [1,\infty )\) and let \(\Omega \) be a measure space. If X has Fourier type \(p\in [1,2]\) then \(L^{r}(\Omega ;X)\) has Fourier type \(\min (p,r,r')\). In particular, \(L^{r}(\Omega )\) has Fourier type \(\min (r,r')\).
2.2 Type and Cotype
We first recall some facts concerning the type and cotype of Banach spaces. For more on these notions and for unexplained results see [1, 17, 29] and [40, Sect. 9.2].
The minimal constants C in (2.1) and (2.2) are called the (Gaussian) type p constant and the (Gaussian) cotype q constant and will be denoted by \(\tau _{p,X}\) and \(c_{q,X}\). We say that X has nontrivial type if X has type \(p\in (1,2]\), and finite cotype if X has cotype \(q\in [2,\infty )\).
Note that it is customary to replace the Gaussian sequence in (2.1) and (2.2) by a Rademacher sequence, i.e. a sequence \((r_{n})_{n\in {{\mathbb N}}}\) of independent random variables on a probability space \((\Omega ,\mathbb {P})\) that are uniformly distributed on \(\{z\in {\mathbb R}\mid z=1\}\). This does not change the class of spaces under consideration, only the minimal constants in (2.1) and (2.2) (see [17, Chap. 12]). We choose to work with Gaussian sequences because the Gaussian constants \(\tau _{p,X}\) and \(c_{q,X}\) occur naturally here.
Each Banach space X has type \(p=1\) and cotype \(q=\infty \), with \(\tau _{1,X}=c_{\infty ,X}=1\). If X has type p and cotype q then X has type r with \(\tau _{r,X}\le \tau _{p,X}\) for all \(r\in [1,p]\) and cotype s with \(c_{s,X}\le c_{q,X}\) for all \(s\in [q,\infty ]\). A Banach space X is isomorphic to a Hilbert space if and only if X has type \(p=2\) and cotype \(q=2\), by Kwapień’s theorem (see [1, Theorem 7.4.1]). Also, a Banach space X with nontrivial type has finite cotype by the Maurey–Pisier theorem (see [1, Theorem 11.1.14]).
Let X be a Banach space, \(r\in [1,\infty )\) and let \(\Omega \) be a measure space. If X has type \(p\in [1,2]\) and cotype \(q\in [2,\infty )\) then \(L^{r}(\Omega ;X)\) has type \(\min (p,r)\) and cotype \(\max (q,r)\) (see [17, Theorem 11.12]).
A Banach space with Fourier type \(p\in [1,2]\) has type p and cotype \(p'\) (see [29]). By a result of Bourgain a Banach space has nontrivial type if and only if it has nontrivial Fourier type (see [45, 5.6.30]).
2.3 Convexity and Concavity
For the theory of Banach lattices we refer the reader to [40]. We repeat some of the definitions which will be used frequently.
Every Banach lattice X is 1convex and \(\infty \)concave. If X is pconvex and qconcave then it is rconvex and sconcave for all \(r\in [1,p]\) and \(s\in [q,\infty ]\). By [40, Proposition 1.f.3], if X is qconcave then it has cotype \(\max (q,2)\), and if X is pconvex and qconcave for some \(q<\infty \) then X has type \(\min (p,2)\).
If X is pconvex and \(p'\)concave for \(p\in [1,2]\) then X has Fourier type p, by [20, Proposition 2.2]. For \((\Omega ,\mu )\) a measure space and \(r\in [1,\infty )\), \(L^{r}(\Omega ,\mu )\) is an rconvex and rconcave Banach lattice. Moreover, if X is pconvex and qconcave and \(r\in [1, \infty )\), then \(L^{r}(\Omega ;X)\) is \(\min (p,r)\)convex and \(\max (q,r)\)concave.
Let \(f=\sum _{k=1}^{n}f_{k}\otimes x_{k}\in L^{p}({\mathbb {R}^{d}})\otimes X\), for \(n\in {{\mathbb N}}\), \(f_{1},\ldots , f_{n}\in L^{p}({\mathbb {R}^{d}})\) and \(x_{1},\ldots , x_{n}\in X\). Then f determines both an element \([t\mapsto \sum _{k=1}^{n}f_{k}(t)x_{k}]\) of \(L^{p}({\mathbb {R}^{d}};X)\) and an element \([\omega \mapsto \sum _{k=1}^{n}x_{k}(\omega )f_{k}]\) of \(X(L^{p}({\mathbb {R}^{d}}))\). Throughout we will identify these and consider f as an element of both \(L^{p}({\mathbb {R}^{d}};X)\) and \(X(L^{p}({\mathbb {R}^{d}}))\). The following lemma, proved as in [60, Theorem 3.9] by using (2.3) and (2.4) on simple Xvalued functions and then approximating, relates the \(L^{p}({\mathbb {R}^{d}};X)\)norm and the \(X(L^{p}({\mathbb {R}^{d}}))\)norm of such an f and will be used later.
Lemma 2.1
 If X is pconvex thenwhere \(C\ge 0\) is as in (2.3).$$\begin{aligned} \Vert f\Vert _{X(L^{p}({\mathbb {R}^{d}}))}\le C\Vert f\Vert _{L^{p}({\mathbb {R}^{d}};X)}, \end{aligned}$$
 If X is pconcave thenwhere \(C\ge 0\) is as in (2.4).$$\begin{aligned} \Vert f\Vert _{L^{p}({\mathbb {R}^{d}};X)}\le C\Vert f\Vert _{X(L^{p}({\mathbb {R}^{d}}))}, \end{aligned}$$
The proof of the following lemma is the same as in [43, Lemma 4] for simple Xvalued functions, and the general case follows by approximation.
Lemma 2.2
2.4 \(\gamma \)Boundedness
2.5 Bessel Spaces
For details on Bessel spaces and related spaces see e.g. [2, 8, 28, 56].
3 Fourier Multipliers Results
In this section we introduce operatorvalued Fourier multipliers acting on various vectorvalued function spaces and discuss some of their properties. We start with some preliminaries and after that in Sect. 3.2 we prove a result that will allow us to transfer boundedness of multipliers on \({\mathbb {R}^{d}}\) to the torus \(\mathbb {T}^{d}\). Then in Sect. 3.3 we present some first simple results under Fourier type conditions. We return to our main multiplier results for spaces with type, cotype, pconvexity and qconcavity in Sects. 3.4 and 3.5.
3.1 Definitions and Basic Properties
In later sections we will use the following lemma about approximation of multipliers, which can be proved as in [23, Proposition 2.5.13].
Lemma 3.1
Proposition 3.2
The Hardy–Littlewood–Sobolev inequality on fractional integration is a typical example where Proposition 3.2 can be applied.
Example 3.3
Let X be a Banach space and \(1<p\le q<\infty \). Let \(m(\xi ) := \xi ^{s}\) for \(s\in [0,d)\) and \(\xi \in {\mathbb {R}^{d}}\). Then \(T_m:L^p({\mathbb R}^d;X)\rightarrow L^q({\mathbb R}^d;X)\) is bounded if and only if \(\frac{1}{p}  \frac{1}{q} = \frac{s}{d}\). In this case \({\mathcal {F}}^{1} m(\cdot ) = C_s \cdot ^{d+s}\) is positive and therefore the result follows from the scalar case (see [24, Theorem 6.1.3]) and Proposition 3.2. The same holds for the multiplier \(m(\cdot ) := (1+\cdot ^{2})^{s/2}\) under the less restrictive condition \(\frac{1}{p}  \frac{1}{q} \le \frac{s}{d}\).
3.2 Transference from \({\mathbb R}^d\) to \({\mathbb T}^d\)
We will mainly consider Fourier multipliers on \({\mathbb R}^d\). However, we want to present at least one transference result to obtain Fourier multiplier results for the torus \({\mathbb T}^d := [0,1]^d\). The transference technique differs slightly from the standard setting of de Leeuw’s theorem where \(p=q\) (see [15, Theorem 4.5] and [28, Chap. 5]), due to the fact that \(\Vert T_{m_a}\Vert _{{\mathcal {L}}(L^p({\mathbb R}^d),L^q({\mathbb R}^d))} = a^{d/r} \Vert T_m\Vert _{{\mathcal {L}}(L^{p}({\mathbb R}^d), L^{q}({\mathbb R}^d))}\), where \(\frac{1}{r} = \frac{1}{p}  \frac{1}{q}\) and \(m_a(\xi ) := m(a\xi )\) for \(a>0\).
Let \(e_k:{\mathbb T}^d\rightarrow {\mathbb C}\) be given by \(e_k(t) := e^{2\pi i k \cdot t}\) for \(k\in {{\mathbb Z}}\) and \(t\in {\mathbb T}^{d}\).
Proposition 3.4
This result seems to be new even in the scalar case \(X=Y={\mathbb C}\).
Proof
The second statement follows from the first since the Xvalued trigonometric polynomials are dense in \(L^{p}({\mathbb T}^{d};X)\). \(\square \)
Remark 3.5
Any Fourier multiplier from \(L^{p}({\mathbb T}^d;X)\) to \(L^{q}({\mathbb T}^d;Y)\) with \(1\le p\le q\le \infty \) trivially yields a multiplier from \(L^u({\mathbb T}^d;X)\) into \(L^v({\mathbb T}^d;Y)\) for all \(p\le u\le v\le q\). Indeed, this follows from the embedding \(L^a({\mathbb T}^d;X)\hookrightarrow L^b({\mathbb T}^d;X)\) for \(a\ge b\). In particular, any boundedness result from \(L^p({\mathbb T}^d;X)\) to \(L^q({\mathbb T}^d;Y)\) implies boundedness from \(L^u({\mathbb T}^d;X)\) into \(L^u({\mathbb T}^d;Y)\).
As an application of Proposition 3.4 and Theorem 1.1 we obtain the following:
Corollary 3.6
Proof
 (1)
for all \(j\in {{\mathbb Z}}\), \(e_{j}\) is an orthogonal projection in H;
 (2)
for all \(j,k\in {{\mathbb Z}}\), \(e_{j} e_{k} = 0\) if \(j\ne k\);
 (3)
for all \(h\in H\), \(\sum _{j\in {{\mathbb Z}}} e_j h = h\).
Corollary 3.7
Proof
Problem 3.8
Can we take \(\frac{1}{r} = \big \frac{1}{a}  \frac{1}{2}\big \) in Corollary 3.7?
If the answer to Problem 3.8 is negative, then the limitations of Theorem 1.1 and Corollary 3.6 are natural. Moreover, from the proof of the latter (see Theorem 3.18 below) it would then follow that the embedding \(H^{\frac{1}{a}\frac{1}{2}}_a({\mathbb R};{\mathscr {C}}^a)\rightarrow \gamma ({\mathbb R};{\mathscr {C}}^a)\) does not hold for \(a\in (1,2)\). Here \(\gamma ({\mathbb R};{\mathscr {C}}^{a})\) is the \({\mathscr {C}}^{a}\)valued \(\gamma \)space used in the proof of Theorem 3.18.
3.3 Fourier Type Assumptions
Before turning to more advanced multiplier theorems, we start with the case where we use the Fourier type of the Banach spaces to derive an analogue of the basic estimate \(\Vert T_m\Vert _{{\mathcal {L}}(L^2({\mathbb R}^d))}\le \Vert m\Vert _{\infty }\).
Proposition 3.9
In Proposition 3.15 we show that this multiplier result characterizes the Fourier type p of X for specific choices of Y, and the Fourier cotype q of Y for specific choices of X.
Proof
Remark 3.10
Remark 3.11
In the scalar setting we noted in (1.1) that the conclusion of Proposition 3.9 holds under the weaker condition \(m\in L^{r,\infty }({\mathbb R}^d)\). In certain cases we can prove such a result in the vectorvalued setting.
Theorem 3.12
Proof
Observe that by real interpolation (see [55, 1.18.6] and [36, (2.33)]) we obtain \({\mathcal {F}}:L^{v', \infty }({\mathbb R}^d;Y)\rightarrow L^{v, \infty }({\mathbb R}^d;Y)\) for all \(v\in (q_0,\infty )\).
The above result provides an analogue of [27, Theorem 1.12]. In general, we do not know the “right” geometric conditions under which such a result holds. We formulate the latter as an open problem.
Problem 3.13
Let \(1<p\le 2\le q<\infty \) and let \(r\in [1,\infty ]\) be such that \(\tfrac{1}{r}=\tfrac{1}{p}\tfrac{1}{q}\). Classify those Banach spaces X and Y for which \(T_{m}\in {\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))\) for all Xstrongly measurable maps \(m:{\mathbb {R}^{d}}\rightarrow {\mathcal {L}}(X,Y)\) such that \(\Vert m(\cdot )\Vert _{{\mathcal {L}}(X,Y)}\in L^{r,\infty }({\mathbb R}^d)\).
A similar question can be asked for the case where \(X = Y\) and m is scalarvalued.
We will now show that the Fourier multiplier result in Proposition 3.9 characterizes the Fourier type of the underlying Banach spaces. To this end we need the following lemma.
Lemma 3.14
Proof
Now we are ready to show that, by letting Y vary, the Fourier multiplier result in Proposition 3.9 characterizes the Fourier type of X, and vice versa.
Proposition 3.15
 (1)
If \(Y={\mathbb C}\) and \(q=2\), then X has Fourier type p.
 (2)
If \(X = {\mathbb C}\) and \(p=2\), then Y has Fourier type \(q'\).
 (3)
If \(Y = X^*\) and \(q = p'\), then X has Fourier type p.
Proof
Remark 3.16
We end this section with a simple example which shows that the geometric limitation in Theorem 3.9 is also natural in the case \(X = Y = \ell ^u\). We will come back to this in Example 3.30, where type and cotype will be used to derive different results.
Example 3.17
We show that this result is sharp in the sense that for \(u\notin [p,p']\) the conclusion is false. This shows that Proposition 3.9 is optimal in the exponent of the Fourier type of the space for \(X=Y=\ell ^{u}\).
3.4 Type and Cotype Assumptions
In Proposition 3.9 and Theorem 3.12 we obtained Fourier multiplier results under Fourier type assumptions on the spaces X and Y. In this section we will present multiplier results under the less restrictive geometric assumptions of type p and cotype q on the underlying spaces X and Y.
First we prove Theorem 1.1 from the Introduction.
Theorem 3.18
It is unknown whether Theorem 3.18 holds with \(p = p_0\) and \(q = q_0\) (see Problem 3.19 below).
Proof
In Theorem 3.21 we provide conditions under which one can take \(p = p_0\) and \(q = q_0\). The general case we state as an open problem:
Problem 3.19
Let \(1\le p\le 2\le q\le \infty \) and \(r\in (1,\infty ]\) be such that \(\tfrac{1}{r}=\tfrac{1}{p}\tfrac{1}{q}\). Classify those Banach spaces X and Y for which \(T_{m}\in {\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))\) for all Xstrongly measurable maps \(m:{\mathbb {R}^{d}}\rightarrow {\mathcal {L}}(X,Y)\) such that \(\{\xi ^{d/r} m(\xi ):\xi \in {\mathbb R}^d\setminus \{0\}\}\) is \(\gamma \)bounded.
The same problem can be formulated in case m is scalarvalued, in which case the \(\gamma \)boundedness reduces to uniform boundedness.
Remark 3.20
Assume X and Y have property \((\alpha )\) as introduced in [46]. (This implies that X has finite cotype, and if X and Y are Banach lattices then property \((\alpha )\) is in fact equivalent to finite cotype.) In the multiplier theorems in this paper where \(\gamma \)boundedness is an assumption, one can deduce a certain \(\gamma \)boundedness result for the Fourier multiplier operators as well. Indeed, assume for example the conditions of Theorem 3.18. Let \(\{m_j:{\mathbb R}^d\setminus \{0\}\rightarrow {\mathcal {L}}(X,Y)\mid j\in \mathcal {J}\}\) be a set of Xstrongly measurable mappings for which there exists a constant \(C\ge 0\) such that for each \(j\in \mathcal {J}\), \(\{\xi ^{\frac{d}{r}}m_j(\xi )\mid \xi \in {\mathbb {R}^{d}}\}\subseteq {\mathcal {L}}(X,Y)\) is \(\gamma \)bounded by C. Note that, since X and Y have finite cotype, \(\gamma \)boundedness and Rboundedness are equivalent. Now we claim that \(\{\widetilde{T_{m_j}}\mid j\in \mathcal {J}\}\subseteq {\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))\) is \(\gamma \)bounded as well. To prove this claim one can use the method of [21, Theorem 3.2]. Indeed, using their notation, it follows from the KahaneKhintchine inequalities that \({\mathrm {Rad}}(X)\) has the same type as X and \({\mathrm {Rad}}(Y)\) has the same cotype as Y. Therefore, given \(j_1, \ldots , j_n\in \mathcal {J}\) and the corresponding \(m_{j_1}, \ldots , m_{j_n}\), one can apply Theorem 3.18 to the multiplier \(M:{\mathbb R}^d\setminus \{0\}\rightarrow {\mathcal {L}}({\mathrm {Rad}}(X),{\mathrm {Rad}}(Y))\) given as the diagonal operator with diagonal \((m_{j_1}, \ldots , m_{j_n})\). In order to check the \(\gamma \)boundedness one now applies property \((\alpha )\) as in [21, Estimate (3.2)].
3.5 Convexity, Concavity and \(L^{p}\)\(L^{q}\) Results in Lattices
In this section we will prove certain sharp results in pconvex and qconcave Banach lattices.
First of all, from the proof of Theorem 3.18 we obtain the following result with the sharp exponents p and q.
Theorem 3.21
Proof
In the case where X is a pconvex and Y is a qconcave Banach lattice, the embeddings in (3.10) can be proved in the same way as in [60, Theorem 3.9], where the inhomogeneous case was considered. Therefore, the result in this case follows from the proof of Theorem 3.18.
Remark 3.22
Note from (3.12) and (3.13) that the constant C in (3.11) depends on X and Y as \(C=\left\ P\right\ _{{\mathcal {L}}(X_{0})}\left\ \iota \right\ _{{\mathcal {L}}(Y,Y_{0})}C_{1}\), where \(P\in {\mathcal {L}}(X_{0})\) is a projection with range X on a pconvex Banach lattice \(X_{0}\) with finite cotype, \(\iota \in {\mathcal {L}}(Y,Y_{0})\) is a continuous embedding of Y in a qconcave Banach lattice \(Y_{0}\) and \(C_{1}\) is a constant that depends only on \(X_{0}\), \(Y_{0}\), p, q and d.
Remark 3.23
By using Theorems 3.18 and 3.21 and by multiplying in the Fourier domain by appropriate powers of \(\xi \), versions of these theorems for multipliers from \(\dot{H}^{\alpha }_p({\mathbb R}^d;X)\) to \(\dot{H}^{\beta }_q({\mathbb R}^d;Y)\) can be derived. Similar results can be derived for the inhomogeneous spaces as well.
So far, in all our results about \((L^{p},L^{q})\)multipliers the indices p and q have been restricted to the range \(p\le 2\le q\), which is necessary when considering general multipliers (see (1.1)). However, we have also seen in Example 3.3 that for the scalar multiplier \(m(\xi )=\xi ^{s}\) such a restriction is not necessary, as follows from Proposition 3.2 since the kernel associated with m is positive. We now show that also for operatorvalued multipliers with positive kernels on pconvex and qconcave Banach lattices, the restriction \(p\le 2\le q\) is not necessary and moreover \(\gamma \)boundedness can be avoided. First we state the result for multipliers between Bessel spaces.
Theorem 3.24
By further approximation arguments one can often avoid the assumptions that \(K(\cdot )x\in L^{1}({\mathbb {R}^{d}};Y)\) for all \(x\in X\). It follows from [50] that the bound in Theorem 3.24 is optimal in a certain sense.
Proof
In terms of \(L^{p}\)\(L^{q}\)multipliers we obtain the following result. Note that below we require that the kernel associated with the multiplicative perturbation \(\xi ^{d/r}m(\xi )\) of m is positive, unlike in Proposition 3.2 where this positivity was required of the kernel associated with m.
Corollary 3.25
Proof
First note that m is of moderate growth at infinity, where we use that \(r>1\). Hence \(T_{m}:\mathcal {S}({\mathbb {R}^{d}})\otimes X\rightarrow \mathcal {S}'({\mathbb {R}^{d}};Y)\) is welldefined. Now the result follows by applying Theorem 3.24 to the symbol \(\xi \mapsto \xi ^{d/r}m(\xi )\in {\mathcal {L}}(X,Y)\), since \(f\mapsto T_{\xi ^{d/r}}(f)\) is an isometric isomorphism \(L^{p}({\mathbb {R}^{d}};X)\rightarrow \dot{H}^{d/r}_{p}({\mathbb {R}^{d}};X)\) and \(T_{m}(f)=T_{\xi ^{d/r}m(\xi )}(T_{\xi ^{d/r}}(f))\) for \(f\in \dot{\mathcal {S}}({\mathbb {R}^{d}};X)\).
3.6 Converse Results and Comparison
In the next result we show that in certain situations the type p of X (or cotype q of Y) is necessary in Theorems 1.1, 3.18 and 3.21. The technique is a variation of the argument of Proposition 3.15 and in particular Lemma 3.14.
Lemma 3.26
At first glance it might seem surprising that we use that X has cotype 2 and Y has type 2. This is to be able to handle the \(\gamma \)bound of \(\{\xi ^{\frac{d}{r}} m(\xi )\mid \xi \in {\mathbb R}^d\}\) in a simple way.
Proof
Proposition 3.27
 (1)
If X has cotype 2, \(Y = {\mathbb C}\), and \(q=2\), then X has type p.
 (2)
If Y has type 2, \(X = {\mathbb C}\), and \(p=2\), then Y has cotype q.
 (3)
If \(Y = X^*\) has type 2, and \(q = p'\), then X has type p.
Proof
Case (2) can be proved in a similar way by reversing the roles of f and g. Indeed, this gives that \(Y^*\) has type \(q'\) and hence Y has cotype q.
In case (3) we let \(f = g\in \mathcal {S}({\mathbb R}^d;X)\) in (3.16) and argue as below (3.17). Here we use that \(X\subseteq X^{**}\) has cotype 2 (see [17, Proposition 11.10]).
Note that Theorem 3.18 and the proof of Proposition 3.27 show that (3.18) holds if X has type \(p_{0}>p\) and cotype 2. Moreover, by the proof above one sees that Pitt’s inequality with \(\beta = 0\) and \(q=2\) implies that X has type p and \(X^*\) has type p. Moreover, in the case \(\alpha = \beta = 0\) and \(q = p'\), Pitt’s inequality is equivalent to X having Fourier type p. It seems that a vectorvalued analogue of Pitt’s inequality has never been studied in detail. This leads to the following natural open problem:
Problem 3.28
Characterize those Banach spaces X for which Pitt’s inequality (3.18) holds.
For pconvex and qconcave Banach lattices, (3.18) can be proved by reducing to the scalar case using the technique of [20, Proposition 2.2].
Next we show that a \(\gamma \)boundedness assumption cannot be avoided in general. In the case where \(p=q\) such a result is due Clément and Prüss (see [28, Chap. 5]). In Proposition 3.9 and Theorem 3.12 we have seen that \(\gamma \)boundedness is not needed for certain \(L^p\)\(L^q\)multiplier theorems. In the following result we derive the necessity of the \(\gamma \)boundedness of \(\{m(\xi )\mid \xi \in {\mathbb R}^d\}\) under special conditions on m.
Proposition 3.29
In Example 3.30 we will provide an example where even the \(\gamma \)boundedness of \(\{\xi ^{d/r} m(\xi )\mid \xi \in {\mathbb R}^d\}\) is necessary. However, in general such a result does not hold (see Remark 3.31).
Proof
Finally, the estimate for the \(\gamma \)bound is wellknown and follows from a randomization argument. \(\square \)
The following example, which is similar to Example 3.17, shows that Theorem 3.21 is sharp in a certain sense. In particular, it shows that the \(\gamma \)boundedness condition is necessary in certain cases.
Example 3.30
Let \(p\in [1,2]\), and for \(q\in [2,\infty )\) let \(r\in (1,\infty ]\) be such that \(\tfrac{1}{r}=\tfrac{1}{p}\tfrac{1}{q}\). Let \(X:=\ell ^{u}\) for \(u\in [1,\infty )\). Let \((e_j)_{j\in {{\mathbb N}}_{0}}\subseteq X\) be the standard basis of X, and for \(k\in {{\mathbb N}}_{0}\) let \(S_{k}\in {\mathcal {L}}(X)\) be such that \(S_{k}(e_{j}):= e_{j+k}\) for \(j\in {{\mathbb N}}_{0}\). Let \(m:{\mathbb R}\rightarrow {\mathcal {L}}(\ell ^{u})\) be given by \(m(\xi ):=\sum _{k=1}^{\infty } c_k {\mathbf {1}}_{(k1,k]}(\xi )S_{k}\) for \(\xi \in {\mathbb R}\), with \(c_k := k^{\alpha } \log (k+1)^{2}\) for \(\alpha \ge 0\) arbitrary but fixed for the moment.
If \(u\in [p, 2]\) then X is a pconvex and qconcave Banach lattice for all \(q\ge p\), hence by Theorem 3.21 we find that with \(\alpha =\tfrac{1}{p}\tfrac{1}{q}+\tfrac{1}{u}  \tfrac{1}{2}\), \(T_m:L^{p}({\mathbb R};X)\rightarrow L^{q}({\mathbb R};X)\) is bounded for all \(q\ge 2\). Note that for \(q=2\) and \(u>p\), m is more singular than in Example 3.17, where we used Proposition 3.9 to obtain the boundedness of \(T_{m}:L^{p}({\mathbb R};X)\rightarrow L^{p'}({\mathbb R};X)\) for \(\alpha =\tfrac{1}{p}\tfrac{1}{p'}> \tfrac{1}{p}+\tfrac{1}{u}1\). In the special case where \(u=p\), both results can be combined using complex interpolation to obtain that \(T_m:L^p({\mathbb R};X)\rightarrow L^q({\mathbb R};X)\) is bounded for all \(q\in [2, p']\) if \(\alpha = \tfrac{2}{p}1\).
Note also that the difference between Proposition 3.9 and Theorem 3.21 is most pronounced when \(p=u=1\). In this case \(X=\ell ^{1}\) has trivial type and trivial Fourier type, but cotype \(q=2\). Hence Proposition 3.9 only yields the boundedness of \(T_{m}:L^{1}({\mathbb R};X)\rightarrow L^{\infty }({\mathbb R};X)\) for \(\alpha \ge 1\), which can also be obtained trivially since in this case m is integrable. On the other hand, Theorem 3.21 yields the nontrivial statement that \(T_{m}:L^{1}({\mathbb R};X)\rightarrow L^{2}({\mathbb R};X)\) is bounded for \(\alpha \ge 1\).
Now fix \(q\in [2,\infty )\) and let \(u\in [2, q]\). Then, similarly, with \(\alpha = 1  \frac{1}{q}  \frac{1}{u}\) the operator \(T_m:L^{2}({\mathbb R};X)\rightarrow L^{q}({\mathbb R};X)\) is bounded. In the special case that \(u = q\), combined with Example 3.17 we find that \(T_m:L^{p}({\mathbb R};X)\rightarrow L^{q}({\mathbb R};X)\) is bounded for all \(p\in [q',2]\) with \(\alpha = \tfrac{2}{q'}1\).
Recall from the last part of Example 3.17 that if \(u\in [1, \infty )\) and \(\alpha = \frac{2}{p}1\) and \(T_m:L^p({\mathbb R};X)\rightarrow L^2({\mathbb R};X)\) is bounded, then \(\frac{1}{u}\le 1\frac{1}{p} + \alpha = \frac{1}{p}\) and thus \(u\ge p\). Similarly, if \(T_m:L^2({\mathbb R};X)\rightarrow L^q({\mathbb R};X)\) is bounded with \(\alpha = 1\frac{2}{q}\), then \(\frac{1}{u'}\le 1\frac{1}{q'} + \alpha = \frac{1}{q'}\), and thus \(u\le q\).
In the following remark we show that one cannot prove the \(\gamma \)boundedness, or even the uniform boundedness, of \(\{\xi ^{d/r} m(\xi )\mid \xi \in {\mathbb R}^d\}\) in general.
Remark 3.31
In the next remark we compare the results obtained in this section with the ones obtained by Fourier type methods.
Remark 3.32
 (i)Consider the case of scalarvalued multipliers m. If \(X = Y\) has Fourier type \(p_0>p\), then Theorem 3.12 states that \(T_m\in {\mathcal {L}}(L^p({\mathbb R}^d;X),L^{p'}({\mathbb R}^d);X))\) for all \(m\in L^{r,\infty }({\mathbb R}^d)\), where \(\frac{1}{r} = \frac{1}{p}  \frac{1}{p'}\). This class of multipliers is larger than the one obtained in Theorem 3.18 sinceOn the other hand, the geometric conditions in Theorem 3.18 are less restrictive. Indeed, Fourier type \(p_0\) implies that X has type \(p_0\) and cotype \(p_0'\), but the converse is false.$$\begin{aligned} \sup \{ \xi ^{\frac{d}{r}} m(\xi )\mid \xi \in {\mathbb R}^d\} \le C_d \Vert m\Vert _{L^{r,\infty }({\mathbb R}^d)}. \end{aligned}$$
 (ii)An important difference between Proposition 3.9 and Theorem 3.12 and the results obtained in Subsections 3.4 and 3.5 is that the former do not require any \(\gamma \)boundedness condition. Of course the assumptions on type and cotype are less restrictive, and furthermore by [30] the \(\gamma \)boundedness can be avoided if X has cotype u and Y has type v and \(\cdot ^{\frac{d}{r}} m(\cdot )\in B^{\frac{d}{w}}_{w,1}({\mathbb R}^d;{\mathcal {L}}(X,Y))\) for \(\frac{1}{w} = \frac{1}{u}\frac{1}{v}\). In this case$$\begin{aligned} \gamma (\{ \xi ^{\frac{d}{r}} m(\xi )\mid \xi \in {\mathbb R}^d\}) \le \Vert \cdot ^{\frac{d}{r}} m(\cdot )\Vert _{{B}^{\frac{d}{w}}_{w,1}({\mathbb R}^d;{\mathcal {L}}(X,Y))}. \end{aligned}$$
4 Extrapolation
In this section we briefly discuss an extension of the extrapolation results of Hörmander in [27].
 \((M1)_{r,\varrho ,n}\) There exists a constant \(M_1\ge 0\) such that for all multiindices \(\alpha \le n\),$$\begin{aligned} R^{\alpha  + \frac{d}{r}  \frac{d}{\varrho }} \Big (\int _{R\le \xi <2R} \Vert \partial ^{\alpha } m(\xi )x\Vert ^{\varrho } \, d\xi \Big )^{1/\varrho }\le M_1\Vert x\Vert \qquad (x\in X, R>0). \end{aligned}$$
 \((M2)_{r,\varrho ,n}\) There exists a constant \(M_2\ge 0\) such that for all multiindices \(\alpha \le n\)$$\begin{aligned} R^{\alpha  + \frac{d}{r}  \frac{d}{\varrho }} \Big (\int _{R\le \xi <2R} \Vert \partial ^{\alpha } m(\xi )^*y^*\Vert ^{\varrho } \, d\xi \Big )^{1/\varrho }\le M_2\Vert y^*\Vert \qquad (y^*\in Y^*, R>0). \end{aligned}$$
Now we can formulate the main result of this section. It extends [27, Theorem 2.5] to the vectorvalued setting and to general exponents \(p,q\in (1, \infty )\).
Theorem 4.1
 (1)Suppose that \(p_0\in (1, \infty ]\), Y has Fourier type \(\varrho \in [1, 2]\) with \(\varrho \le r\), and (M1)\(_{r, \varrho , n}\) holds for \(n := \lfloor \frac{d}{\varrho }  \frac{d}{r} \rfloor +1\). Then \(T_{m}\in {\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))\) andfor all (p, q) such that \(p\in (1, p_0]\) and \(\tfrac{1}{p}\tfrac{1}{q}=\tfrac{1}{r}\), where \(C_{p_0, q_0, p, d}\sim (p1)^{1}\) as \(p\downarrow 1\).$$\begin{aligned} \Vert T_{m}\Vert _{{\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))} \le C_{p_{0},q_{0},p,d} (M_1 + B) \end{aligned}$$(4.1)
 (2)Suppose that \(q_0\in (1, \infty )\), X has Fourier type \(\varrho \in [1, 2]\) with \(\varrho \le r\), and (M2)\(_{r, \varrho , n}\) holds for \(n := \lfloor \frac{d}{\varrho }  \frac{d}{r} \rfloor +1\). Then \(T_{m}\in {\mathcal {L}}(L^{p}({\mathbb {R}^{d}};X),L^{q}({\mathbb {R}^{d}};Y))\) andfor all (p, q) satisfying \(q\in [q_0,\infty )\) and \(\tfrac{1}{p}\tfrac{1}{q}=\tfrac{1}{r}\), where \(C_{p_0, q_0, q, d}\sim q\) as \(q\uparrow \infty \).$$\begin{aligned} \Vert T_m\Vert _{{\mathcal {L}}(L^{p}({\mathbb R}^d;X),L^{q}({\mathbb R}^d;Y))} \le C_{p_{0},q_{0},q,d} (M_2 + B), \end{aligned}$$(4.2)
The proof will be presented in [49]. It is based on the classical argument in the case \(p=q\) (see [23, Theorem 5.2.7]). One of the other ingredients is an operatorvalued analogue of [27, Theorem 2.2].
As a consequence we obtain the following extrapolation result:
Corollary 4.2
In particular, one can always take \(\varrho = 1\) and \(n= \lfloor \frac{d}{r'}\rfloor +1\) in the above results.
Proof
 (i)
\(p_0, q_0\in (1, \infty )\): apply (1) and (2).
 (ii)
\(p_0\in (1, \infty ]\), \(q_0 = \infty \): apply (1).
 (iii)
\(p_0=1\), \(q_0\in (1, \infty )\): apply (2).
 (iv)
\(p_0=1\), \(q_0 =\infty \) is not possible, since \(r\ne 1\).
 (v)
\(p_0=1\), \(q_0 = 1\) is not possible, since \(q_0\ne 1\).\(\square \)
If \(p_0 = q_0=1\), then Theorem 4.1 and Corollary 4.2 are true with \(\varrho =1\) (see [49]).
Next we consider several applications of these extrapolation results.
In [41] an \(L^{p}\)\(L^{q}\)Fourier multiplier result was proved assuming differentiability up to order d. Moreover, in [51] an extension is discussed in the case \(d=1\). We prove a similar result in the Hilbert space case in arbitrary dimensions assuming less differentiability.
Example 4.3
Next let \(r\in (1,2]\). Then all \(p,q\in (1,\infty )\) satisfying \(\tfrac{1}{p}\tfrac{1}{q}=\tfrac{1}{r}\) are such that \(p\in (1,2)\) and \(q\in (2,\infty )\). Hence each m satisfying (4.4) for \(\alpha =0\) yields a bounded operator \(T_{m}:L^{p}({\mathbb {R}^{d}};X)\rightarrow L^{q}({\mathbb {R}^{d}};Y)\) for all such p, q by Theorem 3.12.
Remark 4.4
Thus in the scalar or Hilbertian setting we emphasize that the only new point is that less derivatives are required of the multiplier for \(p<q\).
In the case where X and Y are general Banach spaces, the assertion about \(T_{\xi ^{d/r}}\) remains true. However, the boundedness of \(T_M\) is not as simple to obtain and in general requires geometric conditions on X (even if m is scalarvalued) and an Rboundedness version of the Mihlin condition (see [37]).
Another application of Corollary 4.2 is that we can extrapolate the result of Theorem 3.18 to other values of p and q. A similar result holds for Theorem 3.21.
Corollary 4.5
Let X be a Banach space with type \(p_0\in (1,2]\) and Y a Banach space with cotype \(q_0\in [2,\infty )\), and let \(p_1\in (1, p_0)\) and \(q_1\in (q_0, \infty )\), \(r\in [1,\infty ]\) be such that \(\tfrac{1}{r}=\tfrac{1}{p_1}\tfrac{1}{q_1}\). Let \(m:{\mathbb {R}^{d}}\setminus \{0\}\rightarrow {\mathcal {L}}(X,Y)\) be such that \(\{\xi ^{\frac{d}{r}}m(\xi )\mid \xi \in {\mathbb {R}^{d}}\setminus \{0\}\}\subseteq {\mathcal {L}}(X,Y)\) is \(\gamma \)bounded.
Notes
Acknowledgements
Mark Veraar is supported by the VIDI Subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).
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