Fourier multiplier theorems involving type and cotype
In this paper we develop the theory of Fourier multiplier operators , for Banach spaces and , and an operator-valued symbol. The case has been studied extensively since the 1980’s, but far less is known for . In the scalar setting one can deduce results for from the case . However, in the vector-valued 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 and are UMD spaces and that satisfies a smoothness condition.
We show that for other geometric conditions on and , such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for without any smoothness properties of . Under smoothness conditions the boundedness results can be extrapolated to other values of and as long as remains constant.
Key words and phrases:Operator-valued Fourier multipliers, type and cotype, Fourier type, Hörmander condition, -boundedness
2010 Mathematics Subject Classification:Primary: 42B15; Secondary: 42B35, 46B20, 46E40, 47B38
Fourier multiplier operators play a major role in analysis and in particular in the theory of partial differential equations. Such operators are of the form
where denotes the Fourier transform and is a function on . Usually one is interested in the boundedness of with (the case is trivial by [28, Theorem 1.1]). The class of Fourier multiplier operators coincides with the class of singular integral operators of convolution type , where is a tempered distribution.
The simplest class of examples of Fourier multipliers can be obtained by taking . Then is bounded if and only if , and . For and one obtains only trivial multipliers, namely Fourier transforms of bounded measures. The case where is highly nontrivial. In general only sufficient conditions on are known that guarantee that is bounded, although also here it is necessary that .
In the classical paper  Hörmander studied Fourier multipliers and singular integral operators of convolution type. In particular, he showed that if , then
Here denotes the weak -space. In particular, every with satisfies . It was also shown that the condition is necessary here. More precisely, if there exists a function such that has nonzero measure and for all with , is bounded, then .
Hörmander also introduced an integral/smoothness condition on the kernel which allows one to extrapolate the boundedness of from to for some to boundedness of from to for all satisfying . This led to extensions of the theory of Calderón and Zygmund in . In the case it was shown that the smoothness condition on the kernel can be translated to a smoothness condition on the multiplier 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 [24, 25, 36, 54] and references therein for a treatment and the history of the subject.
In the vector-valued setting it was shown in  that the extrapolation results of Hörmander for still holds. However, there is a catch:
even for one does not have for general unless is a Hilbert space.
In  it was shown that for if satisfies the so-called UMD condition. In  it was realized that this yields a characterization of the UMD property. In , ,  versions of the Littlewood–Paley theorem and the Mihlin multiplier theorem were established in the UMD setting. These are very useful for operator theory and evolution equations (see for example ).
In the vector-valued setting it is rather natural to allow to take values in the space of bounded operators from to . Pisier and Le Merdy showed that the natural analogues of the Mihlin multiplier theorem do not extend to this setting unless has cotype and has type (a proof was published only later on in ). On the other hand there was a need for such extensions as it was realized that multiplier theorems with operator-valued symbols are useful in the stability theory and the regularity theory for evolution equations (see [2, 27, 61]). The missing ingredient for a natural analogue of the Mihlin multiplier theorem turned out to be -boundedness, which is a strengthening of uniform boundedness (see [9, 14]). In  it was shown that Mihlin’s theorem holds for if the sets
are -bounded. Conversely, the -boundedness of is also necessary. These results were used to characterize maximal -regularity, and were then used by many authors in evolution equations, partial differential equations, operator theory and harmonic analysis (see the surveys and lecture notes [2, 16, 34, 38]). A generalization to multipliers on instead of was given in  and , but in some cases one additionally needs the so-called property of the Banach space (which holds for all UMD lattices). Improvements of the multiplier theorems under additional geometric assumptions have been studied in  and  assuming Fourier type and in  assuming type and cotype conditions.
In this article we complement the theory of operator-valued Fourier multipliers by studying the boundedness of from to for . One of our main results is formulated under -boundedness assumptions on . We note that -boundedness implies -boundedness (see Subsection 2.4). The result is as follows (see Theorem 3.18 for the proof):
Let be a Banach space with type and a Banach space with cotype , and let , . Let be such that . If is -strongly measurable and
is -bounded, then is bounded. Moreover, if (or , then one can also take (or ).
The condition cannot be avoided in such results (see below (1.1)). Note that no smoothness on is required. Theorem 1.1 should be compared to the sufficient condition in (1.1) due to Hörmander in the case where . We will give an example which shows that the -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 is scalar-valued and , the -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 -convex and -concave Banach lattices, where one can take and . In  we will deduce multiplier results similar to Theorem 1.1 in the Besov scale, where one can let and for Banach spaces and with type and cotype .
where . We show that in this result the Fourier type assumption is necessary. It should be noted that for many spaces (including all -spaces for ), working with Fourier type yields more restrictive results in terms of the underlying parameters than working with type and cotype (see Subsection 2.2 for a discussion of the differences between Fourier type and (co)type).
The exponents and 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 satisfying and . Here the required smoothness depends on the Fourier type of and and on the number . We note that even in the case where , for we require less smoothness for the extrapolation than in the classical results (see Remark 4.4).
We will mainly consider multiplier theorems on . 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 to the torus . 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 for .
We have pointed out that questions about operator-valued Fourier multiplier theorems were originally motivated by stability and regularity theory. We have already successfully applied our result to stability theory of -semigroups, as will be presented in a forthcoming paper . In  the first-named author has also applied the Fourier multiplier theorems in this article to study the -calculus for generators of -groups.
Other potential applications could be given to the theory of dispersive equations. For instance the classical Strichartz estimates can be viewed as operator-valued --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 vector-valued singular integrals from to .
This article is organized as follows. In Section 2 we discuss some preliminaries on the geometry of Banach spaces and on function space theory. In Section 3 we introduce Fourier multipliers and prove our main results on --multipliers in the vector-valued setting. In Section 4 we present an extension of the extrapolation result under Hörmander–Mihlin conditions to the case .
1.1. Notation and terminology
We write for the natural numbers and .
We denote nonzero Banach spaces over the complex numbers by and . The space of bounded linear operators from to is , and . The identity operator on is denoted by .
For and a measure space, denotes the Bochner space of equivalence classes of strongly measurable, -integrable, -valued functions on . Moreover, is the weak -space of all for which
where for . In the case where we implicitly assume that is the Lebesgue measure. Often we will use the shorthand notations and for the -norm and -norm.
The Hölder conjugate of is denoted by and is defined by . We write for the space of -summable sequences , and denote by the space of -summable sequences .
We say that a function is -strongly measurable if is a strongly measurable -valued map for all . We often identify a scalar function with the operator-valued function given by for .
The class of -valued rapidly decreasing smooth functions on (the Schwartz functions) is denoted by , and the space of -valued tempered distributions by . We write and denote by the -valued duality between and . The Fourier transform of a is denoted by or . If then
A standard complex Gaussian random variable is a random variable of the form , where is a probability space and are independent standard real Gaussians. A Gaussian sequence is a (finite or infinite) sequence of independent standard complex Gaussian random variables on some probability space.
We will use the convention that a constant which appears multiple times in a chain of inequalities may vary from one occurrence to the next.
2.1. Fourier type
A Banach space has Fourier type if the Fourier transform is bounded from to for some (in which case it holds for all) . We then write .
Each Banach space has Fourier type with for all . If has Fourier type then has Fourier type with for all and . We say that has nontrivial Fourier type if has Fourier type for some . In order to make our main results more transparent we will say that has Fourier cotype whenever has Fourier type .
Let be a Banach space, and let be a measure space. If has Fourier type then has Fourier type . In particular, has Fourier type .
2.2. Type and cotype
Let be a Banach space, a Gaussian sequence on a probability space and let and . We say that has (Gaussian) type if there exists a constant such that for all and all ,
We say that has (Gaussian) cotype if there exists a constant such that for all and all ,
with the obvious modification for .
The minimal constants in (2.1) and (2.2) are called the (Gaussian) type constant and the (Gaussian) cotype constant and will be denoted by and . We say that has nontrivial type if has type , and finite cotype if has cotype .
Note that it is customary to replace the Gaussian sequence in (2.1) and (2.2) by a Rademacher sequence, i.e. a sequence of independent random variables on a probability space that are uniformly distributed on . This does not change the class of spaces under consideration, only the minimal constants in (2.1) and (2.2) (see [17, Chapter 12]). We choose to work with Gaussian sequences because the Gaussian constants and occur naturally here.
Each Banach space has type and cotype , with . If has type and cotype then has type with for all and cotype with for all . A Banach space is isomorphic to a Hilbert space if and only if has type and cotype , by Kwapień’s theorem (see [1, Theorem 7.4.1]). Also, a Banach space with nontrivial type has finite cotype by the Maurey–Pisier theorem (see [1, Theorem 11.1.14]).
Let be a Banach space, and let be a measure space. If has type and cotype then has type and cotype (see [17, Theorem 11.12]).
2.3. Convexity and concavity
For the theory of Banach lattices we refer the reader to . We repeat some of the definitions which will be used frequently.
Let be a Banach lattice and . We say that is -convex if there exists a constant such that for all and all ,
with the obvious modification for . We say that is -concave if there exists a constant such that for all and all ,
with the obvious modification for .
Every Banach lattice is -convex and -concave. If is -convex and -concave then it is -convex and -concave for all and . By [41, Proposition 1.f.3], if is -concave then it has cotype , and if is -convex and -concave for some then has type .
If is -convex and -concave for then has Fourier type , by [21, Proposition 2.2]. For a measure space and , is an -convex and -concave Banach lattice. Moreover, if is -convex and -concave and , then is -convex and -concave.
Specific Banach lattices which we will consider are the Banach function spaces. For the definition and details of these spaces we refer to . If is a Banach function space over a measure space and is a Banach space, then consists of all such that , with the norm
If for and then we write for the element of given by
Let , for , and . Then determines both an element of and an element of . Throughout we will identify these and consider as an element of both and . The following lemma, proved as in [60, Theorem 3.9] by using (2.3) and (2.4) on simple -valued functions and then approximating, relates the -norm and the -norm of such an and will be used later.
Let be a Banach function space, and .
The proof of the following lemma is the same as in [44, Lemma 4] for simple -valued functions, and the general case follows by approximation.
Let and be Banach function spaces, a positive operator, and . Then
Let and be Banach spaces. A collection is -bounded if there exists a constant such that
for all , , and each Gaussian sequence . The smallest such is the -bound of and is denoted by . By the Kahane-Khintchine inequalities, we may replace the -norm in (2.5) by an -norm for each .
Every -bounded collection is uniformly bounded with supremum bound less than or equal to the -bound, and the converse holds if and only if has cotype and has type (see ). By the Kahane contraction principle, for each -bounded collection and each , the closure in the strong operator topology of the family is -bounded with
By replacing the Gaussian random variables in (2.5) by Rademacher variables, one obtains the definition of an -bounded collection . Each -bounded collection is -bounded. The notions of -boundedness and -boundedness are equivalent if and only if has finite cotype (see [39, Theorem 1.1]), but the minimal constant in (2.5) may depend on whether one considers Gaussian or Rademacher variables. In this article we work with -boundedness instead of -boundedness because in our results we will allow spaces which do not have finite cotype.
2.5. Bessel spaces
For a Banach space, and the inhomogeneous Bessel potential space consists of all such that . Then is a Banach space endowed with the norm
and lies dense if .
Endow with the subspace topology induced by and set . Let be the space of continuous linear mappings . Then each yields an by restriction, and if and only if . Conversely, one can check that each extends to an element of (see  for the tedious details in the vector-valued setting). Hence for . As in [24, Proposition 2.4.1] one can show that , where is the collection of polynomials on . If is a linear subspace such that if , then we will identify with its image in . In particular, this is the case if for some .
For and , the homogeneous Bessel potential space is the space of all such that , where
Then is a Banach space endowed with the norm
and lies dense if .
3. Fourier multipliers results
In this section we introduce operator-valued Fourier multipliers acting on various vector-valued function spaces and discuss some of their properties. We start with some preliminaries and after that in Subsection 3.2 we prove a result that will allow us to transfer boundedness of multipliers on to the torus . Then in Subsection 3.3 we present some first simple results under Fourier type conditions. We return to our main multiplier results for spaces with type, cotype, -convexity and -concavity in Subsections 3.4 and 3.5.
3.1. Definitions and basic properties
Fix , let and be Banach spaces, and let be -strongly measurable. We say that is of moderate growth at infinity if there exist a constant and a such that
For such an , let be given by
We call the Fourier multiplier operator associated with and we call the symbol of .
Let . We say that is a bounded -Fourier multiplier if there exists a constant such that and
for all . In the case , extends uniquely to a bounded operator from to which will be denoted by , and often just by when there is no danger of confusion. If and then we simply say that is an -Fourier multiplier.
We will also consider Fourier multipliers on homogeneous function spaces. Let and be Banach spaces and let be -strongly measurable. We say that is of moderate growth at zero and infinity if there exist a constant and a such that
For such an , let be given by
where is well-defined by definition of . We use similar terminology as before to discuss the boundedness of . Often we will simply write , to simplify notation.
In later sections we will use the following lemma about approximation of multipliers, which can be proved as in [24, Proposition 2.5.13].
Let and be Banach spaces and . For each let be -strongly measurable, and let be such that for all and almost all . Suppose that there exist and such that
for all and . If is such that for all , and if , then with
The same result holds for if instead we assume that there exist an and such that, for all and ,
The case of positive scalar-valued kernels plays a special role. An immediate consequence of [24, Proposition 4.5.10] is:
Proposition 3.2 (Positive kernels).
Let have moderate growth at zero and infinity. Suppose that is bounded for some and that is positive. Then, for any Banach space , the operator