Jump inequalities via real interpolation
Jump inequalities are the endpoint of Lépingle’s inequality for -variation of martingales. Extending earlier work by Pisier and Xu [MR933985] we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from to for jump estimates for Fourier multipliers.
Lépingle’s inequality [MR0420837] is a refinement of Doob’s martingale maximal inequality in which the supremum over the time parameter is replaced by the stronger -variation norm, (see Section 2.1 for the definition). Since a sequence with bounded -variation norm is necessarily a Cauchy sequence, Lépingle’s inequality is also a quantitative form of the martingale convergence theorem.
We are interested in the endpoint that had been first stated by Pisier and Xu [MR933985] on and in a different formulation by Bourgain [MR1019960, inequality (3.5)] on , . We begin with Bourgain’s formulation.
Throughout the article denotes a Banach space and is a totally ordered set. Unless otherwise stated, we consider only finite totally ordered sets . This will ensure measurability of all functions that we define. All our estimates do not depend on the cardinality of and the passage to the limiting case of infinite index sets will be permitted by the monotone convergence theorem.
For any the -jump counting function of a function is defined by
The quantity is monotonically increasing in and does not change upon replacing by for any fixed .
For a measure space we consider the family of jump quasi-seminorms on functions defined by
for , , and . The quantity is monotonically decreasing in . We omit the index if .
The jump quasi-seminorms are dominated by norms of -variations (which are recalled in (2.1)) in view of (2.3). On the other hand, bounds for -variations can be deduced from estimates for only when , using Lemma 2.12 and interpolation. Since in many situations one does not have -variational control for , having jump control for is then an end-point refinement.
We now briefly highlight the main results of this paper.
We show (in Theorem 1.5) that the end-point results hold for for doubly stochastic operators (hence for symmetric diffusion semi-groups), while previously these results were known only for .
We also have (in Theorem 1.7) an extension of the sampling technique, that allows us to deduce jump inequalities in the discrete case from the continuous case, for appropriate Fourier multipliers. The original sampling theorem (in [MR1888798]) was limited to , while the jump quantities are not equivalent to Banach spaces of this type.
The results sketched above will be applied in future papers [msz-dim-free, msz-discrete].
In more detail, one of our main results is the endpoint Lépingle inequality for vector-valued martingales.
Let and let be a Banach space with martingale cotype . Then for every -finite measure space , every finite totally ordered set , and every martingale indexed by with values in we have
where the implicit constant does not depend on the martingale .
The notion of cotype for Banach spaces is recalled in Section 3, where a more precise version of Theorem 1.3, namely Theorem 3.3, is stated and proved. Hilbert spaces, and in particular the scalar field , have martingale cotype .
In the above formulation, the scalar case of Theorem 1.3 is due to Bourgain [MR1019960, inequality (3.5)]. In the vector-valued case Theorem 1.3 is an endpoint of the -variation, , estimate in [MR933985, Theorem 4.2]. The basic argument that deduces the -variation from -jump estimates appears in [MR2434308, Lemma 2.1]. We record a refined version of that argument in Lemma 2.12.
Prior to [MR1019960] an endpoint Lépingle inequality in the case , , formulated in terms of an estimate in a real interpolation space, appeared in an article by Pisier and Xu [MR933985, Lemma 2.2]. Our Lemma 2.7 shows that the jump quasi-seminorms (1.2) are equivalent to the (quasi-)norms on certain real interpolation spaces which include those used by Pisier and Xu. In particular this shows that the endpoint Lépingle inequalities of Bourgain and of Pisier and Xu are equivalent. While the formulation in terms of the jump quasi-seminorm is convenient for some purposes (e.g. in the proof of Lemma 2.12), the real interpolation point of view, further explained in Section 2, turns out to be crucial in the proofs of Theorem 1.5 and Theorem 1.7.
By Rota’s dilation theorem estimates for martingales can be transferred to doubly stochastic operators. In the case of jump inequalities an additional complication arises. Namely, the quasi-seminorm (1.2) does not seem to define a vector-valued Lorentz space, so it appears unclear whether conditional expectation operators are bounded with respect to it. This obstacle will be overcome by Lemma 2.7 and the Marcinkiewicz interpolation theorem. This allows us to deduce the following result.
Let be a -finite measure space and let be a doubly stochastic operator on , that is,
for all ,
for all ,
Let be a Banach space with martingale cotype . Then for every and every measurable function we have
Here we identify and with the unique contractive extensions of the tensor products and to , respectively.
Theorem 1.5 is new even in the case providing the endpoint to [MR1869071, Proposition 3.1(1)]. In the vector-valued case Theorem 1.5 is an endpoint of [MR3648493, Theorem 5.2] (with ) and [MR3648493, Corollary 6.2]. In [MR3648493] the non-endpoint result was formulated for Banach spaces that are interpolation spaces between a Hilbert space and a uniformly convex space. It was recently extended to general Banach spaces with finite cotype in [arxiv:1803.05107].
Applying Theorem 1.5 with and one can obtain endpoint jump inequalities in a number of situations, e.g. for averages associated to convex bodies (see [arxiv:1708.04639]; we will give a simplified proof in the forthcoming article [msz-dim-free]) or to spheres in the free group (using Bufetov’s proof [MR1923970, MR2923460] of the result from [MR1294672] and its extensions).
Our last result concerns periodic Fourier multipliers for functions on . From [MR1888798, Section 2] we know how sampling can be used to pass from to in Bochner space multiplier estimates, see also Proposition 4.6 (a Bochner space is a vector-valued space , where is a Banach space). These results do not apply to the spaces defined by (1.2) because they are not Bochner spaces. In Proposition 4.7 we extend this sampling technique to real interpolation spaces between Bochner spaces (that are in general not Bochner spaces, see [MR0358326] for counterexamples). This leads to the following result on jump spaces.
Let . Let be a positive integer, a countable ordered set, and a bounded sequence-valued measurable function supported on . Let be the sequence-valued Fourier multiplier operator corresponding to . Define a periodic multiplier by
and denote the associated Fourier multiplier operator over by . Then
Several multipliers to which Theorem 1.7 applies can be found in the article [MR2434308]. Using Theorem 1.7 instead of [MR1888798, Corollary 2.1] one can obtain the jump endpoint of the variational estimates in [arXiv:1512.07523]. Details of this argument will appear in the forthcoming article [msz-discrete].
We follow the convention that () means that () for some absolute constant . If and hold simultaneously then we will write . The dependence of the implicit constants on some parameters is indicated by a subscript to the symbols , and .
The set of non-negative integers will be denoted by .
2. Jump inequalities: abstract theory
The -variation (quasi-)seminorm of a function is defined by
where the former supremum is taken over all finite increasing sequences in .
The quantity is monotonically increasing in and monotonically decreasing in . Moreover,
The quantity vanishes if and only if the function is constant. For the purpose of using interpolation theory it is convenient to factor out the constant functions. The space of sequences with bounded -variation modulo the constant functions is denoted by
For every and we have
2.2. Real interpolation
In this section we recall the definition of Peetre’s -method of real interpolation. The classical reference on this subject is the book [MR0482275].
A pair of quasinormed vector spaces is called compatible if they are both contained in some ambient topological vector space and the intersection is dense both in and in . For any and the -functional is defined by
The function is non-negative and non-decreasing on . It is concave if the quasinorms on and are subadditive (that is, actual norms). Also, for and we have
The real interpolation space for and is defined by the quasi-norm
with the natural modification
in the case . If quasinorms on both spaces are in fact norms and , then (2.5) defines a norm on . From monotonicity of norms it is easy to see that
2.3. Jump inequalities as an interpolation space
The following observation seems to be new and allows us to use standard real interpolation tools to deal with jump inequalities.
For every , , , and there exists a constant such that the following holds. Let be a measure space, a finite totally ordered set, a Banach space, and a measurable function. Then
As already mentioned here and later we consider finite totally ordered sets . This ensures measurability of all functions that we define and allows us to use stopping time arguments. All our estimates do not depend on the cardinality of and therefore the passage to the limiting case of infinite index sets will be permitted.
We begin with the first inequality in (2.8) and normalize
By definition of real interpolation spaces this is equivalent to
for all .
For a fixed we apply (2.9) with replaced by with some small . By definition of the -functional there exists a splitting with
Now, any -jump of corresponds to a -jump of in the sense that for every and any increasing sequence in as in (1.1) we have
Therefore by (2.3) we get
It follows that
This proves the first inequality in (2.8).
We now prove the second inequality in (2.8). Normalizing
we have to show (2.9). Fix and construct stopping times (measurable functions) starting with . Given , let
with the convention that is greater than every element of . For let
With this stopping time we split , where
By construction of the stopping time we have for all and , so that
On the other hand, is constant for , so while estimating its variation norm we can restrict the supremum in (2.1) to sequences taking values in the sequence of stopping times . With to be chosen later we split the jumps according to their size and obtain
By the hypothesis the norm of the highlighted function is at most , and the series can be summed provided that is sufficiently large in terms of the quasimetric constant of the scalar-valued Lorentz space . Hence the splitting witnesses the inequality (2.9) for the -functional. ∎
For every , , and there exists a constant such that for every measure space , finite totally ordered set , and Banach space there exists a (subadditive) seminorm equivalent to the quasi-seminorm on in the sense that
for all measurable functions .
Let . Then the quasinorm is a norm, and the vector-valued Lorentz space admits an equivalent norm (with equivalence constants depending only on and ). Hence the interpolation quasinorm in (2.8) is actually a norm. ∎
2.4. From jump inequalities to -variation
It has been known since Bourgain’s article [MR1019960] that variational estimates can be deduced from jump inequalities. In Bourgain’s article this is accomplished for averaging operators by interpolation with an estimate. More in general, Jones, Seeger, and Wright showed that jump inequalities with different values of can be interpolated to yield variation norm estimates [MR2434308, Lemma 2.1]. Our next result is that a weak type jump inequality implies a weak type estimate for the variation seminorm for a fixed .
For every and there exists such that the following holds. Let be a -finite measure space, a finite totally ordered set, a Banach space, and . Then for every measurable function we have the estimates
In the case we will use log-convexity of . More precisely, let be a measure space, a countable set, and measurable functions with for every , where are positive numbers.
Then from [MR0241685, Lemma 2.3] we know
The same argument shows that the spaces are -convex for .
Given a measure space , let , be a countable set, and let be measurable functions in for every . Then
By scaling it suffices to show that if for all and and are numbers such that , then
Without loss of generality we may assume for all . Let
Proof of Lemma 2.12.
By monotonicity of variation seminorms it suffices to consider . By scaling we may replace the -th power of the left-hand side of (2.13) by
Therefore, it remains to estimate the measure of the set
For we have
We distinguish two cases to estimate (2.17). Suppose first . Then, since admits an equivalent subadditive norm, we get
Suppose now . Then by (2.16) we have
In the case using (2.14) with we obtain
Alternatively, still in the case , we can estimate
3. Endpoint Lépingle inequality for martingales
Let be a -finite measure space and a totally ordered set. A sequence of sub--algebras of is called a filtration if it is increasing and the measure is -finite on each . Let be a Banach space. A -valued martingale adapted to a filtration is a family of functions such that for every with , where denotes the conditional expectation with respect to a sub--algebra .
We recall from [MR3617459, Theorem 10.59] that a Banach space has martingale cotype if and only if for any -valued martingale the “-square function”
satisfies the estimates
where is the martingale maximal function and the implicit constant does not depend on . By Doob’s inequality, see e.g. [MR3617459, Corollary 1.28], we know that
A Banach space has martingale cotype for some if and only if it is uniformly convex, see [MR3617459, Chapter 10].
Now we are in a position to formulate the quantitative version of the endpoint Lépingle inequality for martingales.
Given and , let be a Banach space and a -finite measure space. Suppose that the inequality
holds for arbitrary martingales