Jump inequalities via real interpolation
Abstract.
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 vectorvalued martingales and doubly stochastic operators as well as to pass via sampling from to for jump estimates for Fourier multipliers.
polyjump.bib
1. Introduction
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
(1.1) 
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 quasiseminorms on functions defined by
(1.2) 
for , , and . The quantity is monotonically decreasing in . We omit the index if .
The jump quasiseminorms 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 endpoint refinement.
We now briefly highlight the main results of this paper.

We show (in Theorem 1.5) that the endpoint results hold for for doubly stochastic operators (hence for symmetric diffusion semigroups), 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 [mszdimfree, mszdiscrete].
In more detail, one of our main results is the endpoint Lépingle inequality for vectorvalued martingales.
Theorem 1.3.
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
(1.4) 
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 vectorvalued 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 quasiseminorms (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 quasiseminorm 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 quasiseminorm (1.2) does not seem to define a vectorvalued 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.
Theorem 1.5.
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
(1.6) 
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 vectorvalued case Theorem 1.5 is an endpoint of [MR3648493, Theorem 5.2] (with ) and [MR3648493, Corollary 6.2]. In [MR3648493] the nonendpoint 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 [mszdimfree]) 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 vectorvalued 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.
Theorem 1.7.
Let . Let be a positive integer, a countable ordered set, and a bounded sequencevalued measurable function supported on . Let be the sequencevalued Fourier multiplier operator corresponding to . Define a periodic multiplier by
(1.8) 
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 [mszdiscrete].
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 nonnegative integers will be denoted by .
2. Jump inequalities: abstract theory
2.1. variation
The variation (quasi)seminorm of a function is defined by
(2.1) 
where the former supremum is taken over all finite increasing sequences in .
The quantity is monotonically increasing in and monotonically decreasing in . Moreover,
(2.2) 
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
(2.3) 
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 nonnegative and nondecreasing on . It is concave if the quasinorms on and are subadditive (that is, actual norms). Also, for and we have
(2.4) 
The real interpolation space for and is defined by the quasinorm
(2.5) 
with the natural modification
(2.6) 
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
Note that
so .
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.
Lemma 2.7.
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
(2.8) 
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.
Proof.
We begin with the first inequality in (2.8) and normalize
By definition of real interpolation spaces this is equivalent to
(2.9) 
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
(2.10) 
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
Hence
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 scalarvalued Lorentz space . Hence the splitting witnesses the inequality (2.9) for the functional. ∎
Corollary 2.11.
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 quasiseminorm on in the sense that
for all measurable functions .
Proof.
Let . Then the quasinorm is a norm, and the vectorvalued 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 .
Lemma 2.12.
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
(2.13) 
The previous result [MR2434308, Lemma 2.1] can be recovered by scalarvalued real interpolation. Moreover, Lemma 2.12 reduces Lépingle’s inequality at the endpoint to the jump inequality (3.6).
In the case we will use logconvexity 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
(2.14) 
The same argument shows that the spaces are convex for .
Lemma 2.15.
Given a measure space , let , be a countable set, and let be measurable functions in for every . Then
(2.16) 
Proof.
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
Then
and
so that
Proof of Lemma 2.12.
By monotonicity of variation seminorms it suffices to consider . By scaling we may replace the th power of the lefthand side of (2.13) by
Let
Note that
Therefore, it remains to estimate the measure of the set
For we have
which yields
(2.17) 
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 subalgebras 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 subalgebra .
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
(3.1) 
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
(3.2) 
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.
Theorem 3.3.
Given and , let be a Banach space and a finite measure space. Suppose that the inequality
(3.4) 
holds for arbitrary martingales