Variational estimates for martingale paraproducts
VVARIATIONAL ESTIMATESFOR MARTINGALE PARAPRODUCTS
VJEKOSLAV KOVAČ AND PAVEL ZORIN-KRANICH
Abstract.
We show that bilinear variational estimates of Do, Muscalu, andThiele [DMT12] remain valid for a pair of general martingales with respect to thesame filtration. Our result can also be viewed as an off-diagonal generalization ofthe Burkholder–Davis–Gundy inequality for martingale rough paths by Chevyrevand Friz [CF19]. Introduction If f = ( f n ) ∞ n =0 is a discrete-time real-valued martingale with respect to a filtration F = ( F n ) ∞ n =0 , then Lépingle’s variational inequality [Lép76] claims (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup m ∈ N n Mathematics Subject Classification. Primary 60G42; Secondary 60G44. a r X i v : . [ m a t h . P R ] M a r V. KOVAČ AND P. ZORIN-KRANICH If d f n := f n − f n − for n ≥ denotes martingale differences, then the truncated paraproducts can be written, quiteelegantly, as Π n,n (cid:48) ( f, g ) = (cid:88) n , then Π( f, g ) is again a martingaleadapted to F , so in particular it also satisfies (1.1). However, one still cannot relaxthe condition (cid:37) > for general f and g .It is a bit surprising that there exists a variant of Lépingle’s inequality for truncatedmartingale paraproducts that allows (cid:37) to go below . This is the main result of ourpaper and it is a generalization of Theorem 1.2 from the paper [DMT12] by Do,Muscalu, and Thiele. Theorem 1.1. Take exponents p, q, r satisfying (1.6) . If f = ( f n ) ∞ n =0 and g =( g n ) ∞ n =0 are martingales with respect to the same filtration, then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup m ∈ N n Continuous-time martingales. One benefit of having Theorem 1.1 formu-lated for general discrete-time martingales is that estimates (1.8) and (1.9) immedi-ately transfer to continuous-time martingales X = ( X t ) t ≥ and Y = ( Y t ) t ≥ . It isstandard in stochastic calculus to assume that X and Y almost surely have càdlàgpaths and that their filtration F = ( F t ) t ≥ satisfies “the usual hypotheses” fromProtter’s book [Pro05], i.e. that F is complete and that F is right-continuous. Wefix the exponents p, q, r satisfying (1.6) and additionally assume that (cid:107) X (cid:107) L p := sup t ≥ (cid:107) X t (cid:107) L p < ∞ and (cid:107) Y (cid:107) L q := sup t ≥ (cid:107) Y t (cid:107) L q < ∞ . (1.10)Under more restrictive conditions on X and Y , such as (cid:107) X (cid:107) L ∞ < ∞ and (cid:107) Y (cid:107) L < ∞ ,the papers [BB88] and [KŠ18] proceed by defining the paraproduct as the process Π( X, Y ) = (Π t ( X, Y )) t ≥ given by the stochastic integral Π t ( X, Y ) := (cid:90) (0 ,t ] X s − d Y s . (1.11)Here X s − stands for the left limit lim u → s − X u . The above integral is understood asthe Itô integral and it yields another process with almost surely càdlàg paths. Inorder to extend the definition of Π( X, Y ) to the martingales satisfying (1.10) only,and to enable the application of Theorem 1.1, we prefer to construct the martingaleparaproduct as a limit of certain discrete-time paraproducts, namely the Riemannsums of (1.11).A random partition of [0 , ∞ ) will be any tuple Σ = ( τ , τ , . . . , τ l ) of finite stop-ping times with respect to F such that τ ≤ τ ≤ · · · ≤ τ l . We definethe corresponding Riemann sum of the integral (1.11) as the stochastic process S ( X, Y ; Σ) = ( S t ( X, Y ; Σ)) t ≥ given by S t ( X, Y ; Σ) := l (cid:88) j =1 X min { t,τ j − } (cid:0) Y min { t,τ j } − Y min { t,τ j − } (cid:1) . Following the language of [Pro05], let us say that a sequence of random partitions (Σ n ) ∞ n =1 , Σ n = ( τ ( n )0 , τ ( n )1 , . . . , τ ( n ) l n ) , tends to the identity if lim n →∞ τ ( n ) l n = ∞ a.s.and lim n →∞ max ≤ j ≤ l n | τ ( n ) j − τ ( n ) j − | = 0 a.s. Corollary 1.2. (a) There exists a unique (up to indistinguishability) stochastic pro-ces Π( X, Y ) = (Π t ( X, Y )) t ≥ with a.s. càdlàg paths such that for any sequence ofrandom partitions (Σ n ) ∞ n =1 tending to the identity the Riemann sums S ( X, Y ; Σ n ) converge uniformly on compacts in probability (u.c.p.) towards Π( X, Y ) , i.e. lim n →∞ P (cid:16) sup s ∈ [0 ,t ] | S s ( X, Y ; Σ n ) − Π s ( X, Y ) | > ε (cid:17) = 0 for each ε > and each t > . We say that Π( X, Y ) is the paraproduct ofmartingales X and Y .(b) Truncated paraproducts are now defined as random variables Π t,t (cid:48) ( X, Y ) := Π t (cid:48) ( X, Y ) − Π t ( X, Y ) − X t ( Y t (cid:48) − Y t ) for any ≤ t < t (cid:48) < ∞ . We have (cid:13)(cid:13)(cid:13)(cid:13) sup m ∈ N t Connection with rough paths. One can view the triple H n := ( f n , g n , Π n ( f, g )) as a process with values in a -dimensional Heisenberg group H ∼ = R with the groupoperation ( x, y, z ) · ( x (cid:48) , y (cid:48) , z (cid:48) ) = ( x + x (cid:48) , y + y (cid:48) , z + z (cid:48) + xy (cid:48) ) . Then the truncated martingale paraproducts Π n,n (cid:48) are precisely the z -coordinates ofthe differences of this process. More precisely, for any times n ≤ n (cid:48) we have H n · ( f n (cid:48) − f n , g n (cid:48) − g n , Π n,n (cid:48) ( f, g )) = H n (cid:48) . This corresponds to Chen’s relation [FH14, (2.1)] in the theory of rough paths.On the Heisenberg group we consider the homogeneous box norm (cid:107) ( x, y, z ) (cid:107) :=max( | x | , | y | , | z | / ) and the corresponding distance function d ( H, H (cid:48) ) := (cid:107) H − H (cid:48) (cid:107) .One can verify that (cid:107) H · H (cid:48) (cid:107) ≤ (cid:107) H (cid:107) + (cid:107) H (cid:48) (cid:107) , and this implies the triangle inequality d ( H, H (cid:48)(cid:48) ) ≤ d ( H, H (cid:48) ) + d ( H (cid:48) , H (cid:48)(cid:48) ) . Any other left-invariant homogeneous distance,e.g. the Carnot–Carathéodory distance, would work equally well. Combining (1.9)and (1.3) one can obtain the jump estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) sup m ∈ N n We begin this section with a few words on the notation. Then we review severalknown martingale inequalities that will be needed in subsequent sections. Some ofthem we could not find formulated elsewhere with exactly the same assumptions.However, the proofs of those inequalities are still quite straightforward using theresults available in the existing literature and we include them for completeness.For any two quantities A, B ∈ [0 , ∞ ] we will write A (cid:46) B when there exists anunimportant constant C ∈ [0 , ∞ ) such that A ≤ CB . Furthermore, we will write A ∼ B if both A (cid:46) B and B (cid:46) A hold. Dependencies of the implicit constants onsome parameters will be denoted in the subscripts of (cid:46) and ∼ . For real numbers a and b we will write a ∧ b := min { a, b } , a ∨ b := max { a, b } . We have already encountered the L p -quasinorms h (cid:55)→ (cid:107) h (cid:107) L p in the introductorysection, both for finite p and for p = ∞ . Recall that for a martingale f = ( f n ) ∞ n =0 the quantity (cid:107) f (cid:107) L p is defined by (1.2). Any nonnegative random variable w givesrise to the weighted L p -quasinorms , given for p ∈ (0 , ∞ ) as (cid:107) h (cid:107) L p ( w ) := (cid:0) E ( | h | p w ) (cid:1) /p . ARIATIONAL ESTIMATES FOR MARTINGALE PARAPRODUCTS 5 On the other hand, the weak L p -quasinorm is defined as (cid:107) h (cid:107) L p, ∞ := (cid:16) sup α ∈ (0 , ∞ ) α p P ( | h | > α ) (cid:17) /p for any p ∈ (0 , ∞ ) . Any sequence of random variables h = ( h ( k ) ) ∞ k =1 can be regardedas a vector-valued random element and for p ∈ (0 , ∞ ] and q ∈ (0 , ∞ ) we define the mixed L p ( (cid:96) q ) -quasinorm (cid:107) h (cid:107) L p ( (cid:96) q ) = (cid:13)(cid:13) h ( k ) (cid:13)(cid:13) L p ( (cid:96) qk ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) h ( k ) (cid:12)(cid:12) q (cid:17) /q (cid:13)(cid:13)(cid:13)(cid:13) L p . Finally, p (cid:48) will always denote the conjugated exponent of p ∈ [1 , ∞ ] , i.e. the uniquenumber p (cid:48) ∈ [1 , ∞ ] such that /p + 1 /p (cid:48) = 1 .For any martingale f = ( f n ) ∞ n =0 with respect to a fixed filtration F = ( F n ) ∞ n =0 one defines the maximal function M f := sup n ≥ | f n | and the square function S f := (cid:16) ∞ (cid:88) n =1 | d f n | (cid:17) / . (2.1)Note that M f and S f are two random variables taking values in [0 , ∞ ] . In differentterminology these are the limits of the maximum process of f and the quadraticvariation of f , respectively. If we start merely from a random variable h , then weautomatically assign to it the martingale ( h n ) ∞ n =0 defined by h n := E ( h | F n ) , so M h and S h still make sense.The well known Burkholder–Davis–Gundy inequality claims that (cid:107) M f (cid:107) L p ∼ p (cid:107) S f (cid:107) L p (2.2)for every p ∈ [1 , ∞ ) . Indeed, the case p > is due to Burkholder [Bur66], while thecase p = 1 was shown by Davis [Dav70]. A weighted version of the latter case wasestablished by Osękowski [Osę17]: (cid:107) M f (cid:107) L ( w ) (cid:46) (cid:107) S f (cid:107) L ( M w ) , (2.3)where w is a nonnegative integrable random variable, interpreted as a weight. Theimplicit constant in (2.3) is an absolute one and Osękowski could choose √ .Inequality (2.3) can also be viewed as a probabilistic analogue of a weighted estimateby Fefferman and Stein [FS71].Moreover, Doob’s maximal inequality reads (cid:107) M f (cid:107) L p ≤ p (cid:48) (cid:107) f (cid:107) L p (2.4)for every p ∈ (1 , ∞ ] . It also has a weighted version, formulated for instance as a partof Theorem 3.2.3 in the book [HvNVW16]: (cid:107) M f (cid:107) L p ( w ) ≤ p (cid:48) (cid:107) f ∞ (cid:107) L p ( M w ) (2.5)for p ∈ (1 , ∞ ] . In (2.5) we assume, for convenience, that ( f n ) ∞ n =0 eventually becomesa constant sequence, so that f ∞ := lim n →∞ f n trivially makes sense with respect toevery possible mode of convergence.Suppose that T ≤ T ≤ T ≤ · · · is a sequence of stopping times taking valuesin N with respect to the filtration F and assume that each T k is bounded. Thesestopping times will be used for the purpose of certain “localization.” Boundednessof each individual T k is a convenient assumption for the application of the optionalsampling theorem; see e.g. [GS01]. For every k ∈ N and every n ∈ N we note that ( n ∨ T k − ) ∧ T k is also a stopping time with respect to F and define F ( k ) n := F ( n ∨ T k − ) ∧ T k . (2.6) V. KOVAČ AND P. ZORIN-KRANICH That way, each F ( k ) := ( F ( k ) n ) ∞ n =0 becomes a filtration of the original probabilityspace and each of these sequences of σ -algebras becomes constant for sufficientlylarge indices n . Lemma 2.1. Let ( T k ) ∞ k =0 be an increasing sequence of bounded stopping times, let ( F ( k ) ) ∞ k =1 be a sequence of filtrations defined by (2.6) , and for each k ∈ N let f ( k ) =( f ( k ) n ) ∞ n =0 be a martingale with respect to F ( k ) that eventually becomes a constantsequence. For any p, q ∈ (1 , ∞ ) we have (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) L p ( (cid:96) qk ) (cid:46) p,q (cid:13)(cid:13) f ( k ) ∞ (cid:13)(cid:13) L p ( (cid:96) qk ) . (2.7)Lemma 2.1 can be viewed as an (cid:96) q -valued extension of Doob’s maximal inequality(2.4). The proof of (2.7) is based on (2.5) and it already exists as the proof of[HvNVW16, Theorem 3.2.7]. However, the working assumption in [HvNVW16] isthat f ( k ) are arbitrary martingales with respect to the same filtration, which is notthe case here. For this reason and for the sake of completeness we prefer to repeatthe short argument, rather than just invoke the result from [HvNVW16]. Proof of Lemma 2.1. The case p ≥ q is handled first. Let r ∈ (1 , ∞ ] denote theconjugated exponent of p/q . To an arbitrary random variable w ≥ satisfying (cid:107) w (cid:107) L r = 1 we associate the martingales ( w n ) ∞ n =0 and w ( k ) = ( w ( k ) n ) ∞ n =0 , for each k ∈ N , via w n := E ( w | F n ) , w ( k ) n := E ( w | F ( k ) n ) . (2.8)Consider the expression E (cid:18)(cid:16) ∞ (cid:88) k =1 (cid:0) M f ( k ) (cid:1) q (cid:17) w (cid:19) = ∞ (cid:88) k =1 E (cid:18)(cid:0) M f ( k ) (cid:1) q w (cid:19) = ∞ (cid:88) k =1 (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) q L q ( w ) . By (2.5) this is at most a constant depending on q times ∞ (cid:88) k =1 (cid:13)(cid:13) f ( k ) ∞ (cid:13)(cid:13) q L q ( M w ( k ) ) = ∞ (cid:88) k =1 E (cid:16)(cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) q (cid:0) M w ( k ) (cid:1)(cid:17) ≤ E (cid:16)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) q (cid:17)(cid:0) M w (cid:1)(cid:17) ≤ (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) q (cid:13)(cid:13)(cid:13) L p/q (cid:13)(cid:13) M w (cid:13)(cid:13) L r . Applying (2.4) to (cid:107) M w (cid:107) L r and recalling the freedom that we had in choosing w , weobtain (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:0) M f ( k ) (cid:1) q (cid:13)(cid:13)(cid:13) L p/q (cid:46) p,q (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) q (cid:13)(cid:13)(cid:13) L p/q , which transforms into (2.7) after taking the q -th root of both sides.Turning to the case p ≤ q , we take some r ∈ (1 , p ) and denote a := p/r ∈ (1 , ∞ ) , b := q/r ∈ (1 , ∞ ) . Write (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) r L p ( (cid:96) qk ) = (cid:18) E ( ∞ (cid:88) k =1 (cid:0) M f ( k ) (cid:1) q ) p/q (cid:19) r/p = (cid:13)(cid:13)(cid:0) M f ( k ) (cid:1) r (cid:13)(cid:13) L a ( (cid:96) bk ) . (2.9)We are going to dualize the mixed L a ( (cid:96) b ) -norm above and for this we take a sequence h = ( h ( k ) ) ∞ k =1 of nonnegative random variables such that (cid:107) h (cid:107) L a (cid:48) ( (cid:96) b (cid:48) ) < ∞ . Each h ( k ) defines a martingale ( h ( k ) n ) ∞ n =0 by h ( k ) n := E ( h ( k ) | F ( k ) n ) . Using (2.5) for each fixed k followed by Hölder’s inequality we obtain E ∞ (cid:88) k =1 (cid:0) M f ( k ) (cid:1) r h ( k ) = ∞ (cid:88) k =1 (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) r L r ( h ( k ) ) (cid:46) r ∞ (cid:88) k =1 (cid:13)(cid:13) f ( k ) ∞ (cid:13)(cid:13) r L r ( M h ( k ) ) = E ∞ (cid:88) k =1 (cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) r (cid:0) M h ( k ) (cid:1) ≤ (cid:13)(cid:13)(cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) r (cid:13)(cid:13) L a ( (cid:96) bk ) (cid:13)(cid:13) M h ( k ) (cid:13)(cid:13) L a (cid:48) ( (cid:96) b (cid:48) k ) . ARIATIONAL ESTIMATES FOR MARTINGALE PARAPRODUCTS 7 Then applying the previous case of (2.7) (with p, q replaced by a (cid:48) , b (cid:48) ) to get (cid:13)(cid:13) M h ( k ) (cid:13)(cid:13) L a (cid:48) ( (cid:96) b (cid:48) k ) (cid:46) a,b (cid:13)(cid:13) h ( k ) ∞ (cid:13)(cid:13) L a (cid:48) ( (cid:96) b (cid:48) k ) = (cid:107) h (cid:107) L a (cid:48) ( (cid:96) b (cid:48) ) and using duality we end up with (cid:13)(cid:13)(cid:0) M f ( k ) (cid:1) r (cid:13)(cid:13) L a ( (cid:96) bk ) (cid:46) a,b (cid:13)(cid:13)(cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) r (cid:13)(cid:13) L a ( (cid:96) bk ) . Recall the computation (2.9) and take the r -th root of both sides. (cid:3) Now, let f = ( f n ) ∞ n =0 be a single martingale with respect to F . For every k ∈ N and every n ∈ N we denote, for the rest of the paper, f ( k ) n := f ( n ∨ T k − ) ∧ T k − f T k − , (2.10)i.e. f ( k ) n = for n ≤ T k − ,f n − f T k − for T k − < n ≤ T k ,f T k − f T k − for n > T k . That way, for each k ∈ N we have now defined a particular martingale f ( k ) :=( f ( k ) n ) ∞ n =0 with respect to the filtration F ( k ) given by (2.6). It is “interesting” onlyfor moments between T k − and T k . Consequently, the sequence ( f ( k ) n ) ∞ n =0 eventuallybecomes constant and, in particular, the limit f ( k ) ∞ := lim n →∞ f ( k ) n exists (in everypossible way) and simply equals f T k − f T k − . Many classical inequalities in termsof martingale f have their vector-valued extensions in terms of its “localized pieces” f ( k ) . Our next goal is to formulate and prove a couple of those, as they will beneeded in the next section. Lemma 2.2. Let ( T k ) ∞ k =0 be an increasing sequence of bounded stopping times andlet f be a martingale, both with respect to F . Moreover, let ( f ( k ) ) ∞ k =1 be a sequenceof martingales defined by (2.10) .(a) For any p ∈ (1 , ∞ ) we have (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) L p ( (cid:96) k ) (cid:46) p (cid:107) f (cid:107) L p . (2.11) (b) For any p ∈ (1 , ∞ ) we have (cid:13)(cid:13) S f ( k ) (cid:13)(cid:13) L p ( (cid:96) k ) (cid:46) p (cid:107) f (cid:107) L p . (2.12) Proof of Lemma 2.2. (a) Since the stopping times T k are bounded, using the optionalsampling theorem (see Section 12.4 of the book [GS01]) and applying (2.2) and (2.4)to the “optionally sampled” martingale ( f T n ) ∞ n =0 we get (cid:13)(cid:13) f ( k ) ∞ (cid:13)(cid:13) L p ( (cid:96) k ) = (cid:13)(cid:13)(cid:13)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) f ( k ) ∞ (cid:12)(cid:12) (cid:17) / (cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) f T k − f T k − (cid:12)(cid:12) (cid:17) / (cid:13)(cid:13)(cid:13) L p = (cid:107) S ( f T k ) ∞ k =0 (cid:107) L p (cid:46) p (cid:107) M ( f T k ) ∞ k =0 (cid:107) L p (cid:46) p (cid:107) f (cid:107) L p . Combining this with estimate (2.7) from Lemma 2.1 specialized to q = 2 establishes(2.11).(b) Estimate (2.12) is immediate. We only need to observeS f ( k ) = (cid:16) (cid:88) T k − Multilinear interpolation. We will repeatedly use a multilinear version of theMarcinkiewicz interpolation theorem. We caution the reader that many such resultsexist in the literature, and not every version would be adequate for our purposes. Werefer to [GLLZ12, Corollary 1.1], of which the result below is a special case, althoughit also follows e.g. from the result of [Jan88] on abstract interpolation spaces. Theorem 2.3. Let T be a bisublinear operator, i.e., | T ( f + f , g ) | ≤ | T ( f , g ) | + | T ( f , g ) | and | T ( f, g + g ) | ≤ | T ( f, g ) | + | T ( f, g ) | , initially defined on simplefunctions on a pair of measure spaces with values in measurable functions on a thirdmeasure space. Suppose that the estimate (cid:107) T ( f, g ) (cid:107) L r, ∞ ≤ C (cid:107) f (cid:107) L p (cid:107) g (cid:107) L q (2.13) holds with < p, q ≤ ∞ , /r = 1 /p + 1 /q , and (1 /p, /q ) being the corners of a non-degenerate triangle ∆ ⊂ [0 , ∞ ) . Then for every < p, q ≤ ∞ such that (1 /p, /q ) is in the interior of ∆ and for /r = 1 /p + 1 /q we have (cid:107) T ( f, g ) (cid:107) L r ≤ C (cid:107) f (cid:107) L p (cid:107) g (cid:107) L q , where the constant C depends only on ∆ , p, q , and the constants in (2.13) . A vector-valued estimate for martingale paraproducts The main ingredient in the proof of Theorem 1.1 is the following proposition. Proposition 3.1. Let ( T k ) ∞ k =0 be an increasing sequence of bounded stopping timesand let f and g be martingales, all with respect to the same filtration F . Moreover, let ( f ( k ) ) ∞ k =1 and ( g ( k ) ) ∞ k =1 be sequences of martingales defined from f and g , respectively,via (2.10) . Then for any exponents p, q, r satisfying (1.6) we have the estimate (cid:13)(cid:13) Π ∞ ( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r ( (cid:96) k ) (cid:46) p,q (cid:107) f (cid:107) L p (cid:107) g (cid:107) L q . (3.1)Note that, for each k ∈ N , the paraproduct Π( f ( k ) , g ( k ) ) is a martingale withrespect to the filtration F ( k ) given by (2.6). The sequence (Π n ( f ( k ) , g ( k ) )) ∞ n =0 even-tually becomes constant, so that Π ∞ ( f ( k ) , g ( k ) ) makes sense. A crucial observation,following from (1.4) and needed later, is Π ∞ ( f ( k ) , g ( k ) ) = (cid:88) T k −
Let us first discuss the case r ≥ of estimate (3.1). Webegin by proving the (cid:96) -valued estimate (cid:13)(cid:13) Π ∞ ( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r ( (cid:96) k ) (cid:46) p (cid:13)(cid:13) S Π( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r ( (cid:96) k ) . (3.3)We will reduce it to the weighted estimate (2.3) for martingales Π( f ( k ) , g ( k ) ) . Takean arbitrary nonnegative random variable satisfying (cid:107) w (cid:107) L r (cid:48) = 1 and define ( w n ) ∞ n =0 and w ( k ) = ( w ( k ) n ) ∞ n =0 as in (2.8). We have E (cid:18)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g ( k ) ) (cid:12)(cid:12)(cid:17) w (cid:19) ≤ E (cid:18)(cid:16) ∞ (cid:88) k =1 M Π( f ( k ) , g ( k ) ) (cid:17) w (cid:19) = ∞ (cid:88) k =1 E (cid:18)(cid:0) M Π( f ( k ) , g ( k ) ) (cid:1) w (cid:19) = ∞ (cid:88) k =1 (cid:13)(cid:13) M Π( f ( k ) , g ( k ) ) (cid:13)(cid:13) L ( w ) ARIATIONAL ESTIMATES FOR MARTINGALE PARAPRODUCTS 9 and, by (2.3) applied to martingale Π( f ( k ) , g ( k ) ) for each fixed k , this is at most aconstant times ∞ (cid:88) k =1 (cid:13)(cid:13) S Π( f ( k ) , g ( k ) ) (cid:13)(cid:13) L ( M w ( k ) ) = ∞ (cid:88) k =1 E (cid:18)(cid:0) S Π( f ( k ) , g ( k ) ) (cid:1)(cid:0) M w ( k ) (cid:1)(cid:19) ≤ E (cid:18)(cid:16) ∞ (cid:88) k =1 S Π( f ( k ) , g ( k ) ) (cid:17)(cid:0) M w (cid:1)(cid:19) ≤ (cid:13)(cid:13) S Π( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r ( (cid:96) k ) (cid:13)(cid:13) M w (cid:13)(cid:13) L r (cid:48) . Using Doob’s inequality (2.4) for the martingale ( w n ) ∞ n =0 we end up with E (cid:18)(cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g ( k ) ) (cid:12)(cid:12)(cid:17) w (cid:19) (cid:46) r (cid:13)(cid:13) S Π( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r ( (cid:96) k ) . Recalling the freedom that we had in choosing w we establish (3.3) by dualization.In order to complete the proof of (3.1) in the case r ≥ , observe that the expressionon the right hand side of (3.3) is, by the definition of the paraproduct, equal to (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:16) ∞ (cid:88) n =1 ( f ( k ) n − ) (d g ( k ) n ) (cid:17) / (cid:13)(cid:13)(cid:13) L r ≤ (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 M f ( k ) S g ( k ) (cid:13)(cid:13)(cid:13) L r , which is, by Hölder’s inequality, in turn bounded by (cid:13)(cid:13) M f ( k ) (cid:13)(cid:13) L p ( (cid:96) k ) (cid:13)(cid:13) S g ( k ) (cid:13)(cid:13) L q ( (cid:96) k ) (cid:46) p,q (cid:107) f (cid:107) L p (cid:107) g (cid:107) L q . In the last inequality we used (2.11) and (2.12) for the martingales f and g , respec-tively.We will now prove the weak-type estimate (cid:13)(cid:13) Π ∞ ( f ( k ) , g ( k ) ) (cid:13)(cid:13) L r, ∞ ( (cid:96) k ) (cid:46) p (cid:107) f (cid:107) L p (cid:107) g (cid:107) L (3.4)for any p ∈ (1 , ∞ ) and r ∈ (1 / , such that /p + 1 = 1 /r . This will conclude theproof of (3.1) for r < by real interpolation with the previously established cases(Theorem 2.3). By the homogeneity of (3.4) we can normalize: assume (cid:107) f (cid:107) L p = 1 and (cid:107) g (cid:107) L = 1 . Fix a number ν > and perform Gundy’s decomposition [Gun68] ofthe martingale g at height α = ν r ; see its formulation as Theorem 3.4.1 in the book[HvNVW16]. It splits g as g n = g good n + g bad n + g harmless n , where g good = ( g good n ) ∞ n =0 , g bad = ( g bad n ) ∞ n =0 , and g harmless = ( g harmless n ) ∞ n =0 are mar-tingales with respect to F satisfying g good = g , g bad = g harmless = 0 , (cid:107) g good (cid:107) L ∞ ≤ α, (cid:107) g good (cid:107) L ≤ (cid:107) g (cid:107) L , (3.5) P ( M g bad > ≤ α − (cid:107) g (cid:107) L , (3.6) ∞ (cid:88) n =1 (cid:107) d g harmless n (cid:107) L ≤ (cid:107) g (cid:107) L . (3.7)Construct the martingales g good , ( k ) , g bad , ( k ) , and g harmless , ( k ) for the given sequenceof stopping times via the formula (2.10). Using the previously established case r = 1 of estimate (3.1) and (3.5) we obtain P (cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g good , ( k ) ) (cid:12)(cid:12) > ν (cid:17) (cid:46) ν − (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g good , ( k ) ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13) L (cid:46) p ν − (cid:107) f (cid:107) L p (cid:107) g good (cid:107) L p (cid:48) ≤ ν − (cid:107) f (cid:107) L p (cid:107) g good (cid:107) /p L ∞ (cid:107) g good (cid:107) /p (cid:48) L (cid:46) ν − ν r/p = ν − r . Next, (3.6) yields P (cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g bad , ( k ) ) (cid:12)(cid:12) > (cid:17) ≤ P ( M g bad > (cid:46) ν − r . Finally, by Hölder’s inequality, Doob’s inequality (2.4), and (3.7) we conclude P (cid:16) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g harmless , ( k ) ) (cid:12)(cid:12) > ν (cid:17) (cid:46) r ν − r (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:12)(cid:12) Π ∞ ( f ( k ) , g harmless , ( k ) ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13) r L r ≤ ν − r (cid:13)(cid:13)(cid:13) ∞ (cid:88) k =1 (cid:88) T k − Proof of Theorem 1.1. In the process of proving estimates (1.8) and (1.9) we canconstrain the numbers n , n , . . . , n m to a finite interval of integers { , , , . . . , n max } .Then we only need to take care that the obtained constants do not depend on n max .Afterwards we will be able to let n max → ∞ and use the monotone convergencetheorem, recovering Theorem 1.1 in its full generality.Let us begin with a stopping time argument enabling us to apply Proposition 3.1.We are given two martingales, f = ( f n ) ∞ n =0 and g = ( g n ) ∞ n =0 , with respect to thefiltration F . Fix λ > and recursively define an increasing sequence of stoppingtimes ( S k ) ∞ k =0 by setting S := 0 and S k := min (cid:26) n > S k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) S k −
We plan to apply Proposition 3.1 with the above sequence of stopping times ( T k ) ∞ k =0 . By the definitions of S k and T k we have λ (cid:101) N λ ( f, g ) ≤ (cid:101) N λ ( f,g ) (cid:88) k =1 (cid:12)(cid:12)(cid:12) (cid:88) T k −
Proof of Corollary 1.2. (a) In the particular case (cid:107) X (cid:107) L ∞ < ∞ and (cid:107) Y (cid:107) L < ∞ wealready know that S ( X, Y ; Σ n ) converge u.c.p. as n → ∞ to the stochastic processgiven by (1.11). This is the content of Theorem 21 in Chapter II of the book [Pro05].In the general case, for any δ > we find càdlàg martingales X (cid:48) = ( X (cid:48) t ) t ≥ and Y (cid:48) = ( Y (cid:48) t ) t ≥ with respect to F such that (cid:107) X (cid:48) (cid:107) L ∞ < ∞ , (cid:107) X − X (cid:48) (cid:107) L p < δ , (cid:107) Y (cid:48) (cid:107) L < ∞ , and (cid:107) Y − Y (cid:48) (cid:107) L q < δ . Rewrite the difference S s ( X, Y ; Σ m ) − S s ( X, Y ; Σ n ) as thesum of S s ( X (cid:48) , Y (cid:48) ; Σ m ) − S s ( X (cid:48) , Y (cid:48) ; Σ n ) and S s ( X − X (cid:48) , Y ; Σ m ) + S s ( X (cid:48) , Y − Y (cid:48) ; Σ m )+ S s ( X (cid:48) , Y (cid:48) − Y ; Σ n ) + S s ( X (cid:48) − X, Y ; Σ n ) . (5.1)From the first part of the proof we know lim m,n →∞ P (cid:16) sup s ∈ [0 ,t ] | S s ( X (cid:48) , Y (cid:48) ; Σ m ) − S s ( X (cid:48) , Y (cid:48) ; Σ n ) | > ε (cid:17) = 0 (5.2)for each ε > and each t > . By sampling arbitrary continuous-time martin-gales (cid:101) X and (cid:101) Y at times t ∧ τ ( n ) j we obtain discrete-time martingales such that ( S t ∧ τ ( n ) j ( (cid:101) X, (cid:101) Y ; Σ n )) l n j =0 is their paraproduct. Thus, estimate (1.7) applies and, to-gether with Doob’s inequality for Y , easily gives (cid:13)(cid:13)(cid:13) sup s ∈ [0 ,t ] (cid:12)(cid:12) S s ( (cid:101) X, (cid:101) Y ; Σ n ) (cid:12)(cid:12)(cid:13)(cid:13)(cid:13) L r (cid:46) p,q (cid:13)(cid:13) (cid:101) X (cid:13)(cid:13) L p (cid:13)(cid:13) (cid:101) Y (cid:13)(cid:13) L q , with a constant independent of the partition Σ n . Applying this to each of the fourterms in (5.1), using the Markov–Chebyshev inequality, applying (5.2), and finallyletting δ → + , we obtain lim sup m,n →∞ P (cid:16) sup s ∈ [0 ,t ] | S s ( X, Y ; Σ m ) − S s ( X, Y ; Σ n ) | > ε (cid:17) = 0 for ε, t > . Thus, S ( X, Y ; Σ n ) converge u.c.p. as n → ∞ to some stochastic process,which we denote by Π( X, Y ) . Note that Π( X, Y ) still has càdlàg paths a.s., sincethis property is preserved under taking u.c.p. limits. It is standard to conclude that Π( X, Y ) does not depend on the choice of (Σ n ) ∞ n =0 .(b) We explain how (1.8) implies (1.12); very similarly one can use (1.9) toprove (1.13). It is sufficient to establish a variant of (1.12) in which the numbers t , t , . . . , t m are only taken from a fixed finite set of nonnegative rational numbers Σ , but with a constant that does not depend on Σ . Afterwards, we can let thosesets Σ exhaust [0 , ∞ ) ∩ Q , invoking the monotone convergence theorem. At the veryend one can recall that Π( X, Y ) almost surely has càdlàg paths, so that Π t,t (cid:48) ( X, Y ) is almost surely right-continuous in t and t (cid:48) .Starting with a finite set Σ we take an increasing sequence (Σ n ) ∞ n =0 of finite subsetsof [0 , ∞ ) with the following properties. If we write explicitly Σ n = (cid:8) a ( n )0 , a ( n )1 , . . . , a ( n ) l n (cid:9) , a ( n )0 < a ( n )1 < · · · < a ( n ) l n , then we require Σ = Σ , a ( n )0 = 0 for n ≥ , lim n →∞ a ( n ) l n = ∞ , and lim n →∞ max ≤ j ≤ l n (cid:12)(cid:12) a ( n ) j − a ( n ) j − (cid:12)(cid:12) = 0 . From part (a) applied to deterministic partitions Σ n we know that Π t k − ,t k ( X, Y ) = lim n →∞ (cid:0) S t k ( X, Y ; Σ n ) − S t k − ( X, Y ; Σ n ) − X t k − ( Y t k − Y t k − ) (cid:1) = lim n →∞ (cid:88) j : t k −
V.K. was supported in part by the Croatian Science Foundation under the projectUIP-2017-05-4129 (MUNHANAP). P.Z. was partially supported by the HausdorffCenter for Mathematics (DFG EXC 2047) and DFG SFB 1060. The authors also ac-knowledge support of the bilateral DAAD–MZO grant Multilinear singular integralsand applications . The authors would like to thank P. Friz for turning their attentionto the recent literature on martingale rough paths. References [BB88] R. Bañuelos and A. G. Bennett. “Paraproducts and commutators of martingale trans-forms”. In: Proc. Amer. Math. Soc. mr : (cit. onpp. 2, 3).[Bou89] J. Bourgain. “Pointwise ergodic theorems for arithmetic sets”. In: Inst. Hautes ÉtudesSci. Publ. Math. 69 (1989). With an appendix by the author, Harry Furstenberg,Yitzhak Katznelson and Donald S. Ornstein, pp. 5–45. mr : (cit. on p. 1).[Bur66] D. L. Burkholder. “Martingale transforms”. In: Ann. Math. Statist. 37 (1966), pp. 1494–1504. mr : (cit. on pp. 2, 5). EFERENCES 13 [CF19] I. Chevyrev and P. K. Friz. “Canonical RDEs and general semimartingales as roughpaths”. In: Ann. Probab. . mr : (cit. on pp. 1, 4).[CL92] J.-A. Chao and R.-L. Long. “Martingale transforms with unbounded multipliers”. In: Proc. Amer. Math. Soc. mr : (cit. on p. 2).[Dav70] B. Davis. “On the integrability of the martingale square function”. In: Israel J. Math. mr : (cit. on p. 5).[DMT12] Y. Do, C. Muscalu, and C. Thiele. “Variational estimates for paraproducts”. In: Rev.Mat. Iberoam. . mr : (cit. on pp. 1, 2).[DMT17] Y. Do, C. Muscalu, and C. Thiele. “Variational estimates for the bilinear iteratedFourier integral”. In: J. Funct. Anal. . mr : (cit. on p. 2).[FH14] P. K. Friz and M. Hairer. A course on rough paths . With an introduction to regularitystructures. Universitext. Springer, Cham, 2014, pp. xiv+251. mr : (cit. onp. 4).[FS71] C. Fefferman and E. M. Stein. “Some maximal inequalities”. In: Amer. J. Math. mr : (cit. on p. 5).[FV06] P. Friz and N. Victoir. “The Burkholder-Davis-Gundy inequality for enhanced mar-tingales”. In: Séminaire de probabilités XLI . Vol. 1934. Lecture Notes in Math.Springer, Berlin, 2006, pp. 421–438. arXiv: math / 0608783 . mr : (cit. onp. 4).[GLLZ12] L. Grafakos, L. Liu, S. Lu, and F. Zhao. “The multilinear Marcinkiewicz interpolationtheorem revisited: the behavior of the constant”. In: J. Funct. Anal. mr : (cit. on p. 8).[GS01] G. R. Grimmett and D. R. Stirzaker. Probability and random processes . Third. OxfordUniversity Press, New York, 2001, pp. xii+596. mr : (cit. on pp. 5, 7).[Gun68] R. F. Gundy. “A decomposition for L -bounded martingales”. In: Ann. Math. Statist. 39 (1968), pp. 134–138. mr : (cit. on p. 9).[HvNVW16] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces .Vol. I: Martingales and Littlewood-Paley theory . Cham: Springer, 2016, pp. xvi+614. mr : (cit. on pp. 5, 6, 9).[Jan88] S. Janson. “On interpolation of multilinear operators”. In: Function spaces and ap-plications (Lund, 1986) . Vol. 1302. Lecture Notes in Math. Springer, Berlin, 1988,pp. 290–302. mr : (cit. on p. 8).[KŠ18] V. Kovač and K. A. Škreb. “Bellman functions and L p estimates for paraproducts”.In: Probab. Math. Statist. . mr : (cit. on p. 3).[Lép76] D. Lépingle. “La variation d’ordre p des semi-martingales”. In: Z. Wahrscheinlichkeits-theorie und Verw. Gebiete mr : (cit. on p. 1).[MSZ18] M. Mirek, E. M. Stein, and P. Zorin-Kranich. “Jump inequalities via real interpola-tion”. Preprint. 2018. arXiv: (cit. on pp. 1, 4, 11).[Osę17] A. Osękowski. “A Fefferman-Stein inequality for the martingale square and maximalfunctions”. In: Statist. Probab. Lett. 129 (2017), pp. 81–85. mr : (cit. on p. 5).[Pro05] P. E. Protter. Stochastic integration and differential equations . Vol. 21. StochasticModelling and Applied Probability. Second edition. Version 2.1, Corrected thirdprinting. Springer-Verlag, Berlin, 2005, pp. xiv+419. mr : (cit. on pp. 3,11).(Vjekoslav Kovač) Department of Mathematics, Faculty of Science, University ofZagreb, Bijenička cesta 30, 10000 Zagreb, Croatia E-mail address : [email protected] (Pavel Zorin-Kranich) Mathematical Institute, University of Bonn, Endenicher Allee60, D-53115 Bonn, Germany E-mail address ::