Martingale transforms and their projection operators on manifolds

Transcription

1 artingale transforms and their rojection oerators on manifolds Rodrigo Bañuelos, Fabrice Baudoin Deartment of athematics Purdue University West Lafayette, 4796 Setember 9, Abstract We rove the boundedness on L, < <, of oerators on manifolds which arise by taking conditional exectation of transformations of stochastic integrals. These oerators include various classical oerators such as second order Riesz transforms and oerators of Lalace transform-tye. Contents Introduction artingale inequalities 3 3 Proof of Theorem. and some refinements 7 3. An estimate of the constant in Alications 4 4. ultiliers of Lalace transform-tye for Schrödinger oerators on manifolds Second order Riesz transforms on Lie grous of comact tye Introduction The classical martingale inequalities of Burkholder and Gundy [8] lay a fundamental role in many areas of robability and its alications. These inequalities have their roots in the celebrated 966 martingale transforms inequality of Burkholder [6]. In 984 [7], Burkholder obtained Suorted in art by NSF Grant #637-DS Suorted in art by NSF Grant 9736 DS

2 the shar constants in his 966 martingale inequalities. In recent years, these shar inequalities have had many alications to the study of basic singular integrals and Fourier multiliers on Euclidean sace IR d with the Lebesgue measure the ordinary Lalacian case and with Gaussian measure the Ornstein Uhlenbeck case. For an account of some of this literature we refer the reader to the overview article []. The urose of this aer is to show that these martingale transform techniques aly to wide range of oerators on manifolds, including multiliers arising from Schrödinger oerators and Riesz transforms on Lie grous. For the Lalacian, and other self-adjoint diffusions without a zero order term, Burkholder s shar inequalities can be alied and our L bounds are exactly as those on IR d. However, in the case of Schrödinger oerators the shar Burkholder inequalities do not aly in any direct way and in this case we obtain our results by roving a version of the Burkholder-Gundy inequalities with an exonential weight which does not roduce best constants. Our results in this aer see Remark. below also correct a ga contained in [], [] and [3]. See also [] for details on the correct formulation of some of the results in those aers. To introduce our oerators and state our results, let be a smooth manifold endowed with a smooth measure µ. Let X,, X d be locally Lischitz vector fields defined on. We consider the Schrödinger oerator, L = Xi X i + V, i= where Xi denotes the formal adjoint of X i with resect to µ and where V : R is a nonositive smooth function, often referred to it in this aer as a otential. We assume that L is essentially self-adjoint with resect to µ on the sace C of smooth and comactly suorted functions and that the solution of the Stratonovitch stochastic differential equation associated with L is non-elosive. We denote by P t t the heat semigrou with generator L. We can write L = d i= X i + X + V, for some locally Lischitz vector field X. Let A ij : [, + R, i, j d be bounded and smooth real valued functions and let At, x = A ij be the d d matrix with A ij entries. Set A = At, x L [,+, where At, x is the usual quadratic norm of the d d matrix At, x. That is, At, x = su{ At, xξ ; ξ IR d, ξ = }. Our goal in this aer is to study the continuity on L µ for < < of the oerator. S A f = i,j= P t X i A ij tx j P t fdt and to obtain recise information on the size of their norms. The following two remarks shed some light on the structure of these oerators in two imortant secial cases.

3 Remark.. If L = d i= X i X i + V is subellitic, then the semigrou P t admits a smooth symmetric kernel t, x, y see [5], [7] and in that case, it is easily seen that S A fx = Kx, zfzdµz, with Kx, z = i,j= + A ij t, yx y i t, y, xxy j t, y, zdµydt. Remark.. Let, g be a comlete Riemannian manifold. In this framework the Lalace- Beltrami oerator is essentially self-adjoint on C. We may then consider the Schrödinger oerator L = + V where, as above, V is a non-ositive smooth otential. In this case, the oerator S A can be written as S A f = P t divat P t fdt. For < < let denote the maximum of and q, where + q =. Thus = max{, } and {. =, <,, <. The following is the main result of this aer. Theorem.. For any < < there is a constant C deending only on such that for every f L µ,.3 S A f C A f. If the otential V, then.4 S A f A f, and this bound is shar. As we shall see below, these oerators include the multiliers of Lalace transform-tye and second order Riesz transforms on Lie grous of comact tye. The fact that the bound in.4 is best ossible follows from the fact that the best constant in the L inequality for second order Riesz transforms R j R k on IR d is ; see 4. below and [3] and [3]. The aer is organized as follows. In, we recall various versions of Burkholder s shar inequalities and rove a version of the Burkholder-Gundy inequality needed for.3. The roof of Theorem. is given in 3 where we also give an exlicit bound for the constant in.3. In 4, we give some concrete examles of our oerators. 3

4 artingale inequalities In this section we recall the shar martingale inequalities of Burkholder and rove a version of the Burkholder-Gundy inequalities used in the roof of.3 in Theorem.. Let f = {f n, n } be a martingale with different sequence df = {df k, k }, where df k = f k f k for k and df = f. Given a redictable sequence of random variables {v k, k } uniformly bounded for all k the martingale difference sequence {v k df k, k } generates a new martingale called the martingale transform of f and denoted by v f. We set f = su n f n for < <. Burkholder s 966 result [6] asserts that the oerator f v f = g is bounded on L for all < <. In his 984 seminal aer [7] Burkholder determined the norm of this oerator by roving the following result. Theorem.. Let f = {f n, n } be a martingale and let g = v f be its martingale transform by the redictable sequence v = {v k, k } with v k taking values in [, ] for all k. Then. g f, < <, and the constant is best ossible. In [9], K.P. Choi used the techniques of Burkholder to identify the best constant in the martingale transforms where the redictable sequence v takes values in [, ] instead of [, ]. While Choi s constant is not as exlicit as the constant of Burkholder, one does have a lot of information on it. Theorem.. Let f = {f n, n } be a real-valued martingale and let g = v f be its martingale by a redictable sequence v = {v k, k } with values in [, ] for all k. Then. g c f, < <, with the best constant c satisfying where α = [ log c = + log + e + e ] + log + α + + e e + e. otivated by Theorems. and. the following definition was introduced in [3]. Definition.. Let < b < B < and < < be given and fixed. We define C,b,B as the least ositive number C such that for any real-valued martingale f and for any transform g = v f of f by a redictable sequence v = {v k, k } with values in [b, B], we have.3 g C f. 4

5 Thus, for examle, C, a,a = a by Burkholder s Theorem. and C,,a = a c by Choi s Theorem.. It is also the case that for any b, B as above, { } B b.4 max, max{ B, b } C,b,B max{b, b } For the alications to the above oerators we will need versions of these inequalities for continuous-time martingales. Suose that Ω, F, P is a comlete robability sace, filtered by F t t, a nondecreasing and right-continuous family of sub-σ-fields of F. Assume, as usual, that F contains all the events of robability. Let X, Y be adated, real valued martingales which have right-continuous aths with left-limits r.c.l.l.. Denote by [X, X] the quadratic variation rocess of X: we refer the reader to Dellacherie and eyer [] for details. Following [4] and [7], we say that Y is differentially subordinate to X if the rocess if Y X and [X, X] t [Y, Y ] t t is nondecreasing and nonnegative as a function of t. We have the following extension of the Burkholder inequalities roved in [4] for continuous-ath martingales and in [7] in the general case. Set X = su t X t, < <. Theorem.3. If Y is differentially subordinate to X, then.5 Y X, < <, and the inequality is shar. The case of non-symmetric multiliers is covered by the following result roved in [3]. Theorem.4. Suose < b < B < and X t, Y t are two real valued martingales with which have right-continuous aths with left-limits with Y X and which satisfy [ B b.6 X, B b ] [ X Y b + B X, Y b + B ] X and nondecreasing for all t differential subordination. Then and the inequality is shar. t Y C,b,B X, < <, Note that when B = a > and b = a, we have the case of Theorem.3. As we shall see,.5 will give the bound in.4 and.6 will give some extensions. For the general case when V for the inequality.3 none of the above results aly and we shall need a variation of the Burkholder-Gundy inequalities. While this bound is not shar, it alies to a wide class of rocesses and can even be used to study oerators on manifolds acting on forms. We shall comment more on this a little later. We start by recalling the following domination inequality due to Lenglart [8] See also Revuz-Yor [5], Proosition.. Lenglart Let N t t be a ositive adated right-continuous rocess and A t t be an increasing rocess. Assume that for every bounded stoing time τ, EN τ EA τ. t 5

6 Then, for every k,, E su t T k N t k k E A k T. We shall use this lemma to rove the following Theorem.5. Let T > and t t T be a continuous local martingale. Consider the rocess R t t Z t = e Vsds e R s Vudu d s, where V t t T is a non ositive adated and continuous rocess. For every < <, there is a universal constant C, indeendent of T, t t T and V t t T such that E su Z t C E T. t T Proof. By stoing it is enough to rove the result for bounded. Let q. From Itô s formula we have dz t = Z t V t dt + d t and d Z t q = q Z t q sgnz t dz t + qq Z t q d < > t = q Z t q V t dt + qsgnz t Z t q d t + qq Z t q d < > t. Since V t, as a consequence of the Doob s otional samling theorem, we get that for every bounded stoing time τ, E Z τ q τ qq E Z t q d < > t. From the Lenglart s domination inequality, we deduce then that for every k,, k E su Z t q k k T k qq E Z t q d < > t. t T k We now bound T k kq T E Z t q d < > t E su Z t t T E su Z t t T kq q E k d < > t < > kq T q. 6

7 As a consequence, we obtain E su t T k Z t q k qq k Letting = qk yields the claimed result. k E su Z t t T kq q E < > kq T We should note here that in the above roof the fixed time T can be relaced by a stoing time. ore interesting and useful for alications is the fact that this result admits the following multidimensional generalization whose roof identical to the above roof and which we leave to the interested reader. q. Theorem.6. Let T > and t t T be an R d valued continuous local martingale. Consider the solution of the matrix equation d t = V t t dt, = Id, where V t t T is an adated and continuous rocess taking values in the set of symmetric and non ositive d d matrices. Consider the rocess Z t = t t s d s. For every < <, there is a universal constant C, that is indeendent from d, T, t t T and V t t T such that.7 E su Z t C E T. t T Remark.. Theorem.6 may be used to correct a ga contained in the aers [], [], [3] t of X. D. Li. In these aers the author considers quantities like t s d s but writes them as t t s d s to give L estimates using the classical Burkholder-Gundy inequality or even exlicit exressions for the constants using the Burkholder bounds of Theorem.3. This, however, is not ossible due to the non-adatedness of the rocess t s which revents us from bringing the t inside the integral. Therefore, in those aers, several of the exlicit constants given there involving need to be relaced with less recise constants C deending only on. The necessary changes and correct formulations for Li s results are contained in []. 3 Proof of Theorem. and some refinements Consider the Schrödinger oerator L = d i= X i +X +V. The diffusion Y t t with generator d i= X i + X can then be constructed via the Stratonovitch stochastic differential equation dy t = X Y t dt + 7 X i Y t dbt, i i=

8 which we assume non exlosive. In the sequel we shall denote by Y t t the solution of this equation started with an initial distribution µ. We will use E x to denote the exectation associated with the rocess Y starting at x and E to denote the exectation of the rocess starting with µ. See [] for some literature related to the ossible intricacies associated to the rocess starting with the ossibly infinite measure µ. Let us recall that by the Feynman-Kac formula, the semigrou P t acting on f C can be written as er t P t fx = E x V Ysds fy t. Let us fix T > in what follows. We first consider the relevant martingales associated to our oerators. We have Lemma 3.. Let f C. The rocess e R t V Ysds P T t fy t is a martingale and we have er T V Ysds fy T = P T fy + i= T The roof of the lemma is clear. For t T, set Z t = er t V Ysds, Z t = P T t fy t t T er t V Ysds X i P T t fy t db i t. and aly the Itô formula. The next exression rovides the robabilistic connection to our oerators. For f C, < T <, set SAfx T = E e R T T V Ysds e R t i,j= V Ysds A ij T t, Y t X j P T t fy t dbt i Y T = x. If g is another function in C, the Itô isometry and the lemma give SAfgdµ T R T T = E e V Ysds e R t V Ysds A ij T t, Y t X j P T t fy t dbtgy i T = = = i,j= i,j= T i,j= i,j= T E er V Ysds gx T T e R t V Ysds A ij T t, Y t X j P T t fy t dbt i T E A ij T t, Y t X i P T t gy t X j P T t fy t dt A ij tx i P t gx j P t fdµdt. 8

9 Therefore, we have 3. S T Af = i,j= T P t X i A ij tx j P t fdt. Lemma 3.. For any < <, there is a constant C deending only on such that for every f L µ, 3. S T Af C A f. If the otential V, then 3.3 S T Af A f. In articular these constants do not deend on T. Proof. We first observe that if V then the martingale t i,j= A ij T t, Y t X j P T t fy t db i t is differentially subordinate to A P T t fy t. It Follows from the contraction of the conditional exectation on L, < <, and Theorem.3, that S T Af A f, which is the estimate in 3.3. We now deal with the case of V. We first rove an L estimates for T X i P T t f Y t dt i= which is the quadratic variation of the martingale aearing in Lemma 3.. Using Itô s formula with t T for P T t fy t, we obtain fy T = P T fy + + T i= T Xi + X + P T t f Y t dt t i= X i P T t f Y t db i t. Therefore E T Xi + X + P T t f Y t dt f t. i= 9

10 We now comute Xi + X + P T t f = P T t f t i= P T t f = Xi + X P T t f + i= X i P T t f i= Xi + X + V P T t f i= X i P T t f V P T t f i= X i P T t f. i= This imlies that T T E X i P T t f Y t dt E i= Xi + X + P T t f Y t dt f t. Combining this with Theorem.5 and again with the fact that the conditional exectation is a contraction on L, we obtain S T Af A f, which roves the result for =. Assume now that < <. The above comutations show that P T t f Y t is a submartingale. From the Lenglart-Léingle-Pratelli estimate and Doob s maximal inequality, we deduce that T E i= Xi + X + P T t f Y t dt / E t i= We conclude T T E X i P T t fy t dt E i= su t T PT t f Y t / / / EfY T / / f. Xi + X + P T t f Y t dt t i= / / f Combining this with Theorem.5, roves the result in the range < <. Finally, the adjoint of SA T acting on L µ is SA T acting on Lq µ, where + q =. The result is then obtained by duality.

11 Proof of Theorem.. It remains to show that we can let T in Lemma 3.. To see this observe that by alying the lemma with the matrix A ij t, x = X i P t fxx j P t gx d / i= X d / ip t f x i= X ip t g x we obtain the uniform bound 3.4 T / / X i P t f x X i P t g x dµdt a f g q i= i= with + q =, where a = C in the case of V and otherwise it is. Since we deduce that and moreover that S T Afgdµ = i,j= T + lim SAfgdµ T = T i,j= = S A fgdµ A ij tx i P t gx j P t fdµdt, S A fgdµ a A f g q. A ij tx i P t gx j P t fdµdt This shows that the same bounds in Lemma 3. hold with S T A relaced by S A. This comletes the roof of Theorem.. The following corollary of the above roof generalizes to manifolds the key estimate of Nazarov and Volberg [4] and Dragičević and Volberg [], [], for the Lalacian in IR d ; see [, Corollary 3.9.]. Corollary 3.. For all f, g C, < <, 3.5 / / X i P t f x X i P t g x dµdt a f g q, i= i= with + q =, where a = C as in 3., if V and a =, if V. We now state a result that follows from Theorem.4 when we have some additional information on the matrix A = A ij. Again, this is exactly as on IR d ; see [3].

12 Theorem 3.. Suose V. Let A = A ij be symmetric and suose that there are universal constants < b < B < such that for all t, x [,, b ξ A ij t, xξ i ξ j B ξ, i,j= for all ξ IR d. Then for all < < we have 3.6 S A f C,b,B f, where C,b,B is the constant in Theorem.4. Proof. As above, this result will follow if we can rove the same result for the oerator T SAfx T = E A ij T t, Y t X j P T t fy t dbt i Y T = x. Consider the two martingales Y t = i,j= T i,j= A ij T t, Y t X j P T t fy t db i t, X t = i,j= T X j P T t fy t db i t. It is simle to verify see [3] that under our assumtions on the matrix A, these martingales satisfy the hyothesis of Theorem.4. From this and the contraction of the conditional exectation on L we obtain 3.6 for the oerators S T A. 3. An estimate of the constant in.3 The above aroach based on Theorem.5 to rove.3 is general but does not lead to exlicit constants with useful information. We resent below an alternative argument which rovides exlicit constants with some additional information. We use the notations introduced in the revious section. Proosition 3.. Suose that V and that < <. Then T X i P T t fy t dbt i f i= Proof. From Itô s formula alied to P T t fy t, we find P T t fy t = P T fy + t s + Xi + X P T s fy s dt+ i= i= T X i P T t fy t db i t.

13 But, s + and therefore Xi + X P T s fy s = V Y s P T s fy s i= P T t fy t = P T fy t V Y s P T s fy s dt + i= t X i P T s fy s db i s. Assume now f. In that case, the above comutation shows that P T t fx t is a non negative submartingale. Thus from Lenglart-Léingle-Pratelli see Theorem 3. in [9] and Doob s maximal inequality, we have T P T fy V Y s P T s fy s dt su P T t fy t f. We conclude i= For a general f, we can write and we see that t X i P T s fy s dbs i i= T t T X i P T t fy t dbt i = f = f + f i= t + f. X i P T s f + Y s db i s t i= t i= t i= X i P T s f + Y s dbs i X i P T s f Y s dbs i f + f f. X i P T s f Y s dbs i From this we obtained the following exlicit bound in.3. Corollary 3.. Let < <. For any V, f L µ, 4 S A f 8 A f 3

14 Proof. We first note that the martingale is differentially subordinate to i,j= T A A ij T tx j P T t fy t db i t i= T X i P T t fy t db i t. From Theorem.3 and the above roosition we obtain S T Afgdµ = i,j= = E T i,j= which imlies the corollary. 4 Alications T A ij tx i P t gx j P t fdµdt A ij T tx j P T t fy t db i t i= T A ij T tx j P T t fy t dbt i i,j= i= A T X i P T t fy t dbt i i= A q q q f g q 4 8 A f g q, i= T T T X i P T t gy t dbt i X i P T t gy t dbt i q X i P T t gy t dbt i 4. ultiliers of Lalace transform-tye for Schrödinger oerators on manifolds We work in the same setting as the revious Section. We consider the following multilier for the Schrödinger oerator L = d i= X i X i + V. First, let a L [,. Set T a f = + atlp t fdt. We can observe that since L is essentially self-adjoint, from sectral theorem, there is a measure sace Ω, ν, a unitary ma U : L νω L µ and a non negative measurable function on Ω such that U LUfx = λxfx, x Ω. 4 q

15 The oerator T a acts on these as multilication oerator on L νω in the sense that U T a Ufx = λx + ate tλx dtfx, x Ω. Let us also observe that in several settings the oerators T a can be interreted as multiliers in the Fourier analysis sense. For instance, let G be a comact Lie grou. Let U be a bounded oerator on L G which commutes with left and right translations. Then there exists a bounded function Φm on Ĝ, the sace of equivalence classes of irreducible unitary reresentations of G, such that Uf = Φmd m χ m f, m Ĝ where χ m is the character and d m the dimension of the reresentation. Conversely, for any bounded Φ, the above oerator defines a bounded oerator on L G which commutes with left and right translations. In this framework, if L is an essentially self-adjoint diffusion oerator that commutes with left and right translations like the Lalace-Beltrami oerator for a bi-invariant metric, then we have T a f = m Ĝ λm where λm m Ĝ is the sectrum of L. + With the notations of the revious section, we see that T a f = S Af + ate tλm dtd m χ m f + atp t V P t fdt where At = atid. In the following we denote by Q t the arkovian semigrou with generator d i= X i X i. Proosition 4.. Fix < <. i For any nonnegative otential V that satisfies V m for some m, we have 4. T a f 4 a at e mt Q t V q /q dt f, with q + =. ii Suose V. Let a be such that for all t [,, < b at B <. Then 4. S a f C,b,B f, where the C,b,B is the constant of Theorem.4. 5

16 Proof. Since T a f = S Af + + atp t V P t fdt, using the results of the revious section, we only need to bound in L the oerator + atp t V P t fdt. From Feynman-Kac formula, we have t P t V P t fx = E er V Xx s ds V Xt x P t fxt x, where X x t t is the diffusion with generator d i= X i X i started at x. Thus As a consequence, P t V P t fx E e q R t V Xx s ds V Xt x q /q E Pt fxt x / which yields the exected result. e mt Q t V q /q E Q t f X x t / e mt Q t V q /q Q t f x / P t V P t f e mt Q t V q /q f, For instance, we immediately deduce from the revious roosition: Corollary 4.. Fix < <. i For any non-negative otential V that satisfying V m, for some m,, we have T a f 4 a at e mt dt f. ii Assume that the semigrou Q t is ultracontractive. For any nonnegative otential V that satisfies V L q µ, we have T a f 4 a 4 + V q with q + =. + at Q t, dt f, Remark 4.. Lalace transform-tye oerators have been extensively studied in many settings. For some of this literature, see [6], [5], [6], [], and references contained in those aers. 6

17 4. Second order Riesz transforms on Lie grous of comact tye Let G be a Lie grou of comact tye with Lie algebra g. We endow G with a bi-invariant Riemannian structure and consider an orthonormal basis X,, X d of g. In this setting the Lalace-Beltrami oerator can be written as L = Xi. i= It is essentially self-adjoint on the sace of smooth and comactly suorted functions and the assumtions of the revious section are satisfied. It is remarkable here that Xi = X i and the vector fields X i do commute with the semigrou P t = e tl. In this case, if the matrix A only deends on time, we get therefore S A f = i,j In articular, for a constant A, we obtain + A ij tx i X j P t fdt. S A f = i,j A ij i= X i X i X j f. Defining the Riesz transforms on G by we see that R j f = S A f = / Xi X j f i= A ij R i R j f. i,j= We have the following result which follows from Theorems. and 3.. These inequality are exactly as those roved in IR d for the classical Riesz transforms. Theorem 4.. Fix < <. i For any constant coefficient matrix A, A ij R i R j f A f. i,j= ii Assume that A = A ij d i,j= is symmetric matrix with real entries and eigenvalues λ λ... λ d. Then 4.3 A ij R i R j f C,λ,λ d f i,j= 7

18 and this inequality is shar. In articular, if J {,,..., d}, then 4.4 j J R j f C,, f = c f, where c is the Choi constant in. and this inequality is also shar. The fact that the constants in this theorem are best ossible follows from the fact that they are already best ossible on IR d. These results simly show that the construction in [3] for Riesz transforms on IR d extend to Lie grous of comact tye without change in their norms. We may observe that, slightly more generally, a similar statement holds on Riemannian manifolds for which the gradient commutes with the heat semigrou P t. From the Bakry-Émery criterion see [], such a commutation imlies that the Ricci curvature of is non negative. If G is a semisimle comact Lie grou see for instance the classical reference [4] for an account about structure theory and harmonic analysis on semisimle comact Lie grous, we can deduce from the above result an interesting class of multiliers. Proosition 4.. Let G be a comact semisimle Lie grou. Let Λ + be the set of highest weights and be the set of all roots. For α, β, denote U α,β f = λ, α λ, β d λ λ + ρ ρ χ λ f, λ Λ + where ρ = α α, χ λ is the character of the highest weight reresentation and d λ its dimension. Then, for < <, U α,β is bounded in L G and U α,β f α β f. Proof. Let T be a maximal torus of G and let t be its Lie algebra. Then, the sub-algebra t of g generated by t is a Cartan sub-algebra. Let t denote the dual of t. If α, we denote by H α the element of t such that for every H t, H, H α = αh where, is induced from the Killing form. With these notations, we see that U α,β = C H α H β, where C is the Casimir oerator which is also the Lalace-Beltrami oerator. References [] D. Bakry &. Emery: Diffusions hyercontractives, Sémin. de robabilités XIX, Univ. Strasbourg, Sringer, 983. [] R. Bañuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications, To aear, Illinois Journal of athematics, Volume in honor of D.L. Burkholder. 8

19 [3] R. Bañuelos and A. Osȩkowski, artingales and Shar Bounds for Fourier multiliers, to aear in Annales Academiae Scientiarum Fennicae athematica. [4] R. Bañuelos and G. Wang, Shar inequalities for martingales with alications to the Beurling-Ahlfors and Riesz transforms, Duke ath. J , no. 3, [5] F. Baudoin, An introduction to the geometry of stochastic flows. Imerial College Press, London, 4. x+4. [6] D.L. Burkholder, artingale transforms, Ann. ath. Statist , [7] D.L. Burkholder, A shar inequality for martingale transforms, Ann. Prob 7 979, [8] D.L. Burkholder and R. F. Gundy, Extraolation and interolation of quasi-linear oerators on martingales, Acta ath. 4 97, [9] K. P. Choi, A shar inequality for martingale transforms and the unconditional basis constant of a monotone basis in L,, Trans. Amer. ath. Soc , [] C. Dellacherie and P. A. eyer, Probabilities and Potential B: Theory of martingales, North Holland, Amsterdam, 98. [] O. Dragičević and A. Volberg, Shar estimate of the Ahlfors-Beurling oerator via averaging martingale transforms, ichigan ath. J. 5 3, [] O. Dragičević and A. Volberg, Bellman function, Littlewood-Paley estimates and asymtotics for the Ahlfors-Beurling oerator in L C, Indiana Univ. ath. J. 54 5, no. 4, [3] E. Geiss, S. ontgomery-smith, E. Saksman, On singular integral and martingale transforms, Trans. Amer. ath. Soc. 36, [4] S. Helgason, Grous and Geometric Analysis, Academic Press, Orlando 984 [5] T.P. Hytönen, Littlewood-Paley-Stein theory for semigrous in UD saces, Rev. at. Iberoam. 37, [6] T.P. Hytönen, Asects of robabilistic Littlewood-Paley theory in Banach saces, Banach saces and their alications in analysis, Walter de Gruyter, Berlin, 7, , [7] D. Jerison & A. Sánchez-Calle, Subellitic second order differential oerators, Lecture. Notes in ath., , [8] E. Lenglart, Relation de domination entre deux rocessus, Ann. Inst. H. Poincaré Prob. Statist , [9] E. Lenglart, D. Léingle and. Pratelli, Présentation unifiée de certaines inégalités de la théorie des martingales, Séminaire de robabilités de Strasbourg 4 98,

20 [] X.-D. Li, artingale transform reresentation and L estimates for Beurling-Ahlfors and Riesz transforms on Riemannian manifolds, rerint. [] X.-D. Li, Riesz transforms on forms and L Hodge decomosition on comlete Riemannian manifolds, Rev. at. Iberoam. 6, [] X.-D. Li, On the weak L Hodge decomosition and Beurling Ahlfors transforms on comlete Riemannian manifolds, Probab. Theory Relat. Fields 5, 44. [3] X.-D. Li, artingale transforms and L norm estimates of Riesz transforms on comlete Riemannian manifolds, Probab. Theory Relat. Fields 4 8, [4] F. Nazarov and A. Volberg, Heat extension of the Beurling oerator and estimates for its norm, St. Petersburg ath. J. 5, 4, [5] D. Revuz and. Yor, Continuous martingales and Brownian motion, Sringer, 5. [6] N. Th. Varooulos, Asects of robabilistic Littlewood-Paley theory, J. Funct. Anal , no., 5 6. [7] G. Wang, Differential subordination and strong differential subordination for continuoustime martingales and related shar inequalities, Ann. Probab. 3 no. 995, 5 55.