MATHEMATICS OF INTEREST
MATHEMATICS OF INTEREST MATH 325
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Research Statement Tao MEI 1 My research interests are in functional analysis and harmonic analysis especially in generalizations of classical results from harmonic analysis to operator valued matrix valued functions and related subjects such as matrix valued paraproducts and non commutative martingales Here operator valued77 means functions or other objects with values in the non commutative Lp spaces associated with a semi nite von Neumann algebra M However our results are already signi cant in the n by 72 matrix valued case where we estimate with constants independent of the size 72 when n a These generalizations not only are interesting in themselves but as in the classical case also have applications to other domains such as prediction theory and rational approximation They also have applications to operator theory and operator space theory 1 Recent Work 11 Operator Valued Hardy Spaces The Hardy spaces are very important objects in classical analysis Among several equivalent de nitions one is as follows HPHR f E IPGRLHJ HHP Hle HHfHLp lt 00h for 1 Sp lt 00 where Hf is the Hilbert transform of f Fruitful results on Hardy spaces such as interpo lation results equivalence between Hp and LP for 1 lt p lt 00 have been developed during last century that turned Hp theory into an important branch of classical analysis One of the most remarkable results of Hp theory is the Fe erman Stein duality theorem which says in particular that the dual of HWR is another well known space the BMO space de ned as follows BMoaR f 6 Lime HfHBMO sup i wt 7 mt lt oo CR lIl I where f1 7 f1 ftdt We constructed Hp spaces for operator valued functions by considering the non commutative Littlewood Paley G functions The non commutativity is of course the main di iculty of our study and the main di erence between operator valued Hardy spaces and the vector valued ones One analogue of classical results we proved is that our Hhs are preduals of the non commutative BMO spaces de ned in recent works on matrix valued harmonic analysis and non commutative martingale inequalities see K NPTV NTV PX For convenience I will describe my results only in the matrix valued case Because of the non commutativity there are now two non commutative BMO spaces the column BMO and row BMO Let Mn be the algebra of n X n matrices with its usual trace tr For A 6 Mn denote by HAHMW the operator norm of A on 53 Then the column BMO space is de ned by BMOJRMW g0 JR a Mn llngBMOC lt 00 1Partially supported by the National Science Foundation 0200690 and a Tamu Funded Vigre fellowship Where 1 1 HepllBMoc SUP H7 90t WNW WWW Id l l I and 901 i L Lptdt Similarly7 the row BMO space is BMOTURvMW HWHBMOT WHBMOC lt Note that these two norms are not equivalent uniformly over n Denote by Sf 1 lt p lt 00 the Schatten p classes on 5 For f E L llR7 1ft 7 STILL7 let F denote its Poisson integral We de ne the non commutative G function as GM w iVFaymdy Where lVFtyl and BF 8F 2772 72 8F 2 BF 8F 5 a 5 De ne H5 resp H nl 1 lt p lt 00 to be the space of all f such that GHQ E ZIPR S resp GM d E HORSE and set Hflng llecHLppsg resp Hflng WM When n 17 all these spaces coincide With the classical Hardy spaces Theorem 1 MZ Non commutative generalization of Fe erman s duality theorem aH lcL BMOJRMTL with equivalent norms independent ofn 1 Similarly Hi BMOTURMVITL with equivalent norms independent ofn And as in the classical case the duality between and BMOJRMTL implies an atomic decomposition of Remark 1 Note that the trace class valued Hardy space H1Sl has a di erent dual than the above Theorem 2 MZ Equivalence between Hp and LP H nl H n ZIPRSZ with equivalent norms for all 1 lt p S 2 H nl H n HORSE with equivalent norms for all 2 lt p lt 00 The equivalence constants are independent of n Theorem 3 MZ Interpolation Let 1 lt p lt 00 Then with equivalent norms X7 Y LplR7 85 l p where X BMOJRMTL BMOJRMTL or LOCOR7 Mn Y Hi 71 or L1 lR7 8711 andtheqquot are39dl J ofn 12 Matrix valued dyadic paraproduct Let EEC be the unit circle with the usual dyadic ltration Let b be an Mn valued function on T The matrix valued dyadic paraproduct associated with b7 denoted by 7Tb7 is the operator on HOT 85 de ned as m debEkilf W e m 81 k where Ek is the conditional expectation with respect to 1 and dkb Ekb 7 Ek1b In the classical case when E is a scalar valued function7 it is well known that HITbHL2aL2 3 HbHBMOdv where BMOd denotes the usual dyadic BMO norm Note 7Tb is usually considered as a dyadic singular integral and plays an important role in the proof of the classical T1 theorem Also note its relation with the Hankel operator with symbol b see Pe7 which has a norm equivalent to HbHltH13 in the matrix valued case We may ask two natural questions as follows Q1 Does there exist a constant c gt 0 independent of n such that7 for all 1 lt p7 q lt 007 b Lquszpmmxsz b Lanszpmmsgy HW H lt CHW H 7 Q2 Can we dominate H7rbHL2 msggp 17320 uniformly over 72 by some reasonable BMO norm Note we have various candidates or BMO norms in the matrix valued case Nazarov7 Pisier7 Treil7 Volberg proved that this is not true for the BMOC norm de ned in Section 21 Recently7 we gave a partial positive answer to Q1 and proved that there exists a constant c gt 0 independent of n such that7 for all 1 lt p lt q lt 007 maxfll blqusmanszy HWM Hquszquml S C maxill bllmw pmwgy HWM HLmsgmesalv where b denotes the adjoint of b We still do not know what happens when p gt q We gave a negative answer to Q2 and proved that even HbHmeMn does not dominate H7rbHL2ggsngL2m ggb uniformly over 72 see 13 Dif culties and some useful techniques Noncommutativity We lose some nice classical properties in the operator valued case because of the non commutativity For example7 we will no longer have a good John Nirenberg theorem for operator valued BMO see JM7 PS7 Absence of maximal element A straightforward de nition of the maximal function in the operator valued case is not possible However7 using Pisier7s non commutative vector valued spaces we may partially overcome this problem in many situations In fact7 we proved a non commutative Hardy Littlewood maximal inequality for operator valued functions see Mm Noncommutative Martingale inequalities As in the classical case7 we could borrow some ideas from the study of non commutative martingales when studying operator valued functions ln particular7 Pisier and Xu7s work on the non commutative Burkholder Gundy inequalities see PXD inspired us to consider the non commutative analogue of the clas sical Littlewood Paley G function to de ne our operator valued Hp spaces Moreover7 we used Junge7s work on Doob7s maximal inequality see to prove our non commutative Hardy Littlewood maximal inequality mentioned above However7 it seems dif cult to con vert results from operator valued martingales to operator valued functions by following the classical methods Brownian martingales or distribution functions We found a trick to treat some special situations The following result is an application of this trick Theorem 4 Let T be the unit circle Denote by BMOOI the scalar valued BMO space and denote by BMOdT the scalar valued usual dyadic BMO space on T We have 21 900 3 llWllBMoar S 6llWllBModm 39 BMOdUT 2 Proposed Research problems Operator Valued Singular Integrals Let T be a singular integral associated with a standard kernel It is well known that i If T can be extended into a bounded operator on L2 R7 then T has bounded extension on LPOR for 1 lt p lt 00 and from L lR to BMOOR as well as from HWR to L1lR ii T can be extended into a bounded operator on L2lR7 if and only if T1 and T1 are in BMOOR and T has the weak boundedness property Question Are there some non commutative analogues of these results We have some partial results on this question and will continue to work on it in the future Operator Valued LittlewoodPaley theory for arbitrary sets Let us consider a function on the unit circle with the form f 2 Rue n67 aking7 J L Rubio de Francia and J Bourgain proved see R7 nEIk Denote by Aka 2 Bl llfllLP llAkakllel2gt for arbitrary k C Z As an application of this result7 Rubio de Francia7 Coifman and Semmes proved that the strong 2 variation condition implies the boundedness of Fourier multipliers on LP see RCS Generalization of these results to the operator valued setting would be very interesting and would yield a new suf cient condition of the boundedness of Schur multipliers on the Schatten p classes This was recently attempted in PoS Many other questions in n X 72 matrix valued harmonic analysis deserve further inves tigation only in few cases is the behaviour of the constants when n a 00 completely understood CPHAkal HLPl27 13 p S 2 S S Cpllfllev 2 lt22 lt oo References B J Bourgain On Square Functions on the Trigonometric System Bull Soc Math Belg Ser B 37 1985 no 1 20726 BS 0 Blasco S Pott Operator valued Dyadic BMO Spaces Preprint J M Junge Doob s Inequality for Non commutative Martingales J Reine Angew Math 549 2002 149190 J M M J unge M Musat A Noncommutative Version of the J ohn Nirenberg theorem Trans action of AMS to appear K Nets H Katz Matrix valued paraproducts J Fourier Anal Appl 300 1997 9137921 M1 T Mei Operator Valued Hardy Spaces Memoirs of AMS to appear M2 T Mei BMO is the intersection of two translates of dyadic BMO Comptes Rendus de l Academie des Sciences Paris Series I 336 2003 1003 1006 M3 T Mei Remarks on Non commutative BMO Spaces Preprint M4 T Mei Notes on the Matrix Valued Dyadic Paraproducts Preprint NPTV F Nazarov G Pisier S Treil A Volberg Sharp Estimates in Vector Carleson Imbed ding Theorem and for Vector Paraproducts J Reine Angew Math 542 2002 147 171 NTV F Nazarov S Treil and A Volberg Counterexample to In nite Dimensional Carleson Embedding Theorem CR Ac Sci Paris Ser I Math 325 1997 no 4 383r388 P G Pisier Notes on Banach Space Valued HP Spaces preprint Pe S Petermichl Dyadic Shifts and a Logarithmic Estimate for Hankel Operators with Matrix Symbol CR Acad Sci Paris 330 2000 no 6455 460 Pos D Potapov F Sukochev A Vector valued Littlewood Paley Theorem for Arbitrary inte vals and Its Applications Preprint PS S Pott M Smith Vector Paraproducts and Hankel operators of Schatten Class via p John Nirenberg theorem J Funct Anal 2172004 no 1 38 78 PX G Pisier Q Xu Non commutative Martingale Inequalities Comm Math Phys 189 1997 667698 R J L Rubio de Francia A Littlewood Paley Inequality for Arbitrary Intervals Rev Mat Iberoamericana 11985 no 2 1714 RCS J L Rubio de Francia R Coifman S Semmes Multiplicateurs de Fourier de LP et Estimations Quadratiques Comptes Rendus de l Academie des Sciences Paris Series I 3061988 3517354 S E M Stein Harmonic Analysis Princeton Univ Press Princeton New Jersey 1993