Mathematical Sciences: Harmonic Analysis and Operator-Theory
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work on this project will concentrate on three problems in                 mathematical analysis.  The first, arising from the Calderon-                   Zygmund theory of operators, synthesizes the traditional                        operator theory with methods of real variables and harmonic                     analysis to give sharp criteria for the power-boundedness of                    pseudo-differential operators.  Particular emphasis is to be                    placed on the so-called Kato conjecture which contends that the                 domain of the square root of a general divergence-type operator                 is the fundamental Sobolev space H.  This conjecture has been                   shown to be connected with on-going research on the problem of                  boundedness of the Cauchy-kernel along curves.  Work will be done               on the corresponding multilinear question where very little is                  known at this time except that the one-dimensional proof of the                 conjecture will not carry over.  Applications to boundedness of                 broad classes of differential operators will be developed.                           The second line of investigation is related to the idea that               a function which is infinitely differentiable in vertical and                   horizontal directions is itself infinitely differentiable.  In                  the study of Anosov diffeomorphisms or flows one wants to show                  similarly that certain functions defined on a manifold are                      smooth.  This work will consider the restriction of functions on                stable and unstable foliations in an effort to expand present                   results from infinitely differentiable to the real-analytic                     version.  Results of this nature should have application to the                 study of the Livsic equation for systems with non-zero Lyapounov                exponents.                                                                           A third thrust will be concerned with the time evolution of                a system in quantum theory, if the evolution is determined by a                 one-parameter group.  This work deals with the problem of giving                an infinitesimal characterization of the cocyles, for a given                   one-parameter automorphism group, that is, a structure theorem                  playing for cocyles, the role of Stone's theorem for one-                       parameter unitary groups.  This investigation should provide                    insight into the relationships between methods of classical and                 quantum probabilities.
{'FirstName': 'Jean-Lin', 'LastName': 'Journe', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jean-Lin Journe', 'EmailAddress': None, 'NSF_ID': '000496274', 'StartDate': '11/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Princeton University', 'CityName': 'PRINCETON', 'ZipCode': '085442001', 'PhoneNumber': '6092583090', 'StreetAddress': '1 NASSAU HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'New Jersey', 'StateCode': 'NJ', 'CONGRESSDISTRICT': '12', 'CONGRESS_DISTRICT_ORG': 'NJ12', 'ORG_UEI_NUM': 'NJ1YPQXQG7U5', 'ORG_LGL_BUS_NAME': 'THE TRUSTEES OF PRINCETON UNIVERSITY', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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