Mathematical Sciences: Boundary Behavior of Holomorphic     Functions and Mappings
{'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 to be done continues a program developing the function                theory on domains in n-dimensional complex space using the                      language of metric geometry.  Recent progress made in the                       theories of Hardy spaces, Nevanlinna theory, boundary limits                    along curves and the theories of Lipschitz and Bloch spaces has                 provided considerable impetus to the directions this mathematical               research will take.  In particular studies of the Lusin area                    integral and other aspects of harmonic analysis will be carried                 out using the metric geometry approach.  The long-term goal of                  this work is to develop a unified theory for boundary behavior of               holomorphic functions in all dimensions in which the unifying                   language is that of Riemannian geometry.                                             Specific topics to be undertaken include investigations into               mapping properties of the Lusin area integral for strongly                      pseudoconvex regions using newly defined approach regions and a                 study of the regularity for the d-bar problem - a first order                   partial differential equation.  The latter concerns extremely                   delicate norm estimates which cannot be achieved in conventional                function spaces.  New spaces will have to be developed to obtain                sharp results.                                                                       Additional work will be carried out to establish whether or                not certain rigidity must occur in holomorphic vector-valued maps               which exhibit higher order contact with the image domain.  Such                 results are false in one complex dimension but considerable                     evidence exists suggesting that the situation is different in                   higher dimensions.  Continuing efforts will also be made to                     understand the structure of automorphism groups of biholomorphic                maps between domains.  Powerful results show how these maps                     extend holomorphically across boundaries of strongly pseudoconvex               regions.  The next logical step will be to consider weakly                      pseudoconvex regions, especially those of finite type, to                       determine the extent of such analytic continuations.
{'FirstName': 'Steven', 'LastName': 'Krantz', 'PI_MID_INIT': 'G', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Steven G Krantz', 'EmailAddress': 'sk@math.wustl.edu', 'NSF_ID': '000467439', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Washington University', 'CityName': 'SAINT LOUIS', 'ZipCode': '63110', 'PhoneNumber': '3147474134', 'StreetAddress': 'ONE BROOKINGS DR', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Missouri', 'StateCode': 'MO', 'CONGRESSDISTRICT': '01', 'CONGRESS_DISTRICT_ORG': 'MO01', 'ORG_UEI_NUM': 'L6NFUM28LQM5', 'ORG_LGL_BUS_NAME': 'WASHINGTON UNIVERSITY, THE', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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