Mathematical Sciences: Fourier Analysis and Partial         Differential Equations
{'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 during the term of this project will cover                 several problem areas of mathematical analysis.  They include                   studies of maximal functions and their control of the                           differentiation of integrals, multiple parameter singular                       integrals and the theory of elliptic partial differential                       equations.  Much of the present work continues earlier efforts to               extend fundamental one (complex) variable results to higher                     dimensions.  Basic concepts such as maximal functions and                       parametrized families of rectangles for differentiation do not                  extend directly.  Special classes have been substituted and                     gradually a full Hardy-space environment is being developed.  The               goals are the same: to obtain bounds on the fundamental operators               of analysis such as the Riesz transform and the multidimensional                Hilbert transform.  Connecting maximal operators with classical                 singular integrals much be accomplished before these bounds and                 the resulting applications can be made.                                              The study of Dirichlet problems on half-space relates                      directly to the questions of harmonic analysis related above.                   The boundary function and its nontangential maximal function are                to be in some Lebesque class.  This work is distinguished from                  earlier investigations in that it treats equations defined on                   unbounded domains.  A general theory of existence of solutions                  has yet to be established.  This research will concentrate on                   developing criteria in terms of the coefficients of the matrix                  determining the equation.  Additional work will be done in                      extending classical restriction theorems for the Fourier                        transform to the setting of compact Riemannian manifolds without                boundary and to study smoothness of Radon transforms over lower                 dimensional submanifolds.
{'FirstName': 'Robert', 'LastName': 'Fefferman', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Robert A Fefferman', 'EmailAddress': 'rfefferm@uchicago.edu', 'NSF_ID': '000232417', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Chicago', 'CityName': 'CHICAGO', 'ZipCode': '606375418', 'PhoneNumber': '7737028669', 'StreetAddress': '5801 S ELLIS AVE', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Illinois', 'StateCode': 'IL', 'CONGRESSDISTRICT': '01', 'CONGRESS_DISTRICT_ORG': 'IL01', 'ORG_UEI_NUM': 'ZUE9HKT2CLC9', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF CHICAGO', 'ORG_PRNT_UEI_NUM': 'ZUE9HKT2CLC9'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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