Mathematical Sciences: Geometric Inequalities in Fourier    Analysis
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This work will combine geometric problems with Fourier                     analysis in a concerted effort to exploit the symmetry structure                of manifolds in determining that basic analytic inequalities for                natural operators such as convolution, Fourier transforms,                      fractional integration, Poisson integrals and their generators.                 These are the mathematical instruments which characterize                       analysis on function spaces over a manifold.                                         The thrust of the research will be to understand analysis on               global manifolds defined by the classical Lie groups and their                  associated symmetric spaces, especially the rank one non-compact                symmetric spaces with hyperbolic geometry, defined over real,                   complex or quaterionic numbers.                                                      Using conformal transformations such as the Cayley transform               and stereographic projection, one shows that problems on these                  non-compact domains have equivalent realizations on compact                     domains associated with the rotation and unitary groups as well                 as boundary manifolds such as the Heisenberg group.  The main                   tool in this work is the use of the underlying symmetry structure               of the manifolds to prove sharp fractional integral inequalities.               Such results provide as immediate consequences sharp Sobolev                    estimates for differential operators and other infinitesimal                    generators associated with differentiability.  These inequalities               have wide application to problems in differential geometry, non-                linear partial differential equations, mathematical physics and                 probability.  In addition, connections may be made to variational               problems in string theory associated with conformal metrics.
{'FirstName': 'William', 'LastName': 'Beckner', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'William Beckner', 'EmailAddress': 'beckner@math.utexas.edu', 'NSF_ID': '000398609', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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