Mathematical Sciences: Problems in Fourier Analysis Related to Surfaces and Curves
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Several basic mathematical questions in Fourier analysis on                Euclidean space will be taken up in the project.  The objectives                are to analyze the decay of Fourier transforms of measures                      concentrated on surfaces and power-norm estimates for singular                  integral and maximal operators associated with curves.  This work               derives from fundamental questions concerning partial                           differential operators defined on various function spaces.                           Fourier transforms of functions restricted to hypersurfaces                have controlled asymptotic growth.  Estimates for this growth (or               decay) have been the subject of intense research and are known to               run from absence of decay (for hyperplanes) to reciprocal power                 decay for surfaces of finite type.  This type of decay is not                   restricted to such surfaces although the order of contact of a                  surface with its tangent plane plays some, as yet unknown, major                role in guaranteeing decay at infinity (of the transform).  Work                will be done in finding the correct conditions between finite                   type and "flat" surfaces.  Recent unpublished examples have                     confirmed that convexity is not the right criterion.                                 A related line of investigation concerns the Hilbert                       transform and maximal function along a convex plane curve.  Of                  concern are the power-norms of these functions and their                        relationship to the same norms of the antecedent function.  The                 theory is relatively well developed for convex curves given by                  smooth even functions but breaks down when these conditions are                 weakened.  Efforts to find the exact conditions for boundedness                 of the two transforms will continue.
{'FirstName': 'James', 'LastName': 'Vance', 'PI_MID_INIT': 'T', 'PI_SUFX_NAME': 'Jr', 'PI_FULL_NAME': 'James T Vance', 'EmailAddress': None, 'NSF_ID': '000252784', 'StartDate': '07/15/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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