Mathematical Sciences: Harmonic Analysis, Sobolev           Inequalities and Geometry
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{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Three main themes highlight this mathematical research.  The               first combines ideas of harmonic analysis and probability theory.               They involve infinite series of Haar functions - orthogonal step                functions defined on an interval.  The power norm of any                        alternating sum is always dominated by a fixed multiple of the                  original norm.  Although the sharpest multiplier is known, the                  alternating sums which maximize or minimize the norm are not                    easily obtained.  In this project, work will be done in                         determining optimal choices of sign change based on the                         coefficients of the original series.                                                 A second line of investigation will focus on exponential                   multiples of the Laplace operator on the surface of a sphere.                   The logarithm of the determinant of such an operator contains a                 nonnegative energy term which vanishes only when the exponential                is the potential in the basic conformality equation.  This result               has implications in one lower dimension in that it becomes the                  classical Lebedev-Milin inequality which, in turn, derives from                 Szego's theorem.  Efforts will be made in reversing the argument                in higher dimensions; from a version of the Szego theorem to the                energy inequality.  Underlying this research are basic questions                concerning the spectrum of the Laplacian.                                            Work of a more geometric flavor will continue on the                       question of which functions defined on a sphere can be the                      curvature functions of conformally related metrics.  Sufficient                 topological conditions are known depending on the critical points               of the function.  Necessary and sufficient conditions based on                  analytic properties of the function will be sought.
{'FirstName': 'Alice', 'LastName': 'Chang', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Alice Chang', 'EmailAddress': 'chang@math.princeton.edu', 'NSF_ID': '000471580', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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