Mathematical Sciences: Harmonic Analysis Related To Weight Functions

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Work to be carried out on this project focuses on estimates of differential operators and resulting uniqueness of solutions which can be shown to follow from such estimates. Particular emphasis is to be placed on p-power integral conditions on the lower order coefficients of the operators. Among the specific goals of the research is one of characterizing what has been termed the unique continuation properties of potentials in differential equalities. These a priori measures determine when a smooth function must vanish identically from very local vanishing of its derivatives. Particular interest will be paid to the Laplace operator in which the potential is taken in the largest logical class, known as the Kato class. Related work will also be carried out on questions of global unique continuation in which vanishing of potentials below hyperplanes is to ensure that they vanish everywhere. Results due to C. Fefferman for perturbations of the Laplacian confirm that unique continuation is possible for potentials with high index. This research will seek to lower the condition to integrability for all powers greater than unity. Efforts will also be made to apply any new result to the Dirac operator where unique continuation is only verifiable in specific cases rather than function classes. They key to understanding inequalities needed in the work is a restriction phenomenon which must be established. Basically, these phenomena say that the quadratic norm of a Fourier transform confined to the surface of a sphere is dominated by the weighted quadratic integral of its antecedent. The weight is the potential of the differential inequality. This research will have fundamental implications in the field of elliptic partial differential equations.