Mathematical Sciences: Research in Interpolation Theory and Harmonic Analysis

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Work on this project focuses on two general areas of harmonic analysis. The first concerns stability of Fredholm properties in interpolation scales. It is motivated by a recurring theme inthe theory of partial differential equations defined in domainswith relatively irregular boundaries (Lipschitz domains). Namely, when the operators which associate boundary values with solutions are known to be bounded in a certain Lebesgue p-norm say the quadratic - then they turn out to be bounded across alarger range of values of p. The work to be done involves extensions of 'stability range' to larger classes of operators on Banach spaces. Goals include finding upper bounds for the length of the range and reducing assumptions currently in force regarding the dimension of the null space of the operators. The second line of investigation relates to efforts extending classical results on Rademacher and Walsh series, especially in the context of series with large gaps. It is known that in such circumstances, the quadratic norm the coefficients of a series is bounded by the maximum norm of the sum of the series. If the series involved are trigonometric, then the maximum norm must be replaced by p-norms. Recent results have extended the classical comparisons to include the bounded mean oscillation norm. Work will now be done to obtain more general BMO estimates for the same type of problem where the series are restricted to measurable sets rather than intervals. The first hurdle to be overcome is that of drafting a proper definition of BMO for sets of functions defined on other than intervals. Support for this project will be restricted to graduate student stipends and travel allowance.