Wheeden 9302991 This award continues support for mathematical research into problems arising in the modern theory of partial differential equations and singular integral operators. The form of the analysis which takes place involves critical inequalities measuring the power norms of functions with that of their gradients, especially when the measures are weighted by different functions. Such comparisons fall under the heading of Sobolev- Poincare inequalities. From such results one obtains upper and lower bounds for fundamental solutions of divergence form degenerate parabolic equations in case the degeneracy is time- dependent and measured in terms of the two weight functions. Work will be done establishing similar inequalities when the gradient is replaced by vector fields other than the gradient. A second line of investigation concerns Carleson-type estimates for measuring functions (weighted) in a half-space with their boundary value norms taken against different weights. Good, but incomplete, results have been obtained for harmonic functions. Efforts will be made to determine the extent that integral transforms generated by other than the Poisson kernel produce the same type of inequalities - especially in cases of non- convolution kernels. Finally, work is to be done on characterizing weights which give good bounds on weighted fractional integrals when compared with the original function norms. The measurement of integrals plays a central role in determining both existence and bounds on solutions of differential equations. Many of these results involve parameter bounds which are dependent only on the underlying dimension. Through careful analysis of the integrals and norms of the corresponding derivatives one obtains significant information about the existence and regularity of solutions. ***
