Mathematical Sciences: Research in Classical Harmonic Analysis and Applications to Partial Differential Equations

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Abstract Grafakos The investigator proposes to pursue research in the area of multilinear singular integral operators and partial differential equations in connection with Hardy space techniques. His techniques include compensated compactness methods, atomic decomposition, and estimates for singular integral operators. The investigator also proposes to obtain sharp estimates for maximal operators on Euclidean spaces. In particular, he is interested in computing the operator norm of the Hardy-Littlewood maximal function on different Lebesgue spaces. This proposal has two parts. In the first part the investigator plans to use mathematical analysis techniques, (in particular harmonic analysis), to solve problems in differential equations which naturally arise in the study of dynamics of fluids or in the study of motions of waves. The researcher believes that the methods that he proposes to use will give new insight to some problems that arise in these areas and will strengthen the connection of mathematics with applications. In the second part, he proposes to work on better quantitative understanding of averaging operations. The averaging of a function is an important and useful operation since it regularizes a given set of data by smoothing out its singularities. The investigator proposes to study how sharply are certain quantitative properties of functions transferred to their averages.