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Pseudo-differential operators play important roles in harmonic analysis, several complex variables, partial differential equations and other branches of modern mathematics. We studied some types of multilinear and multiparameter Pseudo-differential operators. They include a class of trilinear Pseudo-differential operators, where the symbols are in the forms of products of Hormander symbols defined on lower dimensions, and we established the Holder type Lp estimate for these operators. They derive from the trilinear Coifman-Meyer type operators with flag singularities. And we also studied a class of bilinear bi-parameter Pseudo-differential operators, where the symbols are taken from the general Hormander class, and we studied the order of symbols which could imply the Holder type Lp estimates. Such types of operators are motivated by the Calderon-Vaillancourt theorem in the single parameter setting.
Trudinger-Moser inequalities can be treated as the limiting case of the Sobolev embeddings. Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogue Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two types of inequalities. Surprisingly, we proved that these critical and subcritical Trudinger-Moser and Adams inequalities are actually equivalent.
Zhang, Lu, "Multilinear And Multiparameter Pseudo-Differential Operators And Trudinger-Moser Inequalities" (2016). Wayne State University Dissertations. 1611.