Leibniz-type rules associated to bilinear pseudodifferential operators

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dc.contributor.author Brummer, Joshua
dc.date.accessioned 2018-07-03T21:20:02Z
dc.date.available 2018-07-03T21:20:02Z
dc.date.issued 2018-08-01 en_US
dc.identifier.uri http://hdl.handle.net/2097/39030
dc.description.abstract Leibniz-type rules associated to bilinear pseudodifferential operators have received considerable attention due to their applications in obtaining fractional Leibniz rules and the study of various partial differential equations. Generally speaking, fractional Leibniz rules provide a way of estimating the size and smoothness of a product of functions in terms of the size and smoothness of the individual functions themselves. Such rules are helpful in determining well-posedness results for solutions of PDEs modeling a variety of real world phenomena, ranging from Euler and Navier-Stokes equations (which model incompressible fluid flow, such as airflow over a wing) to Korteweg-de Vries equations (which model waves on shallow water surfaces). Bilinear pseudodifferential operators act to combine two functions using their Fourier transforms and a symbol, which is a function that assigns different weights to the functions’ frequency components as they are combined. Thus, Leibniz-type rules associated to bilinear pseudodifferential operators serve as a generalization of fractional Leibniz rules by providing estimates on the size and smoothness of some combination of two functions, for which pointwise multiplication is recoverable by choosing a symbol identically equal to one. A variety of function spaces may be used to measure the size and smoothness of functions involved, including Lebesgue spaces, Sobolev spaces, and Besov and Triebel-Lizorkin spaces. Further, bilinear pseudodifferential operators may be considered in association with different classes of symbols, which is to say that the symbol itself (and possibly its derivatives) will possess certain decay properties. New Leibniz-type rules in two different settings will be presented in this manuscript. In the first setting, Leibniz-type rules associated to bilinear pseudodifferential operators with homogeneous symbols in a certain class are proved, where the sizes of the functions involved are measured using a combination of Lebesgue space norms and norms corresponding to function spaces admitting appropriate molecular decompositions, specifically focusing on the case of homogeneous Besov-type and Triebel-Lizorkin-type spaces. In the second setting, Leibniz-type rules and biparameter counterparts are proved in weighted Lebesgue and Sobolev spaces associated to Coifman-Meyer multiplier operators. All of the new Leibniz-type rules proved in the manuscript yield corresponding new fractional Leibniz rules, which are highlighted as appropriate. Various techniques from Fourier analysis serve as important tools in the proofs of these new results, such as obtaining paraproduct decompositions for bilinear pseudodifferential operators and utilizing Littlewood-Paley theory and square function-type estimates. en_US
dc.language.iso en_US en_US
dc.subject Fourier analysis en_US
dc.subject Leibniz-type rules en_US
dc.subject Bilinear pseudodifferential operator en_US
dc.subject Kato-Ponce inequalities en_US
dc.title Leibniz-type rules associated to bilinear pseudodifferential operators en_US
dc.type Dissertation en_US
dc.description.degree Doctor of Philosophy en_US
dc.description.level Doctoral en_US
dc.description.department Department of Mathematics en_US
dc.description.advisor Virginia Naibo en_US
dc.date.published 2018 en_US
dc.date.graduationmonth August en_US

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