H-distributions and compactness by compensation

Mišur, Marin (2017) H-distributions and compactness by compensation. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Abstract

H-measures are matrix Radon measures describing the behaviour of weak limits of quadratic quantities. They proved to be very successful tool in investigations of asymptotic limits of quadratic quantities. However, they turned insufficient for nonlinear problems. Recent investigations resulted with the introduction of variant H-measures, so called H-distributions, which surmount some of the noted inadequacies, and allow the treatment of terms involving sequences of \(L^p\) functions. Basic tools for the construction of aforementioned microlocal objects are pseudodifferential and singular integral operators. In the \(L^2\) case, Fourier transform together with Plancherel’s theorem proved to be a very efficient tool. However, in the \(L^p\) case one needs to use the theory of Fourier multipliers (Marcinkiewicz theorem, Hörmander-Mihlin theorem) which requires a higher regularity of the space of test functions, together with corresponding bounds on derivatives. There are two crucial steps in the construction of the above mentioned microlocal objects. First one is an application of the First commutation lemma to pass from trilinear functional to a bilinear one, while the second one is an application of the Schwartz kernel theorem to identify the obtained bilinear functional with an element from the dual of smooth functions on the product domain. We showed the Krasnoselskij type of result for unbounded domains, and with its help we lowered the regularity of the symbol needed for a variant of the First commutation lemma for the \(L^p\) spaces. We also showed how the same idea can be used to improve the results on existence of H-distributions on the Lebesgue spaces with mixed norm. Furthermore, we studied how further we can lower the regularity of the Sobolev multiplier under the assumption that the symbol of the Fourier multiplier operator satisfies only the Hörmander condition. What’s more, in the case when the symbol of the Fourier multiplier is defined on the unit sphere, Luc Tartar showed that the result remains valid in the \(L^2\) case even when we have coefficients from the VMO space (the space of functions of vanishing mean oscillations). We arrived at the same conclusion in the \(L^p\) space. In the end we showed a variant of the First commutation lemma in the case when we have general pseudodifferential operator instead of the Fourier multiplier operator. For that we used the bounds from Hwang’s results on boundedness of pseudodifferential operators. To give a better description of H-distributions, we refined the notion of distributions by introducing a notion of anisotropic distributions of finite order. Those are distributions which have different order in different coordinate directions. The main obstacle was adjusting the Schwartz kernel theorem to this new notion. We used Dieudonne’s approach which used the structure theorem of distributions. An advantage of this approach is that the order of distribution increases only with respect to one variable, while it remains unchanged with respect to the other. This allowed us to consider partial differential equations with continuous coefficients in the localisation principle of H-distributions. Up to now, we could only consider the smooth case. Let us emphasise that continuous coefficients were optimal in the \(L^2\) case. Motivated by Panov’s approach in the article on ultra-parabolic H-measures, we showed a variant of compactness by compensation. For that we used a variant of H-distributions, which we obtained from a result on the extension of the bilinear functionals to Bôchner spaces. This variant of H-distributions allowed us to consider variable discontinuous coefficients in differential restrictions and quadratic form. What is more, the derivatives in differential restrictions could be of fractional order. Because of that, we do not have symbols defined on the unit sphere, but on a more general manifold. For this reason we had to use the Marcinkiewicz multiplier theorem for continuity of the Fourier multiplier operators. We applied this new variant of compactness by compensation to a nonlinear degenerate equation of parabolic type, for which the known theory of H-measures was not adequate.

Item Type: Thesis (Doctoral thesis)
Keywords: H-measures, H-distributions, Fourier multiplier, kernel theorem, compactness by compensation
Supervisor: Antonić, Nenad
Date: 2017
Number of Pages: 97
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 04 Jul 2017 11:37
Last Modified: 04 Jul 2017 11:37
URI: http://digre.pmf.unizg.hr/id/eprint/5539

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