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1	Klasické operátory harmonické analýzy v Orliczových prostorech / Classical operators of harmonic analysis in Orlicz spaces Musil, Vít January 2018 (has links) Classical operators of harmonic analysis in Orlicz spaces V'ıt Musil We deal with classical operators of harmonic analysis in Orlicz spaces such as the Hardy-Littlewood maximal operator, the Hardy-type integral operators, the maximal operator of fractional order, the Riesz potential, the Laplace transform, and also with Sobolev-type embeddings on open subsets of Rn or with respect to Frostman measures and, in particular, trace embeddings on the boundary. For each operator (in case of embeddings we consider the identity operator) we investigate the question of its boundedness from an Orlicz space into another. Particular attention is paid to the sharpness of the results. We further study the question of the existence of optimal Orlicz domain and target spaces and their description. The work consists of author's published and unpublished results compiled together with material appearing in the literature.
2	Optimalita prostorů funkcí pro klasické integrální operátory / Optimality of function spaces for classical integral operators Mihula, Zdeněk January 2017 (has links) We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1
3	Nerovnosti pro integrální operátory / Nerovnosti pro integrální operátory Holík, Miloslav January 2011 (has links) The presented work contains a survey of the so far known results about the operator inequalities of the type "good λ", "better good λ" and "rearranged good λ" on the function spaces over the Euclidean space with the Lebesgue measure and their corollaries in the form of more complex operator inequal- ities and norm estimates. However, the main aim is to build similar theory for the Riesz potential operator on the function spaces over the quasi-metric space with the so-called "doubling" measure. Combining the corollaries of this theory with the known norm estimates we obtain the boundedness for the Riesz potential operator on the Lebesgue and Lorentz spaces.
4	Théorie non linéaire du potentiel et équations quasilinéaires avec données mesures / Nonlinear potential theory and quasilinear equations with measure data Nguyen, Quoc-Hung 25 September 2014 (has links) Cette thèse concerne l’existence et la régularité de solutions d’équations non-linéaires elliptiques, d’équations paraboliques et d’équations de Hesse avec mesures, et les critères de l’existence de solutions grandes d’équations elliptiques et paraboliques non-linéaires. / This thesis is concerned to the existence and regularity of solutions to nonlinear elliptic, parabolic and Hessian equations with measure, and criteria for the existence of large solutions to some nonlinear elliptic and parabolic equations. Équations quasi-linéaire elliptiques Équations quasi-linéaires paraboliques Solutions renormalisée Solutions maximales Solutions grandes Potentiel de Wolff Potentiel de Riesz Potentiel de Bessel Potentiel maximal Noyau de la chaleur Noyau de Bessel parabolique Mesures de Radon Capacités de Lorentz-Bessel Capacités de Bessel Capacités de Hausdorff Lemme de recouvrement de Vitali Espaces de Lorentz Domaines épais uniformes Domaines plats de Reifenberg Estimations de décroissance Estimations de Lorentz-Morrey Estimations capacitaires Équations de milieu poreux Équations de type Riccati Quasilinear elliptic equations Quasilinear parabolic equations Renormalized solutions Maximal solutions Large solutions Wolff potential Riesz potential Bessel Potential Maximal potential Heat kernel Parabolic Bessel kernel Radon measures Lorentz-Bessel capacities Bessel capacities Hausdorff capacities Vitial covering Lemma Lorentz spaces Uniformly thick domain Reifenberg flat domain Decay estimates Lorentz- Morrey estimates Capacitary estimates Porous medium equations Riccati type Equations

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