This thesis contains work from the author's papers Palsson and Sovine (2020); Iosevich, Palsson, and Sovine (2022); and Palsson and Sovine (2022) with coauthors Eyvindur Palsson and Alex Iosevich. These works establish new $L^p$-improving, quasi-Banach, and sparse bounds for several bilinear and multilinear operators that generalize the linear spherical average to the multilinear setting, and maximal variants of these operators, with an emphasis on the triangle averaging operator and the bilinear spherical averaging operator. / Doctor of Philosophy / This thesis establishes new regularity properties for several mathematical operations that generalize the operation of taking the average of a function over a sphere to operations that average the product of several input functions over a surface to produce a single output function. These operations include the triangle averaging operator, the $k$-simplex averaging operators for $k$ an integer greater than 1, and the bilinear spherical averaging operator, as well as maximal operators obtained by allowing the radius of the averaging surface to vary over some range of values.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-29T12:08:37Z No. of bitstreams: 1 arquivototal.pdf: 967006 bytes, checksum: bd1b76a7b376f5fda6d282d14e851d1a (MD5) / Made available in DSpace on 2016-03-29T12:08:37Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 967006 bytes, checksum: bd1b76a7b376f5fda6d282d14e851d1a (MD5) Previous issue date: 2014-03-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study two generalizations of the well-known concept of absolutely summing operators. The rst one consists of the multiple summing multilinear operators and it is focused on a result of coincidence that is equivalent to the Bohnenblust- Hille inequality. This inequality asserts that, for K = R or C and every positive integer m there exists positive scalars BK;m 1 such that N X i1;:::;im=1 U(ei1 ; : : : ; eim) 2m m+1!m+1 2m BK;m sup z1;:::;zm2DN jU(z1; :::; zm)j for every m-linear mapping U : KN KN ! K and every positive integer N, where (ei)N i=1 denotes the canonical basis of KN: In this line our main goal is the investigation of the best constants BK;m satisfying the above inequality. The second generalization involves the concept of absolutely summing multilinear operators at a given point; we present an abstract version of these operators involving many of their properties. We prove that, considering appropriate sequence spaces, we have other kind of operators as particular cases of our version. / No presente trabalho vamos trabalhar com duas generalizações dos bem conhecidos operadores absolutamente somantes. A primeira envolve os operadores multilineares múltiplo somantes e nos focaremos num resultado de coincidência que é equivalente à desigualdade multilinear de Bohnenblust-Hille. Esta a rma que, para = R ou C, e todo inteiro positivo m 1, existem escalares BK;m 1 tais que N X i1;:::;im=1 U(ei1 ; : : : ; eim) 2m m+1!m+1 2m BK;m sup z1;:::;zm2DN jU(z1; :::; zm)j para toda forma m-linear U : KN KN ! K e todo inteiro positivo N, onde )N i=1 é a base canônica de KN: Nessa linha, nosso objetivo será a investigação das melhores constantes BK;m que satisfazem essa desigualdade. A segunda generalização envolve o estudo dos operadores multilineares absolutamente somantes num ponto; apresentaremos uma versão abstrata destes operadores que engloba várias de suas propriedades. Veremos que, considerando os espaços de sequências adequados, teremos outros tipos de operadores como casos

Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-04-23T21:42:23Z No. of bitstreams: 1 Arquivototal.pdf: 889863 bytes, checksum: feeec177f9d947c98727574b765d8acb (MD5) / Made available in DSpace on 2018-04-23T21:42:23Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 889863 bytes, checksum: feeec177f9d947c98727574b765d8acb (MD5) Previous issue date: 2017-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study the notion of index of summability for pairs of Banach spaces. This index plays the role of a kind of “measure” of how the space of m-homogeneous polynomials from E to F (or the space of multilinear operators of E1×···×Em to F) are far from being the space of absolutely summing m-homogeneous polynomials (or with the space of multiple summing multilinear operators). In some cases the optimal index of summability is presented. / Neste trabalho, estudamos a noção de índice de somabilidade para pares de espaços de Banach. Esse índice desempenha o papel de um tipo de \medida" de como o espaço dos polinômios m-homogêneos de E em F (ou o espaço dos operadores multilineares de E Em em F) está longe de coincidir com o espaço dos polinômios m- homogêneos absolutamente somantes (ou com o espaço dos operadores multilineares multiplo somantes). Em alguns casos o índice ótimo de somabilidade e apresentado. Palavras-chave: Polinômios absolutamente somantes, operadores multilineares absolutamente somantes, espaços de Banach, índice de somabilidade.

The common topic of this thesis is boundedness of integral and supremal operators between weighted function spaces. The first type of results are characterizations of boundedness of a convolution-type operator between general weighted Lorentz spaces. Weighted Young-type convolution inequalities are obtained and an optimality property of involved domain spaces is proved. Additional provided information includes an overview of basic properties of some new function spaces appearing in the proven inequalities. In the next part, product-based bilinear and multilinear Hardy-type operators are investigated. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable. Further objects of study are iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, thus completing the theory of the involved fundamental Hardy-type operators. / Operators acting on function spaces are classical subjects of study in functional analysis. This thesis contributes to the research on this topic, focusing particularly on integral and supremal operators and weighted function spaces. Proving boundedness conditions of a convolution-type operator between weighted Lorentz spaces is the first type of a problem investigated here. The results have a form of weighted Young-type convolution inequalities, addressing also optimality properties of involved domain spaces. In addition to that, the outcome includes an overview of basic properties of some new function spaces appearing in the proven inequalities.  Product-based bilinear and multilinear Hardy-type operators are another matter of focus. It is characterized when a bilinear Hardy operator inequality holds either for all nonnegative or all nonnegative and nonincreasing functions on the real semiaxis. The proof technique is based on a reduction of the bilinear problems to linear ones to which known weighted inequalities are applicable.  The last part of the presented work concerns iterated supremal and integral Hardy operators, a basic Hardy operator with a kernel and applications of these to more complicated weighted problems and embeddings of generalized Lorentz spaces. Several open problems related to missing cases of parameters are solved, completing the theory of the involved fundamental Hardy-type operators. / <p>Artikel 9 publicerad i avhandlingen som manuskript med samma titel.</p>

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-29T11:06:09Z No. of bitstreams: 1 arquivo total.pdf: 821623 bytes, checksum: 520d1fa102a8bdfeb531d12a30d60f61 (MD5) / Made available in DSpace on 2016-03-29T11:06:09Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 821623 bytes, checksum: 520d1fa102a8bdfeb531d12a30d60f61 (MD5) Previous issue date: 2014-03-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Os teoremas de Bohnenblust Hille, demonstrados em 1931 no prestigioso jornal Annals of Mathematics, foram utilizados como ferramentas muito úteis na solução do famoso Problema da convergência absoluta de Bohr. Após um longo tempo esquecidos, estes teoremas têm sido bastante explorados nos últimos anos. Este último quinquê- nio experimentou o surgimento de várias obras dedicadas a estimar as constantes de Bohnenblust Hille ([13, 18, 20, 26, 27, 39, 42, 44, 46, 53]) e também conexões inesperadas com a Teoria da Informação Quântica apareceram (ver, por exemplo, [38]). Há, de fato, quatro casos para serem investigados: polinomial (casos real e complexo) e multilinear (casos real e complexo). Podemos resumir em uma frase as principais informa ções dos trabalhos recentes: as constantes das desigualdades de Bohnenblust Hille são, em geral, extraordinariamente menores do que as primeiras estimativas tinham previsto. Neste trabalho apresentamos algumas das nossas pequenas contribuições ao estudo das constantes nas desigualdades de Bohnenblust-Hille, os quais encontram-se contidos em ([40, 41, 42, 44]).The Bohnenblust Hille theorems, proved in 1931 in the prestigious journal Annals of Mathematics, were used as very useful tools in the solution of the famous "Bohr's absolute convergence problem". After a long time overlooked, these theorems have been explored in the recent years. Last quinquennium experienced the rising of several works dedicated to estimate the Bohnenblust Hille constants ([13, 18, 20, 26, 27, 39, 42, 44, 46, 53]) and also unexpected connections with Quantum Information Theory appeared (see, e.g., [38]). There are in fact four cases to be investigated: polynomial (real and complex cases) and multilinear (real and complex cases). We can summarize in a sentence the main information from the recent preprints: the Bohnenblust Hille constants are, in general, extraordinarily smaller than the rst estimates predicted. In this work, we present some of our small contributions to the study of the constants of the inequalities Bohnenblust-Hille, these are contained in ([40, 41, 42, 44]). / Os teoremas de Bohnenblust Hille, demonstrados em 1931 no prestigioso jornal Annals of Mathematics, foram utilizados como ferramentas muito úteis na solução do famoso Problema da convergência absoluta de Bohr. Após um longo tempo esquecidos, estes teoremas têm sido bastante explorados nos últimos anos. Este último quinquê- nio experimentou o surgimento de várias obras dedicadas a estimar as constantes de Bohnenblust Hille ([13, 18, 20, 26, 27, 39, 42, 44, 46, 53]) e também conexões inesperadas com a Teoria da Informação Quântica apareceram (ver, por exemplo, [38]). Há, de fato, quatro casos para serem investigados: polinomial (casos real e complexo) e multilinear (casos real e complexo). Podemos resumir em uma frase as principais informa ções dos trabalhos recentes: as constantes das desigualdades de Bohnenblust Hille são, em geral, extraordinariamente menores do que as primeiras estimativas tinham previsto. Neste trabalho apresentamos algumas das nossas pequenas contribuições ao estudo das constantes nas desigualdades de Bohnenblust-Hille, os quais encontram-se contidos em ([40, 41, 42, 44])

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-23T16:38:50Z No. of bitstreams: 1 arquivototal.pdf: 1943524 bytes, checksum: 935ea8764b03a0cab23d8c7c772a137d (MD5) / Made available in DSpace on 2017-08-23T16:38:50Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1943524 bytes, checksum: 935ea8764b03a0cab23d8c7c772a137d (MD5) Previous issue date: 2016-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is divided into three parts. In the first part, we investigate the behavior of the constants of the Bohnenblust–Hille and Hardy–Littlewood polynomial and multilinear inequalities. In the second part, we show an optimal spaceability result for a set of non-multiple summing forms on `p and we also generalize a result related to cotype (from 2010) as highlighted by G. Botelho, C. Michels, and D. Pellegrino. Moreover, we prove new coincidence results for the class of absolutely and multiple summing multilinear operators (in particular, we show that the well-known Defant–Voigt theorem is optimal). Still in the second part, we show a generalization of the Bohnenblust–Hille and Hardy–Littlewood multilinear inequalities and we present a new class of summing multilinear operators, which recovers the class of absolutely and multiple summing operators. In the third part, it is proved the existence of large algebraic structures inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of non-constant di↵erentiable real functions vanishing on dense sets, and the family of noncontinuous separately continuous real functions. / Este trabalho est´a dividido em trˆes partes. Na primeira parte, investigamos o comportamento das constantes das desigualdades polinomial e multilinear de Bohnenblust–Hille e Hardy–Littlewood. Na segunda parte, mostramos um resultado ´otimo de espa¸cabilidade para o complementar de uma classe de operadores m´ultiplo somantes em `p e tamb´em generalizamos um resultado relacionado a cotipo (de 2010) devido a G. Botelho, C. Michels e D. Pellegrino. Al´em disso, provamos novos resultados de coincidˆencia para as classes de operadores multilineares absolutamente e m´ultiplo somantes (em particular, mostramos que o famoso teorema de Defant–Voigt ´e ´otimo). Ainda na segunda parte, mostramos uma generaliza¸c˜ao das desigualdades multilineares de Bohnenblust–Hille e Hardy–Littlewood e apresentamos uma nova classe de operadores multilineares somantes, a qual recupera as classes dos operadores multilineares absolutamente e m´ultiplo somantes. Na terceira parte, provamos a existˆencia de grandes estruturas alg´ebricas dentro de certos conjuntos, como, por exemplo, a fam´ılia das fun¸c˜oes mensur´aveis `a Lebesgue que s˜ao sobrejetivas em um sentido forte, a fam´ılia das fun¸c˜oes reais n˜ao constantes e diferenci´aveis que se anulam em um conjunto denso e a fam´ılia das fun¸c˜oes reais n˜ao cont´ınuas e separadamente cont´ınuas.