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Some classical inequalities, summability of multilinear operators and strange functionsAraújo, Gustavo da Silva 08 March 2016 (has links)
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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.
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