Gustavo A. Muñoz-Fernández

Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars

The Bohnenblust-Hille inequality was obtained in 1931 and ( in the case of real scalars) asserts that for every positive integer m there is a constant Cm so that ((N)Sigma(i1 , . . . , im=1)vertical bar T(e(i1) (,...,) e(im))vertical bar(2m/m+1))(m+1/2) <= C-m parallel to T parallel to for all positive integers N and every m-linear mapping T : l(infinity)(N) x...x l(infinity)(N) -> R. Since then, several authors have obtained upper estimates for the values of C-m. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for C-m.

Sierpinski-Zygmund functions and other problems on lineability

We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpinski-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results.

A Characterization of Continuity Revisited

Abstract It is well known that a function f : ℝ → ℝ is continuous if and only if the image of every compact set under f is compact and the image of every connected set is connected. We show that there exist two 2C-dimensional linear spaces of nowhere continuous functions that (except for the zero function) transform compact sets into compact sets and connected sets into connected sets respectively.

Estimates for the asymptotic behaviour of the constants in the Bohnenblust–Hille inequality

A classical inequality due to H.F. Bohnenblust and E. Hille states that for every positive integer n there is a constant C n  > 0 so that for every positive integer N and every n-linear mapping . The original estimates for those constants from Bohnenblust and Hille are In this note we present explicit formulae for quite better constants, and calculate the asymptotic behaviour of these estimates, completing recent results of the second and third authors. For example, we show that, if C ℝ,n and C ℂ,n denote (respectively) these estimates for the real and complex Bohnenblust–Hille inequality then, for every even positive integer n, for a certain sequence {r n } which we estimate numerically to belong to the interval (1, 3/2) (the case n odd is similar). Simultaneously, assuming that {r n } is in fact convergent, we also conclude that

When the Identity Theorem "Seems" to Fail

Abstract The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A ⊂C∞ (ℝ); (iv) every element of A has infinitely many zeros in ℝ; and (v) for every f ε A 0 and n ε N, f(n) (the nth derivative of f) enjoys the same properties as the elements in A0 . This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.

Lineability in sequence and function spaces

We prove the existence of large algebraic structures - including large vector subspaces or infinitely generated free algebras - inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of discontinuous separately continuous real functions. Lineability in special spaces of sequences is also investigated.Programa de Doutorado Sanduiche no Exterior (Coordenacao de aperfeicoamento de pessoal de nivel superior)

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in is exactly . We also give the exact value of the real polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in . Finally, we provide lower estimates for the real polynomial Bohnenblust–Hille constant for polynomials in of higher degrees.

A hierarchy in the family of real surjective functions

Abstract This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in ℝℝ.