Some techniques on nonlinear analysis and applications

Abstract

In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second result, more technical, but also connected to the first one, is an extension of the well-known Pietsch Domination Theorem. The last decade witnessed the birth of different families of Pietsch Domination-type results and some attempts of unification. Our result, that we call "full general Pietsch Domination Theorem" is potentially a definitive Pietsch Domination Theorem which unifies the previous versions and delimits what can be proved in this line. The connections to the recent notion of weighted summability are traced.

Absolutely continuous multilinear operators

“…It can be also extended to the multilinear case. For the proof of this Domination Theorem we use the full general Pietsch Domination Theorem recently presented by Pellegrino et al in [18]. Let X 1 , ..., X m , Y and E 1 , ..., E k be (arbitrary) non-void sets, H be a family of mappings from X 1 × ... × X m to Y .…”

Domination spaces and factorization of linear and multilinear summing operators

“…for all ϕ ∈ B X * j , b l ∈ K, x j ∈ X j with j = 1, ..., m. Then Theorem 4.6 in [17] gives the result.…”

“…Using the full general Pietsch Domination Theorem presented by Pellegrino et all in [17] we obtain the Domination Theorem for our class.…”

“…Actually, even the most abstract approaches to the problem are based in this kind of expressions (see [2,16,17]). In [2] the R-S-abstract p-summing mappings were introduced with the aim of unifying the wide variety of summing classes of mappings that can be found in the literature.…”

Composition results for strongly summing and dominated multilinear operators

“…Without any claim of completeness we mention: absolutely summing multilinear operators, see [1,2,9,24,37]; multiple summing multilinear operators, see [3,8,28,32]; dominated multilinear operators, see [21,27,30,31,33]. For the theory of polynomials in Banach spaces and their applications the interested reader can consult [17,18,29] and for the connection between holomorphy types and ideals of multilinear mappings we recommend the reader [7].…”