A general Pietsch Domination Theorem

Abstract

In this short communication we show that the unified Pietsch Domination Theorem proved in Botelho et al. (2010) [1] remains true even if we remove two of its hypotheses.

Factorization of p-dominated polynomials through Lp-spaces

“…To avoid a repetition of standard arguments to prove it, we will use the abstract and general theorem given in [11] that unifies known domination results in several different linear and nonlinear operator ideals. A simplified version of this general result appears in [28], and a more abstract version has been considered in [30]. …”

When is the Haar measure a Pietsch measure for nonlinear mappings?

“…Next, we describe the general Pietsch Domination Theorem proved in [4,21]. Let X, Y and E be (arbitrary) sets, and H be a family of mappings from Y to X.…”

“…The Unified Pietsch Domination Theorem [4,21] reads as follows: Theorem 2.4. Let R and S be as above, 0 < p < ∞ and f ∈ H. Then f is R-S-abstract p-summing if and only if there is a constant C > 0 and a regular Borel probability measure µ on K such that…”

“…The measure µ above is also called a Pietsch measure for v. The Pietsch Domination Theorem has analogs in different contexts, including versions for classes of absolutely summing nonlinear operators (see, for example, [1,6,7,9,10,15,22]). Recently, in [4,21,23] the concept of abstract R-S-abstract p-summing mapping was introduced in such a way that several previous known versions of the Pietsch Domination Theorem can be regarded as particular instances of one single result.…”

Domination spaces and factorization of linear and multilinear summing operators

“…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.…”