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Abstract Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1 for all sequences {Ij} of intervals in Γ and all sequences {fj} in Lp(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when The present proof uses a transference argument, an approach which shows that any constant Cp,q for which the inequality holds when G = will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space. [ABSTRACT FROM AUTHOR]

Abstract In this article we obtain the boundedness of the periodic, discrete and ergodic bilinear Hilbert transform, from , where 1$, and $p_3\ge 1$" />. The main techniques are a bilinear version of the transference method of Coifman and Weiss and certain discretization of bilinear operators. In the periodic case, we also obtain the boundedness for [ABSTRACT FROM AUTHOR]

A sufficient condition is given under which the sum, product and indeed any polynomial combination of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. This generalizes a result of Gillespie for Hilbert space operators. It is shown in particular that if $X$ is a UMD space, then the sum of finitely many commuting real scalar-type spectral operators acting on $X$ is a well-bounded operator (a result which fails on general reflexive Banach spaces). [ABSTRACT FROM AUTHOR]

$AC$-operators are a generalization in the context of well-bounded\-ness of normal operators on Hilbert space. It was shown by Doust and Walden that compact $AC$-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper. [ABSTRACT FROM AUTHOR]

Suppose that $(\Omega ,\mathcal{M},\mu )$ is a $\sigma $-finite measure space, $1