Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory.Comment: 44 pages, 5 figures, title changed, minor editin

Failure of the trilinear operator space Grothendieck theorem

We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on $\ell_\infty$, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded, answering a question of Pisier. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.Comment: Reformatted for Discrete Analysi

Product mixing in the alternating group

We prove the following one-sided product-mixing theorem for the alternating group: Given subsets $X,Y,Z \subset A_n$ of densities $\alpha,\beta,\gamma$ satisfying $\min(\alpha\beta,\alpha\gamma,\beta\gamma)\gg n^{-1}(\log n)^7$, there are at least $(1+o(1))\alpha\beta\gamma |A_n|^2$ solutions to $xy=z$ with $x\in X, y\in Y, z\in Z$. One consequence is that the largest product-free subset of $A_n$ has density at most $n^{-1/2}(\log n)^{7/2}$, which is best possible up to logarithms and improves the best previous bound of $n^{-1/3}$ due to Gowers. The main tools are a Fourier-analytic reduction noted by Ellis and Green to a problem just about the standard representation, a Brascamp--Lieb-type inequality for the symmetric group due to Carlen, Lieb, and Loss, and a concentration of measure result for rearrangements of inner products.Comment: 19 pages. Reformatted for Discrete Analysis but otherwise identical to the previous versio

The effects of latent variables in the development of comorbidity among common mental disorders

Background: Although numerous studies have examined the role of latent predispositions to internalizing and externalizing disorders in the structure of comorbidity among common mental disorders, none examined latent predispositions in predicting development of comorbidity. Methods: A novel method was used to study the role of latent variables in the development of comorbidity among lifetime DSM-IV disorders in the National Comorbidity Surveys. Broad preliminary findings are briefly presented to describe the method. The method used survival analysis to estimate time-lagged associations among 18 lifetime DSM-IV anxiety, mood, behavior, and substance disorders. A novel estimation approach examined the extent to which these predictive associations could be explained by latent canonical variables representing internalizing and externalizing disorders. Results: Consistently significant positive associations were found between temporally primary and secondary disorders. Within-domain time-lagged associations were generally stronger than between-domain associations. The vast majority of associations were explained by a model that assumed mediating effects of latent internalizing and externalizing variables, although the complexity of this model differed across samples. A number of intriguing residual associations emerged that warrant further investigation. Conclusions: The good fit of the canonical model suggests that common causal pathways account for most comorbidity among the disorders considered. These common pathways should be the focus of future research on the development of comorbidity. However, the existence of several important residual associations shows that more is involved than simple mediation. The method developed to carry out these analyses provides a unique way to pinpoint these significant residual associations for subsequent focused study. Depression and Anxiety, 2011. (c) 2011 Wiley-Liss, Inc