Net spaces and boundedness of integral operators

In this paper we introduce new functional spaces which we call the net spaces. Using their properties, the necessary and sufficient conditions for the integral operators to be of strong or weak-type are obtained. The estimates of the norm of the convolution operator in weighted Lebesgue spaces are presented.

Sharp A₂ inequality for haar shift operators

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Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.

Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

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Weak-type estimates for Haar shift operators: Sharp power on the A(p) characteristic

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Maximal operators and differentiation theorems for sparse sets

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Two weight inequalities for discrete positive operators

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Two weight inequalities for maximal truncations of dyadic Calderón-Zygmund operators

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Coefficients of Drinfeld modular forms and Hecke operators

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Propagation connectivity of random hypergraphs

We study the concept of propagation connectivity on random 3-uniform hypergraphs. This concept is inspired by a simple linear time algorithm for solving instances of certain constraint satisfaction problems. We derive upper and lower bounds for the propagation connectivity threshold, and point out some algorithmic implications.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Reduced fronto-temporal and limbic connectivity in the 22q11.2 deletion syndrome: vulnerability markers for developing schizophrenia?

The 22q11.2 deletion syndrome (22q11DS) is a widely recognized genetic model allowing the study of neuroanatomical biomarkers that underlie the risk for developing schizophrenia. Recent advances in magnetic resonance image analyses enable the examination of structural connectivity integrity, scarcely used in the 22q11DS field. This framework potentially provides evidence for the disconnectivity hypothesis of schizophrenia in this high-risk population. In the present study, we quantify the whole brain white matter connections in 22q11DS using deterministic tractography. Diffusion Tensor Imaging was acquired in 30 affected patients and 30 age- and gender-matched healthy participants. The Human Connectome technique was applied to register white matter streamlines with cortical anatomy. The number of fibers (streamlines) was used as a measure of connectivity for comparison between groups at the global, lobar and regional level. All statistics were corrected for age and gender. Results showed a 10% reduction of the total number of fibers in patients compared to controls. After correcting for this global reduction, preserved connectivity was found within the right frontal and right parietal lobes. The relative increase in the number of fibers was located mainly in the right hemisphere. Conversely, an excessive reduction of connectivity was observed within and between limbic structures. Finally, a disproportionate reduction was shown at the level of fibers connecting the left fronto-temporal regions. We could therefore speculate that the observed disruption to fronto-temporal connectivity in individuals at risk of schizophrenia implies that fronto-temporal disconnectivity, frequently implicated in the pathogenesis of schizophrenia, could precede the onset of symptoms and, as such, constitutes a biomarker of the vulnerability to develop psychosis. On the contrary, connectivity alterations in the limbic lobe play a role in a wide range of psychiatric disorders and therefore seem to be less specific in defining schizophrenia.

White matter maturation reshapes structural connectivity in the late developing human brain.

From toddler to late teenager, the macroscopic pattern of axonal projections in the human brain remains largely unchanged while undergoing dramatic functional modifications that lead to network refinement. These functional modifications are mediated by increasing myelination and changes in axonal diameter and synaptic density, as well as changes in neurochemical mediators. Here we explore the contribution of white matter maturation to the development of connectivity between ages 2 and 18 y using high b-value diffusion MRI tractography and connectivity analysis. We measured changes in connection efficacy as the inverse of the average diffusivity along a fiber tract. We observed significant refinement in specific metrics of network topology, including a significant increase in node strength and efficiency along with a decrease in clustering. Major structural modules and hubs were in place by 2 y of age, and they continued to strengthen their profile during subsequent development. Recording resting-state functional MRI from a subset of subjects, we confirmed a positive correlation between structural and functional connectivity, and in addition observed that this relationship strengthened with age. Continuously increasing integration and decreasing segregation of structural connectivity with age suggests that network refinement mediated by white matter maturation promotes increased global efficiency. In addition, the strengthening of the correlation between structural and functional connectivity with age suggests that white matter connectivity in combination with other factors, such as differential modulation of axonal diameter and myelin thickness, that are partially captured by inverse average diffusivity, play an increasingly important role in creating brain-wide coherence and synchrony.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Index theory for basic Dirac operators on Riemannian foliations

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.