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Abstract. Wave packets emerged in recent years as a very useful tool in the study of nonlinear wave equations. In this article we introduce a phase space transform adapted to the geometry of wave packets, and use it to characterize and study the associated classes of pseudodifferential and Fourier integral operators.

Abstract. For functions F, G on R n , any k-dimensional affine subspace H ⊂ R n , 1 ≤ k < n, and exponents p, q, r ≥ 2 with 1 p + 1 q + 1 r = 1, we prove the estimateHere, the mixed norms on the right are defined in terms of the Fourier transform byLebesgue measure on the affine subspace H ⊥ ξ := ξ + H ⊥ . Dually, one obtains restriction theorems for the Fourier transform on affine subspaces. We use this, and a maximal variant, to prove results for a variety of multilinear convolution operators, including L p -improving bounds for measures; bilinear variants of Stein's spherical maximal theorem; estimates for m-linear oscillatory integral operators; Sobolev trace inequalities; and bilinear estimates for solutions to the wave equation.

The epidemic of overweight and obesity around the world and in the US is a major public health challenge, with 1.5 billion overweight and obese adults worldwide, and 68% of US adults and 31% of US children and adolescents overweight or obese. Obesity leads to serious health consequences, including an increased risk of type 2 diabetes mellitus and heart disease. Current preventive and medical treatments include lifestyle modification, medication, and bariatric surgery in extreme cases; however, they are either not very efficacious or are very expensive. Obesity is a complex condition involving the dysregulation of several organ systems and molecular pathways, including adipose tissue, the pancreas, the gastrointestinal tract, and the CNS. The role of the CNS in obesity is receiving more attention as obesity rates rise and treatments continue to fail. While the role of the hypothalamus in regulation of appetite and food intake has long been recognized, the roles of the CNS reward systems are beginning to be examined as the role of environmental influences on energy balance are explored. Omega-3 polyunsaturated fatty acids are essential nutrients that play a beneficial role in several disease processes due to their anti-inflammatory effects, modulation of lipids, and effects on the CNS. Omega-3 fatty acids, specifically EPA and DHA, have shown promising preliminary results in animal and human studies in the prevention and treatment of obesity. Given their effects on many of the pathways involved in obesity, and specifically in the endocannabinoid and mesocorticolimbic pathways, we hypothesize that EPA and DHA supplementation in populations can reduce the reward associated with food, thereby reduce appetite and food intake, and ultimately contribute to the prevention or reduction of obesity. If these fatty acids do harbor such potential, their supplementation in many parts of the world may hold great promise in reducing the global burden of obesity.

We continue our investigation of a model introduced by Adkins and Nappi, in which omega mesons stabilize chiral solitons. The aim of this article is to show that the energy associated to equivariant solutions does not concentrate.Comment: 12 pages, 2 figure