Sharp LpL^p-LqL^q estimates for generalized kk-plane transforms

In this paper, optimal $L^p-L^q$ estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the $k$-plane transform. An important advance over previous work is that full $L^p-L^q$ estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is lead to make coercivity estimates for certain functionals on $L^p$ for $p < 1$.Comment: 23 pages; 1 figur

Uniform sublevel Radon-like inequalities

This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin, relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new $L^p$-improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger for FIOs (building on the ideas of Sogge and Mockenhaupt, Seeger, and Sogge); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.Comment: 40 page

Uniform estimates for cubic oscillatory integrals

This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs.Comment: 22 pages; v2 added reference

On multilinear determinant functionals

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).Comment: 14 page

H theorem for contact forces in granular materials

A maximum entropy theorem is developed and tested for granular contact forces. Although it is idealized, describing two dimensional packings of round, rigid, frictionless, cohesionless disks with coordination number Z=4, it appears to describe a central part of the physics present in the more general cases. The theorem does not make the strong claims of Edwards' hypothesis, nor does it rely upon Edwards' hypothesis at any point. Instead, it begins solely from the physical assumption that closed loops of grains are unable to impose strong force correlations around the loop. This statement is shown to be a generalization of Boltzmann's Assumption of Molecular Chaos (his \textit{stosszahlansatz}), allowing for the extra symmetries of granular stress propagation compared to the more limited symmetries of momentum propagation in a thermodynamic system. The theorem that follows from this is similar to Boltzmann's $H$ theorem and is presented as an alternative to Edwards' hypothesis for explaining some granular phenomena. It identifies a very interesting feature of granular packings: if the generalized \textit{stosszahlansatz} is correct, then the bulk of homogeneous granular packings must satisfy a maximum entropy condition simply by virtue of being stable, without any exploration of phase space required. This leads to an independent derivation of the contact force statistics, and these predictions have been compared to numerical simulation data in the isotropic case. The good agreement implies that the generalized \textit{stosszahlansatz} is indeed accurate at least for the isotropic state of the idealized case studied here, and that it is the reductionist explanation for contact force statistics in this case.Comment: 15 pages, 8 figures, to appear in Phys. Rev.

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