29,056 research outputs found
Sharp - estimates for generalized -plane transforms
In this paper, optimal estimates are obtained for operators which
average functions over polynomial submanifolds, generalizing the -plane
transform. An important advance over previous work is that full
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 for .Comment: 23 pages; 1 figur
Uniform sublevel Radon-like inequalities
This paper is concerned with establishing uniform weighted -
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 -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 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 -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 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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