75 research outputs found
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
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
Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production
This article provides sharp constructive upper and lower bound estimates for
the non-linear Boltzmann collision operator with the full range of physical non
cut-off collision kernels ( and ) in the trilinear
energy . These new estimates prove that, for
a very general class of , the global diffusive behavior (on ) in the
energy space is that of the geometric fractional derivative semi-norm
identified in the linearized context in our earlier works [2009, 2010, 2010
arXiv:1011.5441v1]. We further prove new global entropy production estimates
with the same anisotropic semi-norm. This resolves the longstanding, widespread
heuristic conjecture about the sharp diffusive nature of the non cut-off
Boltzmann collision operator in the energy space .Comment: 29 pages, updated file based on referee report; Advances in
Mathematics (2011
- β¦