107 research outputs found
Oscillatory and Fourier Integral operators with degenerate canonical relations
We mostly survey results concerning the boundedness of oscillatory and
Fourier integral operators. This article does not intend to give a broad
overview; it mainly focusses on a few topics directly related to the work of
the authors.Comment: 37 pages, to appear in Publicacions Mathematiques (special issue,
Proceedings of the 2000 El Escorial Conference in Harmonic Analysis and
Partial Differential Equations
Oscillatory integral operators with homogeneous polynomial phases in several variables
We obtain decay estimates in for oscillatory integral
operators whose phase functions are homogeneous polynomials of degree m and
satisfy various genericity assumptions. The decay rates obtained are optimal in
the case of (2+2)--dimensions for any m while, in higher dimensions, the result
is sharp for m sufficiently large. The proof for large follows from
essentially algebraic considerations. For cubics in (2+2)--dimensions, the
proof involves decomposing the operator near the conic zero variety of the
determinant of the Hessian of the phase function, using an elaboration of the
general approach of Phong and Stein [1994].Comment: 39 pages, 2 figures; minor corrections; to appear in Journal of
Functional Analysi
On -point configuration sets with nonempty interior
We give conditions for -point configuration sets of thin sets to have
nonempty interior, applicable to a wide variety of configurations. This is a
continuation of our earlier work \cite{GIT19} on 2-point configurations,
extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in
Euclidean spaces. We show that for a general class of -point configurations,
the configuration set of a -tuple of sets, , has
nonempty interior provided that the sum of their Hausdorff dimensions satisfies
a lower bound, dictated by optimizing -Sobolev estimates of associated
generalized Radon transforms over all nontrivial partitions of the points
into two subsets. We illustrate the general theorems with numerous specific
examples. Applications to 3-point configurations include areas of triangles in
or the radii of their circumscribing circles; volumes of pinned
parallelepipeds in ; and ratios of pinned distances in and . Results for 4-point configurations include cross-ratios
on , triangle area pairs determined by quadrilaterals in , and dot products of differences in .Comment: 32 pages, no figure
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