We show that every non-trivial Sobolev bound for generalized Radon transforms which average funct... more We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods. 1
We give a Mikhlin multplier theorem for a class of nonhomegenous dilations in plane, and study si... more We give a Mikhlin multplier theorem for a class of nonhomegenous dilations in plane, and study singular kernels that have a flag type singularity along the parabola y = cx 2 . We show that the multiplier of these operators consists of two distinct parts, one which is essentially the Fourier transform of the flat kernel, and another, which is characterized by its highly oscillatory factor that signifies the curvature of the singular support of the kernel. This characterization can be used to study the behaviour of this class on various function spaces. For example the L p boundedness of the corresponding operator on 1 < p < ∞ will be a trivial corollary..
Transactions of the American Mathematical Society, Jan 1, 2009
We show that every non-trivial Sobolev bound for generalized Radon transforms which average funct... more We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods.
We show that every non-trivial Sobolev bound for generalized Radon transforms which average funct... more We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods. 1
We give a Mikhlin multplier theorem for a class of nonhomegenous dilations in plane, and study si... more We give a Mikhlin multplier theorem for a class of nonhomegenous dilations in plane, and study singular kernels that have a flag type singularity along the parabola y = cx 2 . We show that the multiplier of these operators consists of two distinct parts, one which is essentially the Fourier transform of the flat kernel, and another, which is characterized by its highly oscillatory factor that signifies the curvature of the singular support of the kernel. This characterization can be used to study the behaviour of this class on various function spaces. For example the L p boundedness of the corresponding operator on 1 < p < ∞ will be a trivial corollary..
Transactions of the American Mathematical Society, Jan 1, 2009
We show that every non-trivial Sobolev bound for generalized Radon transforms which average funct... more We show that every non-trivial Sobolev bound for generalized Radon transforms which average functions over families of curves and surfaces yields an incidence theorem for suitably regular discrete sets of points and curves or surfaces in Euclidean space. This mechanism allows us to deduce geometric results not readily accessible by combinatorial methods.
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