TY - JOUR

T1 - Fourier integral operators, fractal sets, and the regular value theorem

AU - Eswarathasan, Suresh

AU - Iosevich, Alex

AU - Taylor, Krystal

PY - 2011/11/10

Y1 - 2011/11/10

N2 - We prove that if EεR≥d, for d⊂2, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by dimH(E), and φis a sufficiently regular function, then the upper Minkowski dimension of the set. does not exceed dimH(E)εm, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

AB - We prove that if EεR≥d, for d⊂2, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by dimH(E), and φis a sufficiently regular function, then the upper Minkowski dimension of the set. does not exceed dimH(E)εm, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

KW - Fourier integral operators

KW - Generalized Radon transforms

KW - Lattice points

KW - Regular value theorem

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U2 - 10.1016/j.aim.2011.07.012

DO - 10.1016/j.aim.2011.07.012

M3 - Article

AN - SCOPUS:80052666812

VL - 228

SP - 2385

EP - 2402

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -