45,981 research outputs found

    On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

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    We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to Z2d\mathbb{Z}_2^d. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, gg-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted LpL^p spaces, 1<p<1 < p < \infty, and from weighted L1L^1 to weighted weak L1L^1. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type

    Average decay estimates for Fourier transforms of measures supported on curves

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    We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.Comment: We have learned about an important reference on the subject matter, and revised the paper accordingl

    Boundary reconstruction for the broken ray transform

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    We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for derivatives this transform has to be weighted by powers of the second fundamental form. The problem studied here is related to Calder\'on's problem with partial data.Comment: 23 pages, 1 figure; final versio
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