45,981 research outputs found
On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
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 . Noteworthy, we admit
negative values of the multiplicity functions. Our investigations include
maximal operators, -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
spaces, , and from weighted to weighted weak .
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
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
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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