630 research outputs found
Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions
We study potential operators associated with Laguerre function expansions of
convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove
qualitatively sharp estimates of the corresponding potential kernels. Then we
characterize those , for which the potential operators
are bounded. These results are sharp analogues of the classical
Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and
Dunkl-Laguerre settings.Comment: 25 pages, 2 figure
Sharp estimates of the potential kernel for the harmonic oscillator with applications
We prove qualitatively sharp estimates of the potential kernel for the
harmonic oscillator. These bounds are then used to show that the
estimates of the associated potential operator obtained recently by Bongioanni
and Torrea are in fact sharp.Comment: 10 pages, 1 figure; v2 (corrections in Section 3 concerning Theorem
3.1 and its proof and Figure 1
Negative powers of Laguerre operators
We study negative powers of Laguerre differential operators in , .
For these operators we prove two-weight estimates, with ranges of
depending on . The case of the harmonic oscillator (Hermite operator) has
recently been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain
Laguerre settings. The procedure is fairly direct for Laguerre function
expansions of Hermite type, due to some monotonicity properties of the kernels
involved. The case of Laguerre function expansions of convolution type is less
straightforward. For half-integer type indices we transfer the desired
results from the Hermite setting and then apply an interpolation argument based
on a device we call the {\sl convexity principle} to cover the continuous range
of . Finally, we investigate negative powers of the
Dunkl harmonic oscillator in the context of a finite reflection group acting on
and isomorphic to . The two weight estimates we
obtain in this setting are essentially consequences of those for Laguerre
function expansions of convolution type.Comment: 30 page
Hankel Multipliers And Transplantation Operators
Connections between Hankel transforms of different order for -functions
are examined. Well known are the results of Guy [Guy] and Schindler [Sch].
Further relations result from projection formulae for Bessel functions of
different order. Consequences for Hankel multipliers are exhibited and
implications for radial Fourier multipliers on Euclidean spaces of different
dimensions indicated
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
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