1,856 research outputs found
Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group
We prove Hardy inequalities for the conformally invariant fractional powers
of the sublaplacian on the Heisenberg group . We prove two
versions of such inequalities depending on whether the weights involved are
non-homogeneous or homogeneous. In the first case, the constant arising in the
Hardy inequality turns out to be optimal. In order to get our results, we will
use ground state representations. The key ingredients to obtain the latter are
some explicit integral representations for the fractional powers of the
sublaplacian and a generalized result by M. Cowling and U. Haagerup. The
approach to prove the integral representations is via the language of
semigroups. As a consequence of the Hardy inequalities we also obtain versions
of Heisenberg uncertainty inequality for the fractional sublaplacian.Comment: 35 pages. Revised versio
Potential operators associated with Jacobi and Fourier-Bessel expansions
We study potential operators (Riesz and Bessel potentials) associated with
classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for
the corresponding potential kernels. Then we characterize those , for which the potential operators are of strong type , of weak
type and of restricted weak type . These results may be thought
of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional
integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient
of our line of reasoning, we also obtain sharp estimates of the Poisson kernel
related to Fourier-Bessel expansions.Comment: 28 pages, 4 figures; v2 (some comments on Bessel potentials added
An extension problem and trace Hardy inequality for the sublaplacian on -type groups
In this paper we study the extension problem for the sublaplacian on a
-type group and use the solutions to prove trace Hardy and Hardy
inequalities for fractional powers of the sublaplacian.Comment: 39 page
Vector-valued extensions for fractional integrals of Laguerre expansions
We prove some vector-valued inequalities for fractional integrals defined in
the context of two different orthonormal systems of Laguerre functions. Our
results are based on estimates of the corresponding kernels with precise
control of the parameters involved. As an application, mixed norm estimates for
the fractional integrals related to the harmonic oscillator are deduced.Comment: 21 pages. Revised versio
The Riesz transform for the harmonic oscillator in spherical coordinates
In this paper we show weighted estimates in mixed norm spaces for the Riesz
transform associated with the harmonic oscillator in spherical coordinates. In
order to prove the result we need a weighted inequality for a vector-valued
extension of the Riesz transform related to the Laguerre expansions which is of
independent interest. The main tools to obtain such extension are a weighted
inequality for the Riesz transform independent of the order of the involved
Laguerre functions and an appropriate adaptation of Rubio de Francia's
extrapolation theorem.Comment: 19 pages. To appear in Constructive Approximatio
Maximal estimates for a generalized spherical mean Radon transform acting on radial functions
We study a generalized spherical means operator, viz. generalized spherical
mean Radon transform, acting on radial functions. As the main results, we find
conditions for the associated maximal operator and its local variant to be
bounded on power weighted Lebesgue spaces. This translates, in particular, into
almost everywhere convergence to radial initial data results for solutions to
certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations.
Moreover, our results shed some new light to the interesting and important
question of optimality of the yet known boundedness results for the
maximal operator in the general non-radial case. It appears that these could
still be notably improved, as indicated by our conjecture of the ultimate sharp
result.Comment: 20 pages, 2 figures. Sharpness results added and minor things
improved or corrected. Accepted for publication in Annali di Matematica Pura
ed Applicat
Quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function
We consider the Rubio de Francia's Littlewood--Paley square function
associated with an arbitrary family of intervals in with finite
overlapping. Quantitative weighted estimates are obtained for this operator.
The linear dependence on the characteristic of the weight turns
out to be sharp for , whereas the sharpness in the range
remains as an open question. Weighted weak-type estimates in the endpoint
are also provided. The results arise as a consequence of a sparse domination
shown for these operators, obtained by suitably adapting the ideas coming from
Benea (2015) and Culiuc et al. (2016).Comment: 18 pages. Revised versio
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