182 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 $\mathbb{H}^n$. 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

### An extension problem and trace Hardy inequality for the sublaplacian on $H$-type groups

In this paper we study the extension problem for the sublaplacian on a
$H$-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

### 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 $\mathbb{R}$ with finite
overlapping. Quantitative weighted estimates are obtained for this operator.
The linear dependence on the characteristic of the weight $[w]_{A_{p/2}}$ turns
out to be sharp for $3\le p<\infty$, whereas the sharpness in the range $2<p<3$
remains as an open question. Weighted weak-type estimates in the endpoint $p=2$
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

### $A_1$ theory of weights for rough homogeneous singular integrals and commutators

Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular
integrals $T_{\Omega}$ and commutators of $BMO$ symbols and $T_{\Omega}$ are
obtained. In particular the following estimates are proved: % $\|T_\Omega
\|_{L^p(w)}\le c_{n,p}\|\Omega\|_{L^\infty}
[w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)}$ %
and % $\| [b,T_{\Omega}]f\|_{L^{p}(w)}\leq
c_{n,p}\|b\|_{BMO}\|\Omega\|_{L^{\infty}}
[w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)},$ % for $1<p<\infty$ and $1/p+1/p'=1$.Comment: 19 page

### Kato–Ponce estimates for fractional sublaplacians in the Heisenberg group

We give a proof of commutator estimates for fractional powers of the sublaplacian
on the Heisenberg group. Our approach is based on pointwise and $L^p$ estimates involving square
fractional integrals and Littlewood--Paley square functionsBERC 2022-2025
RYC2018-025477-I
Ikerbasque
PID2021-123034NB-I00
IT1615-2

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