182 research outputs found

    Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

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    We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group Hn\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 HH-type groups

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

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    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

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    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

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    We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in R\mathbb{R} with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight [w]Ap/2[w]_{A_{p/2}} turns out to be sharp for 3p<3\le p<\infty, whereas the sharpness in the range 2<p<32<p<3 remains as an open question. Weighted weak-type estimates in the endpoint p=2p=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

    A1A_1 theory of weights for rough homogeneous singular integrals and commutators

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    Quantitative A1AA_1-A_\infty estimates for rough homogeneous singular integrals TΩT_{\Omega} and commutators of BMOBMO symbols and TΩT_{\Omega} are obtained. In particular the following estimates are proved: % TΩLp(w)cn,pΩL[w]A11p[w]A1+1pfLp(w) \|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,TΩ]fLp(w)cn,pbBMOΩL[w]A11p[w]A2+1pfLp(w), \| [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<1<p<\infty and 1/p+1/p=11/p+1/p'=1.Comment: 19 page

    Kato–Ponce estimates for fractional sublaplacians in the Heisenberg group

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    We give a proof of commutator estimates for fractional powers of the sublaplacian on the Heisenberg group. Our approach is based on pointwise and LpL^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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