Volume 32, Issue 1
Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group

Y. Liu & G. B. Tang

Anal. Theory Appl., 32 (2016), pp. 78-89.

Published online: 2016-01

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  • Abstract

Let Hn be the Heisenberg group and Q=2n+2 be its homogeneous dimension. In this paper, we consider the Schr ¨odinger operator −∆Hn +V, where ∆Hn is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H ¨older class Bq1for q1 ≥ Q/2. We show that the operators T1 = V(−∆Hn +V)−1 and T2 = V1/2(−∆Hn +V)−1/2 are both bounded from H1L(Hn) into L1(Hn). Our results are also valid on the stratified Lie group.

  • Keywords

Heisenberg group stratified Lie group reverse Hölder class Riesz transform Schrödinger operator

  • AMS Subject Headings

52B10 65D18 68U05 68U07

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COPYRIGHT: © Global Science Press

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@Article{ATA-32-78, author = {Y. Liu and G. B. Tang}, title = {Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group}, journal = {Analysis in Theory and Applications}, year = {2016}, volume = {32}, number = {1}, pages = {78--89}, abstract = {

Let Hn be the Heisenberg group and Q=2n+2 be its homogeneous dimension. In this paper, we consider the Schr ¨odinger operator −∆Hn +V, where ∆Hn is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H ¨older class Bq1for q1 ≥ Q/2. We show that the operators T1 = V(−∆Hn +V)−1 and T2 = V1/2(−∆Hn +V)−1/2 are both bounded from H1L(Hn) into L1(Hn). Our results are also valid on the stratified Lie group.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2016.v32.n1.7}, url = {http://global-sci.org/intro/article_detail/ata/4656.html} }
TY - JOUR T1 - Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group AU - Y. Liu & G. B. Tang JO - Analysis in Theory and Applications VL - 1 SP - 78 EP - 89 PY - 2016 DA - 2016/01 SN - 32 DO - http://doi.org/10.4208/ata.2016.v32.n1.7 UR - https://global-sci.org/intro/article_detail/ata/4656.html KW - Heisenberg group KW - stratified Lie group KW - reverse Hölder class KW - Riesz transform KW - Schrödinger operator AB -

Let Hn be the Heisenberg group and Q=2n+2 be its homogeneous dimension. In this paper, we consider the Schr ¨odinger operator −∆Hn +V, where ∆Hn is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H ¨older class Bq1for q1 ≥ Q/2. We show that the operators T1 = V(−∆Hn +V)−1 and T2 = V1/2(−∆Hn +V)−1/2 are both bounded from H1L(Hn) into L1(Hn). Our results are also valid on the stratified Lie group.

Y. Liu & G. B. Tang. (1970). Hardy Type Estimates for Riesz Transforms Associated with Schrödinger Operators on the Heisenberg Group. Analysis in Theory and Applications. 32 (1). 78-89. doi:10.4208/ata.2016.v32.n1.7
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