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A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group

A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group Let $${\mathbb {H}^n}$$ H n be the Heisenberg group and $$Q=2n+2$$ Q = 2 n + 2 be its homogeneous dimension. The Schrödinger operator is denoted by $$ - {\Delta _{{\mathbb {H}^n}}} + V$$ - Δ H n + V , where $${\Delta _{{\mathbb {H}^n}}}$$ Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $${B_{{q_1}}}$$ B q 1 for $${q_1} \ge \frac{Q}{2}$$ q 1 ≥ Q 2 . Let $$H^p_L(\mathbb {H}^n)$$ H L p ( H n ) be the Hardy space associated with the Schrödinger operator for $$\frac{Q}{Q+\delta _0}<p\le 1$$ Q Q + δ 0 < p ≤ 1 , where $$\delta _0=\min \{1,2-\frac{Q}{q_1}\}$$ δ 0 = min { 1 , 2 - Q q 1 } . In this note we show that the operators $${T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}$$ T 1 = V ( - Δ H n + V ) - 1 and $${T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}$$ T 2 = V 1 / 2 ( - Δ H n + V ) - 1 / 2 are bounded from $$H_L^p({\mathbb {H}^n})$$ H L p ( H n ) into $${L^p}({\mathbb {H}^n})$$ L p ( H n ) . Our results are also valid on the stratified Lie group. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

A note for Riesz transforms associated with Schrödinger operators on the Heisenberg Group

Analysis and Mathematical Physics , Volume 7 (1) – Mar 17, 2016

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0128-6
Publisher site
See Article on Publisher Site

Abstract

Let $${\mathbb {H}^n}$$ H n be the Heisenberg group and $$Q=2n+2$$ Q = 2 n + 2 be its homogeneous dimension. The Schrödinger operator is denoted by $$ - {\Delta _{{\mathbb {H}^n}}} + V$$ - Δ H n + V , where $${\Delta _{{\mathbb {H}^n}}}$$ Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $${B_{{q_1}}}$$ B q 1 for $${q_1} \ge \frac{Q}{2}$$ q 1 ≥ Q 2 . Let $$H^p_L(\mathbb {H}^n)$$ H L p ( H n ) be the Hardy space associated with the Schrödinger operator for $$\frac{Q}{Q+\delta _0}<p\le 1$$ Q Q + δ 0 < p ≤ 1 , where $$\delta _0=\min \{1,2-\frac{Q}{q_1}\}$$ δ 0 = min { 1 , 2 - Q q 1 } . In this note we show that the operators $${T_1} = V{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1}}$$ T 1 = V ( - Δ H n + V ) - 1 and $${T_2} = {V^{1/2}}{( - {\Delta _{{\mathbb {H}^n}}} + V)^{ - 1/2}}$$ T 2 = V 1 / 2 ( - Δ H n + V ) - 1 / 2 are bounded from $$H_L^p({\mathbb {H}^n})$$ H L p ( H n ) into $${L^p}({\mathbb {H}^n})$$ L p ( H n ) . Our results are also valid on the stratified Lie group.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Mar 17, 2016

References