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.
Analysis and Mathematical Physics – Springer Journals
Published: Mar 17, 2016
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