Dirichlet Problem for Schrödinger Operators on Heisenberg Groups

Ji Li; Qingze Lin; Liang Song

doi:10.1007/s11118-024-10169-x

Dirichlet Problem for Schrödinger Operators on Heisenberg Groups

Ji Li, Qingze Lin^*, Liang Song

^*Corresponding author for this work

School of Mathematical and Physical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the Dirichlet problem associated to the Schrödinger operator L=-ΔH_n+V on Heisenberg group Hⁿ: (Formula presented.) with f in L^p(Hⁿ) (1<p<∞) and in H_L¹(Hⁿ), i.e., the Hardy space associated with L. Here ΔH_n is the sub-Laplacian on Hⁿ and the nonnegative potential V belongs to the reverse Hölder class B_Q/2 with Q the homogeneous dimension of Hⁿ. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition V∈B_Q/2. This result is new even back to Rⁿ (the condition will become V∈B_n/2) since the previous known result requires V∈B_(n+1)/2 which went through a Liouville type theorem.

Original language	English
Number of pages	19
Journal	Potential Analysis
DOIs	https://doi.org/10.1007/s11118-024-10169-x
Publication status	E-pub ahead of print - 11 Nov 2024

Keywords

Poisson integral
Schrödinger operator
Dirichlet problem
Heisenberg group
Hardy spaces

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10.1007/s11118-024-10169-x

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title = "Dirichlet Problem for Schr{\"o}dinger Operators on Heisenberg Groups",

abstract = "We investigate the Dirichlet problem associated to the Schr{\"o}dinger operator L=-ΔHn+V on Heisenberg group Hn: (Formula presented.) with f in Lp(Hn) (1L1(Hn), i.e., the Hardy space associated with L. Here ΔHn is the sub-Laplacian on Hn and the nonnegative potential V belongs to the reverse H{\"o}lder class BQ/2 with Q the homogeneous dimension of Hn. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition V∈BQ/2. This result is new even back to Rn (the condition will become V∈Bn/2) since the previous known result requires V∈B(n+1)/2 which went through a Liouville type theorem.",

keywords = "Poisson integral, Schr{\"o}dinger operator, Dirichlet problem, Heisenberg group, Hardy spaces",

author = "Ji Li and Qingze Lin and Liang Song",

year = "2024",

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day = "11",

doi = "10.1007/s11118-024-10169-x",

language = "English",

journal = "Potential Analysis",

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T1 - Dirichlet Problem for Schrödinger Operators on Heisenberg Groups

AU - Li, Ji

AU - Lin, Qingze

AU - Song, Liang

PY - 2024/11/11

Y1 - 2024/11/11

N2 - We investigate the Dirichlet problem associated to the Schrödinger operator L=-ΔHn+V on Heisenberg group Hn: (Formula presented.) with f in Lp(Hn) (1L1(Hn), i.e., the Hardy space associated with L. Here ΔHn is the sub-Laplacian on Hn and the nonnegative potential V belongs to the reverse Hölder class BQ/2 with Q the homogeneous dimension of Hn. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition V∈BQ/2. This result is new even back to Rn (the condition will become V∈Bn/2) since the previous known result requires V∈B(n+1)/2 which went through a Liouville type theorem.

AB - We investigate the Dirichlet problem associated to the Schrödinger operator L=-ΔHn+V on Heisenberg group Hn: (Formula presented.) with f in Lp(Hn) (1L1(Hn), i.e., the Hardy space associated with L. Here ΔHn is the sub-Laplacian on Hn and the nonnegative potential V belongs to the reverse Hölder class BQ/2 with Q the homogeneous dimension of Hn. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition V∈BQ/2. This result is new even back to Rn (the condition will become V∈Bn/2) since the previous known result requires V∈B(n+1)/2 which went through a Liouville type theorem.

KW - Poisson integral

KW - Schrödinger operator

KW - Dirichlet problem

KW - Heisenberg group

KW - Hardy spaces

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U2 - 10.1007/s11118-024-10169-x

DO - 10.1007/s11118-024-10169-x

M3 - Article

AN - SCOPUS:85206697807

SN - 0926-2601

JO - Potential Analysis

JF - Potential Analysis

ER -

Dirichlet Problem for Schrödinger Operators on Heisenberg Groups

Abstract

Keywords

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