Bilinear operators associated with Schrödinger operators

Chin Cheng Lin; Ying Chieh Lin; Heping Liu; Yu Liu

doi:10.4064/sm205-3-4

Bilinear operators associated with Schrödinger operators

Chin Cheng Lin, Ying Chieh Lin, Heping Liu, Yu Liu

Department of Mathematics

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

4 Scopus citations

Abstract

Let L = -Δ +V be a Schrödinger operator in ℝ^d and H_L¹(ℝ^d) be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T ^- defined by T^±(f,g)(x) = (T₁f)(x) (T₂g)(x) ± (T₂f)(x)(T₁g)(x), where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T ^- is bounded from L^p(ℝ) × L^q(ℝ ^d) to H_L¹(ℝ^d) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

Original language	English
Pages (from-to)	281-295
Number of pages	15
Journal	Studia Mathematica
Volume	205
Issue number	3
DOIs	https://doi.org/10.4064/sm205-3-4
State	Published - 2011

Keywords

Bilinear operators
Hardy spaces
Riesz transforms
Schrödinger operators

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title = "Bilinear operators associated with Schr{\"o}dinger operators",

abstract = "Let L = -Δ +V be a Schr{\"o}dinger operator in ℝd and HL1(ℝd) be the Hardy type space associated to L. We investigate the bilinear operators T+ and T - defined by T±(f,g)(x) = (T1f)(x) (T2g)(x) ± (T2f)(x)(T1g)(x), where T1 and T2 are Calder{\'o}n-Zygmund operators related to L. Under some general conditions, we prove that either T+ or T - is bounded from Lp(ℝ) × Lq(ℝ d) to HL1(ℝd) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.",

keywords = "Bilinear operators, Hardy spaces, Riesz transforms, Schr{\"o}dinger operators",

author = "Lin, {Chin Cheng} and Lin, {Ying Chieh} and Heping Liu and Yu Liu",

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TY - JOUR

T1 - Bilinear operators associated with Schrödinger operators

AU - Lin, Chin Cheng

AU - Lin, Ying Chieh

AU - Liu, Heping

AU - Liu, Yu

PY - 2011

Y1 - 2011

N2 - Let L = -Δ +V be a Schrödinger operator in ℝd and HL1(ℝd) be the Hardy type space associated to L. We investigate the bilinear operators T+ and T - defined by T±(f,g)(x) = (T1f)(x) (T2g)(x) ± (T2f)(x)(T1g)(x), where T1 and T2 are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T+ or T - is bounded from Lp(ℝ) × Lq(ℝ d) to HL1(ℝd) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

AB - Let L = -Δ +V be a Schrödinger operator in ℝd and HL1(ℝd) be the Hardy type space associated to L. We investigate the bilinear operators T+ and T - defined by T±(f,g)(x) = (T1f)(x) (T2g)(x) ± (T2f)(x)(T1g)(x), where T1 and T2 are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T+ or T - is bounded from Lp(ℝ) × Lq(ℝ d) to HL1(ℝd) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

KW - Bilinear operators

KW - Hardy spaces

KW - Riesz transforms

KW - Schrödinger operators

UR - http://www.scopus.com/inward/record.url?scp=80054025431&partnerID=8YFLogxK

U2 - 10.4064/sm205-3-4

DO - 10.4064/sm205-3-4

M3 - 期刊論文

AN - SCOPUS:80054025431

SN - 0039-3223

VL - 205

SP - 281

EP - 295

JO - Studia Mathematica

JF - Studia Mathematica

IS - 3

ER -

Bilinear operators associated with Schrödinger operators

Abstract

Keywords

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