On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

Bae Jun Park

doi:10.1007/s11118-020-09877-x

On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

Bae Jun Park

Korea Institute for Advanced Study

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

3 Scopus citations

Abstract

We study a multilinear version of the Hörmander multiplier theorem, namely∥Tσ(f1,…,fn)∥Lp≲supk∈ℤ∥σ(2k⋅,…,2k⋅)ϕ(n)̂∥L(s1,…,sn)2∥f1∥Hp1⋯∥fn∥Hpn.We show that the estimate does not hold in the limiting case min (s₁, … , s_n) = d/ 2 or ∑ _k_∈_J(s_k/ d− 1 / p_k) = − 1 / 2 for some J⊂ { 1 , … , n}. This provides the necessary and sufficient condition on (s₁, … , s_n) for the boundedness of T_σ.

Original language	English
Pages (from-to)	87-96
Number of pages	10
Journal	Potential Analysis
Volume	56
Issue number	1
DOIs	https://doi.org/10.1007/s11118-020-09877-x
State	Published - Jan 2022
Externally published	Yes

Keywords

Hörmander multiplier theorem
Multilinear operator
Sobolev space with sharp regularity conditions

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10.1007/s11118-020-09877-x

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title = "On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions",

abstract = "We study a multilinear version of the H{\"o}rmander multiplier theorem, namely∥Tσ(f1,…,fn)∥Lp≲supk∈ℤ∥σ(2k⋅,…,2k⋅)ϕ(n{\^)}∥L(s1,…,sn)2∥f1∥Hp1⋯∥fn∥Hpn.We show that the estimate does not hold in the limiting case min (s1, … , sn) = d/ 2 or ∑ k∈J(sk/ d− 1 / pk) = − 1 / 2 for some J⊂ \{ 1 , … , n\}. This provides the necessary and sufficient condition on (s1, … , sn) for the boundedness of Tσ.",

keywords = "H{\"o}rmander multiplier theorem, Multilinear operator, Sobolev space with sharp regularity conditions",

author = "Park, \{Bae Jun\}",

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year = "2022",

month = jan,

doi = "10.1007/s11118-020-09877-x",

language = "English",

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pages = "87--96",

journal = "Potential Analysis",

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T1 - On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

AU - Park, Bae Jun

PY - 2022/1

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N2 - We study a multilinear version of the Hörmander multiplier theorem, namely∥Tσ(f1,…,fn)∥Lp≲supk∈ℤ∥σ(2k⋅,…,2k⋅)ϕ(n)̂∥L(s1,…,sn)2∥f1∥Hp1⋯∥fn∥Hpn.We show that the estimate does not hold in the limiting case min (s1, … , sn) = d/ 2 or ∑ k∈J(sk/ d− 1 / pk) = − 1 / 2 for some J⊂ { 1 , … , n}. This provides the necessary and sufficient condition on (s1, … , sn) for the boundedness of Tσ.

AB - We study a multilinear version of the Hörmander multiplier theorem, namely∥Tσ(f1,…,fn)∥Lp≲supk∈ℤ∥σ(2k⋅,…,2k⋅)ϕ(n)̂∥L(s1,…,sn)2∥f1∥Hp1⋯∥fn∥Hpn.We show that the estimate does not hold in the limiting case min (s1, … , sn) = d/ 2 or ∑ k∈J(sk/ d− 1 / pk) = − 1 / 2 for some J⊂ { 1 , … , n}. This provides the necessary and sufficient condition on (s1, … , sn) for the boundedness of Tσ.

KW - Hörmander multiplier theorem

KW - Multilinear operator

KW - Sobolev space with sharp regularity conditions

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DO - 10.1007/s11118-020-09877-x

M3 - Article

AN - SCOPUS:85091045277

SN - 0926-2601

VL - 56

SP - 87

EP - 96

JO - Potential Analysis

JF - Potential Analysis

IS - 1

ER -

On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

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