Weak type estimates for cone type multipliers associated with a convex polygon

Sunggeum Hong; Joonil Kim; Chan Woo Yang

doi:10.1512/iumj.2007.56.2946

Weak type estimates for cone type multipliers associated with a convex polygon

Sunggeum Hong^*, Joonil Kim, Chan Woo Yang

^*Corresponding author for this work

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

Abstract

Let P be a convex polygon in ℝ² which contains the origin in its interior. Let p be the associated Minkowski functional defined by ρ(ξ) = inf{ε > 0: ε^-1 ξ ∈ P), ξ ≠ 0. We consider the family of convolution operators T^δ associated with cone type multipliers (1- ρ(ξ) ²/τ²)^δ₊, (ξ, τ) ∈ ℝ² × ℝ, and show that T^δ is of weak type (p, p) on H^p (ℝ³), 1/2 < p < 1 for the critical value 5 = 2 (1 / p - 1).

Original language	English
Pages (from-to)	1827-1870
Number of pages	44
Journal	Indiana University Mathematics Journal
Volume	56
Issue number	4
DOIs	https://doi.org/10.1512/iumj.2007.56.2946
Publication status	Published - 2007

Keywords

Cone type multipliers
Convex polygons
Hardy spaces
Minkowski functional

ASJC Scopus subject areas

General Mathematics

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10.1512/iumj.2007.56.2946

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title = "Weak type estimates for cone type multipliers associated with a convex polygon",

abstract = "Let P be a convex polygon in ℝ2 which contains the origin in its interior. Let p be the associated Minkowski functional defined by ρ(ξ) = inf\{ε > 0: ε-1 ξ ∈ P), ξ ≠ 0. We consider the family of convolution operators Tδ associated with cone type multipliers (1- ρ(ξ) 2/τ2)δ+, (ξ, τ) ∈ ℝ2 × ℝ, and show that Tδ is of weak type (p, p) on Hp (ℝ3), 1/2 < p < 1 for the critical value 5 = 2 (1 / p - 1).",

keywords = "Cone type multipliers, Convex polygons, Hardy spaces, Minkowski functional",

author = "Sunggeum Hong and Joonil Kim and Yang, \{Chan Woo\}",

year = "2007",

doi = "10.1512/iumj.2007.56.2946",

language = "English",

volume = "56",

pages = "1827--1870",

journal = "Indiana University Mathematics Journal",

issn = "0022-2518",

publisher = "Department of Mathematics, Indiana University",

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TY - JOUR

T1 - Weak type estimates for cone type multipliers associated with a convex polygon

AU - Hong, Sunggeum

AU - Kim, Joonil

AU - Yang, Chan Woo

PY - 2007

Y1 - 2007

N2 - Let P be a convex polygon in ℝ2 which contains the origin in its interior. Let p be the associated Minkowski functional defined by ρ(ξ) = inf{ε > 0: ε-1 ξ ∈ P), ξ ≠ 0. We consider the family of convolution operators Tδ associated with cone type multipliers (1- ρ(ξ) 2/τ2)δ+, (ξ, τ) ∈ ℝ2 × ℝ, and show that Tδ is of weak type (p, p) on Hp (ℝ3), 1/2 < p < 1 for the critical value 5 = 2 (1 / p - 1).

AB - Let P be a convex polygon in ℝ2 which contains the origin in its interior. Let p be the associated Minkowski functional defined by ρ(ξ) = inf{ε > 0: ε-1 ξ ∈ P), ξ ≠ 0. We consider the family of convolution operators Tδ associated with cone type multipliers (1- ρ(ξ) 2/τ2)δ+, (ξ, τ) ∈ ℝ2 × ℝ, and show that Tδ is of weak type (p, p) on Hp (ℝ3), 1/2 < p < 1 for the critical value 5 = 2 (1 / p - 1).

KW - Cone type multipliers

KW - Convex polygons

KW - Hardy spaces

KW - Minkowski functional

UR - https://www.scopus.com/pages/publications/35448962904

U2 - 10.1512/iumj.2007.56.2946

DO - 10.1512/iumj.2007.56.2946

M3 - Article

AN - SCOPUS:35448962904

SN - 0022-2518

VL - 56

SP - 1827

EP - 1870

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 4

ER -

Weak type estimates for cone type multipliers associated with a convex polygon

Abstract

Keywords

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