Endpoint estimates for some maximal operators associated to the circular conical surface

Yaryong R. Heo

doi:10.1016/j.jmaa.2008.10.011

Endpoint estimates for some maximal operators associated to the circular conical surface

Yaryong R. Heo

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

1 Citation (Scopus)

Abstract

For ψ ∈ C₀^∞ (R^d) and m > 0 we consider the maximal operator given byM_m f (x, t) = under(sup, r > 0) | under(∫, R^d) f (x - y, t - | y |^m) frac(1, r^d) ψ (frac(y, r)) d y | . It is well known that M_m is a L^p-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that M_m is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M₁ maps the standard Hardy space H¹ to weak L¹ if d ≥ 4.

Original language	English
Pages (from-to)	152-162
Number of pages	11
Journal	Journal of Mathematical Analysis and Applications
Volume	351
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2008.10.011
Publication status	Published - 2009 Mar 1
Externally published	Yes

Bibliographical note

Funding Information:
This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-357-C00002. E-mail address: [email protected].

Keywords

Maximal functions
Singular integrals

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2008.10.011

Cite this

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title = "Endpoint estimates for some maximal operators associated to the circular conical surface",

abstract = "For ψ ∈ C0∞ (Rd) and m > 0 we consider the maximal operator given byMm f (x, t) = under(sup, r > 0) | under(∫, Rd) f (x - y, t - | y |m) frac(1, rd) ψ (frac(y, r)) d y | . It is well known that Mm is a Lp-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that Mm is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M1 maps the standard Hardy space H1 to weak L1 if d ≥ 4.",

keywords = "Maximal functions, Singular integrals",

author = "Heo, {Yaryong R.}",

note = "Funding Information: This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-357-C00002. E-mail address: [email protected].",

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doi = "10.1016/j.jmaa.2008.10.011",

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T1 - Endpoint estimates for some maximal operators associated to the circular conical surface

AU - Heo, Yaryong R.

N1 - Funding Information: This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-357-C00002. E-mail address: [email protected].

PY - 2009/3/1

Y1 - 2009/3/1

N2 - For ψ ∈ C0∞ (Rd) and m > 0 we consider the maximal operator given byMm f (x, t) = under(sup, r > 0) | under(∫, Rd) f (x - y, t - | y |m) frac(1, rd) ψ (frac(y, r)) d y | . It is well known that Mm is a Lp-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that Mm is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M1 maps the standard Hardy space H1 to weak L1 if d ≥ 4.

AB - For ψ ∈ C0∞ (Rd) and m > 0 we consider the maximal operator given byMm f (x, t) = under(sup, r > 0) | under(∫, Rd) f (x - y, t - | y |m) frac(1, rd) ψ (frac(y, r)) d y | . It is well known that Mm is a Lp-bounded operator for 1 < p ≤ ∞. Also A. Seeger and T. Tao proved that Mm is of weak-type L log log L if m ≠ 1. In this paper we consider the case m = 1 and prove M1 maps the standard Hardy space H1 to weak L1 if d ≥ 4.

KW - Maximal functions

KW - Singular integrals

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DO - 10.1016/j.jmaa.2008.10.011

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SN - 0022-247X

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EP - 162

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Endpoint estimates for some maximal operators associated to the circular conical surface

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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