Properties of the Maximal Functions Associated to Bases of Rectangles in ℝ³

Alexander Stokolos

doi:10.1017/S0013091506001180

Properties of the Maximal Functions Associated to Bases of Rectangles in ℝ³

Alexander Stokolos

Mathematical Sciences

DePaul University

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

6 Scopus citations

Abstract

<div class="line" id="line-151"> This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions ( t , 1/ t , s ) within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps L log² L into L ¹,∞. We give a simple geometric condition that guarantees that the L log² L class cannot be enlarged. The proof develops the author’s methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fe&fflig;erman and Pipher.</div>

Original language	American English
Journal	Proceedings of Edinburgh Mathematical Society
Volume	51
DOIs	https://doi.org/10.1017/S0013091506001180
State	Published - Jun 2008

Disciplines

Mathematics

Keywords

Differentiation bases
Maximal operators
Primary 42B25
Soria bases

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10.1017/S0013091506001180License: Unspecified

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title = "Properties of the Maximal Functions Associated to Bases of Rectangles in ℝ³",

abstract = " This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions ( t , 1/ t , s ) within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps L log² L into L ¹,∞. We give a simple geometric condition that guarantees that the L log² L class cannot be enlarged. The proof develops the author’s methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fe&fflig;erman and Pipher.",

keywords = "Differentiation bases, Maximal operators, Primary 42B25, Soria bases",

author = "Alexander Stokolos",

note = "PROPERTIES OF THE MAXIMAL OPERATORS ASSOCIATED WITH BASES OF RECTANGLES IN $\mathbb{R}^3$ - Volume 51 Issue 2 - Alexander Stokolos",

year = "2008",

month = jun,

doi = "10.1017/S0013091506001180",

language = "American English",

volume = "51",

journal = "Proceedings of Edinburgh Mathematical Society",

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T1 - Properties of the Maximal Functions Associated to Bases of Rectangles in ℝ³

AU - Stokolos, Alexander

N1 - PROPERTIES OF THE MAXIMAL OPERATORS ASSOCIATED WITH BASES OF RECTANGLES IN $\mathbb{R}^3$ - Volume 51 Issue 2 - Alexander Stokolos

PY - 2008/6

Y1 - 2008/6

N2 - This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions ( t , 1/ t , s ) within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps L log² L into L ¹,∞. We give a simple geometric condition that guarantees that the L log² L class cannot be enlarged. The proof develops the author’s methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fe&fflig;erman and Pipher.

AB - This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions ( t , 1/ t , s ) within a framework of more general Soria bases. The Jessen–Marcinkiewicz–Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps L log² L into L ¹,∞. We give a simple geometric condition that guarantees that the L log² L class cannot be enlarged. The proof develops the author’s methods applied previously in the two-dimensional case and is related to theorems of Córdoba, Soria and Fe&fflig;erman and Pipher.

KW - Differentiation bases

KW - Maximal operators

KW - Primary 42B25

KW - Soria bases

UR - https://doi.org/10.1017/S0013091506001180

U2 - 10.1017/S0013091506001180

DO - 10.1017/S0013091506001180

M3 - Article

VL - 51

JO - Proceedings of Edinburgh Mathematical Society

JF - Proceedings of Edinburgh Mathematical Society

ER -

Properties of the Maximal Functions Associated to Bases of Rectangles in ℝ³

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