On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ

Alexander M. Stokolos

On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ

Mathematical Sciences

Research output: Contribution to conference › Presentation

Abstract

<div class="line" id="line-5"> A weak L(log+ L)n−1 type inequality |{Mf(x) > λ}| ≤ C ∫ |f|λ (1 + log+ |f|λ)n-1 is a quantitative version of Jessen-Marcinkiewich-Zygmund theorem. It was established by Miguel de Guzman in 1972. It is easy to see that this is the best possible estimate.</div><div class="line" id="line-15"> <br/></div><div class="line" id="line-18"> By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given in&filig;nite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log+ L)n−1. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rn for any n > 1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend some results known for R2 only.</div>

Original language	American English
State	Published - May 2004
Event	University of Missouri ShowMe Analysis Meeting - Columbia, MO Duration: May 1 2004 → …

Conference

Conference	University of Missouri ShowMe Analysis Meeting
Period	05/1/04 → …

Keywords

Rare Maximal Functions
Weak Type Inequalities

DC Disciplines

Mathematics

Cite this

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title = "On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ",

abstract = " A weak L(log+ L)n−1 type inequality |{Mf(x) > λ}| ≤ C ∫ |f|λ (1 + log+ |f|λ)n-1 is a quantitative version of Jessen-Marcinkiewich-Zygmund theorem. It was established by Miguel de Guzman in 1972. It is easy to see that this is the best possible estimate. By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given in&filig;nite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log+ L)n−1. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rn for any n > 1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend some results known for R2 only.",

keywords = "Rare Maximal Functions, Weak Type Inequalities",

author = "Stokolos, {Alexander M.}",

year = "2004",

month = may,

language = "American English",

note = "University of Missouri ShowMe Analysis Meeting ; Conference date: 01-05-2004",

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TY - CONF

T1 - On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ

AU - Stokolos, Alexander M.

PY - 2004/5

Y1 - 2004/5

N2 - A weak L(log+ L)n−1 type inequality |{Mf(x) > λ}| ≤ C ∫ |f|λ (1 + log+ |f|λ)n-1 is a quantitative version of Jessen-Marcinkiewich-Zygmund theorem. It was established by Miguel de Guzman in 1972. It is easy to see that this is the best possible estimate. By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given in&filig;nite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log+ L)n−1. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rn for any n > 1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend some results known for R2 only.

AB - A weak L(log+ L)n−1 type inequality |{Mf(x) > λ}| ≤ C ∫ |f|λ (1 + log+ |f|λ)n-1 is a quantitative version of Jessen-Marcinkiewich-Zygmund theorem. It was established by Miguel de Guzman in 1972. It is easy to see that this is the best possible estimate. By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given in&filig;nite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log+ L)n−1. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rn for any n > 1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend some results known for R2 only.

KW - Rare Maximal Functions

KW - Weak Type Inequalities

M3 - Presentation

T2 - University of Missouri ShowMe Analysis Meeting

Y2 - 1 May 2004

ER -

On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ

Abstract

Conference

Keywords

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