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Currently displaying 1 – 4 of 4

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Orlicz bounds for operators of restricted weak type

Paul Alton Hagelstein — 2005

Colloquium Mathematicae

It is shown that if T is a sublinear translation invariant operator of restricted weak type (1,1) acting on L¹(𝕋), then T maps simple functions in L log L(𝕋) boundedly into L¹(𝕋).

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein — 2001

Studia Mathematica

Let $M_{S}$ denote the strong maximal operator. Let $M_{x}$ and $M_{y}$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that $\int_{Q} M_{y} M_{x} h < \infty$ but $\int_{Q} M_{x} M_{y} h = \infty$ . It is shown that if f is a function supported on Q such that $\int_{Q} M_{y} M_{x} f < \infty$ but $\int_{Q} M_{x} M_{y} f = \infty$ , then there exists a set A of finite measure in ℝ² such that $\int_{A} M_{S} f = \infty$ .

A note on rare maximal functions

Paul Alton Hagelstein — 2003

Colloquium Mathematicae

A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).

On the uniqueness of the uncentered ergodic maximal function

Paul Alton Hagelstein — 2004

Fundamenta Mathematicae

It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.

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