Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein

Studia Mathematica, Tome 147 (2001), p. 37-50 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $\int_{A} M_{S} f = \infty$ .

Publié le : 2001-01-01

Zbl 0983.42012

EUDML-ID : urn:eudml:doc:284422

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     title = {Local integrability of strong and iterated maximal functions},
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     volume = {147},
     year = {2001},
     pages = {37-50},
     zbl = {0983.42012},
     language = {en},
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Paul Alton Hagelstein. Local integrability of strong and iterated maximal functions. Studia Mathematica, Tome 147 (2001) pp. 37-50. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm147-1-4/