Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas; Ferenc Weisz

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 4, page 1079-1101
  • ISSN: 0011-4642

Abstract

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The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p ( d ) (in the case p > 1 ), but (in the case when 1 / p ( · ) is log-Hölder continuous and p - = inf { p ( x ) : x d } > 1 ) on the variable Lebesgue spaces L p ( · ) ( d ) , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type ( 1 , 1 ) . In the present note we generalize Besicovitch’s covering theorem for the so-called γ -rectangles. We introduce a general maximal operator M s γ , δ and with the help of generalized Φ -functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1 / p ( · ) is log-Hölder continuous and p - > s , where 1 s is arbitrary (or p - s ), then the maximal operator M s γ , δ is bounded on the space L p ( · ) ( d ) (or the maximal operator is of weak-type ( p ( · ) , p ( · ) ) ).

How to cite

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Szarvas, Kristóf, and Weisz, Ferenc. "Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces." Czechoslovak Mathematical Journal 66.4 (2016): 1079-1101. <http://eudml.org/doc/287548>.

@article{Szarvas2016,
abstract = {The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_\{p\}(\mathbb \{R\}^d)$ (in the case $p >1$), but (in the case when $1/p(\cdot )$ is log-Hölder continuous and $p_\{-\} = \inf \lbrace p(x) \colon x \in \mathbb \{R\}^d \rbrace > 1$) on the variable Lebesgue spaces $L_\{p(\cdot )\}(\mathbb \{R\}^d)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma $-rectangles. We introduce a general maximal operator $M_\{s\}^\{\gamma ,\delta \}$ and with the help of generalized $\Phi $-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function $1/p(\cdot )$ is log-Hölder continuous and $p_\{-\} > s$, where $1 \le s \le \infty $ is arbitrary (or $p_\{-\} \ge s$), then the maximal operator $M_\{s\}^\{\gamma ,\delta \}$ is bounded on the space $L_\{p(\cdot )\}(\mathbb \{R\}^d)$ (or the maximal operator is of weak-type $(p(\cdot ),p(\cdot ))$).},
author = {Szarvas, Kristóf, Weisz, Ferenc},
journal = {Czechoslovak Mathematical Journal},
keywords = {variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch’s covering theorem; weak-type inequality; strong-type inequality; variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality},
language = {eng},
number = {4},
pages = {1079-1101},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces},
url = {http://eudml.org/doc/287548},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Szarvas, Kristóf
AU - Weisz, Ferenc
TI - Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 1079
EP - 1101
AB - The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_{p}(\mathbb {R}^d)$ (in the case $p >1$), but (in the case when $1/p(\cdot )$ is log-Hölder continuous and $p_{-} = \inf \lbrace p(x) \colon x \in \mathbb {R}^d \rbrace > 1$) on the variable Lebesgue spaces $L_{p(\cdot )}(\mathbb {R}^d)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma $-rectangles. We introduce a general maximal operator $M_{s}^{\gamma ,\delta }$ and with the help of generalized $\Phi $-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function $1/p(\cdot )$ is log-Hölder continuous and $p_{-} > s$, where $1 \le s \le \infty $ is arbitrary (or $p_{-} \ge s$), then the maximal operator $M_{s}^{\gamma ,\delta }$ is bounded on the space $L_{p(\cdot )}(\mathbb {R}^d)$ (or the maximal operator is of weak-type $(p(\cdot ),p(\cdot ))$).
LA - eng
KW - variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch’s covering theorem; weak-type inequality; strong-type inequality; variable Lebesgue space; maximal operator; $\gamma $-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality
UR - http://eudml.org/doc/287548
ER -

References

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