Abstract
In this paper, we will prove that, for $1< p<\infty$ , the $L^{p}$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the norm of the centered Hardy-Littlewood maximal operator for all $0<\gamma<\infty$ . When $p=1$ , we also find that the weak $(1,1)$ norm of the truncated centered Hardy-Littlewood maximal operator $M^{c}_{\gamma}$ equals the weak $(1,1)$ norm of the centered Hardy-Littlewood maximal operator for $0<\gamma<\infty$ . Moreover, the same is true for the truncated uncentered Hardy-Littlewood maximal operator. Finally, we investigate the properties of the iterated Hardy-Littlewood maximal function.
Highlights
Define the centered Hardy-Littlewood maximal function by Mcf (x) = sup r> |B(x, r)| f (y) dy, B(x,r) ( . )and the uncentered Hardy-Littlewood maximal function by Mf sup Bx |B|
It is well known that the Hardy-Littlewood maximal function plays an important role in many parts of analysis
Define the iterated Hardy-Littlewood maximal function denoted by Mk+ as follows: Mk+ f (x) := M Mkf (x), ( . )
Summary
The basic real-variable construct was introduced by Hardy and Littlewood [ ] for n = , and by Wiener [ ] for n ≥. It is well known that the Hardy-Littlewood maximal function plays an important role in many parts of analysis. It is a classical mean operator, and it is frequently used to majorize other important operators in harmonic analysis
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