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On the domain and range of the maximal operator. (English) Zbl 1039.42015

The authors consider the centered maximal operator given by \[ Mf(x)= \sup_{R> 0}\, {1\over| B_R(x)|} \int_{B_R(x)} | f(y)|\, dy,\quad x\in \mathbb{R}^n\;(f\in L^1_{\text{loc}} (\mathbb{R}^n)), \] where \(B_R(x)\) represents the open ball centered at \(x\) with radius \(R\).
They first investigate the so-called domain of the maximal operator denoted by \(\mathbb{D}\) and constituted by all functions \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\) for which \(Mf\not\equiv\infty\).
A characterization of \(\mathbb{D}\) is established and some examples are given to show that if \(0< r< 1\), then it has \(L^1_{\text{loc}}(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) \not\subset \mathbb{D}\not\subset L^1(\mathbb{R}^n)+ L^\infty(\mathbb{R}^n)\).
The range of the maximal operator is also analyzed. The authors prove some properties of the maximal operator that involves logarithmic Lebesgue spaces and the grand \(L^p\) spaces. They also give some characterizations of the functions \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\) for which \(Mf\in L^1(\mathbb{R}^n)+ L^\infty(\mathbb{R}^n)\).
Finally, it is considered the local maximal operator \(M_\Omega\) on an open bounded set \(\Omega\subset \mathbb{R}^n\) defined by \[ M_\Omega f(x)= \sup_{Q\ni x}\, {1\over | Q|} \int_Q| f(y)\,dy,\quad x\in \Omega, \] where \(Q\subset \Omega\) are cubes whose edges are parallel to coordinate axes. The authors give a characterization of the domain of \(M_\Omega\) and also analyze its range.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

[1] American Mathematical Society 47 (1984)
[2] Lecture Notes in Math 1150 (1985)
[3] Publications de l’Institut de Mathématique de l’Université de Strasbourg X (1957)
[4] Indiana Univ. Math. J 44 pp 305– (1995)
[5] Lecture Notes in Math 481 (1975)
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