A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces. (English) Zbl 1478.42019
Summary: In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator (\(B\)-maximal operator) on \(L_{p(\cdot),\gamma}(\mathbb{R}_{k,+}^n)\) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the \(B\)-maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the \(B\)-maximal operator is not bounded on \(L_{p(\cdot),\gamma}(\mathbb{R}_{k,+}^n)\) variable exponent Lebesgue spaces in the case of \(p_-=1\). We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional \(B\)-maximal function) on \(L_{p(\cdot),\gamma}(\mathbb{R}_{k,+}^n)\) variable exponent Lebesgue spaces.
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42A50 | Conjugate functions, conjugate series, singular integrals |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
\(B\)-maximal operator; generalized translation operator; singular integrals; variable exponent Lebesgue spacesReferences:
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