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Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces. (English) Zbl 1418.42037

Summary: We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space \(H_p\) to the classical Lebesgue space \(L_p\) and from the variable dyadic martingale Hardy space \(H_{p(\cdot)}\) to the variable Lebesgue space \(L_{p(\cdot)}\). Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from \(H_{p(\cdot)}\) to \(L_{p(\cdot)}\) and from the variable Hardy-Lorentz space \(H_{p(\cdot),q}\) to the variable Lorentz space \(L_{p(\cdot),q}\). As a consequence, we can prove theorems about almost everywhere and norm convergence.

MSC:

42B30 \(H^p\)-spaces
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
60G42 Martingales with discrete parameter
60G46 Martingales and classical analysis
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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