We point out that if the Hardy–Littlewood maximal operator is bounded on the space L p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L p(ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L p(t) (ℝn), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝn, if and only if p(t) = const.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.
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Kopaliani, T.S. Littlewood–Paley theorem on spaces L p(t)(ℝn). Ukr Math J 60, 2006–2014 (2008). https://doi.org/10.1007/s11253-009-0186-0
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DOI: https://doi.org/10.1007/s11253-009-0186-0