Littlewood–Paley theorem on spaces L p(t)(ℝ n )

Abstract

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) (ℝ n ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ n , if and only if p(t) = const.