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April 16, 2026Open Access

Weighted BMO-BLO estimates for Littlewood–Paley square operators

Key Points

The research aims to establish the boundedness of Littlewood–Paley square operators on weighted spaces.
Introduced and examined the new weighted space BLO.
Proved that if T(f)(x0) is finite at a point, then it is finite almost everywhere.
Showed boundedness of T(f) from BMO(ω) to BLO(ω) for ω in A1.
Discussed the John–Nirenberg inequality for the space BLO(ω).
Provided an equivalent characterization of the space BLO(ω).
T(f)(x) is finite almost everywhere in ℝⁿ given finiteness at one point.
Boundedness from BMO(ω) to BLO(ω) was established.
The corresponding John–Nirenberg inequality applicable to BLO(ω) was formulated.
An equivalent characterization of BLO(ω) was offered.

Abstract

Let T (f) denote the Littlewood–Paley square operators, including the Littlewood–Paley G-function G (f), Lusin’s area integral S (f) and Stein’s function G^_ (f) with >2. We establish the boundedness of Littlewood–Paley square operators on the weighted spaces BMO () with A₁. The weighted space BLO () (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of BMO (). It is proved that if T (f) (x₀) is finite for a single point x₀ R^n, then T (f) (x) is finite almost everywhere in R^n. Moreover, it is shown that T (f) is bounded from BMO () into BLO (), provided that A₁. The corresponding John–Nirenberg inequality suitable for the space BLO () with A₁ is discussed. Based on this, the equivalent characterization of the space BLO () is also given.

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