Weighted BMO-BLO estimates for Littlewood--Paley square operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910837956935680 |
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| author | Wang, Hua |
| author_facet | Wang, Hua |
| contents | Let $T(f)$ denote the Littlewood--Paley square operators, including the Littlewood--Paley $\mathcal{G}$-function $\mathcal{G}(f)$, Lusin's area integral $\mathcal{S}(f)$ and Stein's function $\mathcal{G}^{\ast}_λ(f)$ with $λ>2$. We establish the boundedness of Littlewood--Paley square operators on the weighted spaces $\mathrm{BMO}(ω)$ with $ω\in A_1$. The weighted space $\mathrm{BLO}(ω)$ (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of $\mathrm{BMO}(ω)$. It is proved that if $T(f)(x_0)$ is finite for a single point $x_0\in\mathbb R^n$, then $T(f)(x)$ is finite almost everywhere in $\mathbb R^n$. Moreover, it is shown that $T(f)$ is bounded from $\mathrm{BMO}(ω)$ into $\mathrm{BLO}(ω)$, provided that $ω\in A_1$. The corresponding John--Nirenberg inequality suitable for the space $\mathrm{BLO}(ω)$ with $ω\in A_1$ is discussed. Based on this, the equivalent characterization of the space $\mathrm{BLO}(ω)$ is also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15125 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted BMO-BLO estimates for Littlewood--Paley square operators Wang, Hua Classical Analysis and ODEs Functional Analysis 42B20, 42B25, 42B35 Let $T(f)$ denote the Littlewood--Paley square operators, including the Littlewood--Paley $\mathcal{G}$-function $\mathcal{G}(f)$, Lusin's area integral $\mathcal{S}(f)$ and Stein's function $\mathcal{G}^{\ast}_λ(f)$ with $λ>2$. We establish the boundedness of Littlewood--Paley square operators on the weighted spaces $\mathrm{BMO}(ω)$ with $ω\in A_1$. The weighted space $\mathrm{BLO}(ω)$ (the space of functions with bounded lower oscillation) is introduced and studied in this paper. This new space is a proper subspace of $\mathrm{BMO}(ω)$. It is proved that if $T(f)(x_0)$ is finite for a single point $x_0\in\mathbb R^n$, then $T(f)(x)$ is finite almost everywhere in $\mathbb R^n$. Moreover, it is shown that $T(f)$ is bounded from $\mathrm{BMO}(ω)$ into $\mathrm{BLO}(ω)$, provided that $ω\in A_1$. The corresponding John--Nirenberg inequality suitable for the space $\mathrm{BLO}(ω)$ with $ω\in A_1$ is discussed. Based on this, the equivalent characterization of the space $\mathrm{BLO}(ω)$ is also given. |
| title | Weighted BMO-BLO estimates for Littlewood--Paley square operators |
| topic | Classical Analysis and ODEs Functional Analysis 42B20, 42B25, 42B35 |
| url | https://arxiv.org/abs/2502.15125 |