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Keywords:
Littlewood-Paley square operator; weighted Morrey-Campanato space; $A_p$ weight, John-Nirenberg-type inequality
Littlewood-Paley square operator; weighted Morrey-Campanato space; $A_p$ weight, John-Nirenberg-type inequality
Summary:
Let $1\leq p<\infty $ and $-n/p<\alpha \leq 1$. A distinguished subset $\mathcal {C}^{\alpha ,p}_{\ast }(\omega )$ of the weighted Morrey-Campanato space $\mathcal {C}^{\alpha ,p}(\omega )$ on $\mathbb R^n$ is introduced and studied. This new class is a proper subset of $\mathcal {C}^{\alpha ,p}(\omega )$. We establish John-Nirenberg-type inequalities suitable for the Morrey-Campanato spaces $\mathcal {C}^{\alpha ,p}(\omega )$ and $\mathcal {C}^{\alpha ,p}_{\ast }(\omega )$ with $\omega \in A_1$. Based on this result, some new equivalent characterizations of the Morrey-Campanato spaces $\mathcal {C}^{\alpha ,p}(\omega )$ and $\mathcal {C}^{\alpha ,p}_{\ast }(\omega )$ are also given. Let $T(f)$ denote the Littlewood-Paley square operators, including the Littlewood-Paley $g$-function $\mathcal {G}_{\psi }(f)$, Lusin's area integral $\mathcal {S}_{\psi }(f)$ and Stein's function $\mathcal {G}^{\ast }_{\lambda ,\psi }(f)$ with $\lambda >2$. Here $\psi $ is a Littlewood-Paley function on $\mathbb R^n$. We establish the boundedness of Littlewood-Paley square operators on weighted Morrey-Campanato spaces. 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 $\mathcal {C}^{\alpha ,p}(\omega )$ into $\mathcal {C}^{\alpha ,p}_{\ast }(\omega )$ for $1\leq p<\infty $ and $0<\alpha \leq 1$, provided that $\omega \in A_1$.
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