Estimates for the product weighted Hardy-Littlewood average and its commutator on product central Morrey spaces (Q2239054)

The author considers the study of the boundedness of the product weighted Hardy-Littlewood average operator \[ \mathcal{H}_{\varphi} f\left(x_{1}, x_{2}\right)=\int_{0}^{1} \int_{0}^{1} f\left(x_{1} t_{1}, x_{2} t_{2}\right) \varphi\left(t_{1}, t_{2}\right) \mathrm{d} t_{1} \mathrm{~d} t_{2} \] on the product central Morrey spaces \(\overrightarrow{\dot{B}^{p, \lambda}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right)\), where \[ \|f\|_{\overrightarrow{\dot{B}^{p, \lambda}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right)}=\sup _{r, s>0}\left(\frac{1}{|R(0, r, s)|^{1+p \lambda}} \int_{R(0, r, s)}|f(z)|^{p} \mathrm{~d} z\right)^{\frac{1}{p}}<\infty \] and \(R(z,r,s) = B(x_1,r)\times B(x_2,s)\), with \(z=(x_1,x_2)\). The main result reads as follows: Let \(1<p<\infty\) and \(-\frac{1}{p} \leq \lambda<0\). Then \(\mathcal{H}_{\varphi}\) is bounded on \(\overrightarrow{\dot{B}^{p, \lambda}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right)\) if and only if \[ C_{p, \lambda}=\int_{0}^{1} \int_{0}^{1} t_{1}^{n \lambda} t_{2}^{m \lambda} \varphi\left(t_{1}, t_{2}\right) d t_{1} d t_{2}<\infty . \] Moreover, \(\left\|\mathcal{H}_{\varphi}\right\|_{\overrightarrow{\dot{B}^{p, \lambda}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right) \rightarrow \overrightarrow{\dot{B}^{p, \lambda}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right)}=C_{p, \lambda} .\) Some previous results can be found in [\textit {Z.W. Fu} et al., Forum Math. 27, No. 5, 2825--2851 (2015; Zbl 1331.42016)]. A similar result also follows for the commutator \( \mathcal{H}_{\varphi}^b f=b\mathcal{H}_{\varphi} f-\mathcal{H}_{\varphi} (bf),\) where \(b\in \overrightarrow{\dot{CMO}^{q}}\left(\mathbb{R}^{n} \times \mathbb{R}^{m}\right)\), the central bounded mean oscillation space.