Some multi-sublinear operators on generalized Morrey spaces with non-doubling measures (Q2914417)

Let \(\mu\) be a Radon measure on \(\mathbb R^d\) which satisfies the condition \(\mu(Q)\leq c\, [\ell(Q)]^n \), \(0<n\leq d\), for any cube \(Q\subset \mathbb R^d\) with the side length \(\ell(Q)>0\).NEWLINENEWLINESuppose that \(k>0\), \(1\leq p <\infty\) and that the function \(\Phi : \mathbb R^+ \mapsto \mathbb R^+\) is increasing. The generalized Morrey space \(\mathcal L^{p,\Phi}(\mu)\) consists of those \(f\) in \(L^p_{loc}(\mu)\) for which the norm NEWLINE\[NEWLINE\|f\|_{L^p_{loc}(\mu)}=\sup_{Q\in \mathcal D(\mu)} \Big(\frac{1}{\Phi(\mu(kQ))}\,\int_Q\,|f|^p\,d\mu\Big)^{1/p}NEWLINE\]NEWLINE is finite; here \(\mathcal D(\mu)\) stands for the family of all doubling cubes with positive \(\mu\)-measure. Note that equivalent norms result from different choices of \(k>0\).NEWLINENEWLINEGiven \(m\in\mathbb N\) and an \(m-\)tuple \((f_1,\dots,f_m)\), let \(\mathcal T\) be a multi-sublinear operator satisfying the size condition NEWLINE\[NEWLINE|\mathcal T(f_1,\dots,f_m)(x)|\leq C\,\int_{(\mathbb R^d)^m} \frac{|f_1(y_1)\cdots f_m(y_m)|}{(|x-y_1|+\cdots + |x-y_m|)^{mn-\alpha}}\,\prod_{i=1}^m d\mu(y_i)NEWLINE\]NEWLINE with \(0\leq\alpha<mn\). The particular cases of the operator \(\mathcal T\) are:NEWLINENEWLINE1) the multilinear fractional integral operator \(I_{\alpha,m}\), \(0<\alpha<mn\), defined by NEWLINE\[NEWLINE I_{\alpha,m}(f_1,\dots,f_m)(x):=\int_{(\mathbb R^d)^m} \frac{f_1(y_1)\cdots f_m(y_m)}{(|x-y_1|+\cdots + |x-y_m|)^{mn-\alpha}}\,\prod_{i=1}^m d\mu(y_i);NEWLINE\]NEWLINENEWLINENEWLINE2) the multilinear Calderón-Zygmund operator given by NEWLINE\[NEWLINET(f_1,\dots,f_m)(x):=\int_{(\mathbb R^d)^m}\,\mathcal K(x,y_1,\dots,y_m)\,f_1(y_1)\cdots f_m(y_m)\,\prod_{i=1}^m d\mu(y_i)NEWLINE\]NEWLINE for functions \(f_i\) with compact supports and \(x\notin\bigcap_{i=1}^m \text{supp} f_i\), where the m-Calderón-Zygmund kernel \(\mathcal K(f_1,\dots,f_m)(x)\) is defined away from the diagonal \(x=y_1=\cdots=y_m\) in \((\mathbb R^d)^{m+1}\) and satisfies the size condition NEWLINE\[NEWLINE|\mathcal K(x,y_1,\dots,y_m)|\leq C/ \Big(\sum_{i=1}^m\,|x-y_i|\Big)^{mn}NEWLINE\]NEWLINE and some smoothness condition;NEWLINENEWLINE3) the multi-sublinear maximal operator NEWLINE\[NEWLINE\mathcal M_{\kappa}(f_1,\dots,f_m)(x):=\sup_{x\in Q\in \mathcal D(\mu)}\,\prod_{i=1}^m \frac{1}{\mu(\kappa Q)}\,\int_Q\,|f_i(y_i)|\,d\mu(y_i)\,,\quad \kappa>1.NEWLINE\]NEWLINENEWLINENEWLINEThe aim of the paper is to prove the boundedness of the operator \(\mathcal T\) from the product of the generalized Morrey spaces \(\prod_{i=1}^m \mathcal L^{p_i,\Phi_i}(\mu)\) to another generalized Morrey space \(\mathcal L^{p,\Phi}(\mu)\). One of the main results of the paper involving the multi-sublinear operator \(\mathcal T\) mentioned above reads as follows:NEWLINENEWLINELet \(0<\alpha<mn\), \(1<p_i<mn/\alpha\) (\(i=1,\dots,m\)), \(1/p=1/p_1+\cdots+1/p_m-\alpha/n\), \(\Phi^{1/p}=\Phi_1^{1/p_1}\cdots \Phi_m^{1/p_m}\), where each increasing function \(\Phi_i:\mathbb R^+\mapsto \mathbb R^+\) (\(i=1,\dots,m\)) satisfies the conditions NEWLINE\[NEWLINE\Phi_i(t)/t\leq C\, \Phi_i(s)/s \quad \text{for} \quad t>sNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sup_{0<r<\infty}\frac{r^{1-\alpha p_i/mn}}{\Phi_i(r)}\,\int_r^\infty\,\frac{\Phi_i(t)}{t^{1-\alpha p_i/mn}}\,\frac{dt}{t}<\infty\,.NEWLINE\]NEWLINE If the operator \(\mathcal T : \prod_{i=1}^m L^{p_i}(\mu)\mapsto L^{p}(\mu)\) is bounded, then the operator \(\mathcal T : \prod_{i=1}^m \mathcal L^{p_i,\Phi_i}(\mu)\mapsto L^{p,\Phi}(\mu)\) is bounded as well.NEWLINENEWLINENEWLINENEWLINE The authors also prove variants of the mentioned result with \(\alpha=0\) involving either the multilinear Calderón-Zygmund operator or the multi-sublinear maximal operator. They obtain as corollaries the corresponding results for the case when generalized Morrey spaces are replaced by Morrey spaces.