New multiple weights and the Adams inequality on weighted Morrey spaces (Q2842088)

Let NEWLINE\[NEWLINE I_{\alpha,m}(\vec{f}\,)(x)=\int_{\mathbb R^{mn}}\frac{f_1(y_1)\cdots f_m(y_m)}{|(x-y_1,\dots, x-y_m)|^{mn-\alpha}}\,d\vec{y} NEWLINE\]NEWLINE be the multilinear fractional integral operator, \(0<\alpha<mn\), and let \(\tilde{A}_{\vec{P},\lambda}(\mathbb R^n)\) be the class of vector weights \(\vec{w}\) for which NEWLINE\[NEWLINE \sup_{B\subset\mathbb R^n}\bigg(\frac1{|B|}\int_B v_{\vec{w}}(x)\,dx\bigg)^{\lambda/p}\prod_{j=1}^m\bigg(\frac1{|B|}\int_B w_j(x)^{-p_j'\lambda/p_j}\,dx\bigg)^{1/p'_j}<\infty, NEWLINE\]NEWLINE where \(v_{\vec{w}}(x)=w_1(x)^{p/p_1}\cdots w_m(x)^{p/p_m}\), \(1<p_j<\infty\), \(0<\lambda<1\), \(\vec{P}=(p_1,\dots,p_m)\), and \(1/p=1/p_1+\cdots+1/p_m\).NEWLINENEWLINEThe main result is the extension of \textit{D. R.~Adams}'s inequality [Duke Math. J. 42, No. 4, 765--778 (1975; Zbl 0336.46038)] to the weighted Morrey spaces:NEWLINENEWLINEIf \(0<\lambda<1-\alpha/(mn)\), \(1/q=1/p-\alpha/(n(1-\lambda))>0\), and \(w\in \tilde{A}_{\vec{P},\lambda}(\mathbb R^n)\), then NEWLINE\[NEWLINE \Big\| I_{\alpha,m}(\vec{f\,})\Big\|_{L^{q,\lambda}(v_{\vec{w}}^\lambda,v_{\vec{w}})}\leq C\prod_{j=1}^m\| f_j\|_{L^{p_j,\lambda}(w_j^\lambda,v_{\vec{w}})}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \| f\|_{L^{p,\lambda}(u,v)}=\sup_{B\subset\mathbb R^n}\bigg(\frac1{v(B)^\lambda}\int_B |f(x)|^pu(x)\,dx\bigg)^{1/p}. NEWLINE\]NEWLINE Similar estimates are also proved in \(\text{BMO}(\mathbb R^n)\) and \(\text{Lip}_\varepsilon(\mathbb R^n)\).