New weighted norm inequalities for certain classes of multilinear operators on Morrey-type spaces
In this paper, the authors investigate the boundedness of some multilinear operators on Morrey-type spaces. To be precise, let \(T\) be a multilinear operator defined on the \(m\)-fold product of Schwartz spaces and take values into the space of tempered distributions: \[T:\, \mathcal S(\mathbb R^n)\times\cdots\times \mathcal S(\mathbb R^n)\to \mathcal S^\prime(\mathbb R^n).\] Moreover, assume that \(T\) satisfies the following conditions. \par (i) There exists a function \(K\) defined off the diagonal \(x=y_1=\cdots=y_m\) in \((\mathbb R^n)^{m+1}\) such that for all \(x\notin \cap_{j=1}^m \mathrm{supp }f_j\), \[T(f_1, \ldots, f_m)(x)=\int_{(\mathbb R^n)^m} K(x, y_1,\dots, y_m)f_1(y_1)\cdots f_m(y_m)dy_1\dots dy_m.\] \par (ii) For any \(N\ge 0\), there exists a positive constant \(C\) such that \[|K(y_0, y_1, \dots, y_m)|\le \frac{C}{(\sum_{k,\,l=0}^m|y_k-y_l|)^{mn}(1+\sum_{k,\,l=0}^m|y_k-y_l|)^N}.\] \par (iii) For some \(\epsilon>0\) and any \(N\ge 0\), there exists a positive constant \(C\) such that \begin{align*} &|K(y_0, y_1, \dots,y_j,\dots, y_m)-K(y_0, y_1, \dots,y^\prime_j,\dots, y_m)|\\ &\quad\le \frac{C|y_j-y^\prime_j|^\epsilon}{(\sum_{k,\,l=0}^m|y_k-y_l|)^{mn+\epsilon}(1+\sum_{k,\,l=0}^m|y_k-y_l|)^N} \end{align*} provided that \(0\le j\le m\) and \(|y_j-y^\prime_j|\le \frac12\max_{0\le k\le m}|y_j-y_k|\). \par (iv) There exist \(1\le q_1,\ldots, q_m<\infty\) and \(\frac1q=\frac1{q_1}+\cdots+\frac1{q_m}\) such that \(T\) is bounded from \(L^{q_1}\times\cdots\times L^{q_m}\to L^q\). On the other hand, for any cube \(Q\) and \(\theta\ge0\), write \(\varphi(Q)=(1+r)\) and \(\varphi_\theta(Q)=(1+r)^\theta\), where \(r\) is the side-length of \(Q\). Let \(\vec p=(p_1,\ldots, p_m)\) such that \(1/p=1/{p_1}+\cdots+ 1/p_m\) with \(1\le p_1,\ldots, p_m<\infty\). Given \(\vec w=(w_1,\ldots, w_m)\) with each \(w_j\) being non-negative measurable function, let \[v_{\vec w}=\prod_{j=1}^mw_j^{p/p_j}.\] Write \(\vec w\in A^\theta_{\vec p}\), if \[\sup_Q\left(\frac1{\varphi_\theta(Q)|Q|}\int_Q v_{\vec w}(x)dx\right)^{1/p}\prod_{j=1}^m\left(\frac1{\varphi_\theta(Q)|Q|}\int_Qw_j(x)^{1-p^\prime_j}dx\right)^{1/{p^\prime_j}}<\infty,\] where the supremum is taken over all cubes \(Q\subset \mathbb R^n\), and the term \(\left(\frac1{|Q|}\int_Qw_j(x)^{1-p^\prime_j}dx\right)^{1/{p^\prime_j}}\) is understood as \((\mathrm{ ess\, inf}_Q w_j)^{-1}\) when \(p_j=1\). For \(1\le p_1,\ldots, p_m<\infty\), let \(A^\infty_{\vec p}=\cup_{\theta\ge 0}A^\theta_{\vec p}\). In this paper, the authors show that for \(\alpha\in (-\infty, \infty)\), \(\lambda\in [0, 1)\), \(p<q\le \infty\), \(p_1,\dots, p_m\in [1, \infty)\) with \(1/p=\sum_{j=1}^m 1/{p_i}\) and \(\vec w=(w_1,\ldots, w_m)\in A^\infty_{\vec p}\), if \(p_i>1\), \(i=1,\ldots, m\), there exists a positive constant \(C\) such that \[\|T(\vec f)\|_{M^{p,\,q}_{\alpha,\,\lambda}(v_{\vec w},w)}\le C \prod_{j=1}^m\|f_i\|_{M^{p_i,\,\tilde q}_{\tilde\alpha,\,\tilde\lambda}(w_i)},\] where \(\tilde q=qp_i/p\), \(\tilde \lambda=p\lambda/p\), \(\tilde\alpha=p(\alpha+p\lambda\theta)/p_i\) with \(\theta\ge 0\).
scientific article; zbMATH DE number 7375763
scientific article