Maximal multilinear Calderón-Zygmund operators with non-doubling measures (Q1046939)

Within the frameworks of the general setting of Lebesgue spaces on \(\mathbb{R}^d\) with non-doubling measure, the authors consider the maximal multilinear singular Calderón-Zygmund type operator. The assumptions on the kernel of the operator are those known as ''standard conditions'' adjusted for the multilinear version of the operator. The measure \(\mu\) is supposed to satisfy the growth condition \(\mu B(x,r)\leq Cr^n, 0<n\leq d.\) They prove the boundedness of the operator from \(L^{p_1}(\mu)\times \cdots \times L^{p_m}(\mu)\) into \(L^p(\mu)\) with \(\frac{1}{p} = \frac{1}{p_1}+\cdots +\frac{1}{p_m}\) when all the \(p_i\) are greater than one. In the case where one of the \(p_i\) is equal to one, the weak type estimate is given. A weighted version of weak type estimates is also proved.