Extrapolation of weighted norm inequalities for multivariable operators and applications
The celebrated Rubio de Francia extrapolation theorem states that if a given operator is bounded on \(L^{p_0}(w)\) for some \(1\leq p_0<\infty\) and all \(A_p\) weights \(w\in A_{p_0}\), then it is bounded on all spaces \(L^p(\mathbb R^n)\) for \(1<p<\infty\). The authors extend this to the case of many functions. More precisely, let \(1\leq q_1, \ldots, q_m<\infty\) with \(1/q=1/q_1+\cdots+1/q_m\leq m\). Suppose that for all \(B>1\) there is a constant \(C_0(B)\) such that \(\| T(f_1,\ldots,f_m)\| _{L^q(w_1^{q}\cdots w_m^q)}\leq C_0 \prod_{j=1}^{m}\| f_j\| _{L^{q_j}(w_j^{q_j})}\) for all tuples of weights \(w_1^{q_1},\ldots, w_m^{q_m}\in (A_{q_1},\ldots, A_{q_m})\) with \([w_j^{q_j}]_{A_{q_j}}\leq B\). Then for all indices \(1<p_1,\ldots, p_m<\infty\) with \(1/p=1/p_1+\cdots+1/p_m<m\), all \(B>1\), and all weights \(w_1^{p_1},\ldots, w_m^{p_m}\in (A_{p_1},\ldots, A_{p_m})\) with \([w_j^{p_j}]_{A_{p_j}}\leq B\), there exists \(C=C(B)\) such that \(\| T(f_1,\ldots,f_m)\| _{L^p(w_1^{p}\cdots w_m^p)}\leq C \prod_{j=1}^{m}\| f_j\| _{L^{p_j}(w_j^{p_j})}\). They give a version of this for which there is only one weight. They also consider the situation for which the initial estimates are of weak type. As applications, they treat multilinear Calderón-Zygmund operators, vector-valued inequalities, a multilinear extension of the Marcinkiewicz and Zygmund theorem on \(\ell^2\)-valued extensions of bounded linear operators.
Calderón-Zygmund singular integral operators
scientific article; zbMATH DE number 2070478
scientific article