Weighted norm inequalities with general weights for the commutator of Calderón
Let \(p_1,\dots, p_{m+1}\in [1,\infty]\), \(p= (\sum^{m+1}_{k=1} p^{-1}_k)^{-1}\in(0, \infty)\). Let the weight and the compactly supported functions be denoted by \(f\) and \(a_1,\dots, a_m\). We denote by \(\vec v_w= \prod^{m+1}_{k=1} w^{p/p_k}_k\), \(m= \min\{p_k: 1\leq k\leq m+1\}\). The authors consider the commutator of Calderón \[ C_{m+1}:= C_{m+1}(f; a_1,\dots, a_m)(x). \] They prove that: Theorem 1. There exists a constant \(C> 0\), such that \[ \| C_{m+1}\|_X\leq C \prod^m_{k=1}\| a_k\|_{Y^k}\| f\|_Z, \] where (i) If \(p\in (0,1)\), \(m> 1\), then \[ X= L^p(\mathbb{R},\vec v_w),\;Y_k= L^{p_k}(\mathbb{R}, M_{L\log L}w_k),\;Z= L^{p_{m+1}}(\mathbb{R}, M_{L\log L}w_{m+1}). \] (ii) If \(p> 1\), \(\delta> 0\) fixed, then \[ X= L^p(\mathbb{R},\vec v_w),\;Y_k= L^{p_k}(\mathbb{R}, M_{L(\log L)^{p+\delta}}w_k),\;Z= L^{p_{m+1}}(\mathbb{R}, M_{L(\log L)^{p+\delta}} w_{m+1}). \] (iii) If \(p+(0,1]\), \(m= 1\), \(\delta> 0\) fixed, then \[ X= L^{p,\infty}(\mathbb{R},\omega),\;Y_k= L^{p_k}(\mathbb{R}, M_{L(\log L)^{1+\delta}}w),\;Z= L^{p_{m+1}}(\mathbb{R}, M_{L(\log L)^{1+\delta}} w). \] They obtain in Theorem 2 weighted estimates for any \(m\)-linear operator with kernel satisfying sufficient conditions. Then the authors get Theorem 1 from Theorem 2.
scientific article; zbMATH DE number 6142992
scientific article