Necessary condition on weights for maximal and integral operators with rough kernels
Tom 263 / 2022
Streszczenie
Let , m\in \mathbb {N} and let T_{\alpha ,m} be an integral operator given by a kernel of the form K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my), where the A_i are invertible matrices and each k_i satisfies a fractional size condition and a generalized fractional Hörmander condition. Ibañez-Firnkorn and Riveros (2018) proved that T_{\alpha ,m} is controlled in L^p(w)-norms, w\in \mathcal {A}_{\infty }, by the sum of maximal operators M_{A_i^{-1},\alpha }. In this paper we present a class \mathcal {A}_{A,p,q} of weights, where A is an invertible matrix. These weights are appropriate for weak-type estimates of M_{A^{-1},\alpha }. For certain kernels k_i we can characterize the weights yielding strong-type estimates of T_{\alpha ,m}. Also, we give a strong-type estimate using testing conditions.
