Weightes inequalities integral operators with almost homogeneous kernels

URCIUOLO, MARTA

Georgian Mathematical Journal

Año: 2006 vol. 13 p. 183 - 191

In this paper we study integral operators of the form    Tf(x)=∫k₁(x-a₁y)k₂(x-a₂y)...k_{m}(x-a_{m}y)f(y)dy,k_{i}(y)=∑_{j∈Z}2^{((jn)/(q_{i}))}ϕ_{i,j}(2^{j}y), 1≤q_{i}<∞, (1/(q₁))+(1/(q₂))+...+(1/(q_{m}))=1. We suppose suppϕ_{i,j}⊂{y∈Rⁿ:2⁻¹≤|y|≤2} and that that there exists p_{i}>q_{i} such that ‖ϕ_{i,j}‖_{p_{i}}≤c, independent of j. If (1/(p₁))+...+(1/(p_{m}))+(1/(p_{m+1})) =1, and max_{1≤i≤m}{((p_{i})/(p_{i}-q_{i}))}<p<∞, we obtain the boundedness of T:L^{p}(w)→L^{p}(w)  for all power weights w in A_{p/p_{m+1}}.