Fatou type convergence of nonlinear m-singular integral operators

Abstract

In the present paper we investigate both the pointwise convergence and the Fatou type convergence of the operatorsTλ[m]f(x)=∫-∞∞Kλt,∑k=1m-1k-1mkf(x+kt)dt,x∈R,λ∈Λ,to f(x0) as λ→λ0 and (x,λ)→x0,λ0 in LpRp⩾1, respectively. Here Λ is a nonempty set of indices with a topology and λ0 is an accumulation point of Λ in this topology.