2015/03/02 by Street, Brian
#42B20 (Primary) 42B25 #44A12 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1503.00751
The purpose of this paper is to study the smoothing properties (in Lp Sobolev spaces) of operators of the form f↦ ψ(x) ∫ f(γt(x)) K(t) dt, where γt(x) is a C^∞ function defined on a neighborhood of the origin in (t,x)∈ℝN× ℝn, satisfying γ0(x)≡ x, ψ is a C^∞ cut-off function supported on a small neighborhood of 0∈ ℝn, and K is a "multi-parameter fractional kernel" supported on a small neighborhood of 0∈ ℝN. When K is a Calderón-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when K is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on γ were given under which the above operator is bounded on Lp (1