Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0,1}$ into itself
Volume 106 / 1993
Studia Mathematica 106 (1993), 1-44
DOI: 10.4064/sm-106-1-1-44
Abstract
We consider operators of the form $(Ωf)(y) = ʃ_{-∞}^∞ Ω(y,u)f(u)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h ∈ L^∞$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ^{0,1}_1$ (= B) into itself. In particular, all operators with $h(y) = e^{i|y|^a}$, a > 0, a ≠ 1, map B into itself.
