Nonconvolution transforms with oscillating kernels that map ${\displaystyle {Ḃ}_1^{0,1}}$ into itself

We consider operators of the form ${\displaystyle (\Omega f)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du}$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and ${\displaystyle h\in L^{\infty }}$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${\displaystyle {Ḃ}^{0,1}\_1}$ (= B) into itself. In particular, all operators with ${\displaystyle h\left(y\right)=e^{i{\left|y\right|}^a}}$, a > 0, a ≠ 1, map B into itself.