Non-convolution operators with oscillating kernels that map \(\dot\mathbb{B}_ 1^{0,1}\) into itself and map \(L^ p\) into itself (Q1895327)

The author considers the kernels \(\Omega_1 (y,u)= K(y,u) e^{i|y- u|^a}\) for \(a>1\). He shows that the operators \(Tf(y)= \int (\Omega_1 (y+ {1\over 2}, u)- \Omega_1 (y,u)) f(u) du\) map \(\mathbb{B} (\mathbb{R}^n)\) into itself. He also shows that the operators \(\int (\Omega_1 (y,u)) f(u) du\) map \(L^p\) into itself for \(p>1\).