Uniform oscillation inequality for the sharply truncated Hilbert transform
Let JH(y)=1[−1,1](y)y−1J_H(y)=\boldsymbol{1}_{[-1,1]}(y)y^{-1}JH(y)=1[−1,1](y)y−1 be the Hilbert transform kernel with a sharp cutoff, and let Oscn(JH;f)\operatorname{Osc}_n(J_H;f)Oscn(JH;f) denote the corresponding oscillation operat…