$$L^p$$ and $$H^1$$ Boundedness of Oscillatory Singular Integral Operators with Hölder Class Kernels

For oscillatory singular integrals with polynomial phases and Holder class kernels, we establish their uniform boundedness on $$L^p$$ spaces as well as a sharp logarithmic bound on the Hardy space $$H^1$$ . These results improve the ones in (Pan in Forum Math 31: 535–542, 2019) by removing the restriction that the phase polynomials be quadratic.