On oscillatory integral Hilbert transformation with trigonometric polynomial phase

摘要

Let $nmathcal{P}_n n$nmathcal{P}_n n denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $nmathop {sup }limits_{p( cdot ) in mathcal{P}_n } left| {p.v.int_mathbb{R} {frac{{e^{ip(x)} }}n{x}dx} } right| leqslant C_n ,n$nmathop {sup }limits_{p( cdot ) in mathcal{P}_n } left| {p.v.int_mathbb{R} {frac{{e^{ip(x)} }}n{x}dx} } right| leqslant C_n ,