Weighted Fourier Inequalities and Boundedness of Variation

We study the trigonometric series $\sum _{n=1}^\infty \lambda _n \cos nx$ and $\sum _{n=1}^\infty \lambda _n \sin nx$ with $\{\lambda _n\}$ being a sequence of bounded variation. Let $\psi $ denote the sum of such a series. We obtain necessary and sufficient conditions for the validity of the weighted Fourier inequality $\left (\int _0^\pi |\psi (x)|^q \omega (x)\,dx\right )^{1/q} \le C\!\left (\sum _{n=1}^\infty u_n\left (\sum _{k=n}^\infty |\lambda _{k}-\lambda _{k+1}|\right )^p \right )^{1/p}$, $0<p\le q<\infty $, in terms of the weight $\omega $ and the weighted sequence $\{u_n\}$. Applications to the series with general monotone coefficients are given.