I am trying to estimate the norm of an L1-multiplier. So, I would like to find an upper bound for the L1 norm of the following trigonometric polynomial:
$$h(x)=\sum_{k=b}^{k=4b}\frac{k^{\alpha}}{\log^{\beta}(|k|+1)}\cos(kx),$$
where $\alpha \in (1,2)$ and $\beta >0$. If I had to guess, I would say that $|h|_{L^1}\lesssim \frac{\log(b)}{\log^{\beta}(b)}b^{\alpha}$ since usually Dirichlet kernel produces an extra $\log(b)$ factor. At last, I underline that this function is related to $\alpha$-fractional derivative of the Dirichlet kernel, up to the logarithmic correction.
Any idea? Thank you in advance.