Unconditional convergence of Fourier series with respect to the Haar system in the spaces $\Lambda_\omega^p$

Criteria for a Haar system to be a basic system and an unconditional basic system in the spaces
$$ \Lambda_\omega^p=\{f\in L^p: \omega_p(\delta, f)=O\{\omega(\delta)\}\}, $$
where $1<p<\infty$ and $\omega$ is a modulus of continuity, are proved.