Geodesic
Unconditional convergence of Fourier series with respect to the Haar system in the spaces $\Lambda_\omega^p$
Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 685-695

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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 $1$ and $\omega$ is a modulus of continuity, are proved. 
@article{MZM_1978_23_5_a4,
     author = {V. G. Krotov},
     title = {Unconditional convergence of {Fourier} series with respect to the {Haar} system in the spaces $\Lambda_\omega^p$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {685--695},
     publisher = {mathdoc},
     volume = {23},
     number = {5},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a4/}
}
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V. G. Krotov. Unconditional convergence of Fourier series with respect to the Haar system in the spaces $\Lambda_\omega^p$. Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 685-695. http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a4/