Geodesic
On the Haar and Franklin series with identical coefficients
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1989), pp. 3-9.

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If $\big\lbrace \chi_n(x)\big\rbrace^\infty_{n=1}$ is Haar system and $\big\lbrace f_n(x)\big\rbrace^\infty_{n=0}$ is Franklin system, then for every $\lbrace a_n\rbrace^\infty_{n=0}$ and $p>0$ the following relation is proved \begin{equation} \left\Vert\left\lbrace\sum\limits^\infty_{n=0}a^2_n f^2_n(x)\right\rbrace^{\frac{1}{2}} \right\Vert_p \sim \left\Vert\left\lbrace\sum\limits^\infty_{n=0}a^2_n \chi^2_{n+1}(x)\right\rbrace^{\frac{1}{2}} \right\Vert_p, \end{equation} (1) has been proved in [2] when $p>l$ and in [4] when $\dfrac{1}{2}$ but the methods of [2] and [4] are not applicable in the case $0$. Some consequences are received from (1) as well. 
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G. G. Gevorkyan. On the Haar and Franklin series with identical coefficients. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1989), pp. 3-9. http://geodesic.mathdoc.fr/item/UZERU_1989_3_a0/

[1] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[2] Z. Ciesielski, P. Simon, P. Sjolin, “Eqwivalance of Haar and Franclin Bases in $L_p$ spaces”, Studia Math., 60:2 (1977), 195–211 | DOI | MR

[3] C. Fefferman, E. M. Stein, “Some maximal inequalities”, Amer. J. Math., 93 (1971), 107–115 | DOI | MR | Zbl

[4] P. Sjölin, J.-O. Strömberg, “Basis properties of Hardy spaces”, Ark. math., 21:1 (1983), 111–125 | DOI | Zbl

[5] G. G. Gevorkyan, “Neogranichennost operatora sdviga po sisteme Franklina v prostranstve $L_1$”, Matem. zametki, 38:4 (1985), 523–533 | MR | Zbl

[6] G. G. Gevorkyan, “Nekotorye teoremy o bezuslovnoi skhodimosti i mazhorante ryadov Franklina i ikh primenenie k prostranstvam $\operatorname{Re}H_p$”, Tr. MIAN, 190, Nauka, M., 1989, 49–74 | MR | Zbl

[7] Z. Ciesielski, “Properties of the orthonormal Franclin system”, Studia Math., 23:2 (1963), 141–157 | DOI | MR | Zbl

[8] Z. Ciesielski, “Propeties of the orthonormal Franklin system. II”, Studia Math., 27 (1966), 289–323 | DOI | MR | Zbl

[9] D. L. Burkholder, R. F. Gundy, “Extrapolation and interpolation of quasi-linear operators on martingales”, Acta mathematica, 124 (1970), 250–304 | DOI | MR | Zbl

[10] Z. Ciesielski, S. Kwapien, Some properties of the Haar, Walsh-Paley, Franklin and the bounded polygonal orthonormal bases in $L_p$ Spaces Commentat, Math. Tom. Spec. Honor. Ladislae Orlicz. Part 2, Warszawa, 1979, 37–42 | Zbl

[11] Z. Ciesielski, Convergence of spline expansions. Linear spaces and approximation, Birkhauser Verlag-Basel, 1978, 443–448

[12] P. Simon, “Remarks on the shift operators with respect to the Haar and Franklin systems”, Acta math. Acad. Sci. Hungar, 39:1-3 (1982), 251–254 | DOI | Zbl