Geodesic
On a theorem of Marcinkiewicz type for $H$-valued functions. A continual form of the Paley–Littlewood theorem
Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 237-243 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this report there are proved theorems on the boundedness of the convolution operator acting from the space $L_p(H')$ ($p$-summable functions on the line with values in the Hilbert space $(H')$ into the space $L_p(H'')$. There is derived a new version of the Paley–Littlewood Theorem. Bibliography: 3 items. 
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     author = {P. I. Lizorkin},
     title = {On a theorem of {Marcinkiewicz} type for $H$-valued functions. {A~continual} form of the {Paley{\textendash}Littlewood} theorem},
     journal = {Sbornik. Mathematics},
     pages = {237--243},
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P. I. Lizorkin. On a theorem of Marcinkiewicz type for $H$-valued functions. A continual form of the Paley–Littlewood theorem. Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 237-243. http://geodesic.mathdoc.fr/item/SM_1972_16_2_a6/

[1] M. K. Gavurin, “Über die Stieltjessche Integration abstracter Functionen”, Fundam. Math., 27 (1936), 255–268

[2] H. Danford, Dzh. Shvarts, Lineinye operatory. II, Mir, Moskva, 1966

[3] A. Benedek, A. P. Calderon, R. Panzone, “Convolution operators on Banach space valued functions”, Proc. Nat. Acad. Sci. USA, 48:3 (1962), 356–365 | DOI | MR | Zbl