On a theorem of Marcinkiewicz type for $H$-valued functions. A continual form of the Paley–Littlewood theorem

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.