Trigonometric Multipliers on $H_{2\pi}$ In this paper we consider multipliers on the real Hardy space $H_{2\pi}$. It is known that the Marcinkiewicz and the H\"ormander--Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on $L_{2\pi}^p$, $1 Keywords:Multipliers, Hardy spaceCategories:42A45, 42A50, 42A85 2. CMB 2002 (vol 45 pp. 265) Nawrocki, Marek  On the Smirnov Class Defined by the Maximal Function H.~O.~Kim has shown that contrary to the case of$H^p$-space, the Smirnov class$M$defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on$M$and multiplier from$M$into$H^p$,$0 < p \leq \infty\$. Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelopeCategories:46E10, 30A78, 30A76