Abstract. Our goal is twofold: (i) We give a unified treatment of the results initiated by Hardy in 1928 when he proved that the space $L^p({\msbm T})$ for any $1\le p< \infty $, is invariant under the $(C, 1)$-transform of the Fourier coefficients. (ii) We prove new results on the harmonic Cesàro operator ${\cal C}$ and the harmonic Copson operator ${\cal C}^*$ applied to functions defined on either the half real line ${\msbm R}_+$, or the whole real line ${\msbm R}$, or the torus ${\msbm T}$. Among others, we prove that the harmonic Copson operator $C^*$ is bounded on BMO, as well as from the subspace of the even functions in the real Hardy space $H^1$ into $L^1$.

AMS Subject Classification (1991): 47B48; 42A16, 42A38

Keyword(s): harmonic Cesàro operator, harmonic Copson operator, adjoint operator, inverse operator, spectrum of an operator, $L^p$-spaces, BMO, $H^1$-space, real Hardy space, Fourier transform, Fourier coefficient

Received January 14, 1998 and in revised form December 9, 1998. (Registered under 2686/2009.)