ACTA issues

## The Cesàro operator is bounded on the Hardy space $H^1$

Dang Vu Giang, Ferenc Móricz

Acta Sci. Math. (Szeged) 61:1-4(1995), 535-544
5703/2009

 Abstract. Our purpose is to give a short proof of the statements that the Cesàro operator is bounded on both $H^1({\msbm R})$ and $H^1({\msbm T})$. The first statement is new in the literature, while the second one is known. The proof of the first statement is based on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm R})$ is such that its Fourier transform $\hat f(t)=0$ for $t\le0$, then $f\in H^1({\msbm R})$. The following reversed statement is also proved: If $f\in H^1({\msbm R})$, then $f$ can be represented in the form $f=f_1 + f_2$, where both $f_1$ and $f_2$ belong to $H^1({\msbm R}), \hat f_1(t)=0$ for $t\le0$, and $\hat f_2(t)=0$ for $t\ge0$. The proof of the second statement also relies on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm T})$ is such that its Fourier coefficient $\hat f(k)=0$ for $k=-1,-2,\ldots$, then $f\in H^1({\msbm T})$. AMS Subject Classification (1991): 47D05 Keyword(s): Hilbert transform, Hardy space, Fourier transform, finite Borel measure, Fourier-Stieltjes transform, Cesàro operator, closed graph theorem, conjugate function, Fourier series, conjugate series Received May 27, 1994, and in revised form January 4, 1995. (Registered under 5703/2009.)