ACTA issues

The maximal Riemann operator is bounded from \$H^1\$(T) into \$L^1\$(T)

Mónika Bagota

Acta Sci. Math. (Szeged) 62:3-4(1997), 557-564
6135/2009

 Abstract. We consider the Riemann means of Fourier series of functions belonging to the Hardy space \$H^1({\msbm T})\$ or \$L^1({\msbm T})\$, respectively. We prove that the maximal Riemann operator as well as the maximal conjugate Riemann operator are bounded from \$H^1({\msbm T})\$ into \$L^1({\msbm T})\$. It is also true that this operator is bounded from \$L^1({\msbm T})\$ into weak-\$L^1({\msbm T})\$. On closing, we formulate a conjecture about the maximal conjugate Riemann operator of a function in \$L^1({\msbm T})\$. AMS Subject Classification (1991): 47B38 Received February 29, 1996 and in revised form July 11, 1996. (Registered under 6135/2009.)