On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

✍ Scribed by Ferenc Móricz

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 213 KB
• Volume: 219
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/1522-2616(200011)219:1<163::aid-mana163>3.0.co;2-1

✦ Synopsis

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L 1 or the real Hardy spaces defined on IR n , where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H 1 (IR) into L 1 (IR) and from L 1 (IR) into weak -L 1 (IR). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H (1,0) IR 2 , H (0,1) IR 2 into weak -L 1 IR 2 . Hence pointwise Riemann summability of Fourier integrals of functions in H (1,0) ∪ H (0,1) IR 2 follows almost everywhere. The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.