## Vector-valued inequalities for Fourier series

HTML articles powered by AMS MathViewer

- by José L. Rubio De Francia
- Proc. Amer. Math. Soc.
**78**(1980), 525-528 - DOI: https://doi.org/10.1090/S0002-9939-1980-0556625-3
- PDF | Request permission

## Abstract:

Denoting by ${S^\ast }$ the maximal partial sum operator of Fourier series, we prove that ${S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots )$ is a bounded operator from ${L^p}({l^r})$ to itself, $1 < p,r < \infty$. Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.## References

- A. Benedek and R. Panzone,
*The space $L^{p}$, with mixed norm*, Duke Math. J.**28**(1961), 301–324. MR**126155**, DOI 10.1215/S0012-7094-61-02828-9 - Lennart Carleson,
*On convergence and growth of partial sums of Fourier series*, Acta Math.**116**(1966), 135–157. MR**199631**, DOI 10.1007/BF02392815 - A. Cordoba and C. Fefferman,
*A weighted norm inequality for singular integrals*, Studia Math.**57**(1976), no. 1, 97–101. MR**420115**, DOI 10.4064/sm-57-1-97-101 - Charles Fefferman,
*On the divergence of multiple Fourier series*, Bull. Amer. Math. Soc.**77**(1971), 191–195. MR**279529**, DOI 10.1090/S0002-9904-1971-12675-7 - Richard A. Hunt,
*On the convergence of Fourier series*, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235–255. MR**0238019** - Richard A. Hunt and Wo Sang Young,
*A weighted norm inequality for Fourier series*, Bull. Amer. Math. Soc.**80**(1974), 274–277. MR**338655**, DOI 10.1090/S0002-9904-1974-13458-0
J. Marcinkiewicz and A. Zygmund, - Benjamin Muckenhoupt,
*Weighted norm inequalities for the Hardy maximal function*, Trans. Amer. Math. Soc.**165**(1972), 207–226. MR**293384**, DOI 10.1090/S0002-9947-1972-0293384-6

*Quelques inégalités pour les opérations linéaires*, Fund. Math.

**32**(1939), 115-121.

## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**78**(1980), 525-528 - MSC: Primary 42A20
- DOI: https://doi.org/10.1090/S0002-9939-1980-0556625-3
- MathSciNet review: 556625