## Weighted norm inequalities for the conjugate function and Hilbert transform

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- by Richard Hunt, Benjamin Muckenhoupt and Richard Wheeden
- Trans. Amer. Math. Soc.
**176**(1973), 227-251 - DOI: https://doi.org/10.1090/S0002-9947-1973-0312139-8
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## Abstract:

The principal problem considered is the determination of all non-negative functions $W(x)$ with period $2\pi$ such that \[ \int _{ - \pi }^\pi {|\tilde f(\theta ){|^p}W(\theta )\;d\theta \leq C} \;\int _{ - \pi }^\pi {|f(\theta ){|^p}W(\theta )\;d\theta } \] where $1 < p < \infty$,*f*has period $2\pi$,

*C*is a constant independent of

*f*, and $\tilde f$ is the conjugate function defined by \[ \tilde f(\theta ) = \lim \limits _{\varepsilon \to {0^ + }} \frac {1}{\pi }\int _{\varepsilon \leq |\phi | \leq \pi } {\frac {{f(\theta - \phi )\;d\phi }}{{2\tan \phi /2}}.} \] The main result is that $W(x)$ is such a function if and only if \[ \left [ {\frac {1}{{|I|}}\int _I {W(\theta )\;d\theta } } \right ]{\left [ {\frac {1}{{|I|}}\int _I {{{[W(\theta )]}^{ - 1/(p - 1)}}d\theta } } \right ]^{p - 1}} \leq K\] where

*I*is any interval, $|I|$ denotes the length of

*I*and

*K*is a constant independent of

*I*. Various related problems are also considered. These include weak type results, the nonperiodic case, the discrete case, an application to weighted mean convergence of Fourier series, and an estimate for one of the functions in the Fefferman and Stein decomposition of functions of bounded mean oscillation.

## References

- A. Benedek and R. Panzone,
*Continuity properties of the Hilbert transform*, J. Functional Analysis**7**(1971), 217–234. MR**0276835**, DOI 10.1016/0022-1236(71)90032-2 - Charles Fefferman,
*Characterizations of bounded mean oscillation*, Bull. Amer. Math. Soc.**77**(1971), 587–588. MR**280994**, DOI 10.1090/S0002-9904-1971-12763-5 - C. Fefferman and E. M. Stein,
*$H^{p}$ spaces of several variables*, Acta Math.**129**(1972), no. 3-4, 137–193. MR**447953**, DOI 10.1007/BF02392215 - Frank Forelli,
*The Marcel Riesz theorem on conjugate functions*, Trans. Amer. Math. Soc.**106**(1963), 369–390. MR**147827**, DOI 10.1090/S0002-9947-1963-0147827-3 - V. F. Gapoškin,
*A generalization of the theorem of M. Riesz on conjugate functions.*, Mat. Sb. (N.S.)**46(88)**(1958), 359–372 (Russian). MR**0099561** - G. H. Hardy and J. E. Littlewood,
*Some more theorems concerning Fourier series and Fourier power series*, Duke Math. J.**2**(1936), no. 2, 354–382. MR**1545928**, DOI 10.1215/S0012-7094-36-00228-4 - Henry Helson and Gabor Szegö,
*A problem in prediction theory*, Ann. Mat. Pura Appl. (4)**51**(1960), 107–138. MR**121608**, DOI 10.1007/BF02410947 - F. John and L. Nirenberg,
*On functions of bounded mean oscillation*, Comm. Pure Appl. Math.**14**(1961), 415–426. MR**131498**, DOI 10.1002/cpa.3160140317 - 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 - Benjamin Muckenhoupt,
*Poisson integrals for Hermite and Laguerre expansions*, Trans. Amer. Math. Soc.**139**(1969), 231–242. MR**249917**, DOI 10.1090/S0002-9947-1969-0249917-9 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Harold Widom,
*Singular integral equations in $L_{p}$*, Trans. Amer. Math. Soc.**97**(1960), 131–160. MR**119064**, DOI 10.1090/S0002-9947-1960-0119064-7 - A. Zygmund,
*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776** - Marvin Rosenblum,
*Summability of Fourier series in $L^{p}(d\mu )$*, Trans. Amer. Math. Soc.**105**(1962), 32–42. MR**160073**, DOI 10.1090/S0002-9947-1962-0160073-1

## Bibliographic Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**176**(1973), 227-251 - MSC: Primary 42A40; Secondary 44A15, 47G05
- DOI: https://doi.org/10.1090/S0002-9947-1973-0312139-8
- MathSciNet review: 0312139