Subspaces of

Author:
Michael Frazier

Journal:
Trans. Amer. Math. Soc. **290** (1985), 101-125

MSC:
Primary 42B20; Secondary 42B30, 46E99, 47G05

MathSciNet review:
787957

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider subspaces of generated by one singular integral transform. We show that the averages along -lines of the th Riesz transform of or satisfy a certain strong regularity property. One consquence of this result is that such functions satisfy a uniform doubling condition on a.e. -line. We give an example to show, however, that the restrictions to -lines of the Riesz transform of do not necessarily have uniformly bounded norm. Also, for a Calderón-Zygmund singular integral operator with real and odd kernel, we show that , where and are the spaces of or functions of compact support, respectively, and the closure is taken in norm.

**[1]**C. Fefferman and E. M. Stein,*𝐻^{𝑝} spaces of several variables*, Acta Math.**129**(1972), no. 3-4, 137–193. MR**0447953****[2]**M. W. Frazier,*Functions of bounded mean oscillation characterized by a restricted set of martingale or Riesz transforms*, Ph.D. Thesis, University of California, Los Angeles, 1983.**[3]**John B. Garnett,*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971****[4]**John B. Garnett and Peter W. Jones,*The distance in BMO to 𝐿^{∞}*, Ann. of Math. (2)**108**(1978), no. 2, 373–393. MR**506992**, 10.2307/1971171**[5]**S. Janson,*Characterization of**by singular integral transforms on martingales and*, Math. Scand.**41**(1977), 140-152.**[6]**J.-P. Kahane,*Trois notes sur les ensembles parfaits linéaires*, Enseignement Math. (2)**15**(1969), 185–192 (French). MR**0245734****[7]**E. M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Univ. Press, Princeton, N. J., 1970.**[8]**A. Uchiyama,*A constructive proof of the Fefferman-Stein decomposition of*, Acta Math.**148**(1982), 215-241.**[9]**-,*A constructive proof of the Fefferman-Stein decomposition of**on simple martingales*, Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II (Beckner, et al., eds.), University of Chicago Press, Chicago, Ill., 1981, pp. 495-505.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
42B20,
42B30,
46E99,
47G05

Retrieve articles in all journals with MSC: 42B20, 42B30, 46E99, 47G05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0787957-0

Keywords:
,
singular integral operator

Article copyright:
© Copyright 1985
American Mathematical Society