Banach Journal of Mathematical Analysis

Square function inequalities for monotone bases in L1

Adam Osękowski

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Abstract

We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in L1. The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong L1L1 and a weak-type L1L1, estimate for square functions.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 3 (2018), 693-708.

Dates
Received: 3 December 2017
Accepted: 28 February 2018
First available in Project Euclid: 16 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1529114493

Digital Object Identifier
doi:10.1215/17358787-2018-0004

Mathematical Reviews number (MathSciNet)
MR3824747

Zentralblatt MATH identifier
06946077

Subjects
Primary: 46B15: Summability and bases [See also 46A35]
Secondary: 46A35: Summability and bases [See also 46B15] 46B20: Geometry and structure of normed linear spaces

Keywords
contractive projection Haar system square function monotone basis

Citation

Osękowski, Adam. Square function inequalities for monotone bases in $L^{1}$. Banach J. Math. Anal. 12 (2018), no. 3, 693--708. doi:10.1215/17358787-2018-0004. https://projecteuclid.org/euclid.bjma/1529114493


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References

  • [1] T. Andô, Contractive projections in $L_{p}$ spaces, Pacific J. Math. 17 (1966), 391–405.
  • [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
  • [3] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702.
  • [4] D. L. Burkholder, A proof of Pełczyński’s conjecture for the Haar system, Studia Math. 91 (1988), no. 1, 79–83.
  • [5] D. L. Burkholder, The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354 (2002), no. 1, 91–105.
  • [6] K. P. Choi, A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L^{p}(0,1)$, Trans. Amer. Math. Soc. 330 (1992), no. 2, 509–529.
  • [7] L. E. Dor and E. Odell, Monotone bases in $L_{p}$, Pacific J. Math. 60 (1975), 51–61.
  • [8] J. Marcinkiewicz, Quelques théoremes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1938), 84–96.
  • [9] A. M. Olevskiĭ, Fourier series and Lebesgue functions (in Russian), Uspekhi Mat. Nauk 22 (1967), no. 3, 237–239.
  • [10] A. M. Olevskiĭ, Fourier Series with Respect to General Orthogonal Systems, Ergeb. Math. Grenzgeb. (3) 86, Springer, New York, 1975.
  • [11] A. Osękowski, Two inequalities for the first moments of a martingale, its square function and its maximal function, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 4, 441–449.
  • [12] A. Osękowski, Sharp inequalities for monotone bases in $L^{1}$, Houston J. Math. 42 (2016), no. 3, 833–851.
  • [13] R. E. A. C. Paley, A remarkable series of orthogonal functions, I, Proc. Lond. Math. Soc. (2) 34 (1932), no. 4, 241–264.