## An upper bound for the Menchov-Rademacher operator for right triangles

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- by Armen Vagharshakyan PDF
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**150**(2022), 3959-3971 Request permission

## Abstract:

The Menchov-Rademacher inequality is an inequality in harmonic analysis that bounds the $L_2$ norm of a certain maximal operator. It was first established in order to prove almost everywhere convergence of a one-parameter series of orthogonal functions. When two-parameter series of orthogonal functions is considered, the exact way the series is grouped becomes essential. We will consider grouping of a two-parameter series, generated by a sequence of right triangles with a vertex at the origin, who might be non-equilateral, and prove almost everywhere convergence when the eccentricity of those triangles is bounded. In order to carry out the proof, we will derive an analogue of the Menchov-Rademacher inequality for right triangles.## References

- Orthogonal series.
*Encyclopedia of mathematics.*http://encyclopediaofmath.org/index.php?title=Orthogonal_series&oldid=49507 - P. Fatou,
*Séries trigonométriques et séries de Taylor*, Acta Math.**30**(1906), no. 1, 335–400 (French). MR**1555035**, DOI 10.1007/BF02418579 - M. Plancherel,
*Sur la convergence des séries de fonctions orthogonales*, C. R. Acad. Sci. Paris**157**(1913), 539–542. - Michel Plancherel,
*Les problèmes de Cantor et de Du Bois-Reymond dans la théorie des séries de polynômes de Legendre*, Ann. Sci. École Norm. Sup. (3)**31**(1914), 223–262 (French). MR**1509177** - P. Ul’yanov,
*Development of D.E. Men’shov’s results in the theory of orthogonal series*, Russian Math.**47**, no. 5 (1992). - Heinz König,
*Measure and integration*, Springer-Verlag, Berlin, 1997. An advanced course in basic procedures and applications. MR**1633615** - Hans Rademacher,
*Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen*, Math. Ann.**87**(1922), no. 1-2, 112–138 (German). MR**1512104**, DOI 10.1007/BF01458040 - D. Menchoff,
*Sur les series de fonctions orthogonales*, Fund. Math.**4**(1923), 82–105. - Witold Bednorz,
*A note on the Men′shov-Rademacher inequality*, Bull. Pol. Acad. Sci. Math.**54**(2006), no. 1, 89–93. MR**2270798**, DOI 10.4064/ba54-1-9 - E. G. Kounias,
*A note on Rademacher’s inequality*, Acta Math. Acad. Sci. Hungar.**21**(1970), 447–448. MR**270414**, DOI 10.1007/BF01894790 - S. A. Chobanyan,
*Some remarks on the Men′shov-Rademacher functional*, Mat. Zametki**59**(1996), no. 5, 787–790 (Russian); English transl., Math. Notes**59**(1996), no. 5-6, 571–574. MR**1445464**, DOI 10.1007/BF02308830 - Adam Paszkiewicz,
*A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures*, Invent. Math.**180**(2010), no. 1, 55–110. MR**2593277**, DOI 10.1007/s00222-009-0226-2 - Charles Fefferman,
*On the convergence of multiple Fourier series*, Bull. Amer. Math. Soc.**77**(1971), 744–745. MR**435724**, DOI 10.1090/S0002-9904-1971-12793-3 - N. R. Tevzadze,
*The convergence of the double Fourier series at a square summable function*, Sakharth. SSR Mecn. Akad. Moambe**58**(1970), 277–279 (Russian, with Georgian and English summaries). MR**0298338** - Per Sjölin,
*Convergence almost everywhere of certain singular integrals and multiple Fourier series*, Ark. Mat.**9**(1971), 65–90. MR**336222**, DOI 10.1007/BF02383638 - N. Antonov,
*“Almost everywhere convergence over cubes of multiple trigonometric Fourier series,”*Izv. Math.**68**(2004), 223. - Mieczysław Mastyło and Luis Rodríguez-Piazza,
*Convergence almost everywhere of multiple Fourier series over cubes*, Trans. Amer. Math. Soc.**370**(2018), no. 3, 1629–1659. MR**3739187**, DOI 10.1090/tran/7172 - 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 - J. L. Bentley, D. Haken, and J. B. Saxe,
*A general method for solving divide-and-conquer recurrences*, SIGACT News**12**(1980), no. 3, 36–44.

## Additional Information

**Armen Vagharshakyan**- Affiliation: Institute of Mathematics, Armenian National Academy of Sciences, Baghramyan Avenue 24/5, Yerevan 0019, Armenia
- MR Author ID: 776347
- ORCID: 0000-0003-4262-1019
- Email: armensci@yahoo.com
- Received by editor(s): November 11, 2020
- Received by editor(s) in revised form: September 20, 2021, and December 10, 2021
- Published electronically: April 7, 2022
- Additional Notes: Research was supported by the Science Committee of Armenia, grant 18T-1A081
- Communicated by: Dmitriy Bilyk
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 3959-3971 - MSC (2020): Primary 26D15, 42B25
- DOI: https://doi.org/10.1090/proc/15950
- MathSciNet review: 4446244