Email this article Maximal Calderon–Zygmund operators and Weyl multipliers
- Authors: Karagulyan G.A.1,2, Lacey M.T.3, Navoyan K.V.1
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Affiliations:
- Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia
- Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Republic of Armenia
- School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
- Issue: Vol 216, No 10 (2025)
- Pages: 42-61
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/331245
- DOI: https://doi.org/10.4213/sm10277
- ID: 331245
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Abstract
Let $T_k$ , $k=1,2,…,N$ , be a sequence of bounded operators on $L^p$ , $1, and $T^*(f)=\max_{1\le k\le N}|T_k(f)|$ . For some choices of $T_k$ it is of interest the problem of finding the optimal constant $c(N)$ for the bound
$$
\|T^*\|_{L^p\to L^p}\lesssim c(N) \max_{1\le k\le N}\|T_k\|_{L^p\to L^p}.
$$
We consider this problem for Calderon–Zygmund operators. It was prove by first two authors that$c(N)\lesssim \log N$ when $T_k$ are general Calderon–Zygmund operators with uniformly bounded parameters. In this note we consider Calderon–Zygmund operators with kernels, having certain dyadic decomposition. We prove for such operators $c(N)\lesssim\sqrt{\log N}$ . Applying this bound, we prove that the sequence $\log n$ is an almost everywhere convergence Weyl multiplier for any rearranged dyadic block trigonometric polynomials.
$$
\|T^*\|_{L^p\to L^p}\lesssim c(N) \max_{1\le k\le N}\|T_k\|_{L^p\to L^p}.
$$
We consider this problem for Calderon–Zygmund operators. It was prove by first two authors that
About the authors
Grigori Artashesovich Karagulyan
Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia; Faculty of Mathematics and Mechanics, Yerevan State University, Yerevan, Republic of Armenia
Email: g.karagulyan@ysu.am
Doctor of physico-mathematical sciences, Professor
Michael Thoreau Lacey
School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Email: lacey@math.gatech.edu
Doctor of physico-mathematical sciences, Professor
Khazhakanoush V.. Navoyan
Institute of Mathematics of National Academy of Sciences of RA, Yerevan, Republic of Armenia
Email: khvnavoyan@gmail.com
References
- A. Alfonseca, F. Soria, A. Vargas, “An almost-orthogonality principle in $L^2$ for directional maximal functions”, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003, 1–7
- M. Bateman, “Kakeya sets and directional maximal operators in the plane”, Duke Math. J., 147:1 (2009), 55–77
- S. Y. A. Chang, J. M. Wilson, T. H. Wolff, “Some weighted norm inequalities concerning the Schrödinger operators”, Comment. Math. Helv., 60:2 (1985), 217–246
- C. Demeter, F. Di Plinio, “Logarithmic $L^p$ bounds for maximal directional singular integrals in the plane”, J. Geom. Anal., 24:1 (2014), 375–416
- L. Grafakos, P. Honzik, A. Seeger, “On maximal functions for Mikhlin–Hörmander multipliers”, Adv. Math., 204:2 (2006), 363–378
- A. Kamont, G. A. Karagulyan, “On wavelet polynomials and Weyl multipliers”, J. Anal. Math., 150:2 (2023), 529–545
- G. A. Karagulyan, “On unboundedness of maximal operators for directional Hilbert transforms”, Proc. Amer. Math. Soc., 135:10 (2007), 3133–3141
- G. A. Karagulyan, “On systems of non-overlapping Haar polynomials”, Ark. Mat., 58:1 (2020), 121–131
- G. A. Karagulyan, M. T. Lacey, “On logarithmic bounds of maximal sparse operators”, Math. Z., 294:3-4 (2020), 1271–1281
- N. H. Katz, “Maximal operators over arbitrary sets of directions”, Duke Math. J., 97:1 (1999), 67–79
- A. Kolmogoroff, D. Menchoff, “Sur la convergence des series de fonctions orthogonales”, Math. Z., 26:1 (1927), 432–441
- D. Menchoff, “Sur les series de fonctions orthogonales. I”, Fund. Math., 4 (1923), 82–105
- F. Moricz, “On the convergence of Fourier series in every arrangement of the terms”, Acta Sci. Math. (Szeged), 31 (1970), 33–41
- S. Nakata, “On the divergence of rearranged Fourier series of square integrable functions”, Acta Sci. Math. (Szeged), 32 (1971), 59–70
- S. Nakata, “On the divergence of rearranged trigonometric series”, Tohoku Math. J. (2), 27:2 (1975), 241–246
- S. Nakata, “On the unconditional convergence of Walsh series”, Anal. Math., 5:3 (1979), 201–205
- S. Nakata, “On the divergence of rearranged Walsh series”, Tohoku Math. J. (2), 24:2 (1972), 275–280
- W. Orlicz, “Zur Theorie der Orthogonalreihen”, Bull. Intern. Acad. Sci. Polon. Cracovie, 1927, 81–115
- S. N. Poleščuk, “On the unconditional convergence of orthogonal series”, Anal. Math., 7:4 (1981), 265–275
- H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen”, Math. Ann., 87:1-2 (1922), 112–138
- A. Nagel, E. M. Stein, S. Wainger, “Differentiation in lacunary directions”, Proc. Nat. Acad. Sci. U.S.A., 75:3 (1978), 1060–1062
- E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Monogr. Harmon. Anal., III, Princeton Univ. Press, Princeton, NJ, 1993, xiv+695 pp.
- K. Tandori, “Über die orthogonalen Funktionen. X. (Unbedingte Konvergenz.)”, Acta Sci. Math. (Szeged), 23 (1962), 185–221
- K. Tandori, “Beispiel der Fourierreihe einer quadratisch-integrierbaren Funktion, die in gewisser Anordnung ihrer Glieder überall divergiert”, Acta Math. Acad. Sci. Hungar., 15 (1964), 165–173
- K. Tandori, “Über die Divergenz der Walshschen Reihen”, Acta Sci. Math. (Szeged), 27 (1966), 261–263
- Z. Zahorski, “Une serie de Fourier permutee d'une fonction de classe $L^{2}$ divergente presque partout”, C. R. Acad. Sci. Paris, 251 (1960), 501–503
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