Abstract
We prove boundedness of Calderön–Zygmund operators acting in Banach function spaces on domains, defined by the L1 Carleson functional and Lq (1 < q < ∞) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
Article PDF
Similar content being viewed by others
References
H. Aimar, L. Forzani and F. J. Martín-Reyes, On weighted inequalities for singular integrals, Proc. Amer. Math. Soc. 125 (1997), 62–65.
K. Astala and M. J. González, Chord-arc curves and the Beurling transform, Invent. Math. 205 (2016), 62–65.
P. Auscher and A. Axelsson, Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math. 184 (2011), 62–65.
P. Auscher, A. Axelsson and A. McIntosh, Solvability of elliptic systems with square integrable boundary data, Ark. Mat. 48 (2010), 253–287.
P. Auscher, C. Kriegler, S. Monniaux and P. Portal, Singular integral operators on tent spaces, J. Evol. Equ. 12 (2012), 62–65.
P. Auscher and C. Prisuelos-Arribas, Tent space boundedness via extrapolation, Math. Z. 286 (2017), 62–65.
P. Auscher and A. Rosén, Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II, Anal. PDE 5 (2012), 62–65.
A. Axelsson, S. Keith and A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), 62–65.
F. Bernicot, D. Frey and S. Petermichl, Sharp weighted norm estimates beyond Calderón–Zygmund theory, Anal. PDE 9 (2016), 62–65.
W. Chen, R. Han and M.-T. Lacey, Weighted estimates for one-sided martingale transforms, Proc. Amer. Math. Soc. 148 (2020), 62–65.
R. Chill and S. Król, Weighted inequalities for singular integral operators on the half-line, Studia Math. 243 (2018), 62–65.
R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 62–65.
J. M. Conde-Alonso, A. Culiuc, F. Di Plinio and Y. Ou, A sparse domination principle for rough singular integrals, Anal. PDE 10 (2017), 62–65.
Y. Huang, Weighted tent spaces with Whitney averages: factorization, interpolation and duality, Math. Z. 282 (2016), 62–65.
T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. Vol. I. Martingales and Littlewood—Paley Theory, Springer, Cham, 2016.
T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces. Vol. II. Probabilistic Methods and Operator Theory, Springer, Cham, 2017.
T. Hytönen and A. Rosén, On the Carleson duality. Ark. Mat. 51 (2013), 293–313.
T. Hytönen and A. Rosén, Bounded variation approximation of Lpdyadic martingales and solutions to elliptic equations, J. Eur. Math. Soc. (JEMS) 20 (2018), 62–65.
C. E. Kenig and J. Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), 62–65.
M. T. Lacey, An elementary proof of the A2bound, Israel J. Math. 217 (2017), 62–65.
A. K. Lerner, A simple proof of the A2conjecture, Int. Math. Res. Not. IMRN 14 (2013), 62–65.
A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 62–65.
A. K. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics, Expo. Math. 37 (2019), 62–65.
A. K. Lerner and S. Ombrosi, Some remarks on the pointwise sparse domination, J. Geom. Anal. 30 (2020), 62–65.
F. J. Martín-Reyes and A. de la Torre, Sharp weighted bounds for one-sided maximal operators, Collect. Math. 66 (2015), 62–65.
Y. Meyer and R. Coifman, Wavelets, Cambridge University Press, Cambridge, 1997.
K. Nyström and A. Rosén, Cauchy integrals for the p-Laplace equation in planar Lipschitz domains, Ann. Acad. Sci. Fenn. Math. 39 (2014), 62–65.
E. Sawyer, Weighted inequalities for the one-sided Hardy–Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 62–65.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
Funding
Open access funding provided by Chalmers Tekniska Högskola.
Author information
Authors and Affiliations
Corresponding author
Additional information
T. H. was supported by the Academy of Finland (grant No. 314829 “Frontiers of singular integrals”).
Formerly Andreas Axelsson.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made (https://creativecommons.org/licenses/by/4.0/)
About this article
Cite this article
Hytönen, T., Rosén, A. Causal sparse domination of Beurling maximal regularity operators. JAMA 150, 645–672 (2023). https://doi.org/10.1007/s11854-023-0285-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-023-0285-0