-
Previous Article
Theory of $(a,b)$-continued fraction transformations and applications
- ERA-MS Home
- This Volume
-
Next Article
Multifractal formalism derived from thermodynamics for general dynamical systems
Sharp weighted estimates for approximating dyadic operators
1. | Dept. of Mathematics, Trinity College, Hartford, CT 06106-3100, United States |
2. | Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Serrano 121, E-28006 Madrid, Spain |
3. | Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain |
$ |\|T\||_{L^p(w)} \leq C_{n,T}[w]_{A_p}^{\max(1,\frac{1}{p-1})}, $
where $T$ is the Hilbert transform, a Riesz transform, the
Beurling-Ahlfors operator or any operator that can be approximated
by Haar shift operators. Our proof avoids the Bellman function
technique and two weight norm inequalities. We use instead a recent
result due to A. Lerner [15] to estimate the
oscillation of dyadic operators.
The method we use is flexible enough to obtain the sharp one-weight
result for other important operators as well as a very sharp
two-weight bump type result for $T$ as can be found in
[5].
[1] |
Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086 |
[2] |
Radjesvarane Alexandre, Lingbing He. Integral estimates for a linear singular operator linked with Boltzmann operators part II: High singularities $1\le\nu<2$. Kinetic and Related Models, 2008, 1 (4) : 491-513. doi: 10.3934/krm.2008.1.491 |
[3] |
Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591 |
[4] |
Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435 |
[5] |
Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241 |
[6] |
Sasikarn Yeepo, Wicharn Lewkeeratiyutkul, Sujin Khomrutai, Armin Schikorra. On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2915-2939. doi: 10.3934/cpaa.2021071 |
[7] |
Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011 |
[8] |
Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213 |
[9] |
Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049 |
[10] |
Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299 |
[11] |
Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations and Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004 |
[12] |
Pablo Blanc, Juan J. Manfredi, Julio D. Rossi. Games for Pucci's maximal operators. Journal of Dynamics and Games, 2019, 6 (4) : 277-289. doi: 10.3934/jdg.2019019 |
[13] |
Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487 |
[14] |
Mohammad Safdari. The regularity of some vector-valued variational inequalities with gradient constraints. Communications on Pure and Applied Analysis, 2018, 17 (2) : 413-428. doi: 10.3934/cpaa.2018023 |
[15] |
Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3703-3718. doi: 10.3934/dcdss.2021020 |
[16] |
JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure and Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042 |
[17] |
Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040 |
[18] |
Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3043-3054. doi: 10.3934/dcdss.2020463 |
[19] |
Antonio G. García. Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $. Mathematical Foundations of Computing, 2021, 4 (4) : 281-297. doi: 10.3934/mfc.2021019 |
[20] |
Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 |
2020 Impact Factor: 0.929
Tools
Metrics
Other articles
by authors
[Back to Top]