On Entropy Bumps for Calderón-Zygmund Operators

• Autorzy: Michael T. Lacey , Scott Spencer
• Języki publikacji: EN

Abstrakty

EN

We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ℇ be a monotonic increasing function on (1,∞) which satisfy [...] Let σ and w be two weights on Rd. If this supremum is finite, for a choice of 1 < p < ∞, [...] then any Calderón-Zygmund operator T satisfies the bound [...]

Słowa kluczowe

EN

weighted inequality; Ap; bumps; entropy

• Wydawca: De Gruyter Open
• Czasopismo: Concrete Operators
• Rocznik: 2015
• Tom: 2
• Numer: 1

Daty

otrzymano

2015-03-11

zaakceptowano

2015-05-19

online

2015-07-08

Twórcy

autor

Michael T. Lacey

• School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA

autor

Scott Spencer

• School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA

Bibliografia

• [1] Cruz-Uribe, D., Pérez, C., Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators. Indiana Univ. Math. J., 49, 2000, no. 2, 697–721. DOI:10.1512/iumj.2000.49.1795
• [2] Cruz-Uribe, David, Reznikov, Alexander, Volberg, Alexander, Logarithmic bumpconditions and the two-weight boundedness of Calderón–Zygmund operators. Adv. Math., 255, 2014, 706–729. DOI:10.1016/j.aim.2014.01.016
• [3] Ding, Shusen, Two-weight Caccioppoli inequalities for solutions of nonhomogeneous A-harmonic equations on Riemannian manifolds. Proc. Amer. Math. Soc., 132, 2004, no. 8, 2367–2375. DOI:10.1090/S0002-9939-04-07347-2
• [4] Hunt, Richard, Muckenhoupt, Benjamin, Wheeden, Richard, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc., 176, 1973, 227–251.
• [5] Hytönen, Tuomas P., The A2 theorem: remarks and complements. Contemp. Math., 612, Amer. Math. Soc., Providence, RI, 2014, 91–106. DOI:10.1090/conm/612/12226
• [6] Lacey, Michael T., On the Separated Bumps Conjecture for Calderon-Zygmund Operators . HokkaidoMath J, to appear, 2013, 1310.3507
• [7] Lacey, Michael T., An elementary proof of the A2 Bound. 2015, 1501.05818
• [8] Hytönen, Tuomas P., Lacey, Michael T., Martikainen, Henri, Orponen, Tuomas, Reguera, Maria Carmen, Sawyer, Eric T., Uriarte-Tuero, Ignacio, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on Ap weighted spaces. J. Anal. Math., 118, 2012, no. 1, 177–220. DOI:10.1007/s11854-012-0033-3
• [9] Hytönen, Tuomas P., Lacey, Michael T., The Ap-A1 inequality for general Calderón-Zygmund operators. Indiana Univ. Math. J., 61, 2012, no. 6, 2041–2092. DOI:10.1512/iumj.2012.61.4777
• [10] Hytönen, Tuomas, Pérez, Carlos, Sharp weighted bounds involving A1. Anal. PDE, 6, 2013, no. 4, 777–818. DOI:10.2140/apde.2013.6.777
• [11] Hytönen, Tuomas, Pérez, Carlos, Treil, Sergei, Volberg, Alexander, Sharp weighted estimates for dyadic shifts and the A2 conjecture. J. Reine Angew. Math., 687, 2014, 43–86. DOI:10.1515/crelle-2012-0047
• [12] Lacey, Michael T., An Ap-A1 inequality for the Hilbert transform. Houston J. Math., 38, 2012, no. 3, 799–814.
• [13] Lacey, Michael T., Petermichl, Stefanie, Reguera, Maria Carmen, Sharp A2 inequality for Haar shift operators. Math. Ann., 348, 2010, no. 1, 127–141. DOI:10.1007/s00208-009-0473-y
• [14] Lacey, Michael T., Sawyer, Eric T., Uriarte-Tuero, Ignacio, Two Weight Inequalities for Discrete Positive Operators. 2009, Submitted, 0911.3437
• [15] Lerner, Andrei K., On an estimate of Calderón-Zygmund operators by dyadic positive operators. J. Anal. Math., 121, 2013, 141–161. DOI:10.1007/s11854-013-0030-1
• [16] Lerner, Andrei K., A simple proof of the A2 conjecture. Int. Math. Res. Not. IMRN, 2013, no. 14, 3159–3170.
• [17] Lerner, Andrei K., Mixed Ap-Ar inequalities for classical singular integrals and Littlewood-Paley operators. J. Geom. Anal., 23, 2013, no. 3, 1343–1354. DOI:10.1007/s12220-011-9290-0
• [18] Lerner, Andrei K., Moen, Kabe, Mixed Ap-A1 estimates with one supremum. Studia Math., 219, 2013, no. 3, 247–267. DOI:10.4064/sm219-3-5
• [19] Muckenhoupt, Benjamin, Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165, 1972, 207–226.
• [20] Nazarov, Fedor, Reznikov, Alexander, Treil, Sergei, Volberg, ALexander, A Bellman function proof of the L2 bump conjecture. J. Anal. Math., 121, 2013, 255–277. DOI:10.1007/s11854-013-0035-9
• [21] Neugebauer, C. J., title=Inserting Ap-weights, Proc. Amer. Math. Soc., 87, 1983, no. 4, 644–648. DOI:10.2307/2043351
• [22] Pérez, C., Weighted norm inequalities for singular integral operators. J. London Math. Soc. (2), 49, 1994, no. 2, 296–308. DOI:10.1112/jlms/49.2.296
• [23] Sawyer, Eric T., A characterization of a two-weight norm inequality for maximal operators. Studia Math., 75, 1982, no. 1, 1–11.
• [24] Sawyer, Eric T., A characterization of two weight norm inequalities for fractional and Poisson integrals. Trans. Amer. Math. Soc., 308, 1988, no. 2, 533–545. DOI:10.2307/2001090
• [25] Treil, Sergei, Volberg, Alexander, Entropy conditions in two weight inequalities for singular integral operators. 1408.0385 2014,
• [26] Zheng, Dechao, The distribution function inequality and products of Toeplitz operators and Hankel operators. J. Funct. Anal., 138, 1996, no. 2, 477–501. DOI:10.1006/jfan.1996.0073

Identyfikatory

DOI

10.1515/conop-2015-0002