5. Muckenhoupt weights and quantitative weighted norm estimates 14
5.3. Paper [D]
5.3.1. The A 2 theorem and domination by sparse operators
(w)Ap:= max{[w]A∞,[w1−p]A∞}.
5.3.1. The A2 theorem and domination by sparse operators. In [68], M. Lacey ex-tended the A2 theorem for Calderón-Zygmund operators with Dini-continuous ker-nels by showing that for every such operator T we have the weighted norm bound TLp(ω)→Lp(ω) [w]max{1,1/(p−1)}
Ap , 1 < p <∞. However, his result does not specify how the bound depends on the dierent constants related to the operator T. In Pa-per [D], we revisit this extension and as the rst main result, we prove it in a fully quantitative form:
Theorem 5.15 (Theorem 1.3 in [D]). Let T be an ω-CalderónZygmund operator with a Dini-continuous kernel. Let 1< p <∞. Then, for every w∈Ap, we have
TLp(w)→Lp(w)≤cd,p
TL2→L2+CK+ωDini {w}Ap. In particular,
TL2(w)→L2(w)≤cd
TL2→L2+CK+ωDini[w]A2.
This quantitative version of the theorem requires more than simply keeping a close track on the operator characteristics TL2→L2, CK and ωDini in Lacey's proof.
This is mostly due to the way the assumption about the Dini-continuity is used. In Lacey's proof, it is used to make the quantity´ρ
0 ω(s)dss small by choosing a suitable small ρ > 0. This makes certain calculations straightforward but it does not give information about how the bound TLp(ω)→Lp(ω) [w]max{1,1/(p−1)}
Ap depends on the quantityωDini. Hence, we have to add a new twist to the argument.
The basic idea of our proof, however, is based on Lacey's techniques which, in turn, are based on the approach of A. Lerner [70]: dominating Calderón-Zygmund operators poitwise with the help of a bounded number of sparse operators. Recall the collectionsDαfrom (4.1).
Denition 5.16. LetSα⊂Dα. We say that the operator ASα is sparse if ASαf(x) =
Q∈Sα
1Q|f|Q
and the collectionSαsatises the sparseness condition: for each Q∈Sα we have
Q∈Sα QQ
Q≤1 2|Q|.
We note that the sparseness condition is equivalent with a certain Carleson con-dition [73, Section 6]. Sparse operators are very useful operators for dierent dom-ination results because they are considerably more simple than general Calderón-Zygmund operators and they satisfy the following weighted norm estimate:
Theorem 5.17 ([83, Theorem 3.1]). Let S ⊂Dα be a sparse collection of dyadic cubes,1< p <∞and w∈Ap. Then
ASαLp(w)→Lp(w)≤cp[w]Ap, where
cp:=pp2max
p p,pp
.
Theorem 5.15 follows from Theorem 5.17 and the following quantitative version of the Lerner-type domination theorem of Lacey [68, Theorem 4.2]:
Theorem 5.18 (Theorem 2.4 in [D]). Let T be an ω-CalderónZygmund operator with a Dini-continuous kernel. Then for any compactly supported integrable function f ∈L1(Rd)there exist sparse collections Sα⊆Dα,α∈ {0,1,2}d, such that
Tf(x) ≤ cd
TL2→L2+CK+ωDini
α∈{0,1,2}d
ASαf(x) (5.19) for almost everyx∈Rd, where the constantcd depends only on the dimension.
20
In the proof of this theorem, we use adjacent dyadic techniques in a way that exploits the structure of the Euclidean space. In particular, we prove and use the following lemma:
The proof of this lemma is completely elementary: we can simply choose QB =Q0
or shift one of the dyadic descendants of Q0. However, we cannot use the same arguments in metric spaces. The dierence between the structures of the metric and Euclidean adjacent dyadic systems can make proving even some simple claims a challenge: since the Euclidean intuition can be very misleading, nding the key idea can be dicult. This does not mean that the actual proofs are particularly long or challenging but they can be very dierent from their Euclidean counterparts (compare e.g. [52, Lemma 2.5] and [A, Theorem 5.9]).
5.3.2. Weighted estimates for rough homogeneous singular integrals. We apply The-orem 5.15 to investigate weighted theory of rough homogeneous singular integrals.
Denition 5.21. Suppose that Ω an L∞(Sd−1) function such that ´
is a rough homogeneous singular intergral (associated with the function Ω).
The kernel Ksatises the size estimate (5.13) for almost every x∈Rd but it does not have angular smoothness. For x∈Rd, we can write
TΩf(x) = p.v.
Weighted norm estimates for these operators have been studied previously by J.
Duoandikoetxea, J. L. Rubio de Francia and D. K. Watson [32, 31, 104] but their investigations were qualitative, i.e. the results do not specify the dependence of the bounds on the Ap weight characteristic [w]Ap. However, the techniques in [31]
and [104] are suitable also for quantitative exploration once we combine them with Theorem 5.15.
Our second main result in [D] is the following quantitative weighted norm estimate:
Theorem 5.22 (Theorem 1.4 in [D]). For every w∈Ap, we have
TΩLp(w)→Lp(w) ≤ cd,pΩL∞{w}Ap(w)Ap. (5.23) In particular,
TΩL2(w)→L2(w) ≤ cdΩL∞[w]2A
2.
The central idea of the proof of Theorem 5.22 is to decompose the operator TΩinto a series of ωjN-Calderón-Zygmund operators T˜jN with Dini-continuous kernels such that the satisfy ωjNDiniΩL∞(1 +N(j)). This way we can apply Theorem 5.15 as a black box for the operators T˜jN and obtain the desired bounds by using suitable Reverse Hölder arguments and an interpolation result with change of measures by E.
Stein and G. Weiss [100, Theorem 2.11].
As an application of Theorem 5.22, it is straightforward to prove following result for the composition ofmBeurling-Ahlfors transforms B. For simplicity, let us denote Bm :=B◦B◦ · · · ◦B.
Corollary 5.24 (Corollary 4.2 in [D]). For everyw∈Ap, we have BmLp(w)→Lp(w)≤Cpm· {w}Ap·min
1 + logm,(w)Ap , and in particular
BmL2(w)→L2(w)≤C m·[w]A2·min1 + logm,[w]A2 . This result improves an earlier result by O. Dragi£evi¢ [29, Theorem 1].
5.3.3. Related developments and open problems. Although the bounds in Theorem 5.22 and Corollary 5.24 are the best known weighted bounds for these operators, the sharpness of these bounds is an open question. In Paper [D], we present two conjectures:
Conjecture 5.25 (Conjectures 4.5 and 4.6 in [D]). For everyw∈A2 we have TΩfL2(w)≤cdΩL∞[w]A2fL2(w) (5.26) and
BmfL2(w)≤C m·[w]A2fL2(w).
As we note in [D, Remark 3.20], the techniques we use in the proof of Theorem 5.22 are not sucient for solving this conjecture.
The proof of Theorem 5.18 was recently shortened by A. Lerner [72, Theorem 3.1].
His key idea is to use the operatorMT, MTf(x) := sup
Qxess sup
ξ∈Q
T(f1Rd\3Q)(ξ),
and its localized versions to analyze the behaviour ofT and its truncations. He notes in [71, Remark 4.7] that it would suce to show that MTΩ is of weak type (1,1) to prove the bound (5.26). An analogue of the bound (5.23) for the Marcinkiewicz integral operator was proved very recently by G. Hu and M. Qu [47, Theorem 1.2].
Although Lacey's results in [68] concern Calderón-Zygmund operators, the tech-niques in his proofs have already turned out to be useful also for other questions.
In a recent preprint by F. Bernicot, D. Frey and S. Petermichl [8], Lacey's domina-tion technique is applied to prove sharp weighted estimates for operators outside the class of Calderón-Zygmund operators in certain metric spaces. Instead of proving a pointwise bound for the operators, they prove an integral inequality with the help of the adjacent dyadic systems of T. Hytönen and A. Kairema [51] and suitable sparse collections of cubes. A. Lerner considers related questions in [72, Section 4].
Recently, W. Damián, M. Hormozi and K. Li generalized Theorem 5.18 for bilinear Calderón-Zygmund operators [25, Theorem 3.6] by modifying the techniques that we used in [D, Section 2]. A. Lerner, S. Ombrosi and I. P. Rivera-Ríos, in turn, proved an analogue of Theorem 5.18 for commutators [b, T] where b ∈ BMO and T is a Calderón-Zygmund operator with a Dini-continuous kernel [74, Theorem 1.2]. It is an open problem to generalize Theorem 5.18 for a metric setting.
22
References
[1] Hugo Aimar and Roberto A. Macías. Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type. Proc. Amer. Math. Soc., 91(2):213216, 1984.
[2] Alex Amenta and Mikko Kemppainen. Non-uniformly local tent spaces. Publ. Mat., 59(1):245 270, 2015.
[3] Theresa C. Anderson, David Cruz-Uribe, and Kabe Moen. Logarithmic bump conditions for Calderón-Zygmund operators on spaces of homogeneous type. Publ. Mat., 59(1):1743, 2015.
[4] Kari Astala, Tadeusz Iwaniec, and Eero Saksman. Beltrami operators in the plane. Duke Math.
J., 107(1):2756, 2001.
[5] Pascal Auscher and Tuomas Hytönen. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. Comput. Harmon. Anal., 34(2):266296, 2013.
[6] Pascal Auscher and Tuomas Hytönen. Addendum to Orthonormal bases of regular wavelets in spaces of homogeneous type [Appl. Comput. Harmon. Anal. 34(2) (2013) 266296]. Appl.
Comput. Harmon. Anal., 39(3):568569, 2015.
[7] Frédéric Bernicot and Dorothee Frey. Riesz transforms through reverse Hölder and Poincaré inequalities. preprint, 2015. arxiv:1503.02508.
[8] Frédéric Bernicot, Dorothee Frey, and Stefanie Petermichl. Sharp weighted norm estimates beyond Calderón-Zygmund theory. preprint, 2015. arXiv:1510.00973.
[9] O. Beznosova and A. Reznikov. Sharp estimates involvingA∞andLlogLconstants, and their applications to PDE. Algebra i Analiz, 26(1):4067, 2014.
[10] Oleksandra Beznosova and Alexander Reznikov. Equivalent denitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadicLlogLandA∞constants. Rev. Mat. Iberoam., 30(4):11911236, 2014.
[11] Oleksandra V. Beznosova. Linear bound for the dyadic paraproduct on weighted Lebesgue spaceL2(w). J. Funct. Anal., 255(4):9941007, 2008.
[12] Jean Bourgain. Some remarks on Banach spaces in which martingale dierence sequences are unconditional. Ark. Mat., 21(2):163168, 1983.
[13] Stephen M. Buckley. Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc., 340(1):253272, 1993.
[14] Donald L. Burkholder. A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pages 270286.
Wadsworth, Belmont, CA, 1983.
[15] S.-Y. A. Chang, J. M. Wilson, and T. H. Wol. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv., 60(2):217246, 1985.
[16] Chuang Chen, Ji Li, and Fanghui Liao. Some function spaces via orthonormal bases on spaces of homogeneous type. Abstr. Appl. Anal., pages Art. ID 265378, 13, 2014.
[17] Michael Christ. AT(b)theorem with remarks on analytic capacity and the Cauchy integral.
Colloq. Math., 60/61(2):601628, 1990.
[18] Daewon Chung, M. Cristina Pereyra, and Carlos Perez. Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Amer. Math. Soc., 364(3):11631177, 2012.
[19] R. R. Coifman and C. Feerman. Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51:241250, 1974.
[20] Ronald R. Coifman and Guido Weiss. Analyse harmonique non-commutative sur certains es-paceshomogènes. Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin-NewYork, 1971. Étude de certaines intégrales singulières.
[21] Ronald R. Coifman and Guido Weiss. Extensions of Hardy spaces and their use in analysis.
Bull. Amer. Math. Soc., 83(4):569645, 1977.
[22] Jose M. Conde. A note on dyadic coverings and nondoubling Calderón-Zygmund theory. J.
Math. Anal. Appl., 397(2):785790, 2013.
[23] David Cruz-Uribe. Two weight norm inequalities for fractional integral operators and commu-tators. preprint, 2014. arXiv:1412.4157.
[24] Stephan Dahlke, Wolfgang Dahmen, Ilona Weinreich, and Eberhard Schmitt. Multiresolution analysis and wavelets onS2andS3. Numer. Funct. Anal. Optim., 16(1-2):1941, 1995.
[25] Wendolín Damián, Mahdi Hormozi, and Kangwei Li. New bounds for bilinear calderón-zygmund operators and applications. preprint, 2015. arXiv:1512.02400.
[26] Guy David. Morceaux de graphes lipschitziens et intégrales singulières sur une surface. Rev.
Mat. Iberoamericana, 4(1):73114, 1988.
[27] Guy David and Jean-Lin Journé. A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math. (2), 120(2):371397, 1984.
[28] Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. Uni-versity of Chicago Press, Chicago, IL, 2000.
[29] Oliver Dragi£evi¢. Weighted estimates for powers and the Ahlfors-Beurling operator. Proc.
Amer. Math. Soc., 139(6):21132120, 2011.
[30] Oliver Dragi£evi¢, Loukas Grafakos, María Cristina Pereyra, and Stefanie Petermichl. Extrap-olation and sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ.
Mat., 49(1):7391, 2005.
[31] Javier Duoandikoetxea. Weighted norm inequalities for homogeneous singular integrals. Trans.
Amer. Math. Soc., 336(2):869880, 1993.
[32] Javier Duoandikoetxea and José L. Rubio de Francia. Maximal and singular integral operators via Fourier transform estimates. Invent. Math., 84(3):541561, 1986.
[33] R. A. Feerman, C. E. Kenig, and J. Pipher. The theory of weights and the Dirichlet problem for elliptic equations. Ann. of Math. (2), 134(1):65124, 1991.
[34] Tadeusz Figiel. On equivalence of some bases to the Haar system in spaces of vector-valued functions. Bull. Polish Acad. Sci. Math., 36(3-4):119131 (1989), 1988.
[35] Tadeusz Figiel. Singular integral operators: a martingale approach. In Geometry of Banach spaces (Strobl, 1989), volume 158 of London Math. Soc. Lecture Note Ser., pages 95110.
Cambridge Univ. Press, Cambridge, 1990.
[36] Xing Fu, Dachun Yang, and Yiyu Liang. Products of functions in BMO(X)andHat1(X)via wavelets over spaces of homogeneous type. preprint, 2015. arXiv:1506.05910.
[37] Nobuhiko Fujii. Weighted bounded mean oscillation and singular integrals. Math. Japon., 22(5):529534, 1977/78.
[38] José García-Cuerva and José L. Rubio de Francia. Weightednorm inequalities andrelatedtop-ics, volume 116 of North-HollandMathematics Studies. North-Holland Publishing Co., Ams-terdam, 1985. Notas de Matemática [Mathematical Notes], 104.
[39] John B. Garnett and Peter W. Jones. BMO from dyadic BMO. Pacic J. Math., 99(2):351371, 1982.
[40] Mikhael Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci.
Publ. Math., (53):5373, 1981.
[41] Yongsheng Han, Detlef Müller, and Dachun Yang. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal., pages Art. ID 893409, 250, 2008.
[42] Toni Heikkinen and Janne Korvenpää. Finitely randomized dyadic systems and BMO on metric measure spaces. Nonlinear Anal., 120:3042, 2015.
[43] Juha Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
[44] Juha Heinonen, Tero Kilpeläinen, and Olli Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications.
[45] Steve Hofmann and José María Martell. Uniform rectiability and harmonic measure I: Uni-form rectiability implies Poisson kernels inLp. Ann. Sci. Éc. Norm. Supér. (4), 47(3):577654, 2014.
[46] Sergei V. Hru²£ev. A description of weights satisfying theA∞condition of Muckenhoupt. Proc.
Amer. Math. Soc., 90(2):253257, 1984.
[47] Guoen Hu and Meng Qu. WeightedLpbounds for the Marcinkiewicz integral. preprint, 2016.
arXiv:1602.00549.
[48] Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc., 176:227251, 1973.
[49] Tuomas Hytönen. The sharp weighted bound for general Calderón-Zygmund operators. Ann.
of Math. (2), 175(3):14731506, 2012.
[50] Tuomas Hytönen. The two-weight inequality for the Hilbert transform with general measures.
preprint, 2013. arXiv:1312.0843.
24
[51] Tuomas Hytönen and Anna Kairema. Systems of dyadic cubes in a doubling metric space.
Colloq. Math., 126(1):133, 2012.
[52] Tuomas Hytönen, Michael T. Lacey, and Carlos Pérez. Sharp weighted bounds for the q -variation of singular integrals. Bull. London Math. Soc., 45(3):529540, 2013.
[53] Tuomas Hytönen, Suile Liu, Dachun Yang, and Dongong Yang. Boundedness of Calderón-Zygmund operators on non-homogeneous metric measure spaces. Canad. J. Math., 64(4):892 923, 2012.
[54] Tuomas Hytönen and Henri Martikainen. Non-homogeneousT btheorem and random dyadic cubes on metric measure spaces. J. Geom. Anal., 22(4):10711107, 2012.
[55] Tuomas Hytönen and Carlos Pérez. Sharp weighted bounds involving A∞. Anal. PDE, 6(4):777818, 2013.
[56] Tuomas Hytönen, Carlos Pérez, and Ezequiel Rela. Sharp reverse Hölder property for A∞ weights on spaces of homogeneous type. J. Funct. Anal., 263(12):38833899, 2012.
[57] Tuomas Hytönen and Lutz Weis. A T1theorem for integral transformations with operator-valued kernel. J. Reine Angew. Math., 599:155200, 2006.
[58] Tuomas P. Hytönen. An operator-valuedT btheorem. J. Funct. Anal., 234(2):420463, 2006.
[59] Tuomas P. Hytönen. Vector-valued singular integrals revisitedwith random dyadic cubes.
Bull. Pol. Acad. Sci. Math., 60(3):269283, 2012.
[60] Antti Käenmäki, Tapio Rajala, and Ville Suomala. Existence of doubling measures via gener-alised nested cubes. Proc. Amer. Math. Soc., 140(9):32753281, 2012.
[61] Jean-Pierre Kahane. Some random series offunctions, volume 5 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1985.
[62] Anna Kairema. Two-weight norm inequalities for potential type and maximal operators in a metric space. Publ. Mat., 57(1):356, 2013.
[63] Anna Kairema. Sharp weighted bounds for fractional integral operators in a space of homoge-neous type. Math. Scand., 114(2):226253, 2014.
[64] Anna Kairema, Ji Li, M. Cristina Pereyra, and Lesley Ward. Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of ho-mogeneous type. preprint, 2014. arXiv:1402.2886.
[65] Mikko Kemppainen. On vector-valued tent spaces and Hardy spaces associated with non-negative self-adjoint operators. preprint, 2014. arXiv:1402.2886.
[66] Riikka Korte and Outi Elina Kansanen. StrongA∞-weights areA∞-weights on metric spaces.
Rev. Mat. Iberoam., 27(1):335354, 2011.
[67] Michael Lacey, Eric T. Sawyer, and Ignacio Uriarte-Tuero. A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure. Anal. PDE, 5(1):160, 2012.
[68] Michael T. Lacey. An elementary proof of theA2bound. preprint, 2015. arxiv:1501.05818.
[69] Richard Lechner and Markus Passenbrunner. Adaptive deterministic dyadic grids on spaces of homogeneous type. Bull. Pol. Acad. Sci. Math., 62(2):139160, 2014.
[70] Andrei K. Lerner. On an estimate of Calderón-Zygmund operators by dyadic positive opera-tors. J. Anal. Math., 121:141161, 2013.
[71] Andrei K. Lerner. A simple proof of theA2conjecture. Int. Math. Res. Not. IMRN, (14):3159 3170, 2013.
[72] Andrei K. Lerner. On pointwise estimates involving sparse operators. preprint, 2015.
arXiv:1512.07247.
[73] Andrei K. Lerner and Fedor Nazarov. Intuitive dyadic calculus: the basics. preprint, 2015.
arXiv:1508.05639.
[74] Andrei K. Lerner, Sheldy Ombrosi, and Israel P. Rivera-Ríos. On weighted weak type estimates of the commutators of Calderón-Zygmund operators. preprint, 2016. arXiv:1604.01334.
[75] Kangwei Li. A newAp−A∞estimate for Calderón-Zygmund operators in spaces of homoge-neous type. J. Math. Anal. Appl., 428(2):11831192, 2015.
[76] Jouni Luukkainen. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc., 35(1):2376, 1998.
[77] Jouni Luukkainen and Eero Saksman. Every complete doubling metric space carries a doubling measure. Proc. Amer. Math. Soc., 126(2):531534, 1998.
[78] Roberto A. Macías and Carlos Segovia. Lipschitz functions on spaces of homogeneous type.
Adv. in Math., 33(3):257270, 1979.
[79] Henri Martikainen. Vector-valued non-homogeneous T b theorem on metric measure spaces.
Rev. Mat. Iberoam., 28(4):961998, 2012.
[80] Tao Mei. BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci.
Paris, 336(12):10031006, 2003.
[81] Yves Meyer. Wavelets and operators, volume 37 of Cambridge Studies in Advanced Mathemat-ics. Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger.
[82] Dorina Mitrea, Irina Mitrea, Marius Mitrea, and Sylvie Monniaux. Groupoid metrization the-ory. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2013. With applications to analysis on quasi-metric spaces and functional analysis.
[83] Kabe Moen. Sharp weighted bounds without testing or extrapolation. Arch. Math. (Basel), 99(5):457466, 2012.
[84] Andrew J. Morris. The Kato square root problem on submanifolds. J. Lond. Math. Soc. (2), 86(3):879910, 2012.
[85] Benjamin Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans.
Amer. Math. Soc., 165:207226, 1972.
[86] Paul F. X. Müller and Markus Passenbrunner. A decomposition theorem for singular integral operators on spaces of homogeneous type. J. Funct. Anal., 262(4):14271465, 2012.
[87] C. Muscalu, T. Tao, and C. Thiele. The bi-Carleson operator. Geom. Funct. Anal., 16(1):230 277, 2006.
[88] Camil Muscalu, Terence Tao, and Christoph Thiele. Multi-linear operators given by singular multipliers. J. Amer. Math. Soc., 15(2):469496, 2002.
[89] F. Nazarov, S. Treil, and A. Volberg. Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces. Internat. Math. Res. Notices, (15):703726, 1997.
[90] F. Nazarov, S. Treil, and A. Volberg. TheT b-theorem on non-homogeneous spaces. Acta Math., 190(2):151239, 2003.
[91] Fedor Nazarov, Alexander Reznikov, and Alexander Volberg. The proof ofA2conjecture in a geometrically doubling metric space. Indiana Univ. Math. J., 62(5):15031533, 2013.
[92] Kate Okikiolu. Characterization of subsets of rectiable curves in Rn. J. London Math. Soc.
(2), 46(2):336348, 1992.
[93] S. Petermichl. The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classicalApcharacteristic. Amer. J. Math., 129(5):13551375, 2007.
[94] Stefanie Petermichl and Alexander Volberg. Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J., 112(2):281305, 2002.
[95] José L. Rubio de Francia. Factorization theory andApweights. Amer. J. Math., 106(3):533 547, 1984.
[96] E. Sawyer and R. L. Wheeden. Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math., 114(4):813874, 1992.
[97] Eric T. Sawyer. A characterization of a two-weight norm inequality for maximal operators.
Studia Math., 75(1):111, 1982.
[98] Eric T. Sawyer. Two weight norm inequalities for certain maximal and integral operators. In Harmonic analysis (Minneapolis, Minn., 1981), volume 908 of Lecture Notes in Math., pages 102127. Springer, Berlin-New York, 1982.
[99] Eric T. Sawyer, Richard L. Wheeden, and Shiying Zhao. Weighted norm inequalities for oper-ators of potential type and fractional maximal functions. Potential Anal., 5(6):523580, 1996.
[100] E. M. Stein and G. Weiss. Interpolation of operators with change of measures. Trans. Amer.
Math. Soc., 87:159172, 1958.
[101] Krzysztof Stempak. On some structural properties of spaces of homogeneous type. Taiwanese J. Math., 19(2):603613, 2015.
[102] Jan-Olov Strömberg and Alberto Torchinsky. Weighted Hardy spaces, volume 1381 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1989.
[103] Igor E. Verbitsky and Richard L. Wheeden. Weighted norm inequalities for integral operators.
Trans. Amer. Math. Soc., 350(8):33713391, 1998.
[104] David K. Watson. Weighted estimates for singular integrals via Fourier transform estimates.
Duke Math. J., 60(2):389399, 1990.
[105] J. Michael Wilson. Weighted inequalities for the dyadic square function without dyadicA∞. Duke Math. J., 55(1):1950, 1987.
26
[106] Thomas H. Wol. Two algebras of bounded functions. Duke Math. J., 49(2):321328, 1982.
[107] Dachun Yang and Yuan Zhou. Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications. Math. Ann., 346(2):307333, 2010.
[108] Anna Zatorska-Goldstein. Very weak solutions of nonlinear subelliptic equations. Ann. Acad.
Sci. Fenn. Math., 30(2):407436, 2005.