Let $\mu$ be a non-negative Borel measure on $\mathbb{R}^d$ which only satisfies some growth condition, we study two-weight norm inequalities for fractional maximal functions associated to such $\mu$. A necessary and sufficient condition for the maximal operator to be bounded from $L^p(v)$ into weak $L^{q}(u)$ $(1 \leq p \leq q \lt \infty)$ is given. Furthermore, by using certain Orlicz norm, a strong type inequality is obtained.
References
R. Coifman and C. Fefferman, Weighter norm inequalities for maximal furctions and singualar integrals, Studia Math., 51 (1974), 241-250. 0291.44007 10.4064/sm-51-3-241-250 R. Coifman and C. Fefferman, Weighter norm inequalities for maximal furctions and singualar integrals, Studia Math., 51 (1974), 241-250. 0291.44007 10.4064/sm-51-3-241-250
D. Cru-Uribe, New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J., 7(1) (2000), 33-42. 0987.42019 D. Cru-Uribe, New proofs of two-weight norm inequalities for the maximal operator, Georgian Math. J., 7(1) (2000), 33-42. 0987.42019
X. Duong and L. Yan, Weak type $($1,1$)$ estimates of maximal truncated singular operators, International conference on harmonic analysis and related topics, Proc. Centre Math. Appl., Vol. 41, Austral. Nat. Univ., Canberra, 2002, pp. 46-56. X. Duong and L. Yan, Weak type $($1,1$)$ estimates of maximal truncated singular operators, International conference on harmonic analysis and related topics, Proc. Centre Math. Appl., Vol. 41, Austral. Nat. Univ., Canberra, 2002, pp. 46-56.
J. Garc\ipref a-Cuerva and J. M. Martell, Two-weight norm inequalities for maximal operators and fractional integrals on non-homogencous spaces, Indiana Univ. Math. J., 50(3) (2001), 1241-1280. 1023.42012 J. Garc\ipref a-Cuerva and J. M. Martell, Two-weight norm inequalities for maximal operators and fractional integrals on non-homogencous spaces, Indiana Univ. Math. J., 50(3) (2001), 1241-1280. 1023.42012
J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for non doubling measures, Duke Math. J., 102(3) (2000), 533-565. MR1756109 0964.42009 10.1215/S0012-7094-00-10238-4 euclid.dmj/1092749342
J. Mateu, P. Mattila, A. Nicolau and J. Orobitg, BMO for non doubling measures, Duke Math. J., 102(3) (2000), 533-565. MR1756109 0964.42009 10.1215/S0012-7094-00-10238-4 euclid.dmj/1092749342
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. 0236.26016 10.1090/S0002-9947-1972-0293384-6 B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. 0236.26016 10.1090/S0002-9947-1972-0293384-6
F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 1997(15) (1997), 703-726. F. Nazarov, S. Treil and A. Volberg, Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 1997(15) (1997), 703-726.
F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 1998(9) (1998), 463-487. 0918.42009 10.1155/S1073792898000312 F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, 1998(9) (1998), 463-487. 0918.42009 10.1155/S1073792898000312
F. Nazarov, S. Treil and A. Volberg, Accretive system Tb-theorems on nonhomogeneous spaces, Duke Math. J., 113 (2002), 259-312. 1055.47027 10.1215/S0012-7094-02-11323-4 F. Nazarov, S. Treil and A. Volberg, Accretive system Tb-theorems on nonhomogeneous spaces, Duke Math. J., 113 (2002), 259-312. 1055.47027 10.1215/S0012-7094-02-11323-4
C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $L^p$-spaces with different weights, Proc. London Math. Soc., 71 (1995), 135-157. C. Pérez, On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted $L^p$-spaces with different weights, Proc. London Math. Soc., 71 (1995), 135-157.
E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75(1) (1982), 1-11. 0508.42023 10.4064/sm-75-1-1-11 E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math., 75(1) (1982), 1-11. 0508.42023 10.4064/sm-75-1-1-11
E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal funtions, Potential Anal., 5 (1996), 523-580. 0873.42012 10.1007/BF00275794 E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal funtions, Potential Anal., 5 (1996), 523-580. 0873.42012 10.1007/BF00275794
X. Tolsa, $L^2$-boundedness of Cauchy integral operator for continuous measures, Duke Math. J., 98(2) (1999), 269-304. 0945.30032 10.1215/S0012-7094-99-09808-3 X. Tolsa, $L^2$-boundedness of Cauchy integral operator for continuous measures, Duke Math. J., 98(2) (1999), 269-304. 0945.30032 10.1215/S0012-7094-99-09808-3
X. Tolsa, Cotlar's inequality without the doubling condition and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math., 502 (1998), 199-235. MR1647575 0912.42009 X. Tolsa, Cotlar's inequality without the doubling condition and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math., 502 (1998), 199-235. MR1647575 0912.42009
J. Verdera, The fall of the doubling condition in Calderón-Zygmund theory, Publ. Mat., Extra Volume, (2002), 275-292. 1025.42008 J. Verdera, The fall of the doubling condition in Calderón-Zygmund theory, Publ. Mat., Extra Volume, (2002), 275-292. 1025.42008
