In this paper, we introduce weak Hardy spaces $H^{p,\infty}$ on spaces of homogeneous type. We establish an atomic decomposition characterization of these spaces, show the boundedness of fractional integral operators and provide an $H^{p,\infty}$ interpolation theorem. Applications to the Nagel-Stein's singular integral operators and fractional integral operators are also discussed.
References
G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc., 120 (1994), 973-979. 0794.43003 10.1090/S0002-9939-1994-1172944-4 G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc., 120 (1994), 973-979. 0794.43003 10.1090/S0002-9939-1994-1172944-4
D.-C. Chang, A. Nagel and E. M. Stein, Estimates for the $\overline{\partial}$-Neumann problem in pseudoconvex domains of finite type in $\cc^2$, Acta Math., 169 (1992), 153-228. 0821.32011 10.1007/BF02392760D.-C. Chang, A. Nagel and E. M. Stein, Estimates for the $\overline{\partial}$-Neumann problem in pseudoconvex domains of finite type in $\cc^2$, Acta Math., 169 (1992), 153-228. 0821.32011 10.1007/BF02392760
R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. 0358.30023 10.1090/S0002-9904-1977-14325-5R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. 0358.30023 10.1090/S0002-9904-1977-14325-5
R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes., Lecture Notes in Math. Vol. 242, Springer-Verlag, Berlin, 1971. 0224.43006R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes., Lecture Notes in Math. Vol. 242, Springer-Verlag, Berlin, 1971. 0224.43006
Y. Ding and S. Lan, Weak Anisotropic Hardy spaces and interpolation theorems, Sci. in China $($in Chinese$)$, 37 (2007), 1403-1416. 1153.42010 10.1007/s11425-008-0009-zY. Ding and S. Lan, Weak Anisotropic Hardy spaces and interpolation theorems, Sci. in China $($in Chinese$)$, 37 (2007), 1403-1416. 1153.42010 10.1007/s11425-008-0009-z
Y. Ding and X. Wu, Weak Hardy space and endpoint estimates for singular integrals on space of homogeneous type, Turk. J. Math., 34 (2010), 235-247. MR2665991 1191.42009Y. Ding and X. Wu, Weak Hardy space and endpoint estimates for singular integrals on space of homogeneous type, Turk. J. Math., 34 (2010), 235-247. MR2665991 1191.42009
Y. Ding and X. Wu, Fractional integrals on product manifolds, Potential Anal., 30 (2009), 371-383. 1185.42009 10.1007/s11118-009-9120-1Y. Ding and X. Wu, Fractional integrals on product manifolds, Potential Anal., 30 (2009), 371-383. 1185.42009 10.1007/s11118-009-9120-1
R. Fefferman and F. Soria, The spaces weak $H^1$, Studia Math., 85 (1987), 1-16. 0626.42013R. Fefferman and F. Soria, The spaces weak $H^1$, Studia Math., 85 (1987), 1-16. 0626.42013
Y. Han, D. Müller and D. Yang, Littlewood-Paley charicterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr., 279 (2006), 1505-1573. MR2269253 1179.42016 10.1002/mana.200610435Y. Han, D. Müller and D. Yang, Littlewood-Paley charicterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr., 279 (2006), 1505-1573. MR2269253 1179.42016 10.1002/mana.200610435
Y. Han and E. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc., 110 (1994), vi+126pp. 0806.42013Y. Han and E. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc., 110 (1994), vi+126pp. 0806.42013
J. Heinonen, Lectures on Analysis on Metric spaces, Springer-Verlag, New York, 2001. 0985.46008J. Heinonen, Lectures on Analysis on Metric spaces, Springer-Verlag, New York, 2001. 0985.46008
K. Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math., 124 (2002), 129-197. MR1879002 1014.32031 10.1353/ajm.2002.0003K. Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math., 124 (2002), 129-197. MR1879002 1014.32031 10.1353/ajm.2002.0003
A. Nagel and E. M. Stein, The $\square_b$-heat equation on pseudoconvex manifolds of finite type in $\cc^2$, Math. Z., 238 (2001), 37-88. 1039.32051 10.1007/PL00004901A. Nagel and E. M. Stein, The $\square_b$-heat equation on pseudoconvex manifolds of finite type in $\cc^2$, Math. Z., 238 (2001), 37-88. 1039.32051 10.1007/PL00004901
A. Nagel and E. M. Stein, Differentiable control metrics and scaled bump functions, J. Differential Geom., 57 (2001), 465-492. 1041.58006 10.4310/jdg/1090348130A. Nagel and E. M. Stein, Differentiable control metrics and scaled bump functions, J. Differential Geom., 57 (2001), 465-492. 1041.58006 10.4310/jdg/1090348130
A. Nagel and E. M. Stein, On the product theory of singular integrals. Rev. Mat. Iberoamericana, 20 (2004), 531-561. 1057.42016A. Nagel and E. M. Stein, On the product theory of singular integrals. Rev. Mat. Iberoamericana, 20 (2004), 531-561. 1057.42016
A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I. Basic properties, Acta Math., 155 (1985), 103-147. 0578.32044 10.1007/BF02392539A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I. Basic properties, Acta Math., 155 (1985), 103-147. 0578.32044 10.1007/BF02392539
E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. 0821.42001 E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. 0821.42001
E. M. Stein, M. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes. Suppl. Rend. Circ. Mat. Palermo, 1 (1981), 81-97. E. M. Stein, M. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes. Suppl. Rend. Circ. Mat. Palermo, 1 (1981), 81-97.
N. Th. Varopoulos, Analysis on Lie groups, J. Func. Anal., 76 (1988), 346-410. 0634.22008 10.1016/0022-1236(88)90041-9 N. Th. Varopoulos, Analysis on Lie groups, J. Func. Anal., 76 (1988), 346-410. 0634.22008 10.1016/0022-1236(88)90041-9
N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, 100, Cambridge tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1992. N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, 100, Cambridge tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1992.
