Mapping properties for the homogeneous fractional integral operator $T_{{\mit \Omega},\alpha}$ on the Hardy spaces $H^p(R^n)$ are studied. Our results give the extension of Stein-Weiss and Taibleson-Weiss's results for the boundedness of the Riesz potential operator $I_{\alpha}$ on the Hardy spaces $H^p(R^n)$.
References
[1] S CHANILLO, D WATSON AND R. L WHEEDEN, Some integral and maximal operators related to star-like, Studia Math 107 (1993), 223-255 MR1247201 0809.42008[1] S CHANILLO, D WATSON AND R. L WHEEDEN, Some integral and maximal operators related to star-like, Studia Math 107 (1993), 223-255 MR1247201 0809.42008
[2] Y DING AND S. Z Lu, Weighted norm inequalitiesfor fractional integral operators with rough kernel, Cana J Math 50 (1998), 29-39 MR1618714 0905.42010[2] Y DING AND S. Z Lu, Weighted norm inequalitiesfor fractional integral operators with rough kernel, Cana J Math 50 (1998), 29-39 MR1618714 0905.42010
[3] Y DING AND S Z Lu, The L? x LP2 xx Lpk boundedness for some rough operators, J Math Ana Appl 203 (1996), 166-186 MR1412487 0882.42010 10.1006/jmaa.1996.0373[3] Y DING AND S Z Lu, The L? x LP2 xx Lpk boundedness for some rough operators, J Math Ana Appl 203 (1996), 166-186 MR1412487 0882.42010 10.1006/jmaa.1996.0373
[4] Y. DING AND S Z Lu, Hardy spaces estimates for a class of multilinearhomogeneous operators, Sci. Chin Ser A 42 (1999), 1270-1278 MR1749937 0958.42013 10.1007/BF02876027[4] Y. DING AND S Z Lu, Hardy spaces estimates for a class of multilinearhomogeneous operators, Sci. Chin Ser A 42 (1999), 1270-1278 MR1749937 0958.42013 10.1007/BF02876027
[5] Y DING, Weak type bounds for a class of rough operators with power weights, Proc Amer Math Soc 12 (1997), 2939-2942. MR1401735 0887.42008 10.1090/S0002-9939-97-03914-2 0002-9939%28199710%29125%3A10%3C2939%3AWTBFAC%3E2.0.CO%3B2-F[5] Y DING, Weak type bounds for a class of rough operators with power weights, Proc Amer Math Soc 12 (1997), 2939-2942. MR1401735 0887.42008 10.1090/S0002-9939-97-03914-2 0002-9939%28199710%29125%3A10%3C2939%3AWTBFAC%3E2.0.CO%3B2-F
[6] J DUOANDIKOETXEAANDJ L RuBio DE FRANCIA, Maximal and singular integral operators via Fourie transform estimates, Invent Math 84 (1986), 541-561 MR837527 0568.42012 10.1007/BF01388746[6] J DUOANDIKOETXEAANDJ L RuBio DE FRANCIA, Maximal and singular integral operators via Fourie transform estimates, Invent Math 84 (1986), 541-561 MR837527 0568.42012 10.1007/BF01388746
[7] D KURTZ AND R L WHEEDEN, Results on weighted norm inequalities for multipliers, Trans Amer. Mat Soc 255 (1979), 343-362 MR542885 0427.42004 10.2307/1998180[7] D KURTZ AND R L WHEEDEN, Results on weighted norm inequalities for multipliers, Trans Amer. Mat Soc 255 (1979), 343-362 MR542885 0427.42004 10.2307/1998180
[8] B MUCKENHOUPT AND R L WHEEDEN, Weighted norm inequalities for singular and fractional integrals, Trans Amer Math Soc 161 (1971), 249-258 MR285938 0226.44007 10.2307/1995940[8] B MUCKENHOUPT AND R L WHEEDEN, Weighted norm inequalities for singular and fractional integrals, Trans Amer Math Soc 161 (1971), 249-258 MR285938 0226.44007 10.2307/1995940
[9] E M STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeto Univ Press, Princeton, N J, 1993 MR1232192 0821.42001[9] E M STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeto Univ Press, Princeton, N J, 1993 MR1232192 0821.42001
[10] E M STEIN AND G WEISS, On the theory of harmonic functions of several variables I: The theory of H spaces, Acta Math 103 (1960), 25-62 MR121579 0097.28501 10.1007/BF02546524[10] E M STEIN AND G WEISS, On the theory of harmonic functions of several variables I: The theory of H spaces, Acta Math 103 (1960), 25-62 MR121579 0097.28501 10.1007/BF02546524
[11] M H TAIBLESON AND G WEISS, The molecular characterization of certain Hardy spaces, Asterisque 7 (1980), 67-149 MR604370 0472.46041[11] M H TAIBLESON AND G WEISS, The molecular characterization of certain Hardy spaces, Asterisque 7 (1980), 67-149 MR604370 0472.46041
