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$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations
1. | Department of Mathematics and Department of Computer Science, Georgetown University, Washington D.C. 20057 |
2. | Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada |
References:
[1] |
D. R. Adams, On the existence of capacitary strong type estimates in $\mathbf R^n$, Ark. Mat., 14 (1976), 125-140.
doi: 10.1007/BF02385830. |
[2] |
D. R. Adams, Capacity and blow-up for the $3+1$ dimensional wave opartor, Forum Math. 20 (2008), 341-357.
doi: 10.1515/FORUM.2008.017. |
[3] |
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, A Series of Comprehensive Studies in Mathematics, Springer, Berlin, 1996.
doi: 10.1007/978-3-662-03282-4. |
[4] |
D. R. Adams and J. Xiao, Strong type estimates for homogeneous Besov capacities, Math. Ann. 325 (2003), 695-709.
doi: 10.1007/s00208-002-0396-3. |
[5] |
J. Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Physical Review E, 72 (2005), 011109.
doi: 10.1103/PhysRevE.72.011109. |
[6] |
R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[7] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Caffarelli and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
D. Chamorro, Remarks on a Fractional Diffusion Transport Equation with Applications to the Dissipative Quasi-Geostrophic Equation, hal-00505027, version 4 - 12 Apr 2011. |
[10] |
J. Chen, Q. Deng, Y. Ding and D. Fan, Estimates on fractional power dissipatve equations in function spaces, Nonlinear Anal., 75 (2012), 2959-2974.
doi: 10.1016/j.na.2011.11.039. |
[11] |
Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.
doi: 10.1007/s002080050232. |
[12] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[13] |
A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commmun. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[14] |
E. B. Fabes, B. F. Jones and N. M. Riviere, The initial value problem for the Navier-Stokes equations, Arch. Rational Mech. Anal., 45 (1972), 222-240.
doi: 10.1007/BF00281533. |
[15] |
H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. |
[16] |
R. Jiang, J. Xiao, D. Yang and Z. Zhai, Regularity and capacity for the fractional dissipative operator,, , ().
|
[17] |
D. Li, On a frequency localized Bernstein inequality and some generalized Poincare-type inequalities, Math. Res. Lett., 20 (2013), 933-945. arXiv:1212.0183v1 [math.AP]2Dec2012.
doi: 10.4310/MRL.2013.v20.n5.a9. |
[18] |
V. G. Maz'ya and Yu. V. Netrusov, Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Anal., 4 (1995), 47-65.
doi: 10.1007/BF01048966. |
[19] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[20] |
M. Nishio, K. Shimomura and N. Suzuki, $\alpha$-parabolic Bergman spaces, Osaka J. Math., 42 (2005), 133-162. |
[21] |
M. Nishio, N. Suzuki and M. Yamada, Toeplitz operators and Carleson type measures on parabolic Bergman spaces, Hokkaido Math. J., 36 (2007), 563-583.
doi: 10.14492/hokmj/1277472867. |
[22] |
M. Nishio, N. Suzuki and M. Yamada, Compact Toeplitz operators on parabolic Bergman spaces, Hiroshima Math. J., 38 (2008), 177-192. |
[23] |
M. Nishio, N. Suzuki and M. Yamada, Carleson inequalities on parabolic Bergman spaces, Tohoku Math. J., 62 (2010), 269-286.
doi: 10.2748/tmj/1277298649. |
[24] |
M. Nishio and M. Yamada, Carleson type measures on parabolic Bergman spaces, J. Math. Soc. Japan, 58 (2006), 83-96.
doi: 10.2969/jmsj/1145287094. |
[25] |
L. Vázquez, J. J. Trujillo and M. P. Velasco, Fractional heat equation and the second law of thermodynamics, Frac. Calc. Appl. Anal., 14 (2011), 334-342.
doi: 10.2478/s13540-011-0021-9. |
[26] |
G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl., 340 (2008), 1326-1335.
doi: 10.1016/j.jmaa.2007.09.060. |
[27] |
J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1-13. |
[28] |
J. Wu, Lower Bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263 (2006), 803-831.
doi: 10.1007/s00220-005-1483-6. |
[29] |
J. Xiao, Carleson embeddings for Sobolev spaces via heat equation, J. Diff. Equ., 224 (2006), 277-295.
doi: 10.1016/j.jde.2005.07.014. |
[30] |
J. Xiao, Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation, Adv. Math., 207 (2006), 828-846.
doi: 10.1016/j.aim.2006.01.010. |
[31] |
L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operator,, , ().
|
[32] |
Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658.
doi: 10.1016/j.jmaa.2009.03.051. |
[33] |
Z. Zhai, Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces, Nonlinear Anal., 73 (2010), 2611-2630.
doi: 10.1016/j.na.2010.06.040. |
show all references
References:
[1] |
D. R. Adams, On the existence of capacitary strong type estimates in $\mathbf R^n$, Ark. Mat., 14 (1976), 125-140.
doi: 10.1007/BF02385830. |
[2] |
D. R. Adams, Capacity and blow-up for the $3+1$ dimensional wave opartor, Forum Math. 20 (2008), 341-357.
doi: 10.1515/FORUM.2008.017. |
[3] |
D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, A Series of Comprehensive Studies in Mathematics, Springer, Berlin, 1996.
doi: 10.1007/978-3-662-03282-4. |
[4] |
D. R. Adams and J. Xiao, Strong type estimates for homogeneous Besov capacities, Math. Ann. 325 (2003), 695-709.
doi: 10.1007/s00208-002-0396-3. |
[5] |
J. Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Physical Review E, 72 (2005), 011109.
doi: 10.1103/PhysRevE.72.011109. |
[6] |
R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[7] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
L. Caffarelli and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
D. Chamorro, Remarks on a Fractional Diffusion Transport Equation with Applications to the Dissipative Quasi-Geostrophic Equation, hal-00505027, version 4 - 12 Apr 2011. |
[10] |
J. Chen, Q. Deng, Y. Ding and D. Fan, Estimates on fractional power dissipatve equations in function spaces, Nonlinear Anal., 75 (2012), 2959-2974.
doi: 10.1016/j.na.2011.11.039. |
[11] |
Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.
doi: 10.1007/s002080050232. |
[12] |
P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[13] |
A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Commmun. Math. Phys., 249 (2004), 511-528.
doi: 10.1007/s00220-004-1055-1. |
[14] |
E. B. Fabes, B. F. Jones and N. M. Riviere, The initial value problem for the Navier-Stokes equations, Arch. Rational Mech. Anal., 45 (1972), 222-240.
doi: 10.1007/BF00281533. |
[15] |
H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. |
[16] |
R. Jiang, J. Xiao, D. Yang and Z. Zhai, Regularity and capacity for the fractional dissipative operator,, , ().
|
[17] |
D. Li, On a frequency localized Bernstein inequality and some generalized Poincare-type inequalities, Math. Res. Lett., 20 (2013), 933-945. arXiv:1212.0183v1 [math.AP]2Dec2012.
doi: 10.4310/MRL.2013.v20.n5.a9. |
[18] |
V. G. Maz'ya and Yu. V. Netrusov, Some counterexamples for the theory of Sobolev spaces on bad domains, Potential Anal., 4 (1995), 47-65.
doi: 10.1007/BF01048966. |
[19] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal., 68 (2008), 461-484.
doi: 10.1016/j.na.2006.11.011. |
[20] |
M. Nishio, K. Shimomura and N. Suzuki, $\alpha$-parabolic Bergman spaces, Osaka J. Math., 42 (2005), 133-162. |
[21] |
M. Nishio, N. Suzuki and M. Yamada, Toeplitz operators and Carleson type measures on parabolic Bergman spaces, Hokkaido Math. J., 36 (2007), 563-583.
doi: 10.14492/hokmj/1277472867. |
[22] |
M. Nishio, N. Suzuki and M. Yamada, Compact Toeplitz operators on parabolic Bergman spaces, Hiroshima Math. J., 38 (2008), 177-192. |
[23] |
M. Nishio, N. Suzuki and M. Yamada, Carleson inequalities on parabolic Bergman spaces, Tohoku Math. J., 62 (2010), 269-286.
doi: 10.2748/tmj/1277298649. |
[24] |
M. Nishio and M. Yamada, Carleson type measures on parabolic Bergman spaces, J. Math. Soc. Japan, 58 (2006), 83-96.
doi: 10.2969/jmsj/1145287094. |
[25] |
L. Vázquez, J. J. Trujillo and M. P. Velasco, Fractional heat equation and the second law of thermodynamics, Frac. Calc. Appl. Anal., 14 (2011), 334-342.
doi: 10.2478/s13540-011-0021-9. |
[26] |
G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl., 340 (2008), 1326-1335.
doi: 10.1016/j.jmaa.2007.09.060. |
[27] |
J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 2001 (2001), 1-13. |
[28] |
J. Wu, Lower Bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Commun. Math. Phys., 263 (2006), 803-831.
doi: 10.1007/s00220-005-1483-6. |
[29] |
J. Xiao, Carleson embeddings for Sobolev spaces via heat equation, J. Diff. Equ., 224 (2006), 277-295.
doi: 10.1016/j.jde.2005.07.014. |
[30] |
J. Xiao, Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation, Adv. Math., 207 (2006), 828-846.
doi: 10.1016/j.aim.2006.01.010. |
[31] |
L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operator,, , ().
|
[32] |
Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658.
doi: 10.1016/j.jmaa.2009.03.051. |
[33] |
Z. Zhai, Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces, Nonlinear Anal., 73 (2010), 2611-2630.
doi: 10.1016/j.na.2010.06.040. |
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