L^q-Extensions of L^p-spaces by fractional diffusion equations

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May 2015, 35(5): 1905-1920. doi: 10.3934/dcds.2015.35.1905

$L^q$-Extensions of $L^p$-spaces by fractional diffusion equations

Der-Chen Chang ^1, and Jie Xiao ^2,

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

Received May 2014 Revised September 2014 Published December 2014

Abstract
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Based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in the upper-half Euclidean space, this note characterizes a nonnegative Randon measure $\mu$ on $\mathbb R^{1+n}_+$ such that the extension $R_\alpha L^p(\mathbb R^n)\subseteq L^q(\mathbb R^{1+n}_+,\mu)$ holds for $(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$ where $u=R_\alpha f$ is the weak solution of the fractional diffusion equation $(\partial_t + (-\Delta_x)^\alpha)u(t, x) = 0$ in $\mathbb R^{1+n}_+$ subject to $u(0,x)=f(x)$ in $\mathbb R^n$.

Keywords: $L^p$-type capacities, fractional diffusion equations, heat kernel., $L^q$-extensions.

Mathematics Subject Classification: Primary: 31C15, 35K08, 35K9.

Citation: Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905

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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