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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

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 & Continuous Dynamical Systems - A, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905
References:
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D. R. Adams, On the existence of capacitary strong type estimates in $\mathbf R^n$,, Ark. Mat., 14 (1976), 125.  doi: 10.1007/BF02385830.  Google Scholar

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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.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

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J. Chen, Q. Deng, Y. Ding and D. Fan, Estimates on fractional power dissipatve equations in function spaces,, Nonlinear Anal., 75 (2012), 2959.  doi: 10.1016/j.na.2011.11.039.  Google Scholar

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Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes,, Math. Ann., 312 (1998), 465.  doi: 10.1007/s002080050232.  Google Scholar

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P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Anal., 30 (1999), 937.  doi: 10.1137/S0036141098337333.  Google Scholar

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H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

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R. Jiang, J. Xiao, D. Yang and Z. Zhai, Regularity and capacity for the fractional dissipative operator,, , ().   Google Scholar

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D. Li, On a frequency localized Bernstein inequality and some generalized Poincare-type inequalities,, Math. Res. Lett., 20 (2013), 933.  doi: 10.4310/MRL.2013.v20.n5.a9.  Google Scholar

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[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.  doi: 10.1016/j.na.2006.11.011.  Google Scholar

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M. Nishio, K. Shimomura and N. Suzuki, $\alpha$-parabolic Bergman spaces,, Osaka J. Math., 42 (2005), 133.   Google Scholar

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M. Nishio, N. Suzuki and M. Yamada, Toeplitz operators and Carleson type measures on parabolic Bergman spaces,, Hokkaido Math. J., 36 (2007), 563.  doi: 10.14492/hokmj/1277472867.  Google Scholar

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M. Nishio, N. Suzuki and M. Yamada, Compact Toeplitz operators on parabolic Bergman spaces,, Hiroshima Math. J., 38 (2008), 177.   Google Scholar

[23]

M. Nishio, N. Suzuki and M. Yamada, Carleson inequalities on parabolic Bergman spaces,, Tohoku Math. J., 62 (2010), 269.  doi: 10.2748/tmj/1277298649.  Google Scholar

[24]

M. Nishio and M. Yamada, Carleson type measures on parabolic Bergman spaces,, J. Math. Soc. Japan, 58 (2006), 83.  doi: 10.2969/jmsj/1145287094.  Google Scholar

[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.  doi: 10.2478/s13540-011-0021-9.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.09.060.  Google Scholar

[27]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron. J. Differential Equations, 2001 (2001), 1.   Google Scholar

[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.  doi: 10.1007/s00220-005-1483-6.  Google Scholar

[29]

J. Xiao, Carleson embeddings for Sobolev spaces via heat equation,, J. Diff. Equ., 224 (2006), 277.  doi: 10.1016/j.jde.2005.07.014.  Google Scholar

[30]

J. Xiao, Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation,, Adv. Math., 207 (2006), 828.  doi: 10.1016/j.aim.2006.01.010.  Google Scholar

[31]

L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operator,, , ().   Google Scholar

[32]

Z. Zhai, Strichartz type estimates for fractional heat equations,, J. Math. Anal. Appl., 356 (2009), 642.  doi: 10.1016/j.jmaa.2009.03.051.  Google Scholar

[33]

Z. Zhai, Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces,, Nonlinear Anal., 73 (2010), 2611.  doi: 10.1016/j.na.2010.06.040.  Google Scholar

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.  doi: 10.1007/BF02385830.  Google Scholar

[2]

D. R. Adams, Capacity and blow-up for the $3+1$ dimensional wave opartor,, Forum Math. 20 (2008), 20 (2008), 341.  doi: 10.1515/FORUM.2008.017.  Google Scholar

[3]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory,, A Series of Comprehensive Studies in Mathematics, (1996).  doi: 10.1007/978-3-662-03282-4.  Google Scholar

[4]

D. R. Adams and J. Xiao, Strong type estimates for homogeneous Besov capacities,, Math. Ann. 325 (2003), 325 (2003), 695.  doi: 10.1007/s00208-002-0396-3.  Google Scholar

[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).  doi: 10.1103/PhysRevE.72.011109.  Google Scholar

[6]

R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes,, Trans. Amer. Math. Soc., 95 (1960), 263.  doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Diff. Equ., 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[9]

D. Chamorro, Remarks on a Fractional Diffusion Transport Equation with Applications to the Dissipative Quasi-Geostrophic Equation,, hal-00505027, (2011).   Google Scholar

[10]

J. Chen, Q. Deng, Y. Ding and D. Fan, Estimates on fractional power dissipatve equations in function spaces,, Nonlinear Anal., 75 (2012), 2959.  doi: 10.1016/j.na.2011.11.039.  Google Scholar

[11]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes,, Math. Ann., 312 (1998), 465.  doi: 10.1007/s002080050232.  Google Scholar

[12]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations,, SIAM J. Math. Anal., 30 (1999), 937.  doi: 10.1137/S0036141098337333.  Google Scholar

[13]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Commmun. Math. Phys., 249 (2004), 511.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[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.  doi: 10.1007/BF00281533.  Google Scholar

[15]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

[16]

R. Jiang, J. Xiao, D. Yang and Z. Zhai, Regularity and capacity for the fractional dissipative operator,, , ().   Google Scholar

[17]

D. Li, On a frequency localized Bernstein inequality and some generalized Poincare-type inequalities,, Math. Res. Lett., 20 (2013), 933.  doi: 10.4310/MRL.2013.v20.n5.a9.  Google Scholar

[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.  doi: 10.1007/BF01048966.  Google Scholar

[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.  doi: 10.1016/j.na.2006.11.011.  Google Scholar

[20]

M. Nishio, K. Shimomura and N. Suzuki, $\alpha$-parabolic Bergman spaces,, Osaka J. Math., 42 (2005), 133.   Google Scholar

[21]

M. Nishio, N. Suzuki and M. Yamada, Toeplitz operators and Carleson type measures on parabolic Bergman spaces,, Hokkaido Math. J., 36 (2007), 563.  doi: 10.14492/hokmj/1277472867.  Google Scholar

[22]

M. Nishio, N. Suzuki and M. Yamada, Compact Toeplitz operators on parabolic Bergman spaces,, Hiroshima Math. J., 38 (2008), 177.   Google Scholar

[23]

M. Nishio, N. Suzuki and M. Yamada, Carleson inequalities on parabolic Bergman spaces,, Tohoku Math. J., 62 (2010), 269.  doi: 10.2748/tmj/1277298649.  Google Scholar

[24]

M. Nishio and M. Yamada, Carleson type measures on parabolic Bergman spaces,, J. Math. Soc. Japan, 58 (2006), 83.  doi: 10.2969/jmsj/1145287094.  Google Scholar

[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.  doi: 10.2478/s13540-011-0021-9.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.09.060.  Google Scholar

[27]

J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data,, Electron. J. Differential Equations, 2001 (2001), 1.   Google Scholar

[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.  doi: 10.1007/s00220-005-1483-6.  Google Scholar

[29]

J. Xiao, Carleson embeddings for Sobolev spaces via heat equation,, J. Diff. Equ., 224 (2006), 277.  doi: 10.1016/j.jde.2005.07.014.  Google Scholar

[30]

J. Xiao, Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation,, Adv. Math., 207 (2006), 828.  doi: 10.1016/j.aim.2006.01.010.  Google Scholar

[31]

L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operator,, , ().   Google Scholar

[32]

Z. Zhai, Strichartz type estimates for fractional heat equations,, J. Math. Anal. Appl., 356 (2009), 642.  doi: 10.1016/j.jmaa.2009.03.051.  Google Scholar

[33]

Z. Zhai, Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces,, Nonlinear Anal., 73 (2010), 2611.  doi: 10.1016/j.na.2010.06.040.  Google Scholar

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