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

References:
[1]

Ark. Mat., 14 (1976), 125-140. doi: 10.1007/BF02385830.  Google Scholar

[2]

Forum Math. 20 (2008), 341-357. doi: 10.1515/FORUM.2008.017.  Google Scholar

[3]

A Series of Comprehensive Studies in Mathematics, Springer, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.  Google Scholar

[4]

Math. Ann. 325 (2003), 695-709. doi: 10.1007/s00208-002-0396-3.  Google Scholar

[5]

Physical Review E, 72 (2005), 011109. doi: 10.1103/PhysRevE.72.011109.  Google Scholar

[6]

Trans. Amer. Math. Soc., 95 (1960), 263-273. doi: 10.1090/S0002-9947-1960-0119247-6.  Google Scholar

[7]

Comm. Partial Diff. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[8]

Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[9]

hal-00505027, version 4 - 12 Apr 2011. Google Scholar

[10]

Nonlinear Anal., 75 (2012), 2959-2974. doi: 10.1016/j.na.2011.11.039.  Google Scholar

[11]

Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232.  Google Scholar

[12]

SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333.  Google Scholar

[13]

Commmun. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.  Google Scholar

[14]

Arch. Rational Mech. Anal., 45 (1972), 222-240. doi: 10.1007/BF00281533.  Google Scholar

[15]

Springer-Verlag, New York, 1969.  Google Scholar

[16]

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

[17]

Math. Res. Lett., 20 (2013), 933-945. arXiv:1212.0183v1 [math.AP]2Dec2012. doi: 10.4310/MRL.2013.v20.n5.a9.  Google Scholar

[18]

Potential Anal., 4 (1995), 47-65. doi: 10.1007/BF01048966.  Google Scholar

[19]

Nonlinear Anal., 68 (2008), 461-484. doi: 10.1016/j.na.2006.11.011.  Google Scholar

[20]

Osaka J. Math., 42 (2005), 133-162.  Google Scholar

[21]

Hokkaido Math. J., 36 (2007), 563-583. doi: 10.14492/hokmj/1277472867.  Google Scholar

[22]

Hiroshima Math. J., 38 (2008), 177-192.  Google Scholar

[23]

Tohoku Math. J., 62 (2010), 269-286. doi: 10.2748/tmj/1277298649.  Google Scholar

[24]

J. Math. Soc. Japan, 58 (2006), 83-96. doi: 10.2969/jmsj/1145287094.  Google Scholar

[25]

Frac. Calc. Appl. Anal., 14 (2011), 334-342. doi: 10.2478/s13540-011-0021-9.  Google Scholar

[26]

J. Math. Anal. Appl., 340 (2008), 1326-1335. doi: 10.1016/j.jmaa.2007.09.060.  Google Scholar

[27]

Electron. J. Differential Equations, 2001 (2001), 1-13.  Google Scholar

[28]

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[29]

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[30]

Adv. Math., 207 (2006), 828-846. 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]

J. Math. Anal. Appl., 356 (2009), 642-658. doi: 10.1016/j.jmaa.2009.03.051.  Google Scholar

[33]

Nonlinear Anal., 73 (2010), 2611-2630. doi: 10.1016/j.na.2010.06.040.  Google Scholar

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