# American Institute of Mathematical Sciences

• Previous Article
Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
• DCDS Home
• This Issue
• Next Article
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces
May  2015, 35(5): 1905-1920. doi: 10.3934/dcds.2015.35.1905

## $L^q$-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

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 $\mathbb R^n$.
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:
 [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

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
 [1] Samer Dweik. $L^{p, q}$ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134 [2] Damiano Foschi. Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals. Communications on Pure & Applied Analysis, 2005, 4 (3) : 569-588. doi: 10.3934/cpaa.2005.4.569 [3] Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483 [4] Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090 [5] Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080 [6] Ming Wang, Yanbin Tang. Attractors in $H^2$ and $L^{2p-2}$ for reaction diffusion equations on unbounded domains. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1111-1121. doi: 10.3934/cpaa.2013.12.1111 [7] Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145 [8] L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure & Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9 [9] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715 [10] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667 [11] Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113 [12] Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989 [13] Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167 [14] Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 [15] Yong Xia, Yu-Jun Gong, Sheng-Nan Han. A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 185-195. doi: 10.3934/naco.2015.5.185 [16] C. García Vázquez, Francisco Ortegón Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure & Applied Analysis, 2005, 4 (3) : 589-612. doi: 10.3934/cpaa.2005.4.589 [17] Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230 [18] Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037 [19] Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 [20] Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053

2018 Impact Factor: 1.143