By using Fourierʼs transform and Fefferman–Steinʼs theorem, we investigate the ${L}^{p}$-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric Lévy operators, and obtain the unique strong solvability of the corresponding nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. As a consequence, a characterization for the domain of pseudo-differential operators of Lévy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we also show that a large class of non-symmetric Lévy operators generates an analytic semigroup in ${L}^{p}$-spaces. Moreover, as applications, we prove Krylovʼs estimate for stochastic differential equations driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the global well-posedness for a class of quasi-linear first order parabolic systems with critical diffusions. In particular, critical Hamilton–Jacobi equations and multidimensional critical Burgerʼs equations are uniquely solvable and the smooth solutions are obtained.

Keywords: ${L}^{p}$-regularity, Lévy process, Krylovʼs estimate, Sharp function, Critical Burgerʼs equation

@article{AIHPC_2013__30_4_573_0, author = {Zhang, Xicheng}, title = {$ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {30}, number = {4}, year = {2013}, pages = {573-614}, doi = {10.1016/j.anihpc.2012.10.006}, zbl = {1288.35152}, mrnumber = {3082477}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2013__30_4_573_0} }

Zhang, Xicheng. $ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, pp. 573-614. doi : 10.1016/j.anihpc.2012.10.006. http://www.numdam.org/item/AIHPC_2013__30_4_573_0/

[1] Linear and Quasilinear Parabolic Problems, vol. I. Abstract Linear Theory, Monogr. Math. vol. 89, Birkhäuser, Boston, MA (1995) | MR 1345385 | Zbl 0819.35001

,[2] Lévy Processes and Stochastic Calculus, Camb. Stud. Adv. Math. vol. 93, Cambridge University Press (2004) | MR 2072890 | Zbl 1073.60002

,[3] Lipschitz regularity of solutions for mixed integro-differential equations, arXiv:1107.3228v1 | MR 2911421 | Zbl 1298.35033

, , , ,[4] Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. 13 (2011), 1-26 | MR 2735074 | Zbl 1207.35277

, , ,[5] Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance Stoch. 5 no. 3 (2001), 275-303 | MR 1849422 | Zbl 0978.91039

, , ,[6] Interpolation Spaces, Grundlehren Math. Wiss. vol. 223, Springer-Verlag (1976) | Zbl 0128.35104

, ,[7] Fractal Burgers equations, J. Differ. Equ. 148 (1998), 9-46 | MR 1637513 | Zbl 0911.35100

, , ,[8] Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math. 62 no. 5 (2009), 597-638 | MR 2494809 | Zbl 1170.45006

, ,[9] Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math. 171 no. 3 (2010), 1903-1930 | MR 2680400 | Zbl 1204.35063

, ,[10] Euler equations, Navier–Stokes equations and turbulence, Mathematical Foundation of Turbulent Flows, Lect. Notes Math. vol. 1871 (2006), 1-43 | MR 2196360 | Zbl 1190.76146

,[11] On ${L}_{p}$-estimates for a class of nonlocal elliptic equations, J. Funct. Anal. 262 no. 3 (2012), 1166-1199 | MR 2863859 | Zbl 1232.35182

, ,[12] Fractal first order partial equations, Arch. Ration. Mech. Anal. 182 no. 2 (2006), 299-331 | MR 2259335 | Zbl 1111.35144

, ,[13] Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces, Diss. Math. 393 (2001) | MR 1840499

, , ,[14] Stochastic Differential Equations and Applications, vol. 1, Academic Press, New York (1975) | MR 494490 | Zbl 0323.60056

,[15] Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. vol. 44, Springer-Verlag (1983) | MR 710486 | Zbl 0516.47023

,[16] Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math. 49 no. 2 (2012), 421-447 | MR 2945756 | Zbl 1254.60063

,[17] Pseudo Differential Operators, Markov Processes, vol. I. Fourier Analysis and Semigroups, Imperical College Press, World Scientific Publishing, Singapore (2001)

,[18] Solving a nonlinear pseudo-differential equation of Burgers type, Stoch. Dyn. 8 no. 4 (2008), 613-624 | MR 2463738 | Zbl 1166.35389

, , ,[19] Foundations of Modern Probability, Springer-Verlag, New York (1997) | MR 1464694 | Zbl 0892.60001

,[20] Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 no. 3 (2008), 211-240 | MR 2455893 | Zbl 1186.35020

, , ,[21] Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), 445-453 | MR 2276260 | Zbl 1121.35115

, , ,[22] A generalization of the Littlewood–Paley inequality and some other results related to stochastic partial differential equations, Ulam Quaterly 2 no. 4 (1994), 16-26 | MR 1317805 | Zbl 0870.42005

,[23] The heat equation in ${L}_{q}((0,T),{L}_{p})$-spaces with weights, SIAM J. Math. Anal. 32 no. 5 (2001), 1117-1141 | MR 1828321 | Zbl 0979.35060

,[24] Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Grad. Stud. Math. vol. 96, AMS (2008) | MR 2435520 | Zbl 1147.35001

,[25] A note on ${L}_{2}$-estimates for stable integrals with drift, Trans. Am. Math. Soc. 360 (2008), 925-938 | MR 2346477 | Zbl 1137.60029

,[26] Linear and Quasi-Linear Parabolic Equations, Nauka, Moscow (1967), AMS (1968) | Zbl 0174.15403

, , ,[27] On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lith. Math. J. 32 no. 2 (1992) | MR 1246036 | Zbl 0795.45007

, ,[28] On ${L}_{p}$ theory for Zakai equation with discontinuous observation process, arXiv:1012.5816v1 | Zbl 1229.93154

, ,[29] Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press (1999) | MR 1739520 | Zbl 0973.60001

,[30] Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 no. 3 (2006), 1155-1174 | MR 2244602 | Zbl 1101.45004

,[31] On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion, Adv. Math. 226 no. 2 (2011), 2020-2039 | MR 2737806 | Zbl 1216.35165

,[32] Hölder estimates for advection fractional-diffusion equations, arXiv:1009.5723v2 (2011), Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2011) | MR 3060702

,[33] On the differentiability of the solution to an equation with drift and fractional-diffusion, arXiv:1012.2401 (2011), Indiana Univ. Math. J. (2011) | MR 3043588

,[34] Optimal control with state-space constraint II, SIAM J. Control Optim. 24 no. 6 (2006), 1110-1122 | MR 861089 | Zbl 0619.49013

,[35] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ (1970) | MR 290095 | Zbl 0207.13501

,[36] Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press (1993) | MR 1232192

,[37] Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam (1978) | MR 503903 | Zbl 0387.46033

,[38] Lévy processes in the physical sciences, Lévy Processes, Birkhäuser, Boston, MA (2001), 241-266 | MR 1833700 | Zbl 0982.60043

,[39] X. Zhang, Stochastic differential equations with Sobolev drifts and driven by α-stable processes, Ann. Inst. Henri Poincaré B, Probab. Stat., http://dx.doi.org/10.1214/12-AIHP476. | Numdam | MR 3127913

[40] X. Zhang, Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations, Ann. Appl. Probab., http://dx.doi.org/10.1214/12-AAP851. | MR 3024975