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.

@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}, pages = {573--614}, publisher = {Elsevier}, volume = {30}, number = {4}, year = {2013}, doi = {10.1016/j.anihpc.2012.10.006}, mrnumber = {3082477}, zbl = {1288.35152}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.006/} }

TY - JOUR AU - Zhang, Xicheng TI - $ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 573 EP - 614 VL - 30 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.006/ DO - 10.1016/j.anihpc.2012.10.006 LA - en ID - AIHPC_2013__30_4_573_0 ER -

%0 Journal Article %A Zhang, Xicheng %T $ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 573-614 %V 30 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2012.10.006/ %R 10.1016/j.anihpc.2012.10.006 %G en %F 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/articles/10.1016/j.anihpc.2012.10.006/

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