# Higher order fractional laplacian

when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order laplacian ($\alpha>1$ ) ,unlike the lower case, has little connection with stochastic process.(lack of positivy.) Since most paper i have met is the case of $0<\alpha<1$.And i wonder how things change when considering the higer order laplacian?

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The fractional Laplacian can be represented as a hypersingular integral operator, an integral operator with a singularity higher than the space dimension. Its convergence is provided by a regularization whose form depends on the order. From its form the difference in positivity properties is immediately clear. See S. G. Samko, Hypersingular Integrals and their applications, Taylor and Francis, London, 2002.

On the other hand, there exists a theory of parabolic equations with fractional Laplacians (and more general pseudodifferential operators) of any order. See S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser, Basel, 2004.

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thanks for the useful reference.i'm considering the fractional shrodinger equation and it's $L^p$ estimate ,so i first want to know more about it'e kernel –  user23078 Apr 20 '12 at 16:45
i try to look for that book,but it's not available.. I'm really appreciated that if you can send me a copy of the pdf file of your book mentioned above .here is my adress: hsl_lovemath@163.com. thank you very much! –  user23078 Apr 24 '12 at 15:14

Well, the answer is in your question. When $0<\alpha<1$, and only then, $(-\Delta)^\alpha$ has a positive inverse. In particular, equations like $$(-\Delta)^\alpha u=f$$ or $$\partial_tu+(-\Delta)^\alpha u=0$$ satisfy a maximum principle. You get immediately an $L^p$ estimate for every $1\le p\le\infty$. If the equation is quasi-linear instead, it is likely that you have a $BV$-estimate for free. This helps a lot in proving well-posedness of the Boundary-value problem or the Cauchy problem.

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