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Is it possible to calculate the $L^2$ norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$

$$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}dxdy$$

If $u(x)=(1+ |x|^2)^{-\frac{N-2s}{2}},$ can we calculate $\|(-\Delta)^{s/2} u\|_2$ just by using the above formula. I am aware $u$ is the some type of solution to the fractional Yamabe problem but I want a result independent proof. My main goal is to estimate $u$ where $u$ may not belong to $D^{s,2}(\mathbb R^N)$ but is smooth. This can be done using a cut off function multiplied with $u.$ Any reference is welcome.

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  • $\begingroup$ There's a constant missing in the right-hand side. $\endgroup$ – Mateusz Kwaśnicki May 14 at 20:08
  • $\begingroup$ Yes, I am aware of it. $\endgroup$ – GabS May 14 at 20:34
  • $\begingroup$ You should elaborate a bit on what you mean by "calulate". One may argue that $L^p$ norms are already not calculable, for example can you tell me the precise value of $\|u\|_{L^5(\mathbb R^N)}$ of $u(x)=e^{-|x|^2+\sin(2\log(1+|x|))}$? If you're only interested in $u(x)=(1+|x|^2)^{-\frac{N-2s}{2}}$ then your post should say so clearly, right now your question is not well-defined $\endgroup$ – leo monsaingeon May 15 at 0:57
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If your question is whether it is possible to explicitly evaluate the integral in the right-hand side of your equation when $u(x) = (1 + |x|^2)^{-p}$, then I do not know the answer, but I would not be surprised if it be affirmative. The closest things that comes to my mind are:

  1. The explicit expression for $(-\Delta)^s u$ when $p = \tfrac{N + 1}{2}$ or $p = \tfrac{N - s}{2} + n$ ($n = 1, 2, \ldots$), found by S. Samko (see [1]).

  2. The explicit expression for $(-\Delta)^s (1 - |x|^2)_+^{-p}$, found independently by B. Dyda [2] and P. Biller, C. Imbert, G. Karch [3].

The methods for doing integrals of this form, involving Kelvin transformation, were already developed by M. Riesz.


If, however, you are looking for an arbitrary explicit way to evaluate $\|(-\Delta)^{s/2} u\|_2 = \langle (-\Delta)^s u, u \rangle$, then there are other approaches. The most natural thing to do would be to find the Fourier transform of $u$, and use Plancherel's theorem. Since $u(x) = (1 + |x|^2)^{-p}$ is a radial function, its Fourier transform is given by the Hankel transform of the profile $(1 + r^2)^{-p}$. I did not attempt to search for the explicit expression, but I bet it is given in one of the standard tables of integrals.


Yet another approach is to use Mellin transform (rather than Fourier transform) to find an explicit expression for $(-\Delta)^s u$. For $u(x) = (1 + |x|^2)^{-p}$ this is indeed possible, and the result involves the hypergeometric function $_2F_1$; see Corollary 2 (or Corollary 1) in my joint paper with B. Dyda and A. Kuznetsov [4]. The last step would be to evaluate the inner product of $(-\Delta)^s u$ with $u$ (or the $L^2$ norm of $(-\Delta)^{s/2} u$, whichever turns out simpler); again, this is likely to be found in standard integral tables.


EDIT: One more thought: if one is able to evaluate the convolution of $u$ with the Gauss–Weierstrass kernel, or, even better, the value of $$q_t = \int_{\mathbb{R}^N} \int_{\mathbb{R}^N} (u(x) - u(y))^2 (4 \pi t)^{-N/2} \exp(-|x - y|^2 / (4 t)) dx dy,$$ then it can be convenient to use the subordination formula: $$\|(-\Delta)^{s/2} u\|_2^2 = \frac{1}{|\Gamma(-s)|} \int_0^\infty q_t t^{-1 - s} dt.$$


EDIT: As suggested by leo monsaingeon, if the convolution of $u$ with $(y^2 + |\cdot|^2)^{-(N + 2 s)/2}$ is known, then $(-\Delta)^s u$ and $\|(-\Delta)^{s/2} u\|_2$ can be evaluated using the Caffarelli–Silvestre extension technique; see [5].

(One could also consider, for example, Balakhrishnan's formula, but I doubt this is ever useful in calculations: the resolvent kernel for $\Delta$ is not the simplest convolution factor).


References:

[1] Samko, S.: Hypersingular Integrals and Their Applications, Analytical Methods and Special Functions, vol. 5. Taylor & Francis, Ltd., London (2002)

[2] Dyda, B.: Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15(4), 536–555 (2012)

[3] Biler, P., Imbert, C., Karch, G.: Barenblatt profiles for a nonlocal porous medium equation. C. R. Math. Acad. Sci. Paris 349(11–12), 641–645 (2011)

[4] Dyda, B., Kuznetsov, A., Kwaśnicki, M.: Fractional Laplace operator and Meijer G-function. Constructive Approx. 45(3), 427–448 (2017)

[5] Caffarelli, L., Silvestre, L.: An Extension Problem Related to the Fractional Laplacian. Comm. PDE 32(8), 1245–1260 (2007)

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