# $L^2$ norm of fractional Laplacian

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.

• There's a constant missing in the right-hand side. – Mateusz Kwaśnicki May 14 at 20:08
• Yes, I am aware of it. – GabS May 14 at 20:34
• 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 – leo monsaingeon May 15 at 0:57

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)