1
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ There's a constant missing in the right-hand side. $\endgroup$ May 14, 2019 at 20:08
  • $\begingroup$ Yes, I am aware of it. $\endgroup$
    – GabS
    May 14, 2019 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 $\endgroup$ May 15, 2019 at 0:57

1 Answer 1

3
$\begingroup$

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)

$\endgroup$
2

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.