# Fractional laplacian of $(a-x)_+^\alpha$ in $(0,1)$

How can I compute the spectral fractional Laplacian of $$(a-x)_+^\alpha$$ on $$\Omega = (0,1)$$? Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - s} dt ,$$ where $$e^{t\Delta_N}u$$ is the solution of the heat equation with homogeneus Neumann boundary conditions on $$\Omega$$.

• I would be surprised if a nice closed-form expression existed. A natural approach would be to expand $(a-x)_+^\alpha$ in the Fourier cosine series $\sum a_n \cos(n \pi x)$ and use the fact that $(-\Delta_N)^s [\cos(n \pi x)] = (n \pi)^{2s} \cos(n \pi x)$. I do not know if there is a simple expression for $a_n$, though. Feb 16 at 1:23
• Right hand integral looks like an operator with respect to $t$: $\int^\infty_0(u(x,t)-u(x,0))t^{-1-s}dt$ with parameter $x$.
– user171871
Feb 16 at 8:53

The operator $$(-∆)^s$$ is the Fourier multiplier $$\vert\xi\vert^{2s}$$ and thus deserves to be denoted by $$\vert D\vert^{2s}$$. Now the distribution $$u_\alpha(x)=x_+^\alpha$$ is homogeneous with degree $$\alpha$$ and thus has a Fourier transform homogeneous with degree $$-1-\alpha$$. We have in fact $$\hat{u}_\alpha(\xi)=c_\alpha (\xi-i0)^{-1-\alpha}.$$ You are interested in $$v_\alpha(x)=u_\alpha(x-t)$$ so that $$\vert\xi\vert^{2s}\hat{v}_\alpha(\xi)=e^{-it \xi}c_\alpha (\xi-i0)^{-1-\alpha}\vert\xi\vert^{2s},$$ and $$(\vert D\vert^{2s} v_\alpha)(x)=\int e^{i(x-t) \xi}c'_\alpha (\xi-i0)^{-1-\alpha}\vert\xi\vert^{2s}d\xi =(\vert D\vert^{2s} u_\alpha)(x-t).$$ Translations commute with derivations, including fractional versions.
• I think Jay is asking about a different operator: the fractional power of the Neumann Laplacian on $(0,1)$. Feb 16 at 1:21
Let us denote the corresponding eigensystem $$\{\lambda_n,\phi_n\}$$. If $$u(x)=\sum\limits_nu_n\phi_n(x)$$, then we have $$u(x,t)=\sum\limits_n\exp(-\lambda_n t)u_n\phi_n(x)$$. Since $$\int\limits^\infty_0(\exp(-\lambda_n t)-1)t^{-1-s}=-\lambda^s_n\frac{\Gamma(1-s)}{s}$$ we obtain $$\text{R.H.I.}=-C_{s,N}\frac{\Gamma(1-s)}{s}\sum\limits_n\lambda^s_nu_n\phi_n(x)$$