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For the record, the result requested in the question is given in Theorem 1.3.3 in: E.B. Davies, Heat Kernels and Spectral Theory, DOI:10.1017/CBO9780511566158. The assumptions are: $L$ is a positive …

If true, this would follow from the integral expression for the fractional Laplacian: $$(-\Delta)^{s/2} f(x) = \int_{-\pi}^\pi (f(x) - f(y)) \nu(x - y) dy$$ for an appropriate kernel $\nu$. …

Let us approximate the "Laplacian" (second derivative operator) in the interval $[0,1]$ using graph Laplacians on "chain graphs" with vertices $\tfrac in$, $i=0,1,\ldots,n$. … Then graph Laplacian applied to a function $f$, normalised by a factor $n^{-2}$, converges to $f''$, except at the boundary. …

This is more of a comment, but way too long. First, two remarks on the initial part of the question: The Kelvin transform of a harmonic polynomial is of course harmonic, but it is not a polynomial. F …

This means that $\sin x$ is in the domain of $(-\Delta)^s$ for any $s > 0$ (I understand that $(-\Delta)^{s/2}$ is a power of the Dirichlet Laplacian), but it does not belong to $H^s_0(\Omega)$ for any …

Yes, the result is true, in arbitrary dimension $d$. Once I needed this and I could not find a reference; the following is the relevant part of the proof of Proposition 2.1 in my paper Ten equivalent …

Sure one can! Just note that $f$ can be written as the "convolution" of $-g$ with an appropriate integrable kernel, the Green function $G(x)$ for the flat torus: $$f(x) = -\int_\Omega g(y) G(x - y) dy …

I do not there is a strong relation between the two notions in the general case. Curvature is obviously a local object. On the other hand, the behaviour of the first Dirichlet eigenfunction near a bou …

Indeed: the eigenvalue of the periodic $N\times N$ lattice with one vertex removed (and zero "boundary" condition a this vertex) should be approximately equal to the first eigenvalue of the usual Laplacian …

Just a short addendum: long ago Stefan Samko observed this connection between Euclidean and spherical fractional Laplacians, see item d) in page 360 of his conference paper from 2000.

On the probability side, this operator was extensively studied by Zhen-Qing Chen, Panki Kim, Renming Song and Zoran Vondraček. I always thought that Harnack inequality was proved in already in their f …

Bogdan, The boundary Harnack principle for the fractional Laplacian, Stud. Math., 123(1) (1997), 43–80, available at http://matwbn.icm.edu.pl/ksiazki/sm/sm123/sm12313.pdf. [2] M. …

Edit: I was thinking about $(-\Delta)^{\alpha/2}$ rather than $(-\Delta)^\alpha$, so the $\alpha$ in the following answer is equal to $2\alpha$ with the notation of the statement of the question. As s …