# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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### Precise decay of solution fo fractional Schroedinger equations

Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$. ...
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### Fractional partial derivatives and integrals of De Bruijn's $H_t(z)$-function. Does a simpler form exist for the $z$ derivative/integral?

$\newcommand\KummerU{\text{KummerU}}$ $\newcommand\Hypergeom{\text{Hypergeom}}$ During 2018/2019, the polymath 15 project managed to successfully reduce the upper bound of the De Bruijn-Newman ...
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### Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that \begin{equation} \sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
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### Computing the fractional laplacian of a logarithm function

Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
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### Is the spectral fractional Sobolev norm equivalent to other norms (e.g. Gagliardo...)?

Let $s \in (0, 1)$ and $\Omega$ be a bounded subdomain of $\mathbb R^n$ with polygonal/polyhedral boundary. Let $\Delta$ be the Laplace-Dirichlet operator on $\Omega$ (i.e. the Laplace operator with ...
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### Relation between the eigenvalues of the weighted laplacian and fractional laplacian?

Consider the eigenvalue problem $-\Delta u = \lambda u\rho$ for $u\in \dot{H}^{1}(\mathbb{R}^n)$ with $n\geq 3$ and weight $\rho\in L^{n/2}(\mathbb{R}^n).$ Let $(\lambda_k, \psi_k)$ be the increasing ...
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### Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?

I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
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### How to check that $f(t,x)\ge 0$ for any $x\ge 0$?

Suppose $f(t,x)$ satisfices $\partial_t f=\Lambda^{-\alpha}f\partial_xf$, for $0<\alpha<1$ and where $\Lambda=(-\partial_{xx})^{1/2}$, $f_0(x)=f(0,x)$ is odd, and $f_0(x)\ge 0$, $\forall x\ge 0$....
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### Fractional derivative notation in wave turbulence

This is my first question in MathOverflow and I will do my best to format it correctly and make it clear. I am reading a paper on dispersive wave turbulence which introduces the following family of ...
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