# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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### Characterization of locality in Fourier multiplier

Obviously Laplacian $-\Delta$ is a local operator, and it can be written as a Fourier multiplier $|\xi|^2$. Similarly fractional Laplacian $(-\Delta)^s$ is with Fourier multiplier $|\xi|^{2s}$ and is ...
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### What is the relationship between Hölder spaces and differentiability?

I'm porting this question over from MSE as it did not get any responses other than one comment on there. Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...
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### Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?

This question aims to extend this question to (automorphic) Dirichlet L-functions. Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$ with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
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### Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?

The well-known integral expression for the entire function: $$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$ ...
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### A question of interpolation space on homogeneous Carnot group

Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows: A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
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### Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms

Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
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### Poisson Kernel and solution formula for fractional elliptic problem

$$k (-\Delta)^s u + u = 0, \qquad x \in U, \\ u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,$$ with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
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Is it true $$\frac{1}{\Gamma(1-\nu)}\frac{1}{\Gamma(\nu)} \int_{0}^{x}(x-y)^{-\nu}dy\int_0^y (y-t)^{\nu-1}f(t)dt = \int_0^x f(u)du$$ for any continuous function $f(x)$ such that $f(0)=0$ and $0<\... • 419 1 vote 1 answer 470 views ### Fourier transform of the fractional Poisson kernel Recall that the extension of function from$u:\mathbb{R}^n\to \mathbb{R}$can be defined using the Poisson Kernel as follows: $$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+... • 655 2 votes 0 answers 43 views ### Nonlocal perimeter of level sets Let u \in W^{s,1}(B) be given and k < l be two numbers, then I am looking for a way to bound the following term from above. Here B is the euclidean ball.$$ \int_{B: u < k} \int_{B:u>l} ... • 455 0 votes 0 answers 118 views ### Fractional Laplacian of smooth cut off functions Suppose we have a smooth compactly supported function$\phi\in C^{\infty}_c(B_\epsilon(0))$such that$0\leq \phi \leq 1$,$\phi\equiv 1$on the unit ball and$\phi$vanishes outside$B_\epsilon(0).$... • 655 1 vote 0 answers 199 views ### Questions about iterating the Euler-Maclaurin summation formula Introduction The Euler–Maclaurin summation formula is as follows for a positive integer$p$and a continuous function$f(\cdot)$that is$p$times continuously differentiable on the interval$[m,n]$: ... • 4,655 1 vote 0 answers 133 views ### Multivariable fractional calculus I recently started to learn about nonlocal PDEs. Sorry if I am missing something really standard in this field. I know Riesz derivative is defined through Fourier transform: $$\mathcal{F}(\partial^\... 0 votes 0 answers 59 views ### How to prove this estimate involving the Stein Derivative? Recall the Stein Derivative,$$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$I want to show that,$$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \... • 655 0 votes 0 answers 65 views ### Normal vector to a level set and fractional Laplacian Let$U=\{u\le0\}$and$\partial U=\{u=0\}$. Suppose$\nabla u$does not vanish on$\partial U$. Then the (canonical extension of the) normal vector field to$\partial U$(pointing to the interior of$\...
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Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if \$f:[-\pi,\pi] \subset \mathbb{R} \...