User:Sławomir Biały/Sandbox

For Riemann–Liouville integral. The article needs to be adjusted so that the operator is defined on various domains.

bleh

Semigroup properties

The Riemann–Liouville integral I^α is well-defined for functions in L^p for all p ≥ 1, and defines a bounded linear operator from L^p to itself, which follows from an application of the integral Minkowski inequality. Moreover, also from elementary considerations, one may show that I^αƒ tends to ƒ in L^p as α tends to zero along the positive real axis. The operator norm of I^α : L^p → L^p satisfies the estimate

$\|I_{\alpha }\|_{p}\leq {\frac {1}{\operatorname {re} (\alpha )|\Gamma (\alpha )|}}.$

It follows that the spectral radius of the infinitesimal generator A of the semigroup is reduced to zero, since

$\log \rho (A)=\lim _{\operatorname {re} (\alpha )\to +\infty }{\frac {\log \|I^{\alpha }f\|_{p}}{\operatorname {re} (\alpha )}}=-\infty .$

Explicitly, the resolvent of A is given by

$[R(\lambda ;A)f](x)=\int _{0}^{x}E(x-t;\lambda )f(t)\,dt$

where

$E(s;\lambda )=\int _{0}^{\infty }e^{-\lambda \xi }s^{\xi -1}{\frac {d\xi }{\Gamma (\xi )}}$

where we have taken, for simplicity, the basepoint a = 0.

The infinitesimal generator A can be determined formally by an application of the Taylor series for the logarithm:

$Af=\sum _{n=1}^{\infty }(-1)^{n-1}{\frac {1}{n}}(I-\operatorname {id} )^{n}f$