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&alpha; is well-defined for functions in Lp for all p ≥ 1, and defines a bounded linear operator from Lp to itself, which follows from an application of the integral Minkowski inequality. Moreover, also from elementary considerations, one may show that I&alpha;&fnof; tends to &fnof; in Lp as &alpha; tends to zero along the positive real axis. The operator norm of I&alpha; : Lp &rarr; Lp satisfies the estimate
 * $$\|I_\alpha\|_p\le \frac{1}{\operatorname{re}(\alpha)|\Gamma(\alpha)|}.$$

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:
 * $$Af = \sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}(I-\operatorname{id})^nf$$