On relating two approaches for fractional calculus
In this paper two methods are discussed for defining the fractional powers \(L^{\alpha}\) of the operator \(L=x^{a_ 1}Dx^{a_ 2}D...x^{a_ n}Dx^{a_{n+1}},\) that acts on the function space \(F_{p,\mu}\). Here \(\alpha,\mu,\quad a_ i(i=1,2,...,n+1)\) are complex numbers and \(D=d/dx,\) \(F_{p,\mu}=\{\phi \in C^{\infty}(0,\infty):\quad \| x\quad kD\quad k(x^{- \mu}\phi)\|_ p<\infty \quad (resp.\quad x\quad kD\quad k(x^{- \mu}\phi)\to 0\quad as\quad x\to 0+\quad and\quad as\quad x\to \infty)\quad for\quad k=0,1,2,...\}\) with \(1\leq p<\infty\) (resp. with \(p=\infty)\). Equipped with the family \(\{\| x\quad kD\quad k(x^{- \mu}\phi)\|_ p\}^{\infty}_{k=0}\) of seminorms, \(F_{p,\mu}\) becomes a Fréchet space. In case the parameter \(a=\sum^{n+1}_{i=1}a_ i\) is real and \(m=n-a\neq 0\), \(L^{\alpha}\) has already been studied in terms of the Erdélyi-Kober operators (Direct Approach) [the second author, Proc. Lond. Math. Soc., III. Ser. 45, 519-546 (1982; Zbl 0487.44010)]. The critical case \(m=0\) is settled here as follows. Letting \(\eta_ k=k-\sum^{k}_{i=1}a_ i\) and \(- A_ k=x^{-\eta_ k}\delta x^{\eta_ k}\) with \(\delta =xD\), we observe \(L=(-A_ 1)(-A_ 2)...(-A_ n)\) and then define \(L^{\alpha}=(-A_ 1)^{\alpha}(-A_ 2)^{\alpha}...(-A_ n)^{\alpha},\) where \((-A)^{\alpha}\) is given as \[ (- A)^{\alpha}\phi =-\pi^{-1}\sin (\pi \alpha)\int^{\infty}_{0}\lambda^{\alpha -1}[R(\lambda;A)\quad - \quad \lambda /(1+\lambda \quad 2)]A\phi d\lambda \quad -\quad \sin (\pi \alpha /2)A\phi \] for \(0<Re \alpha <2\) together with \(((- A)^{\alpha})\phi =(-A)^{\alpha -\ell}(-A)^{\ell}\phi\) for \(\ell <Re \alpha <\ell +2\), \(\ell =\pm 1,\pm 2,..\). (Spectral Approach). Such a procedure is seen to be a particular case of the more general fractional power \(T^{\alpha}\) of the operator \(T=(-A_ 1)(-A_ 2)...(-A_ n)\) studied on the abstract Fréchet space. It is also demonstrated that the expressions of \(L^{\alpha}\) for \(m=0\) can be obtained as limiting cases, in some sense, of \(L^{\alpha}\) given for \(m>0\).
scientific article; zbMATH DE number 4061567
scientific article