Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces

For $\mu>\beta>0$, the generalized Stieltjes operators $$\mathcal{S}_{\beta,\mu} f(t):={t^{\mu-\beta}}\int_0^\infty {s^{\beta-1}\over (s+t)^{\mu}}f(s)ds, \qquad t>0,$$ defined on Sobolev spaces $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ (where $\alpha\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < \beta - \pp < \mu$, then operators $\mathcal{S}_{\beta,\mu}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Ces\'{a}ro operator on $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ . We calculate and represent explicitly their spectrum set $\sigma (\mathcal{S}_{\beta,\mu})$. The main technique is to subordinate these operators in terms of $C_0$-groups and transfer new properties from some special functions to Stieltjes operators. We also prove some similar results for generalized Stieltjes operators $\mathcal{S}_{\beta,\mu}$ in the Sobolev-Lebesgue $\mathcal{T}_p^{(\alpha)}(\vert t\vert^\alpha)$ defined on the real line $\R$. We show connections with the Fourier and the Hilbert transform and a convolution product defined by the Hilbert transform.