Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces

Abstract

For $μ>β>0$, the generalized Stieltjes operators $$ \mathcal{S}_{β,μ} f(t):={t^{μ-β}}\int_0^\infty {s^{β-1}\over (s+t)^μ}f(s)ds, \qquad t>0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(α)}(t^α)$ (where $α\ge 0$ is the fractional order of derivation and these spaces are embedded in $L^p(\RR^+)$ for $p\ge 1$) are studied in detail. If $0 < β- \pp < μ$, then operators $\mathcal{S}_{β,μ}$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(α)}(t^α)$ . We calculate and represent explicitly their spectrum set $σ(\mathcal{S}_{β,μ})$. 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}_{β,μ}$ in the Sobolev-Lebesgue $\mathcal{T}_p^{(α)}(\vert t\vert^α)$ 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.