Generalized Stieltjes and other integral operators on Sobolev-Lebesgue spaces

Abstract

For $\mu >\beta >0$, the generalized Stieltjes operators $${\mathcal{S}}_{\beta ,\mu }f(t):=t^{\mu -\beta }{\int }_0^{\infty }\frac{s^{\beta -1}}{(s+t{)}^{\mu }}f(s)ds,t>0,$$ 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 $0 < β- \pp < μ$, then operators ${\mathcal{S}}_{\beta ,\mu }$ are bounded (and we compute their operator norms which depend on $p$); commute and factorize with generalized Cesáro 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 ${\mathcal{T}}_p^{(\alpha )}(|t{|}^{\alpha })$ defined on the real line $\mathbb{R}$. We show connections with the Fourier and the Hilbert transform and a convolution product defined by the Hilbert transform.

45 pages, 6 figures