Integral operators on Sobolev–Lebesgue spaces

Abstract

For $\mu , \beta \in {\mathbb {R}}$, we introduce and study in detail the generalized Stieltjes operators

$$\begin{aligned} {\mathcal {S}}_{\beta ,\mu } f(t):={t^{\mu -\beta }}\int _0^\infty {s^{\beta -1}\over (s+t)^{\mu }}f(s)\mathrm{d}s, \qquad t>0, \end{aligned}$$

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 $L^p({\mathbb {R}}^+)$ for $p\ge 1$). If $0< \beta - \frac{1}{p} < \mu$, then operators ${\mathcal {S}}_{\beta ,\mu }$ are bounded, commute and factorize with generalized Cesàro operator on ${{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })$ . We give their norm, and 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 which allows to transfer new properties from some exponential functions to these 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 ${\mathbb {R}}$. We show connections of this family of operators with the Fourier and the Hilbert transform, and a convolution product defined by the Hilbert transform.

Appendix A: Mathematica code

For the interested reader, we include below the Mathematica code of Fig. 1. The code of the remaining figures is analogous to this one.