On the boundedness of generalized Cesàro operators on Sobolev spaces

For β>0β>0 and p≥1p≥1, the generalized Cesàro operatorCβf(t):=βtβ∫0t(t−s)β−1f(s)ds and its companion operator Cβ⁎ defined on Sobolev spaces Tp(α)(tα) and Tp(α)(|t|α) (where α≥0α≥0 is the fractional order of derivation and are embedded in Lp(R+)Lp(R+) and Lp(R)Lp(R) respectively) are studied. We prove that if p>1p>1, then CβCβ and Cβ⁎ are bounded operators and commute on Tp(α)(tα) and Tp(α)(|t|α). We calculate explicitly their spectra σ(Cβ)σ(Cβ) and σ(Cβ⁎) and their operator norms (which depend on p  ). For 1