On the boundedness of generalized Cesàro operators on Sobolev spaces

For β > 0 and p ≥ 1 , the generalized Cesàro operator C β f ( t ) : = β t β ∫ 0 t ( t − s ) β − 1 f ( s ) d s and its companion operator C β ⁎ defined on Sobolev spaces T p ( α ) ( t α ) and T p ( α ) ( | t | α ) (where α ≥ 0 is the fractional order of derivation and are embedded in L p ( R + ) and L p ( R ) respectively) are studied. We prove that if p > 1 , then C β and C β ⁎ are bounded operators and commute on T p ( α ) ( t α ) and T p ( α ) ( | t | α ) . We calculate explicitly their spectra σ ( C β ) and σ ( C β ⁎ ) and their operator norms (which depend on p). For 1 < p ≤ 2 , we prove that C β ( f ) ˆ = C β ⁎ ( f ˆ ) and C β ⁎ ( f ) ˆ = C β ( f ˆ ) where f ˆ denotes the Fourier transform of a function f ∈ L p ( R ) .