On the Boundedness of Generalized Cesàro Operators on Sobolev Spaces

For ? > 0 and p? 1, the generalized Cesàro operator. C?f(t):=?t??0t(t-s)?-1f(s)ds and its companion operator C?* defined on Sobolev spaces Tp(?)(t?) and Tp(?)(|t|?) (where ? ? 0 is the fractional order of derivation and are embedded in Lp(R+) and Lp(R) respectively) are studied. We prove that if p> 1, then C? and C?* are bounded operators and commute on Tp(?)(t?) and Tp(?)(|t|?). We calculate explicitly their spectra ?(C?) and ?(C?*) and their operator norms (which depend on p). For 1 < p? 2, we prove that C?(f)?=C?*(f?) and C?*(f)?=C?(f?) where f? denotes the Fourier transform of a function f?Lp(R). © 2014 Elsevier Inc.