On the boundedness of generalized Cesaro operators on Sobolev spaces

Abstract

For beta > 0 and p >= 1, the generalized Cesaro operator

l(beta)f(t) := beta/t(beta)integral(t)(0)(t - s)(beta-1) f(s)ds

and its companion operator l beta* defined on Sobolev spaces J(p)((alpha))(t(alpha)) and Jp((alpha))(vertical bar t vertical bar(alpha)) (where alpha >= 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 l(beta) and l(beta)* are bounded operators and commute on J(p)((alpha))(t(alpha)) and J(p)((alpha))(vertical bar t vertical bar(alpha)) . We calculate explicitly their spectra sigma(l(beta)) and sigma(l(beta)(*)) and their operator norms (which depend on p). For 1 < p <= 2, we prove that <(l(beta)(f))over cap> = l(beta)*((f) over cap) and <(l(beta)*(f))over cap> = l(beta)((f) over cap) where (f) over cap denotes the Fourier transform of a function f is an element of L-p (R). (C) 2014 Elsevier Inc. All rights reserved.