On Ulyanov inequalities in Banach spaces and semigroups of linear operators Original Research Article

Let image be Banach spaces and image be a consistent, equibounded semigroup of linear operators on image as well as on image it is assumed that image satisfies a Nikolskii type inequality with respect to image and image. Then an abstract Ulyanov type inequality is derived between the (modified) image-functionals with respect to image and image, where image is the infinitesimal generator of image Particular choices of image are Lorentz–Zygmund spaces, of image are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the image-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case image, are partly covered.