Kolmogorov-type inequalities in semilinear metric spaces

Abstract

For functions that take values in an isotropic semilinear metric space we prove two sharp Kolmogorov-type inequalities. In the first one we obtain an estimate for the uniform norm of the derivative (in the Rådström sense) of a function using the uniform norm of the function and the $H^\omega$-norm of the function's derivative; here $\omega$ is an arbitrary modulus of continuity. The second one gives an estimate of the uniform norm of a generalized fractional derivative of a function via its uniform norm and its $H^\omega$-norm.