Sharp Ul’yanov-type inequalities using fractional smoothness

AbstractIn this paper we obtain the following sharp (p,q) inequalities for moduli of smoothness of fractional order: ωk(f,δ)Lq(X)⩽C(∫0δ(t−θωk+θ(f,t)Lp(X))qdtt)1q,1<p<q<∞,θ=d(1/p−1/q), and X=Td or Rd. Corresponding estimates for K-functionals and corresponding embedding theorems for the Lipschitz spaces Lip(α,k,p)={ωk(f,δ)p⩽Cδα} are investigated