Characterization of a rearrangement-invariant hull of a Besov space via interpolation

Abstract

Let X be a rearrangement-invariant Banach function space over a Lipschitz domain Ω⊂ℝn. We characterize the K-functionals for the pairs (X,V1X) and (X,SX), where V1X is the reduced Sobolev space built upon X and SX is the class of measurable functions on Ω such that \(\|t\sp{-\frac{1}{n}}(f^{**}(t)-f^{*}(t))\|_{\overline{X}}<\infty\), \(\overline{X}\) being the representation space of X. Using this result, we obtain an estimate of rearrangements of a function in terms of moduli of continuity and prove its sharpness. Finally we establish sharp embeddings of general Besov spaces into Lorentz spaces and characterize the rearrangement-invariant hull of a general Besov space.