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

Let X be a rearrangement-invariant Banach function space over a Lipschitz domain Ω⊂ℝ n . We characterize the K -functionals for the pairs ( X , V 1 X ) and ( X , S X ), where V 1 X is the reduced Sobolev space built upon X and S X is the class of measurable functions on Ω such that t-1/n(f^**(t)-f^*(t))_X<∞ , 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.