Sobolev type inequalities for rearrangement invariant spaces

Abstract

Let V n be a vectorspace of complex-valued functions defined on R of dimension n + 1 over C. We say that V n is shift invariant (on R) if f ∈ V n implies that f a ∈ V n for every a ∈ R, where f a (x) := f (x − a) on R. In this note we prove the following. Theorem. Let V n ⊂ C[a, b] be a shift invariant vectorspace of complex-valued functions defined on R of dimension n + 1 over C. Let p ∈ (0, 2]. Then f L ∞ [a+δ,b−δ] ≤ 2 2/p 2 n + 1 δ 1/p f L p [a,b] for every f ∈ V n and δ ∈ 0, 1 2 (b − a) .

Classical and approximate Taylor expansions of weakly differentiable functions

“…The analysis of these spaces has been the object of various contributions, especially in recent years, including [11,14,15,16,18,20,21]. However, apart from the specific results cited above in Lorentz and Orlicz spaces, this aspect of the theory seems to be still untouched in a general framework.…”