On the interpolation constant for Orlicz spaces

Abstract

Abstract. In this paper we deal with the interpolation from Lebesgue spaces L p and L q , into an Orlicz space L ϕ , where 1 ≤ p < q ≤ ∞ and ϕ −1 (t) = t

The Orlicz Weiss conjecture

“…Next, we introduce a class of Young functions with polynomial behavior at zero and infinity. This class of Young functions has been considered in [12] in the context of interpolation spaces and it guarantees that associated Orlicz spaces are interpolation spaces between L p spaces. Lemma 2.6.…”

“…Hence, f possesses a inverse f −1 : (0, ∞) → (0, ∞), which is again continuous and strictly increasing. For the convexity of f −1 we refer to [12]. It follows from (7)…”

“…For r = 0 and r = 1 the corresponding functions ρ 0 (t) = 1 and ρ 1 (t) = t can be seen as the extreme cases for ρ when talking about the slope of increasing concave functions. (iii) The following example can be found in [12]. Let Φ −1 be given by (8) with ρ(t) = min(1, t), t ≥ 0 and any choice of 1 < p < q < ∞.…”

“…In[12] the authors state the following result on the representation of Φ and Φ, see[12, Lem. 3.2]: If Φ ∈ P is characterized by(8), then…”

Bilinear embedding in Orlicz spaces for divergence-form operators with complex coefficients

“…Strictly speaking, complex Orlicz-space interpolation [22,21] is not applicable simply because the left hand side of (2.26) is only bi-sublinear and not bilinear in f and g. Various linearization tricks on C d would necessarily blow up the constant as d → ∞. However, in many applications of bilinear embeddings we need to control a bilinear form, which, in turn, often appears by dualizing a linear operator L as (f, g) → Lf, g L 2 (R d ) .…”