A note on norms of compression operators on function spaces

Characterization of the restricted type spaces R(X)

“…177-178]. When X is a Lorentz space, it is easy to show that, in fact, the equality always holds (examples of non-Lorentz r.i. spaces with strict inequality are known [15]). This result agrees with the following remark: We are now going to see a couple of examples for which the upper Boyd index is equal to 1:…”

Weighted norm inequalities and indices

“…is bounded on decreasing functions. For example, in [39] the author exhibits a rearrangement invariant Banach space X such that z X = z X = 1 2 , but β X = 0 and β X = 1, therefore while Theorem 1 implies that P and Q are χ − X bounded, by Boyd's theory P and Q are not bounded on X , (cf.…”

Extrapolation of Lp Data from a Modular Inequality

“…In [11], Shimogaki constructed an r.i. space L ρ (0, 1), m for which τ ρ (t) = τ ρ (t) = √ t, 0 < t < 1 (so that i(ρ) = I(ρ) = i(ρ ) = I(ρ ) = 2), yet p(ρ) = 1 and q(ρ) = ∞. Since the Lorentz space L 21 = L 21 (0, 1), m is the smallest r.i. space on (0, 1) with fundamental function √ t [1, p. 79], we have the continuous embedding…”