By Zbigniew Semadini
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It turns out that Orlicz spaces have many properties so long as F obeys an additional condition. Deﬁnition Let F be a weak Young function. 3 F (x) = |x|p obeys the Δ2 condition since F (2x) = 2p F (x). ∞ F (x) = exp(|x|) − |x| − 1 = n =2 |x|n /n! is a Young function for which the Δ2 condition is not obeyed. Indeed, if F obeys the Δ2 condition, for large n, F (2n ) ≤ DC n for suitable D. Thus, if 2n −1 ≤ x ≤ 2n , we have, with α = log C/ log 2, F (x) ≤ F (2n ) ≤ DC exp((n − 1) log C) ≤ DC exp(α(log 2)(n − 1)) = DC 2α (n −1) ≤ DC xα 36 Convexity so that the Δ2 condition implies that F is polynomially bounded.
73) that is known as Jensen’s inequality. The special case F (y) = ey , that is, f (x) dμ(x) ≤ log is also sometimes called Jensen’s inequality. 74) Convex functions and sets 25 2. 73) is intended in the sense that either |F (f (x))| dμ(x) < ∞ or the interval diverges to +∞. n 3. 3). 3). Proof Let λ0 = f (x) dμ(x). 30, F (λ) − F (λ0 ) ≥ α(λ − λ0 ) for some α. 75) F (f (x)) ≥ F (λ0 ) + α(f (x) − λ0 ) so x → F (f (x)) is bounded below by an L1 function. It follows that either F (f (x)) ∈ L1 or F (f (λ)) dμ = ∞.
96) and is also called the conjugate convex function to G. 45 Let F be a steep convex function on Rν . 98) F ∗ is given by and, in particular, F ∗ (y) < ∞. F ∗ is a steep convex function and (F ∗ )∗ (x) = F (x). 99) (sometimes called Young’s inequality). 98), for each y, there is an x0 (y) where equality holds, and for each x, there is a y0 (x) where equality holds. Proof Given y, by the steepness hypothesis, F (x) ≥ ( y + 1) x − C for some constant C. Thus, x · y − F (x) ≤ − x + C and so x · y − F (x) < −F (0) if x > C + F (0).
Banach Spaces of Continuous Functions by Zbigniew Semadini