Isomorphic and isometric structure of the optimal domains for Hardy-type operators

Quotients, $\ell_\infty$ and abstract Cesàro spaces

“…However, it is also clear † † that X ∩ L ∞ ≡ L ∞ . Therefore, using Theorem 4.6 (via Corollary 4.8) we re-prove Proposition 4.9 from [KKM21] which states that the space Ces ∞ contains, as one would expect, a lattice isometric copy of ℓ ∞ . (b) Let X = L F be an Orlicz space generated by an Orlicz function F such that b F := sup{x > 0 : F (x) < ∞} = 1, F (1) 1 and the left derivative of F at x = 1 is finite.…”

“…(d) The Cesàro operator C is not bounded on the Zygmund space L log L but the Cesàro space C(L log L) is non-trivial (cf. [KKM21]) and the space L log L is order continuous. On the other hand, the Cesàro operator is bounded on L ∞ but the ideal (L ∞ ) a is trivial.…”

Quotients, $$\ell _\infty $$ and abstract Cesàro spaces

“…However, it is also clear 8 that X ∩ L ∞ ≡ L ∞ . Therefore, using Theorem 4.6 (via Corollary 4.8) we re-prove Proposition 4.9 from [45] which states that the space Ces ∞ contains, as one would expect, a lattice isometric copy of ∞ . (b) Let X = L F be an Orlicz space generated by an Orlicz function F such that b F := sup{x > 0 : F(x) < ∞} = 1, F(1) ≤ 1 and the left derivative of F at x = 1 is finite.…”

“…The classical Cesàro spaces ces p and Ces p are only a special case of this general construction X C X, when X = p or X = L p , respectively. We will not say more about the history and the current development of the Cesaro space theory, referring to [6] and the references given there (see also [4,5,8,17,19,21,36,39,42,45,51] and [61]; cf. [49]).…”