The weak maximizing property for (𝐿_{𝑝}[0,1],𝐿_{𝑞}[0,1])
Vinícius Miranda
Source abstract
We solve [J. Math. Anal. Appl. 504 (2021), Paper No. 125433, 14 pp.] (Question 4.2) left open by Dantas, Jung and Martínez-Cervantes by providing a complete characterization for the pairs ( L p [ 0 , 1 ] , L q [ 0 , 1 ] ) (L_p[0,1], L_q[0,1]) having the weak maximizing property. More precisely, we prove that, for 1 > p > ∞ 1>p>\infty and 1 ≤ q > ∞ 1\leq q>\infty , the pair ( L p [ 0 , 1 ] , L q [ 0 , 1 ] ) (L_p[0,1],L_q[0,1]) has the weak maximizing property if and only if p = q = 2 p=q=2 . A key step in the proof is a Gaussian construction that yields the failure of the compact perturbation property for the pair ( ℓ 2 , L r [ 0 , 1 ] ) (\ell _2,L_r[0,1]) whenever r > 2 r>2 .
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