On the action of the Hilbert transform on ℓ1N-valued functions

Publication date: Available online 24 November 2018Source: Statistics & Probability LettersAuthor(s): Adam OsękowskiAbstractLet H be a separable Hilbert space. The periodic Hilbert transform H is bounded as an operator on Lp(T;ℓ1N(H)), 1<p<∞, since ℓ1N(H) is a UMD space. We prove that there is a finite constant cp depending only on p such that cp−1(lnN+1)≤||H||Lp(T;ℓ1N(H))→Lp(T;ℓ1N(H))≤cp(lnN+1).The proof uses probabilistic methods and exploits bounds for differentially subordinate martingales.

Source: Statistics and Probability Letters - November 24, 2018 Category: Statistics Source Type: research