Poincaré inequality 3/2 on the Hamming cube

Abstract

For any n ≥ 1, and any f :where z 3/2 for z = x + iy is taken with principal branch and ℜ denotes the real part.

Square functions and the Hamming cube: duality

“…where z 3/2 is taken in the sense of the principal brunch in the upper half-plane. Inequality (40) improves Beckner's bound for a particular exponent [11]. Consider…”

“…In fact, the reverse implication also holds, i.e., one can derive (10) from (11) for this special U. This was done in the PhD thesis of Wang [21] but we will present a short proof in Section A.2, which partly follows the Davis argument.…”

“…Indeed, the reader can find in [11] the passage from (20) to (21). In fact, inequality (20) is the same as…”

“…Roughly speaking one can take a valid estimate for a square function, dualize it by a certain double Legendre transform, and one can write its corresponding dual estimate on the Hamming cube and vice versa. To illustrate another example of our duality approach, in Section 3.4 we present a short proof of the following theorem which improves a well-known inequality of Beckner Theorem 1.3 (see [11]). For any n ≥ 1, and any f :…”

Convolution Estimates and Number of Disjoint Partitions

“…Inductive step is the same as in [1] without any modifications. This is a standard argument for obtaining estimates on the Hamming cube (see for example [2]). In order to make the paper self contained we decided to repeat the argument.…”