Weighted square function inequalities

Oşekowski, Adam

doi:10.5565/publmat6211804

Cited by 15 publications

(3 citation statements)

References 25 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for A 1 weights w. This improves the main result of [15], where a similar estimate (with √ 5 in place of 2) was proved for dyadic martingales. In view of [1, Theorem 1.3], it seems unlikely that the A 1 characteristic in (1.5) can be replaced by a function of any A p characteristic with p > 1, although for dyadic martingales even the A ∞ characteristic suffices [10, Theorem 2].…”

Section: Consequences Of the Weighted Estimatesupporting

confidence: 80%

Weighted Davis Inequalities for Martingale Square Functions

Wollgast

Zorin-Kranich

2022

J Theor Probab

View full text Add to dashboard Cite

For a Hilbert space-valued martingale $$(f_{n})$$ ( f n ) and an adapted sequence of positive random variables $$(w_{n})$$ ( w n ) , we show the weighted Davis-type inequality $$\begin{aligned} {\mathbb {E}}\left( {|}f_{0}{|} w_{0} + \frac{1}{4} \sum _{n=1}^{N} \frac{{|}df_{n}{|}^{2}}{f^{*}_{n}} w_{n} \right) \le {\mathbb {E}}( f^{*}_{N} w^{*}_{N}). \end{aligned}$$ E | f 0 | w 0 + 1 4 ∑ n = 1 N | d f n | 2 f n ∗ w n ≤ E ( f N ∗ w N ∗ ) . More generally, for a martingale $$(f_{n})$$ ( f n ) with values in a $$(q,\delta )$$ ( q , δ ) -uniformly convex Banach space, we show that $$\begin{aligned} {\mathbb {E}}\left( {|}f_{0}{|} w_{0} + \delta \sum _{n=1}^{\infty } \frac{{|}df_{n}{|}^{q}}{(f^{*}_{n})^{q-1}} w_{n} \right) \le C_{q} {\mathbb {E}}( f^{*} w^{*}). \end{aligned}$$ E | f 0 | w 0 + δ ∑ n = 1 ∞ | d f n | q ( f n ∗ ) q - 1 w n ≤ C q E ( f ∗ w ∗ ) . These inequalities unify several results about the martingale square function.

show abstract

Section: Consequences Of the Weighted Estimatesupporting

confidence: 80%

Weighted Davis Inequalities for Martingale Square Functions

Wollgast

Zorin-Kranich

2022

J Theor Probab

View full text Add to dashboard Cite

show abstract

“…for A 1 weights w. This improves the main result of [Osę18], where a similar estimate (with √ 5 in place of 2) was proved for dyadic martingales. In view of [BO21, Theorem 1.3], it seems unlikely that the A 1 characteristic in (1.5) can be replaced by a function of any A p characteristic with p > 1, although for dyadic martingales even the A ∞ characteristic suffices [GW74,Theorem 2].…”

Section: Introductionsupporting

confidence: 80%

Weighted Davis inequalities for martingale square functions

Wollgast¹,

Zorin-Kranich²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We will exploit the concavity of B in appropriate directions; to this end, we need the following auxiliary geometrical fact, taken from [24]. We provide an easy proof for the sake of completeness.…”

Section: Necessity Of the θ-Regularity Conditionmentioning

confidence: 99%