Best constants in Muckenhoupt's inequality

Oşekowski, Adam

doi:10.5186/aasfm.2017.4256

Cited by 8 publications

(5 citation statements)

References 20 publications

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“…Fix an integer m, an exponent p ∈ (m −1 , ∞) and a constant c ∈ [1, ∞) . By the result of [9], for any 𝜀 > 0 there is a weight w ∈ A mp with [w] A mp = c and a function f ∈ L mp (w) such that the (one-dimensional) maximal function M satisfies…”

Section: Proofmentioning

confidence: 99%

Weighted estimates for the multilinear maximal operator

Oşekowski

2023

Collect. Math.

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The paper contains the study of the weighted $$L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\rightarrow L^p$$ L p 1 × L p 2 × … × L p m → L p estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants in the symmetric case $$p_1=p_2=\ldots =p_m$$ p 1 = p 2 = … = p m , and a tight constant for remaining choices of exponents.

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Section: Proofmentioning

confidence: 99%

Weighted estimates for the multilinear maximal operator

Oşekowski

2023

Collect. Math.

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“…Our starting point is the sharp dimension‐free weighted

L^{p}

estimate for maximal operators established in [11]. Namely, for any

1 < p < \infty

and any probability space

(X, μ)

with the tree structure

scriptT

and any

A_{p}

weight w on X , we have

false∥ M^{X, T} {false∥}_{L^{p} false(w false) \to L^{p} false(w false)} ⩽ \frac{p}{p - 1 - d (p, {[w]}_{A_{p}})} .

Here, for a given

1 < p < \infty

and

c ⩾ 1

, the constant

d (p, c)

is the unique number in

[0, p - 1)

satisfying the equation

c false(1 + d false) {(p - 1 - d)}^{p - 1} = {(p - 1)}^{p - 1} .

We will need the more explicit bound

\begin{matrix} true \\ right ∥ M^{X, scriptT} ∥_{L^{p} (w) \to L^{p} (w)} & left ⩽ \frac{p}{p - 1 - d false(p, {false[w false]}_{A_{p}} false)} \\ left = \frac{p}{p - 1} \end{matrix}

…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…Proof of Theorem 1.3. Our starting point is the sharp dimension-free weighted L p estimate for maximal operators established in [11]. Namely, for any 1 < p < ∞ and any probability space (X, μ) with the tree structure T and any A p weight w on X , we have…”

mentioning

confidence: 99%

A Weighted Maximal Weak‐type Inequality

Oşekowski

Rapicki

2020

Mathematika

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“…As shown in [2], we have the estimate C 2 ≤ 1109. Now, it follows from the extrapolation theorems of Duoandikoetxea [7] and the sharp weighted bounds for the dyadic maximal operator established in [12], that if r ≥ 2, then…”

Section: Introductionmentioning

confidence: 99%