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Andrei K. Lerner

Publications and source records attributed to Andrei K. Lerner.

At least 19 recordsLinked to original sources
arXiv2026

Failure of the linear $A_2$ bound for the strong maximal operator

We show that the strong maximal operator on ${\mathbb R}^2$ does not satisfy a linear weighted $L^2$ estimate in terms of the rectangular $A_2$ characteristic. More precisely, we construct weights with arbitrarily large characteristic for which the operator norm is bounded below by $[w]_{A_2^{\mathrm{str}}}\sqrt{\log [w]_{A_2^{\mathrm{str}}}}$. The proof is elementary. It uses an explicit two-parameter mass recurrence on a product of geometric interval partitions, together with a direct comparison of arbitrary rectangular averages with anchored averages.

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arXiv2026

Dual boundedness of the maximal operator on concave Banach function spaces

We prove a conjecture of Nieraeth asserting that if the Hardy--Littlewood maximal operator $M$ is bounded on an $s$-concave Banach function space $X$ for some $s\in (1,\infty)$, then $M$ is also bounded on the associate space $X'$. The proof is based on a new argument establishing the Fefferman--Stein inequality on $X$.

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arXiv2026

Sparse domination of Calderón-Zygmund operators by mean oscillations

We show that if $T$ is a Dini-continuous Calderón--Zygmund operator satisfying $T(1)=0$, then the usual sparse domination for $T$ can be sharpened by replacing local averages with local mean oscillations. This extends a result of Benea and Bernicot for smoother kernels to the more general Dini-continuous setting. As an application, we characterize the Calderón--Zygmund operators for which a pointwise Sobolev-type inequality holds: this is the case if and only if $T(1)\in L^\infty$. This answers a recent question of Hoang, Moen and Pérez.

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arXiv2026

On an improved restricted reverse weak-type bound for the maximal operator

We obtain an improved lower bound for the restricted reverse weak-type estimate of the Hardy-Littlewood maximal operator $M$. This result is applied to the $λ$-median maximal operator $m_λ$ acting on a Banach function space $X$. We show that under certain assumptions on $X$, the boundedness properties of $m_λ$ and $M$ are equivalent.

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arXiv2024

On a non-standard characterization of the $A_p$ condition

The classical Muckenhoupt's $A_p$ condition is necessary and sufficient for the boundedness of the maximal operator $M$ on $L^p(w)$ spaces. In this paper we obtain another characterization of the $A_p$ condition. As a result, we show that some strong versions of the weighted $L^p(w)$ Coifman--Fefferman and Fefferman--Stein inequalities hold if and only if $w\in A_p$. We also give new examples of Banach function spaces $X$ such that $M$ is bounded on $X$ but not bounded on the associate space $X'$.

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arXiv2024

A note on the maximal operator on Banach function spaces

In this note we answer positively to two conjectures proposed by Nieraeth (2023) about the maximal operator on rescaled Banach function spaces. We also obtain a new criterion saying when the maximal operator bounded on a Banach function space $X$ is also bounded on the associate space $X'$.

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arXiv2024

On some improved weighted weak type inequalities

In this paper we obtain the sharp quantitative matrix weighted weak type bounds for the Christ--Goldberg maximal operator $M_{W,p}$ in the case $1<p<2$, improving a recent result by Cruz-Uribe and Sweeting. Also, in the scalar setting, we improve a weak type bound obtained in the aforementioned work for Calderón--Zygmund operators.

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arXiv2023

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner-Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón--Zygmund operators, the previously known quantitative Bloom weighted estimates are not sharp for the second and higher order commutators.

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arXiv2023

A boundedness criterion for the maximal operator on variable Lebesgue spaces

We obtain a necessary and sufficient condition on an exponent $p(\cdot)$ for which the Hardy--Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. It is formulated in terms of the Muckenhoupt-type condition $A_{p(\cdot)}$, responsible for a local control of $p(\cdot)$, and a certain integral condition on $p(\cdot)$, responsible for the behaviour of $p(\cdot)$ at infinity. Our approach is based on an earlier characterization established by L. Diening and on non-increasing rearrangements.

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arXiv2022

A note on the maximal operator on weighted Morrey spaces

In this paper we consider weighted Morrey spaces ${\mathcal M}_{λ, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{λ, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^λ}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{λ, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of the local Morrey spaces (when ${\mathcal F}$ is the family of all cubes centered at the origin) this question was answered positively in a recent work of Duoandikoetxea--Rosenthal \cite{DR21}. We obtain an extension of \cite{DR21} by showing that the answer is positive when ${\mathcal F}$ is the family of all cubes centered at a sequence of points in ${\mathbb R}^n$ satisfying a certain lacunary-type condition.

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arXiv2022

BMO with respect to Banach function spaces

For every cube $Q \subset \mathbb{R}^n$ we let $X_Q$ be a quasi-Banach function space over $Q$ such that $\|χ_Q\|_{X_Q} \simeq 1$, and for $X= \{X_Q\}$ define \begin{align*} \|f\|_{\mathrm{BMO}_X} &:=\sup_Q \,\|f-{\textstyle\frac{1}{|Q|}\int_Qf} \|_{X_Q},\\ \|f\|_{\mathrm{BMO}_X^*} &:=\sup_Q \,\inf_c\, \|f-c\|_{X_Q}. \end{align*} We study necessary and sufficient conditions on $X$ such that $$ \mathrm{BMO} = \mathrm{BMO}_X = \mathrm{BMO}_{X}^*. $$ In particular, we give a full characterization of the embedding $\mathrm{BMO} \hookrightarrow \mathrm{BMO}_X$ in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding $\mathrm{BMO}_X^* \hookrightarrow \mathrm{BMO}$. Our main theorems recover and improve all previously known results in this area.

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arXiv2022

Operator-free sparse domination

We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$, where $x\in {\mathbb R}^n$ and $Q$ is a cube in ${\mathbb R}^n$. When applied to operators, this result recovers our recent works. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalized Poincaré-Sobolev inequalities, tent spaces, and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localizable in the sense of our previous works, as we will demonstrate in an application to vector-valued square functions.

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arXiv2020

On two weight estimates for iterated commutators

In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators $T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new bump type necessary condition for the two-weighted boundedness of $T_b^m$ is obtained as well. We also provide some results related to a converse to Bloom's theorem.

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arXiv2019

Some remarks on the pointwise sparse domination

We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator $T$ for which it admits a sparse domination.

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