arXiv subjects
Publications and source records attributed to Andrei K. Lerner.
Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence on $[w]_{A_p}$ for the $L^p(w)$ operator norm of $S$ for $1<p\le 2$.
We construct an example showing that the upper bound $[w]_{A_1}\log({\rm{e}}+[w]_{A_1})$ for the $L^1(w)\to L^{1,\infty}(w)$ norm of the Hilbert transform cannot be improved in general.
We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_Ω$, where $M$ is the maximal operator and $T_Ω$ is a rough homogeneous singular integral with angular part $Ω\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
We obtain a weak type $(1,1)$ estimate for a maximal operator associated with the classical rough homogeneous singular integrals $T_Ω$. In particular, this provides a different approach to a sparse domination for $T_Ω$ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou.
In recent years, it has been well understood that a Calderón-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.
L. Diening \cite{D1} obtained the following dual property of the maximal operator $M$ on variable Lebesque spaces $L^{p(\cdot)}$: if $M$ is bounded on $L^{p(\cdot)}$, then $M$ is bounded on $L^{p'(\cdot)}$. We extend this result to weighted variable Lebesque spaces.
We obtain an alternative approach to recent results by M. Lacey \cite{La} and T. Hytönen {\it et al.} \cite{HRT} about a pointwise domination of $ω$-Calderón-Zygmund operators by sparse operators. This approach is rather elementary and it also works for a class of non-integral singular operators.
This book is a short introduction into dyadic analysis with applications to classical weighted norm inequalities.
We show that the Hilbert transform does not map $L^1(M_Φw)$ to $L^{1,\infty}(w)$ for every Young function $Φ$ growing more slowly than $t\log\log ({\rm e}^{\rm e}+t)$. Our proof is based on a construction of M.C. Reguera and C. Thiele.
We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the general context of maximally modulated Calderón-Zygmund operators.
We obtain $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. Our approach works in the general context of maximally modulated Calderón-Zygmung operators.
We establish several mixed $A_p$-$A_\infty$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_\infty$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, of a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller than the previously known bounds (both one supremum and two suprema).
We consider the one-dimensional John-Nirenberg inequality: $$ |\{x\in I_0:|f(x)-f_{I_0}|>\a\}|\le C_1|I_0|\exp\Big(-\frac{C_2}{\|f\|_{*}}\a\Big). $$ A. Korenovskii found that the sharp $C_2$ here is $C_2=2/e$. It is shown in this paper that if $C_2=2/e$, then the best possible $C_1$ is $C_1= \frac{1}{2}e^{4/e}$.
Let $S_{\a,ψ}(f)$ be the square function defined by means of the cone in ${\mathbb R}^{n+1}_{+}$ of aperture $\a$, and a standard kernel $ψ$. Let $[w]_{A_p}$ denote the $A_p$ characteristic of the weight $w$. We show that for any $1<p<\infty$ and $\a\ge 1$, $$\|S_{\a,ψ}\|_{L^p(w)}\lesssim \a^n[w]_{A_p}^{\max(1/2,\frac{1}{p-1})}.$$ For each fixed $\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\a$. Previously this estimate was proved in the case $\a=1$ using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on $\a$. Hence we give a different proof suitable for all $\a\ge 1$ and avoiding the notion of the intrinsic square function.
In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function $\Mm$ introduced in \cite{LOPTT} and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed "$A_p-A_{\infty}$" bound for $\Mm$, some partial results related to a Buckley-type estimate for $\Mm$, and a sufficient condition for the boundedness of $\Mm$ between weighted $L^p$ spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work. Then we obtain a multilinear version of the "$A_2$ conjecture". Several open problems are posed.
We give a simple proof of the $A_2$ conjecture proved recently by T. Hytönen. Our proof avoids completely the notion of the Haar shift operator, and it is based only on the "local mean oscillation decomposition". Also our proof yields a simple proof of the "two-weight conjecture" as well.
Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{\mathscr{D},{\mathcal S}}$ by $${\mathcal A}_{\mathscr{D},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}χ_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calderón-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{\mathscr{D},{\mathcal S}}\|{\mathcal A}_{\mathscr{D},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-weight conjecture" by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the "$A_2$ conjecture"; (iii) an extension of certain mixed $A_p$-$A_r$ estimates to general Calderón-Zygmund operators; (iv) an extension of sharp $A_1$ estimates (known for $T$) to the maximal Calderón-Zygmund operator $T_{\natural}$.