Anastasios D. Delis, Eleftherios N. Nikolidakis
We prove a sharp integral inequality for the dyadic maximal operator, connecting integrals of $ϕ$, and of the dyadic maximal function of $ϕ$.
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Two weight trace inequalities for positive dyadic operators are characterized in terms of discrete Wolff's potentials in the upper triangle case.
In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$.
We consider L^p two weight inequalities for maximal truncations of dyadic Calderon-Zygmund operators. In the case of one weight being doubling, a characterization is given, and for the general case, sufficient conditions are given, including standard and non-standard testing conditions. The arguments of this paper parallel the authors prior work arXiv:0805.4711, but in this paper are much easier.
We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. It is shown that if a so-called quadratic $\mathscr{A}_{p,q}$-condition related to the measures holds, then a family of dyadic shifts satisfies the two weight estimate in an $\mathcal{R}$-bounded sense if and only if it satisfies the direct- and the dual quadratic testing condition. In the case $p=q=2$ this reduces to the result by T. Hytönen, C. Pérez, S. Treil and A. Volberg. The dyadic square function satisfies the two weight estimate if and only if it…
The following deals with the $T(b)$ theorems of David, Journé, and Semmes \cite{DJS} considered in a dyadic setting. We find sharp growth estimates for a global and a local dyadic $T(b)$ Theorem. We use multiscale analysis and Haar wavelets in the local case.
We define a generalized dyadic maximal operator involving the infinite product and discuss weighted inequalities for the operator. A formulation of the Carleson embedding theorem is proved. Our results depend heavily on a generalized Hölder's inequalities.
This book is a short introduction into dyadic analysis with applications to classical weighted norm inequalities.
We find the exact Bellman function for the weak $L^1$ norm of local positive dyadic shifts. We also describe a sequence of functions, self-similar in nature, which in the limit extremize the local weak-type (1,1) inequality.
Dyadic harmonic analysis and weighted inequalities: the sparse revolution
本文探讨了双变量调和分析在获得经典Calderón-Zygmund奇异积分算子及其交换子加权估计中的应用，并总结了稀疏算子主导方法的发展历程及主要贡献。
Using Wilson's Haar basis in $\R^n$, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in $\R^n$. We can then extend "trivially" Beznosova's Bellman function proof of the linear bound in $L^2(w)$ with respect to $[w]_{A_2}$ for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman function proof has an $n$-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Furthermore the argum…
We prove sharp $L^1$ inequalities for the dyadic maximal function $M_Tϕ$ when $ϕ$ satisfies certain $L^1$ and $L^{\infty}$ conditions
We give an alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables, is possible. This last mentioned inequality, which was first noticed in [3],also generalizes in a certain direction the results of [7].
We study trace inequalities of the type $$ \| T_k f\|_{L^q(dμ)}\leq C \|f\|_{L^p(dσ)}, \qquad f \in L^p(dσ), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $dσ$ and $dμ$ are positive Borel measures on $\R^n$. Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy ${\mathcal E}_{k, σ} [μ]=\int_{\R^n} (T_k [μ])^{p'} d σ$ through the $L^1(dμ)$-norm of an appropriate nonlinear potential $W_{k, σ}[μ]$ associated with the kernel $k$ and measures $dμ$, $d σ$. We initially work with a dyadic in…
We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in the complexity of the shift -, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two…
Sharp upper bounds for integral quantities related to the Bellman function of three variables of the dyadic maximal operator
本文研究二进极大算子三积分变量 Bellman 函数相关积分量，给出其精确上界。
A simple shortcut to proving sharp weighted estimates for the Martingale Transform and for the dyadic shift of order 1 (and so for the Hilbert transform) is presented. It is a unified proof for these both transforms. Key words: Calder\on--Zygmund operators, $A_2$ weights, $A_1$ weights, Carleson embedding theorem, Bellman function, dyadic shifts, nonhomogeneous Harmonic Analysis.
We develop technical tools that enable the use of Bellman functions for BMO defined on $α$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $α$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be…
Let $σ$ and $ω$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(fσ) \rVert_{L^q(ω)}\leq C \lVert f \rVert_{L^p(σ)}, \quad \text{for all} \, \, f \in L^p(σ), $$ for both the positive summation operators $T=T_λ(\cdot σ)$ and positive maximal operators $T=M_λ(\cdot σ)$. Here, for a family $\{λ_Q\}$ of non-negative reals indexed by the dyadic cubes $Q$, these operators are defined by $$ T_λ(fσ):=\sum_Q λ_Q \langle f\rangle^σ_Q 1_Q \quad\text{ and } \quad M_λ(fσ):=\sup_Q λ_Q \langle f\rangle^σ_Q 1…
On a sharp upper bound related to the Bellman function of three integral variables of the dyadic maximal operator
该研究聚焦于二进极大算子的三积分变量Bellman函数相关积分量的精确上界，探究其性质，该上界已由文献[11]确定。
In this paper, we prove a sharp Mei's Lemma with assuming the bases of the underlying general dyadic grids are different. As a byproduct, we specify all the possible cases of adjacent general dyadic systems with different bases. The proofs have connections with certain number-theoretic properties.
We give new and elementary proofs of one weight norm inequalities for fractional integral operators and commutators. Our proofs are based on the machinery of dyadic grids and sparse operators used in the proof of the A2 conjecture.
We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.
We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.
We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More precisely, given any measurable set $E_0$ the estimate \[w(\{x\in \mathbb{R}^n: M^{+,d}(\mathcal{X}_{E_0})(x)>t\})\leq \frac{C[(w,v)]_{A_p^{+,d}(\mathcal{R})}^p}{t^p}v(E_0)\] holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal{R})$, that is \[\frac{|E|}{|Q|}\leq [(w,v)]_{A_p^{+,d}(\mathcal{R})}\left(\frac{v(E)}{w(Q)}\right)^{1/p}\] for every dyadic …
We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in A. Reznikov, A. Volberg, "Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture", arXiv:1103.5246, and with decomposition of Calderón-Zygmund operators to dyadic shifts from Hytönen's "The sharp weighted bound for general Calderón-Zygmund operators", arXiv:1007.4330 (and later T. Hytönen, C. Pérez, S. Treil, A. Volb…
For a general Calderon-Zygmund operator $T$ on $R^N$, it is shown that $\|Tf\|_{L^2(w)}\leq C(T)\|w\|_{A_2}\|f\|_{L^2(w)}$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Perez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without bad parts; (ii) a resulting representation of a general Calderon-Zygm…
We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.
Two-weight dyadic Hardy's inequalities
本文研究无限树上的双权二进Hardy不等式，梳理已知迹测度刻画与证明，提出多种新刻画及新证明方法，引入共形不变版本，刻画算子紧性并列出开放问题。
Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets
本文研究了dyadic square functions的下界，针对S₂和S₁，给出了基于布朗运动首次退出时间和Takagi函数的精确下界，改进了经典Burkholder-Davis-Gundy下界。
We demonstrate and develop dyadic-probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view. The main result is a new global $Tb$ theorem for Calderón-Zygmund operators in this setting. Our main tools include maximal truncations, adapted Cotlar type inequalities and suppression and big piece methods. While proving our bilinear results we also advance and refine the linear theory of Calderón-Zygmund operators by improving techn…
In this work, sharp Wirtinger type inequalities for double integrals are established. As applications, two sharp Čebyšev type inequalities for absolutely continuous functions whose second partial derivatives belong to $L^2$ space are proved.
We provide an alternating proof of sharp inequalities related with Burnside's formula for $n!$
This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journé $…
Sharp quadrature formulas for integrals of complex rational functions on circles, real axis and its segments are obtained. We also find sharp quadrature formulas for calculation of $L_2$-norms of rational functions on such sets. Basing on quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive sharp inequalities for different metrics (Nikol'skii type inequalities).
Weighted Inequalities for $t$-Haar multipliers
本文给出三元权重下$t$-Haar乘子一致有界性的充要条件，推广了鞅变换的双权重结果，并简化了单权重情形，同时获得Sawyer型检验条件。
We give a characterization of the two-weight inequality for a simple vector-valued operator. Special cases of our result have been considered before in the form of the weighted Carleson embedding theorem, the dyadic positive operators of Nazarov, Treil, and Volberg in the square integrable case, and Lacey, Sawyer, Uriarte-Tuero in the L^p case. The main technique of this paper is a Sawyer-style argument and the characterization is for 1 < p < \infty. We are unaware of instances where this operator has been given attention in the two-weight setting before.
Sharp constants for an inequality of Poincaré type is studied. The problem is solved by using optimal control theory.
Let $V$ be a symmetric convex body in $\R^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities with sharp constants for algebraic polynomials on $V$ and certain non-symmetric convex bodies are proved as well.
In this paper we propose and prove some generalizations and sharpenings of certain inequalities of Wilker;'s and Shafer-Fink's type. Application of the Wu-Debnath theorem enabled us to prove some double sided inequalities.
We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical Lusin area integral $S(f)$, the Littlewood-Paley $g$-function, and their continuous analogs $S_ψ$ and $g_ψ$. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón-Zygmund operator for all $1<p\le 3/2$ and $3\le p<\infty$, and for its maximal truncations for $3\le p<\infty$.
The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure $dx$ with the Haar measure $dx/x.$ There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information conc…
We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.
We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an $m$-dimensional cube and ball and the corresponding constants for entire functions of exponential type.
In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.
In this paper, we study the weighted inequality for multilinear fractional maximal operators and fractional integrals. We give sharp weighted estimates for both operators.
In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, both in the centered and uncentered versions. In particular, we solve a question originally posed by Bober, Carneiro, Hughes and Pierce.
In this paper, the index groups for which the weighted Young's inequalities hold in both continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, we also obtain the weakest conditions for the boundedness of bilinear Fourier multipliers on modulation spaces in some sense. For the fractional integral operator, the sharp conditions for the boundedness of power weighted Lp-Lq estimates in both continuous case and discrete case are obtained. By a quite different approach from others, our theorems opti…
We prove that the sharp constant in the univariate Bernstein--Nikolskii inequality for entire functions of exponential type is the limit of the sharp constant in the V. A. Markov type inequality with an exponential weight for coefficients of an algebraic polynomials of degree n as $n\to\iy$.
Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.
In the paper, the authors find the best possible constants appeared in two inequalities for bounding the Seiffert mean by the linear combinations of the arithmetic, centroidal, and contra-harmonic means.
In the paper, by establishing the monotonicity of some functions involving the sine and cosine functions, the authors provide concise proofs of some known inequalities and find some new sharp inequalities involving the Seiffert, contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square means of two positive real numbers $a$ and $b$ with $a\ne b$.
In this paper, two sharp inequalities for bounding the psi function $ψ$ and the harmonic numbers $H_n$ are established respectively, some results in [I. Muqattash and M. Yahdi, \textit{Infinite family of approximations of the Digamma function}, Math. Comput. Modelling \textbf{43} (2006), 1329\nobreakdash--1336.] are improved, and some remarks are given.
The Toader-Qi mean of positive numbers $a$ and $b$ defined by \begin{equation*} TQ\left( a,b\right) =\frac{2}{π}\int_{0}^{π/2}a^{\cos ^{2}θ}b^{\sin ^{2}θ}dθ\end{equation*} is related to the modified Bessel function of the first kind. In this paper, we present several properties of this mean, and establish some sharp inequalities for this mean in terms of power and logarithmic means. From these a nice chain of inequalities involving Gauss compound mean, Toader mean and Toader-Qi mean is presented.
In the context of radial weights we study the dimension dependence of some weighted inequalities for maximal operators. We study the growth of the $A_1$-constants for radial weights and show the equivalence between the uniform boundedness of these constants, a dimension-free weak $L^1$ estimate for the maximal operator on annuli and the condition on the weight to be decreasing and essentially constant over dyadic annuli. Each one of these conditions is shown to provide dimension-free weighted weak type $L^1$ estimates for the centred maximal Hardy-Littlewood operator acting on radial functions…
We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in $\R^m$ and the corresponding constants for entire functions of spherical type on $\R^m$. Certain relations in the univariate weighted spaces are discussed as well.
Sharp Bernstein inequalities on simplex
本文在单纯形上建立了两类新的Bernstein不等式：一类基于谱算子得到Jacobi权$L^2$范数下的尖锐不等式，另一类针对倍增权给出$L^p$范数不等式，并指出当$d\ge3$时前者不一定是后者的特例。
Let $Ω$ be a collection of disjoint dyadic squares $ω$, let $π_ω$ denote the non-smooth bilinear projection onto $ω$ \[ π_ω(f,g)(x):=\int\int \mathbf{1}_ω(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} d ξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \Big(\sum_{ω\inΩ} |π_ω (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $Ω$ whenever $1/p + 1/q = 1/s$, $r'<p,q<r$, $r'/2 < s < r/2$.
Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $ρ$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,ρ)$ admits a Calderón--Zygmund decomposition which involves a particular family of sets, called Calderón--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of th…
We give necessary and sufficient conditions in order that inequalities of the type $$ \| T_K f\|_{L^q(dμ)}\leq C \|f\|_{L^p(dσ)}, \qquad f \in L^p(dσ), $$ hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x, y) f(y) d σ(y)$ with nonnegative kernels, and measures $d μ$ and $dσ$ on $\R^n$, in the case where $p>q>0$ and $p>1$. An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y) \sum_{Q\in{\mathcal D}} K(Q) χ_Q(x) χ_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K(Q)$ are arbitrary nonnegative constants a…
We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Bekolle constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Perez.
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