The sharp $A_p$ constant for weights in a reverse-Hölder class

Sharp weak type estimates for weights in the class $A_{p_1, p_2}$

We get sharp estimates for the distribution function of nonnegative weights, which satisfy so called $A_{p_1, p_2}$ condition. For particular choices of parameters $p_1$, $p_2$ this condition becomes an $A_p$-condition or Reverse Hölder condition. We also get maximizers for these sharp estimates. We use the Bellman technique and try to carefully present and motivate our tactics. As an illustration of how these results can be used, we deduce the following result: if a weight $w$ is in $A_2$ then it self-improves to a weight, which satisfies a Reverse Hölder condition.

BMO中非增函数的严格反向Hardy不等式

The sharp reverse Hardy inequality in BMO for nonincreasing functions

本文针对正实数集上的Hardy算子，证明了非增局部可积函数的反向Hardy不等式中Korenovskii提出的因子e/4可去除，得到最优常数α₀，解决了相关猜想。

Schatten范数的严格拟逆Minkowski不等式

Sharp Quasi-Reverse Minkowski Inequality for Schatten Norms

该论文针对Schatten p-范数，证明2≤p<∞时存在严格拟逆Minkowski不等式，对1<p<2则说明最优常数公式不成立并给出反例与解析构造。

超立方体上最优的 Young 卷积不等式及其反向形式

Optimal Young's convolutions inequality and its reverse form on the hypercube

本文在离散超立方体上建立了对角情形下 Young 卷积不等式及其反向的精确形式，并应用于加性能量和和集，同时探讨了非对角情形的必要条件与部分结果。

On the extension of the Reverse Hölder Inequality for power functions on the real axis

We consider the class of all non-negative on $\mathbb{R_+}$ functions such that each of them satisfies the Reverse Hölder Inequality uniformly over all intervals with some constant the minimum value of which can be regarded as the corresponding "norm" of a function. We compare this "norm" with the "norm" of an even extension of a function from $\mathbb{R_+}$ on $\mathbb{R}.$ In this paper the upper estimate for the ratio of such "norms" has been obtained. In the particular case of power functions on $\mathbb{R_+}$ the precise value of the increase of the "norm" of its even extension is given. …

Equivalent definitions of dyadic Muckenhoupt and Reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic $L\log L$ and $A_\infty$ constants

In the dyadic case the union of the Reverse Hölder classes, $RH_p^d$ is strictly larger than the union of the Muckenhoupt classes $ A_p^d$. We introduce the $RH_1^d$ condition as a limiting case of the $RH_p^d$ inequalities as $p$ tends to 1 and show the sharp bound on $RH_1^d$ constant of the weight $w$ in terms of its $A_\infty^d$ constant. We also take a look at the summation conditions of the Buckley type for the dyadic Reverse Hölder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. Our lemmata also al…

New sharp Jordan type inequalities and their applications

In this paper, we prove that for x\in(0,π/2) (cos p_0x)^{1/p_0}<((sin x)/x)<(cos(x/3))^3 with the best constants p_0=0.347307245464... and 1/3. Moreover, if p\in (0,1/3] then the double inequality β_{p}(cos px)^{1/p}<((sin x)/x)<(cos px)^{1/p} holds for x\in(0,π/2), where β_{p}=2π^-1(cos((pπ)/2))^{-1/p} and 1 are the best possible. Its reverse one holds if p\in[1/2,1]. As applications, some new inequalities are established.

Exponential Decay Estimates for Fundamental Matrices of Generalized Schrödinger Systems

In this article, we investigate systems of generalized Schrödinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay estimates for them. The Schrödinger operators that we consider have leading coefficients that are bounded and uniformly elliptic, while the zeroth-order terms are assumed to be nondegenerate and belong to a reverse Hölder class of matrices. In particular, our operators need not be self-adjoint. The exponential bounds are governed by the so-called upper and lo…

Control of the Bilinear Indicator Cube Testing property

We show that the α-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure σ x ω is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding 2(n-α). Moreover, this control is sharp within the class of diagonally reverse doubling product measures. When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of BICT_{T^{α}} for α>0 yields a two weight T1 theorem for doubling weights with appropriate diag…

Improved Buckley's theorem on LCA groups

We present sharp quantitative weighted norm inequalities for the Hardy-Littlewood maximal function in the context of Locally Compact Abelian Groups, obtaining an improved version of the so-called Buckley's Theorem. On the way, we prove a precise reverse Hölder inequality for Muckenhoupt $A_\infty$ weights and provide a valid version of the "open property" for Muckenhoupt $A_p$ weights.

Sharp inequalities for one-sided Muckenhoupt weights

Let $A_\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\mathsf M^+:L^p(w)\to L^{p,\infty}(w)$ for some $p>1$, where $\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\in A_\infty ^+$ if and only if there exist numerical constants $γ\in(0,1)$ and $c>0$ such that $$ w(\{x \in \mathbb{R} : \, \mathsf M ^+\mathbf 1_E (x)>γ\})\leq c w(E) $$ for all measurable sets $E\subset \mathbb R$. Furthermore, letting $$ \mathsf C_w ^+(α):= \sup_{0<w(E)<+\infty} \frac{1}{w(E)} w(\{x\in\mathbb R:\,\mathsf M^+\mathbf 1_E (x)>α\}) $$…