Related papers: Sharp inequalities for linear combinations of orth…

Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_p$, $C_p$ and $\mathfrak{c}_p$ such that $$ \,\,\,\,\sup_{t\geq 0}\left|\left|X_t\right|\right|_p\leq…

\begin{abstract} In this paper we address the problem of finding the best constants in inequalities of the form: $$ \|\big(|P_+f|^s+|P_-f|^s\big)^{\frac{1}{s}}\|_{L^p({\mathbb{T}})}\leq A_{p,s} \|f\|_{L^p({\mathbb{T}})},$$ where $P_+f$ and…

In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a…

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert…

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky \cite{HKV}: let $H$ be the Hilbert transform and let $a,b$ be real constants. Then for $1<p<\infty$ the norm of the operator $aI+bH$ from $L^p(\mathbb…

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in…

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry_Labordere et al. (2013) and provide an upper bound on the expectation of a function of the…

In this paper, we employ the ABP method developed by Brendle to establish the optimal $L^p$ logarithmic Sobolev inequality on manifolds with nonnegative Ricci curvature, as well as a sharp $L^2$ logarithmic Sobolev inequality for…

We give an alternate proof of one of the inequalities proved recently for martingales (=sums of martingale differences) in a non-commutative $L_p$-space, with $1<p<\infty$, by Q. Xu and the author. This new approach is restricted to $p$ an…

The purpose of the paper is to establish weighted maximal $L_p$-inequalities in the context of operator-valued martingales on semifinite von Neumann algebras. The main emphasis is put on the optimal dependence of the $L_p$ constants on the…

The $L^p$ maximal inequalities for martingales are one of the classical results in the theory of stochastic processes. Here we establish the sharp moderate maximal inequalities for one-dimensional diffusion processes, which include the…

A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$…

Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the…

Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a…

The $L^p$ maximal inequalities for martingales are one of the classical results in probability theory. Here we establish the sharp moderate maximal inequalities for upward skip-free Markov chains, which include the $L^p$ maximal…

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form,…

We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^\infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A…

In this paper we revisit the remainder terms of $L^p$-Hardy inequalities for magnetic $p$-Laplacians. In particular, we will give an integral representation of the sharp constant for a crucial algebraic inequality established by C. Cazacu,…

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value $\bf{C}_{G,p}$ such that the inequality $$\text{Var}_{p}(M_{G}f)\leq {\textbf{C}}_{G,p}\text{Var}_{p}(f)$$…

We present a few techniques for proving $L^p$ estimates for martingales. Basic applications to It\^o integration and rough paths are included.