Related papers: A sharp weighted Wirtinger inequality

We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic…

The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty.$ We also calculate the precise value of…

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.

We find the sharp constant $C=C(\tau,p, \mathbb{E}G, \mathbb{E}F)$ of the following inequality $\|(G^{2}+ \tau^{2} F^{2})^{1/2} \|_{p} \leq C \|F\|_{p},$ where $G$ is the transform of a martingale $F$ under a predictable sequence…

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…

For an integral $2$-varifold $V=\underline{v}(\Sigma,\theta_{\ge 1})$ in $\mathbb{R}^n$ with generalized mean curvature $H\in L^2$ such that $\mu(\mathbb{R}^n)=4\pi$ and $\int_{\Sigma}|H|^2d\mu\le 16\pi(1+\delta^2)$ , we show that $\Sigma$…

We present some classical and weighted Poincar\'e inequalities for some one-dimensional probability measures. This work is the one-dimensional counterpart of a recent study achieved by the authors for a class of spherically symmetric…

We obtain sharp estimate on $p$-spectral gaps, or equivalently optimal constant in $p$-Poincar\'e inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp $p$-spectral gap.

Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. More generally, optimal constants are obtained for the full Stein-Weiss potential as…

In this paper, we study the optimal constant in the nonlocal nonlinear Poincar\'e-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} \lambda_\alpha(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac…

We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $\|\psi(|\nabla|) \exp(it\phi(|\nabla|)f \|_{L^2(w)} \leq C\|f\|_{L^2}$, where the weight $w$ is radial and depends only…

We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$. For Hamming cube the sharp constant is not known,…

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy \cite{BuGu70} and Davis \cite{Da70} for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}] \leq C_p \mathbb{E}[(B^*(\tau))^p]$…

We consider a monomial Caffarelli-Kohn-Nirenberg inequality, find the optimal constant and classify the optimizers under an integrated curvature dimension condition. We take advantage of the $\Gamma$-calculus to exploit geometrical…

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…

Using the log-convexity of the Gamma function and Euler's reflection formula, we give a new proof of a classical weighted sine product inequality. Two different parameter choices yield two competing upper bounds for the same product. We…

We give a proof of the Lieb-Thirring inequality in the critical case $d=1$, $\gamma= 1/2$, which yields the best possible constant.

We discuss the value of the best constant in Gaffney inequality namely $$ \lVert \nabla \omega \rVert_{L^{2}}^{2}\leq C\left( \lVert d\omega\rVert_{L^{2}}^{2}+\lVert \delta\omega\rVert_{L^{2}% }^{2}+\lVert \omega\rVert_{L^{2}}^{2}\right) $$…

The optimality of the integral inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds>2\pi$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive…

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e…