Related papers: Improved quadratic Gowers uniformity for the M\"ob…

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

In the conformal class of the standard metric on the $3$-sphere, we prove a quantitative refinement of the Andrews-De Lellis-Topping inequality in terms of a two-term distance to the set of minimizing conformal factors. This inequality is…

The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{\alpha\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where $c$ and $\alpha$ are absolute constants, $T$ is a Calder\'on--Zygmund…

A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $\mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case.…

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…

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger's inequality resolves an open question raised by Weidl from 2006. The improvements…

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$,…

Let $M$ be a relatively compact $C^2$ domain in a complex manifold $\mathcal M$ of dimension $n$. Assume that $H^{1}(M,\Theta)=0$ where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $M$. Suppose that the Levi-form of the…

We study in this article the Improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms…

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…

We show that natural noncommutative gauge theory models on $\mathbb{R}^3_\lambda$ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of $\mathbb{R}^3_\lambda$ and the components of…

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $\Gamma \cong \mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\| \;\, \|_{st}$ and the distance…

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, the universal spectral cluster estimates of Sogge \cite{MR930395} for the…

We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq…

We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then…

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in…

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Sigma x [0,1] x R, where Sigma is the Heegaard surface, instead of Sym^g(Sigma). We then show that the entire invariance proof can be carried…

We introduce a new technique to establish Hungarian multivariate theorems. In this article we apply this technique to the strong approximation bivariate theorems of the uniform empirical process. It improves the Komlos, Major and Tusn\'ady…

We show that M\"uller's $L\log L$ bound $$F(Du)\geq 0,\,Du\in L^p_{\mathrm{loc}}(\mathbb{R}^n)\implies F(Du)\in L\log L_{\mathrm{loc}}(\mathbb{R}^n)$$ for $F =\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$.…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…