Related papers: Sharp Pitt inequality and logarithmic uncertainty …

We observe that the ratio of determinants of $2d$ Laplacians which appear in the duality transformation relating two sigma models with abelian isometries can be represented as a torsion of an elliptic (DeRham) complex. As a result, this…

We obtain sharp inequalities for the k-plane transform, the "j-plane to k-plane" transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some…

For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of…

The operator of double differentiation, perturbed by the composition of the differentiation operator and a convolution one, on a finite interval with Dirichlet boundary conditions is considered. We obtain uniform stability of recovering the…

Let $L$ be a linear operator in $L^2(\mathbb{R}^n)$ which generates a semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy the Gaussian upper bound. In this paper, we investigate several kinds of weighted norm inequalities for the conical…

Uncertainty principle is one of the fundamental principles of quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak…

The Pinsker inequality lower bounds the Kullback--Leibler divergence $D_{\textrm{KL}}$ in terms of total variation and provides a canonical way to convert $D_{\textrm{KL}}$ control into $\lVert \cdot \rVert_1$-control. Motivated by…

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…

In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we…

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…

We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the…

We prove sharp L^2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first order space derivatives, using only the Hardy inequality. We then deduce bounds on…

The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ diffusion coefficients with uniformly bounded…

We establish fractional Hardy inequality on bounded domains in $\mathbb{R}^{d}$ with inverse of distance function from smooth boundary of codimension $k$, where $k=2, \dots,d$, as weight function. The case $sp=k$ is the critical case, where…

The entropic formulation of the inertia and the gravity relies on quantum, geometrical and informational arguments. The fact that the results are completly classical is missleading. In this paper we argue that the entropic formulation…

We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it…

We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our…

We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of…

The Generalized Uncertainty Principle and the related minimum length are normally considered in non-relativistic Quantum Mechanics. Extending it to relativistic theories is important for having a Lorentz invariant minimum length and for…

Uncertainty is an essential consideration for time series forecasting tasks. In this work, we specifically focus on quantifying the uncertainty of traffic forecasting. To achieve this, we develop Deep Spatio-Temporal Uncertainty…