Related papers: Sharp Eigenfunction Bounds on the Torus for large …

We study a problem with three equivalent formulations: describing Gibbs measures for five-vertex model in quadrant; classifying coherent systems on a p-deformation of the Gelfand-Tsetlin graph related to Grothendieck polynomials; finding…

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in…

This paper presents a comprehensive analysis of the spectral properties of the connection Laplacian for both real and discrete tori. We introduce novel methods to examine these eigenvalues by employing parallel orthonormal basis in the…

We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kova\v{c}'s maximal restriction principle to show…

We prove the range of exponents in the general $L^2$ Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class of measures on $\mathbb{R}^d$. This extends to higher dimensions the sharpness result of…

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is…

We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the…

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into…

We consider the equation $-\Delta_p u=f(u)$ in a smooth bounded domain of $\mathbb{R}^n $, where $\Delta_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\geq p+4p/(p-1)$. Instead, when…

We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. For dimensions $3 \leq m \leq 6$, we generalize the work of…

We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this…

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction…

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\Om \subset \R^m$ satisfies formally the equation $-\Delta_p =1$ in $\Om$ and $|\nabla u|^{p-2}…

In the recent work arXiv:1311.3999, the authors proved that real analytic manifolds $(M, g)$ with maximal eigenfunction growth must have a self-focal point p whose first return map has an invariant L1 measure on $S^*_p M$. In this addendum…

We establish a new a priori bound for $ L^2 $-bounded sequences of solutions to the mKdV equations on the torus. This first enable us to construct weak solutions in $ L^2$ for this equation and to check that the "solutions" constructed by…

We prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Lapacian operator with $1< p< \infty$ on a compact Bakry-Emery manifold $(M^n,g,f)$ satisfying $\Ric+\nabla^2 f \geq \kappa \, g$, provided that…

We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in $\mathbb Z^d$ with dyadic radii have $\ell^p(\mathbb Z^d)$ bounds for all $p\in[2, \infty]$…

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound…

Motivated by pioneering works of Bandle and Wagner, given a bounded Lipschitz domain $\Omega \subset \mathbb R^d$ with $d\ge3$, we consider the Robin-Laplacian torsional rigidity $\tau_\alpha(\Omega)$ with negative boundary parameter…

We explore Liouville's theorem and the Strong Liouville Property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the…