Related papers: Vafa-Witten bound on the complex projective space

We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$.…

Let $\Omega$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality…

We compute the Hausdorff dimension of the set of simultaneously $q^{-\lambda}$-well approximable points on the Veronese curve in $\mathbb{R}^n$ for $\lambda$ between $\frac{1}{n}$ and $\frac{2}{2n-1}$. For $n=3$, the same result is given…

For perturbed Stark operators $Hu=-u^{\prime\prime}-xu+qu$, the author has proved that $\limsup_{x\to \infty}{x}^{\frac{1}{2}}|q(x)|$ must be larger than $\frac{1}{\sqrt{2}}N^{\frac{1}{2}}$ in order to create $N$ linearly independent…

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting…

Several inequalities for eigenvalues involving convex combinations and compressions are given. These inequalities are matrix version of the basic convexity inequality f((a+b)/2) < (f(a)+f(b))/2.

The main aim of this paper is to prove that when $0<p<1/2$ the maximal operator $\overset{\sim }{\sigma }_{p}^{\ast }f:=\underset{n\in \mathbb{N}}{% \sup }\frac{\left\vert \sigma_{n}f\right\vert }{\left( n+1\right) ^{1/p-2}}$ is bounded…

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type,…

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth…

We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain $\Omega$ in Euclidean space in terms of the…

We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-\Delta+|x|^2$ on $\mathbb R^d$. Let $\Pi_\lambda$ denote the projection operator to the vector space spanned by the eigenfunctions…

Let M be a closed spin manifold of dimension at least three with a fixed topological spin structure. For any Riemannian metric, we can construct the associated Dirac operator. The spectrum of this Dirac operator depends on the metric of…

We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$,…

We calculate the low-lying eigenvalues and eigenvectors of the hermitian domain wall Dirac operator on various gauge backgrounds by Ritz minimization. The mass dependence of these eigenvalues is studied to extract the physical 4 dimensional…

The main aim of this paper is to prove that the maximal operator $\overset{% \sim }{\sigma }^{*}f:=\underset{n\in P}{\sup }\frac{\left| \sigma_{n}f\right| }{\log ^{2}\left( n+1\right) }$ is bounded from the Hardy space $H_{1/2}$ to the…

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green…

We show that if $\mu$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $$ \dim_{\rm H} \mu…

We provide a lower bound for the first eigenvalue of the Laplace-Beltrami operator on a closed orientable hypersurface minimally embedded in an orientable compact Riemannian manifold with Ricci curvature bounded below by a positive…

The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions:…

Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields…