Kato's Inequality Research Articles

Refined Kato inequality and applications to norm and numerical radius inequalities

Refined Kato inequality and applications to norm and numerical radius inequalities

Finite-key analysis for coherent one-way quantum key distribution

Coherent-one-way (COW) quantum key distribution (QKD) is a significant communication protocol that has been implemented experimentally and deployed in practical products due to its simple equipment requirements. However, existing security analyses of COW-QKD either provide a short transmission distance or lack immunity against coherent attacks in the finite-key regime. In this paper, we present a tight finite-key security analysis within the universally composable framework for a variant of COW-QKD, which has been proven to extend the secure transmission distance in the asymptotic case. We combine the quantum leftover hash lemma and entropic uncertainty relation to derive the key rate formula. When estimating statistical parameters, we use the recently proposed Kato's inequality to ensure security against coherent attacks and achieve a higher key rate. Our paper confirms the security and feasibility of COW-QKD for practical application and lays the foundation for further theoretical study and experimental implementation. Published by the American Physical Society 2024

On Landau–Kato inequalities via semigroup orbits

On Landau–Kato inequalities via semigroup orbits

Regularity of Minimizing p-Harmonic Maps Into Spheres and Sharp Kato Inequality

Abstract We study regularity of minimizing $p$-harmonic maps $u \colon B^{3} \to \mathbb{S}^{3}$ for $p$ in the interval $[2,3]$. For a long time, regularity was known only for $p = 3$ (essentially due to Morrey [ 24]) and $p = 2$ (Schoen–Uhlenbeck [ 29]), but recently Gastel [ 7] extended the latter result to $p \in [2,2+\frac{2}{15}]$ using a version of Kato inequality. Here, we establish regularity for a small interval $p\in [2.961,3]$ by combining Morrey’s methods with Hardt and Lin’s Extension Theorem [ 11]. We also improve on the other result by obtaining regularity for $p \in [2,p_{0}]$ with $p_{0} = \frac{3+\sqrt{3}}{2} \approx 2.366$. In relation to this, we address a question posed by Gastel and prove a sharp Kato inequality for $p$-harmonic maps in two-dimensional domains, which is of independent interest.

Finite-key security analysis of differential-phase-shift quantum key distribution

Differential-phase-shift (DPS) quantum key distribution (QKD) is one of the major QKD protocols that can be implemented with a simple setup using a laser source and a passive detection unit. Recently, an information-theoretic security proof of this protocol has been established in [npj Quant. Inf. 5, 87 (2019)] assuming the infinitely large number of emitted pulses. To implement the DPS protocol in a real-life world, it is indispensable to analyze the security with the finite number of emitted pulses. The extension of the security proof to the finite-size regime requires the accommodation of the statistical fluctuations to determine the amount of privacy amplification. In doing so, Azuma's inequality is often employed, but unfortunately we show that in the case of the DPS protocol, this results in a substantially low key rate. This low key rate is due to a loose estimation of the sum of probabilities regarding three-photon emission whose probability of occurrence is very small. The main contribution of our work is to show that this obstacle can be overcome by exploiting the recently found novel concentration inequality, Kato's inequality. As a result, the key rate of the DPS protocol is drastically improved. For instance, assuming typical experimental parameters, a 3 Mbit secret key can be generated over 77 km for 8.3 hours, which shows the feasibility of DPS QKD under a realistic setup.

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Some domination inequalities for spectral zeta kernels on closed Riemannian manifolds

We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-type operator on smooth functions on closed Riemannian manifolds. We then apply the result to establish some new domination inequalities for spectral zeta functions and their related spectral zeta kernels on $n$-dimensional unit spheres using Kato's inequalities and majorisation techniques. Our results are the generalisations of Kato's comparison inequalities for Riemannian surfaces to $n$-dimensional closed Riemannian manifolds.

Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

<abstract><p>This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.</p></abstract>

On the fractional relativistic Schrödinger operator

We collect some interesting results for equations driven by the fractional relativistic Schrödinger operator (−Δ+m2)s with s∈(0,1) and m>0. More precisely, for the linear theory, we prove Hölder-Schauder-Zygmund regularity results and a Kato's inequality. For the nonlinear theory, we obtain L∞-regularity, exponential decay, a Pohozaev-type identity, and a symmetry result for solutions of certain nonlinear fractional problems.

Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first‐order stochastic conservation law with a multiplicative source term involving a ‐Brownian motion. After having defined a measure‐valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure‐valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure‐valued entropy solution of the stochastic conservation law, which proves the existence of the measure‐valued entropy solution.

Heinz–Kato inequality in Banach spaces

It is observed that in Banach spaces, sectorial operators having bounded imaginary powers satisfy a Heinz–Kato inequality.

KATO'S INEQUALITIES UP TO THE BOUNDARY FOR A QUASILINEAR ELLIPTIC OPERATOR

In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator

Multiplicity and Concentration of Solutions for a Fractional Kirchhoff Equation with Magnetic Field and Critical Growth

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.

Einstein four-manifolds of three-nonnegative curvature operator

In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to $$(S^4, g_0)$$ or $$({\mathbb {C}}P^2, g_{FS})$$ , and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to $$(S^4, g_0)$$ , $$({\mathbb {C}}P^2, g_{FS})$$ , or $$(S^2\times S^2, g_0\oplus g_0)$$ , up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with $$\mathrm {Ric}=g$$ and nonnegative sectional curvature is bounded above by $$\frac{4}{3}+4^{\frac{1}{3}}$$ . The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenbock formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenbock formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdzinski’s argument.

Kato's inequalities for admissible functions to quasilinear elliptic operators <i>A </i>

Kato's inequalities for admissible functions to quasilinear elliptic operators <i>A </i>

Finsler Hardy–Kato's inequality

We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by [3] in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.

Uniqueness and comparison principles for semilinear equations and inequalities in Carnot groups

Abstract Variants of the Kato inequality are proved for distributional solutions of semilinear equations and inequalities on Carnot groups. Various applications to uniqueness, comparison of solutions and Liouville theorems are presented.

Kato's Inequality for Magnetic Relativistic Schrödinger Operators

Kato's inequality is shown for the magnetic relativistic Schrödinger operator H_{A,m} defined as the operator theoretical square root of the selfadjoint, magnetic nonrelativistic Schrödinger operator (-i\nabla-A(x))^2+m^2 with an L^{2}_{\mathrm {loc}} vector potential A(x) .

Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.