In this paper we prove a new variant of the Herz majorizing principle for operators defined by K-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove L p-boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hörmander-Michlin type. We also find sharp bounds on the L p-norm of large imaginary ...

Abstract. Assume that f is Dunkl polyharmonic in R (i.e. (∆h) f = 0 for some integer p, where ∆h is the Dunkl Laplacian associated to a root system R and to a multiplicity function κ, defined on R and invariant with respect to the finite Coxeter group). Necessary and successful condition that f is a polynomial of degree ≤ s for s ≥ 2p− 2 is proved. As a direct corollary, a Dunkl harmonic functi...

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y, σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y, σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y, σ) is not the one in algebraic geometry.

We consider an equivariant analogue of a conjecture of Borcherds. Let (Y,σ) be a real K3 surface without real points. We shall prove that the equivariant determinant of the Laplacian of (Y,σ) with respect to a σ-invariant Ricci-flat Kähler metric is expressed as the norm of the Borcherds Φ-function at the “period point”. Here the period of (Y,σ) is not the one in algebraic geometry.

Consider 3 dimensional Brownian motion started on the unit sphere {|x| = 1} with initial density ρ. Let ρt be the first hitting density on the sphere {|x| = t + 1}, t > 0. Then the linear operators Tt defined by Tt ρ = ρt form a semigroup with an infinitesimal generator which is approximately the square root of the Laplacian. This paper studies the analogous situation for Brownian motion with a...

• Rn has a natural measure space structure; namely, Lebesgue measure m on the Borel σalgebra. The most important property of Lebesgue measure is that it is invariant under translation. This leads to nice interactions between differentiation and integration, such as integration by parts, and it gives nice functional-analytic properties to differentiation operators: for instance, the Laplacian ∆ ...

The rotational dimension is a minor-monotone graph invariant related to the of Euclidean space containing spectral embedding corresponding first nonzero eigenvalue Laplacian, which introduced by Göring, Helmberg and Wappler. In this paper, we study dimensions graphs contain large complete graphs. characterized its dimension. It will be obtained that chordal may made while keeping constant.

We study a Laplace operator on semidiscrete surfaces which is defined by variation of the Dirichlet energy functional. We show existence and its relation to the mean curvature normal, which is itself defined via variation of area. We establish several core properties like linear precision (closely related to the mean curvature of flat surfaces), and pointwise convergence. It is interesting to o...

Swarm stability is concerned for descriptor compartmental networks with linear time-invariant protocol. Compartmental network is a specific type of dynamical multi-agent system. Necessary and sufficient conditions for both consensus and critical swarm stability are presented, which require a joint matching between the interactive dynamics of nearest neighboring vertices and the Laplacian spectr...