Riesz transforms associated with the Neumann Laplacian

In this paper we study spectral stability of the $$\bar{\partial }$$ -Neumann Laplacian under the Kohn–Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the $$\bar{\partial }$$ -Neumann Laplacian when either the operator or the underlying domain is perturbed.

We study spectral stability of the {bar{partial }}-Neumann Laplacian on a bounded domain in {mathbb {C}}^n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the {bar{partial }}-Neumann Laplacian on bounded pseudoconvex domains in {mathbb {C}}^n, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in {mathbb {C}}^n.

We demonstrate how to approximate one-dimensional Schrödinger operators with δ-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with ‘room-and-passage’ geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in generalised norm resolvent sense to the above singular Schrödinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.

The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.

We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in Pólya’s conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by Pólya’s original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value $\alpha$ of the boundary parameter, Pólya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the $k^{\textrm{}}$th eigenvalue in regions of the $(k,\alpha )$-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.

We prove heat kernel estimates for the $${\bar{\partial}}$$ -Neumann Laplacian $${\square}$$ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.

We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems.

Consider the Neumann Laplacian in the region below the graph of [Formula: see text], for a positive smooth function [Formula: see text] with both [Formula: see text] and [Formula: see text] bounded. As [Formula: see text] such region collapses to [Formula: see text] and an effective operator is found, which has Robin boundary conditions at [Formula: see text]. Then, we recover (under suitable assumptions in the case of unbounded [Formula: see text]) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for [Formula: see text] diverging less than exponentially.

Hearing the type of a domain in [formula omitted] with the [formula omitted]-Neumann Laplacian

Spectrum of the [formula omitted]-Neumann Laplacian on the Fock space

This article has three aims. First, we study Hardy spaces, h^p_L(\Omega) , associated with an operator L which is either the Dirichlet Laplacian \Delta_{D} or the Neumann Laplacian \Delta_{N} on a bounded Lipschitz domain \Omega in {\mathbb{R}}^n , for 0 < p \leq 1 . We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decompositions. Second, we establish regularity results for the Green operators, regarded as the inverses of the Dirichlet and Neumann Laplacians, in the context of Hardy spaces associated with these operators on a bounded semiconvex domain \Omega in {\mathbb{R}}^n . Third, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces h^p_r(\Omega) and h^p_z(\Omega) , then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain \Omega in {\mathbb{R}}^n , and for the Neumann problem on a bounded convex domain \Omega in {\mathbb{R}}^n , in the context of the standard Hardy spaces h^p_r(\Omega) and h^p_z(\Omega) . This gives a new solution to the conjecture made by D.-C. Chang, S. Krantz and E. M. Stein regarding the regularity of Green operators for the Dirichlet and Neumann problems on h^p_r(\Omega) and h^p_z(\Omega) , respectively, for all \frac{n}{n+1} < p\leq 1 .

In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $$\Delta _N$$ , written $${\text {VMO}}_{\Delta _N}({\mathbb {R}^n})$$ . We first describe it with the classical $${\text {VMO}}({\mathbb {R}^n})$$ and certain $${\text {VMO}}$$ on the half-spaces. Then we demonstrate that $${\text {VMO}}_{\Delta _N}({\mathbb {R}^n})$$ is actually $${\text {BMO}}_{\Delta _N}({\mathbb {R}^n})$$ -closure of the space of the smooth functions with compact supports. Beyond that, it can be characterized in terms of compact commutators of Riesz transforms and fractional integral operators associated to the Neumann Laplacian. By means of the functional analysis, we also obtain the duality between certain $${\text {VMO}}$$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for $${\text {BMO}}$$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe howthesemay differ fromeach other, characterize when they agree, and study connections to essential selfadjointness and stochastic completeness. Finally, we study basic common features of all Laplacians associated to a graph. In particular, we characterize when the associated semigroup is positivity improving and present some basic estimates on its long term behavior. We also discuss some situations in which the Laplacian associated to a graph is unique and, in this context, characterize its boundedness.

Essential self-adjointness, generalized eigenforms, and spectra for the [formula omitted]-Neumann problem on G-manifolds

We consider a family of non-compact manifolds Xε (“graph-like manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian $$\Delta _{X_{\epsilon}}$$ and the generalized Neumann (Kirchhoff) Laplacian $$\Delta _{X_0 } $$ on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations.