Hodge Dirac Research Articles

Spectral metric and Einstein functionals for the Hodge–Dirac operator

We examine the metric and Einstein bilinear functionals of differential forms introduced by Dąbrowski et al. (2023), for the Hodge–Dirac operator d+\delta on an oriented, closed, even-dimensional Riemannian manifold. We show that they are equal (up to a numerical factor) to these functionals for the canonical Dirac operator on a spin manifold. Furthermore, we demonstrate that the spectral triple for the Hodge–Dirac operator is spectrally closed, which implies that it is torsion-free.

The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

Dabrowski et al. [Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877] gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator [Formula: see text] on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.

Continuum limit for a discrete Hodge–Dirac operator on square lattices

Continuum limit for a discrete Hodge–Dirac operator on square lattices

2D Discrete Hodge–Dirac Operator on the Torus

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups.

Bounds on the Phillips Calculus of Abstract First Order Differential Operators

For an operator generating a group on \(L^p\) spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on \(L^2\) rather than \(L^p\). We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge–Dirac operator has such calculus.

Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains

This paper concerns Hodge–Dirac operators D_{{}^\Vert}=d+\underline{\delta} acting in L^p(\Omega, \Lambda) where \Omega is a bounded open subset of {\mathbb{R}}^n satisfying some kind of Lipschitz condition, \Lambda is the exterior algebra of {\mathbb{R}}^n , d is the exterior derivative acting on the de Rham complex of differential forms on \Omega , and \underline{\delta} is the interior derivative with tangential boundary conditions. In L^2(\Omega,\Lambda) , \underline{\delta} = {d}^* and D_{{}^\Vert} is self-adjoint, thus having bounded resolvents \{({\rm I}+itD_{{}^\Vert})^{-1}\}_{t\in{\mathbb{R}}} as well as a bounded functional calculus in L^2(\Omega,\Lambda) . We investigate the range of values p_H < p < p^H about p=2 for which D_{{}^\Vert} has bounded resolvents and a bounded holomorphic functional calculus in L^p(\Omega,\Lambda) . On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L^p(\Omega,\Lambda) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian \Delta_{{{}^\Vert}} is the square of the Hodge–Dirac operator, i.e., -\Delta_{{}^\Vert}={D_{{}^\Vert}}^2 , so it also has a bounded functional calculus in L^p(\Omega,\Lambda) when p_H < p < p^H . But the Stokes operator with Hodge boundary conditions, which is the restriction of -\Delta_{{}^\Vert} to the subspace of divergence free vector fields in L^p(\Omega,\Lambda^1) with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of p , namely for max \{1,{p_H}_S\} < p < p^H where {p_H}_S is the Sobolev exponent below p_H , given by 1/{{p_H}_S} =1/{p_H}+1/n , so that {{p_H}_S} < 2n/(n+2) . In 3 dimensions, {p_H}_S < 6/5 . We show also that for bounded strongly Lipschitz domains \Omega , p_H < 2n/(n+1) < 2n/(n-1) < p^H , in agreement with the known results that p_H < 4/3 < 4 < p^H in dimension 2, and p_H < 3/2 < 3 < p^H in dimension 3. In both dimensions 2 and 3, {p_H}_S<1 , implying that the Stokes operator has a bounded functional calculus in L^p(\Omega,\Lambda^1) when \Omega is strongly Lipschitz and 1 < p < p^H .

L^p-Analysis of the Hodge\u2013Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the H^infty -functional calculus in L^p for all 1<p<infty for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.

Calderón reproducing formulas and applications to Hardy spaces

We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with D through functional calculus based on the Fourier transform. We apply these to prove the embedding H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M) , 1 \leq p \leq 2 , for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where D=d+d^* is the Hodge–Dirac operator on a complete Riemannian manifold M that has doubling volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of H^1_D(\wedge T^*M) . The embedding H^p_L \subseteq L^p , 1 \leq p \leq 2 , where L is either a divergence form elliptic operator on \mathbb R^n , or a nonnegative self-adjoint operator that satisfies Davies–Gaffney estimates on a doubling metric measure space, is also established in the case when the semigroup generated by the adjoint -L^* is ultracontractive.

Finite speed of propagation and off-diagonal bounds for Ornstein–Uhlenbeck operators in infinite dimensions

We study the Hodge–Dirac operators $${\mathscr {D}}$$ associated with a class of non-symmetric Ornstein–Uhlenbeck operators $${\mathscr {L}}$$ in infinite dimensions. For $$p\in (1,\infty )$$ we prove that $$i{\mathscr {D}}$$ generates a $$C_0$$ -group in $$L^p$$ with respect to the invariant measure if and only if $$p=2$$ and $${\mathscr {L}}$$ is self-adjoint. An explicit representation of this $$C_0$$ -group in $$L^2$$ is given, and we prove that it has finite speed of propagation. Furthermore, we prove $$L^2$$ off-diagonal estimates for various operators associated with $${\mathscr {L}}$$ , both in the self-adjoint and the non-self-adjoint case.

Local Hardy Spaces of Differential Forms on Riemannian Manifolds

We define local Hardy spaces of differential forms \(h^{p}_{\mathcal{D}}(\wedge T^{*}M)\) for all p∈[1,∞] that are adapted to a class of first-order differential operators \(\mathcal{D}\) on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge–Dirac operator on M and Δ=D 2 is the Hodge–Laplacian, then the local geometric Riesz transform D(Δ+aI)−1/2 has a bounded extension to \(h^{p}_{D}\) for all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of \(h^{1}_{\mathcal{D}}\) in terms of local molecules is also obtained. These results can be viewed as the localization of those for the Hardy spaces of differential forms \(H^{p}_{D}(\wedge T^{*}M)\) introduced by Auscher, McIntosh, and Russ.

Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds

We study boundary-value problems with mixed boundary conditions in weakly Lipschitz domains of compact boundaryless Riemannian Lipschitz manifolds. These include the case of the Maxwell system, the Hodge–Dirac operator, and the Hodge–Laplacian. Our approach brings to bear tools from functional analysis, differential geometry, harmonic analysis, and topology. Bibliography: 45 titles.

Transmission problems for Maxwell's equations with weakly Lipschitz interfaces

We prove sufficient conditions on material constants, frequency and Lipschitz regularity of interface for well posedness of a generalized Maxwell transmission problem in finite energy norms. This is done by embedding Maxwell's equations in an elliptic Dirac equation, by constructing the natural trace space for the transmission problem and using Hodge decompositions for operators d and δ on weakly Lipschitz domains to prove stability. We also obtain results for boundary value problems and transmission problems for the Hodge–Dirac equation and prove spectral estimates for boundary singular integral operators related to double layer potentials. Copyright © 2005 John Wiley & Sons, Ltd.

Finite Energy Solutions of Maxwell's Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of Lp forms in Lipschitz domains, we show that both are well posed provided that 2−ε<p<2+ε, for some ε>0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.

Hodge—dirac operators for hyperbolic spaces

In this article we extend the concept of hyperbolically harmonic functions for the upper half plane, first introduced by Leutwiler, to an arbitrary conformally flat manifold. We shall consider the special cases of the two standard models for hyperbolic spaces, namely the Poincaré model for the upper half space and the spherical model. For these two manifolds, an analogue of the Cauchy—Riemann equations, obtained via the Euclidean Dirac operator, is studied, along with particular decompositions of its solutions.

Monogenic vector fields on the poincaré manifold

In this paper a study is made of vector valued functions satisfying the equation , where D E is the Euclidean Dirac operator, and is the (anti-Euclidean) inner product. These functions, called hypermonogenic, were introduced by Leutwiler (see [10-13]). In the first part of this paper we explore the relation between these functions and the Poincaré metric on half space . Indeed, up to a scalar correction factor, these functions are monogenic, in the sense that they are solutions of the Hodge Dirac operator. As a result, they are a logical generalisation of classical holomorphic functions on C - clearly the equation considered reduces to the Cauchy-Riemann equation for n= 2 – in a way similar to, but not equal to, the theory of monogenic functions on Euclidean space. Furthermore, a natural isomorphism between hypermonogenic functions in a half space and in a ball (the second model of hyperbolic space) is given. This isomorphism shows that the series expansion of a hyperharmonic function in the neighbourhood of a point of the manifold can be most easily expressed in terms of the spherical harmonics of Euclidean space. Then two important results for these functions are given. First, a result by Leutwiler, stating that a hypermonogenic function near the ‘boundary’ can be described by its values on and its ‘characteristic function’ on , is generalised from the case n = 3 to arbitrary n> 2. Second, it is proved that a hypermonogenic function can be developed in a neighbourhood of a boundary point in a series of cyhndrical waves. Notice that both results are remarkable in that they deal with an inhomogeneous space, because of the singular hyperplane . Finally, we introduce a similar theory for a metric associated with a subspace of lower dimension, and indicate how it may be Hnked with a higher dimensional analogue of the theory of Padé approximation.