Related papers: A Functional Analytic Perspective to the div-curl …

We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that…

The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting…

This survey is a chapter of a forthcoming book. This chapter recalls the classical formulation of the Div-Curl lemma along with its proof, and presents some possible generalizations in the fractional setting, within the framework of the…

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the $b-c$ systems…

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth…

We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz…

We generalize the Bartsch-Li's splitting lemma at infinity for $C^2$-functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow…

We give some theoretical as well as computational results on Laplace and Maxwell constants. Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general…

We present new examples of complexes of differential operators of order $k$ (any given positive integer) that satisfy div-curl and/or $L^1$-duality estimates.

We consider the integral and derivative operators of tempered fractional calculus, and examine their analytic properties. We discover connections with the classical Riemann-Liouville fractional calculus and demonstrate how the operators may…

In this paper we introduce the concept of \emph{multivector functionals.} We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.g., the $A$-directional derivative and the…

We present a streamlined account of a recent theorem on the classification of the $L$-functions of degree 2 and conductor 1 from the extended Selberg class. We also present a more general new result dealing with functional equations…

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We…

We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate…

In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal M-L^\infty$, where $\mathcal{M}$ is the set of Radon measures on $\mathbb{R}^d$. The proof follows the classical approach by defining here…

This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups,…

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces.…

The well-known identity involving the expression presented in the above title is considered in Riemannian and in Euclidean space without restriction on the coordinate system adopted therein. The Riemann and Ricci tensors intrinsically…

We consider two operators $A_{0},B_{0}$ between two Hilbert spaces satisfying $A_{0}\subseteq-B_{0}^{\ast}$ and $B_{0}\subseteq-A_{0}^{\ast}$ and inspect extensions $A^{\#}$ and $B^{\#}$ of $A_{0}$ and $B_{0}$, respectively, whose domain…

We develop a new approach to the study of spectral asymmetry. Working with the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry…