A Lichnerowicz estimate for the spectral gap of the sub-Laplacian

For a second order operator on a compact manifold satisfying the strong H\"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.Comment: 13 pages. To appear in Proceedings of the AM

Eigenvalues of the Laplacian on balls with spherically symmetric metrics

In this article we will explore Dirichlet Laplace eigenvalues of balls with spherically symmetric metrics. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radii. As a special case we shall show that the Dirichlet Laplace eigenvalues of balls with small radii on the sphere are smaller than the corresponding eigenvalues of the Euclidean balls with the same radii. The opposite correspondence is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces

Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $\mathbf{R}^3.$ Moreover, we give a new proof of the convexity result for harmonic functions on a Riemannian manifold when integrating over spheres. The inequality we obtain for the case of positively curved Riemannian manifolds with non-constant curvature is slightly better than the one previously known.Comment: 22 pages, Accepted for publication in The Journal of Geometric Analysi

On the three ball theorem for solutions of the Helmholtz equation

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $\Delta u_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.$ The following three ball inequality $M_{u_k}(2r)\le C(k,r,\alpha)M_{u_k}(r)^{\alpha}M_{u_k}(4r)^{1-\alpha}$ is well known, it holds for some $\alpha\in (0,1)$ and $C(k,r,\alpha)>0$ independent of $u_k$. We show that the constant $C(k,r,\alpha)$ grows exponentially in $k$ (when $r$ is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.Comment: 17 page

Humoral immunity to SARS-CoV-2 mRNA vaccination in multiple sclerosis: the relevance of time since last rituximab infusion and first experience from sporadic revaccinations

Introduction The effect of disease-modifying therapies (DMT) on vaccine responses is largely unknown. Understanding the development of protective immunity is of paramount importance to fight the COVID-19 pandemic. Objective To characterise humoral immunity after mRNA-COVID-19 vaccination of people with multiple sclerosis (pwMS). Methods All pwMS in Norway fully vaccinated against SARS-CoV-2 were invited to a national screening study. Humoral immunity was assessed by measuring anti-SARS-CoV-2 SPIKE RBD IgG response 3–12 weeks after full vaccination, and compared with healthy subjects. Results 528 pwMS and 627 healthy subjects were included. Reduced humoral immunity (anti-SARS-CoV-2 IgG <70 arbitrary units) was present in 82% and 80% of all pwMS treated with fingolimod and rituximab, respectively, while patients treated with other DMT showed similar rates as healthy subjects and untreated pwMS. We found a significant correlation between time since the last rituximab dose and the development of humoral immunity. Revaccination in two seronegative patients induced a weak antibody response. Conclusions Patients treated with fingolimod or rituximab should be informed about the risk of reduced humoral immunity and vaccinations should be timed carefully in rituximab patients. Our results identify the need for studies regarding the durability of vaccine responses, the role of cellular immunity and revaccinations. This article is made freely available for use in accordance with BMJ’s website terms and conditions for the duration of the covid-19 pandemic or until otherwise determined by BMJ. You may use, download and print the article for any lawful, non-commercial purpose (including text and data mining) provided that all copyright notices and trade marks are retained.publishedVersio

Altimetry for the future: Building on 25 years of progress

In 2018 we celebrated 25 years of development of radar altimetry, and the progress achieved by this methodology in the fields of global and coastal oceanography, hydrology, geodesy and cryospheric sciences. Many symbolic major events have celebrated these developments, e.g., in Venice, Italy, the 15th (2006) and 20th (2012) years of progress and more recently, in 2018, in Ponta Delgada, Portugal, 25 Years of Progress in Radar Altimetry. On this latter occasion it was decided to collect contributions of scientists, engineers and managers involved in the worldwide altimetry community to depict the state of altimetry and propose recommendations for the altimetry of the future. This paper summarizes contributions and recommendations that were collected and provides guidance for future mission design, research activities, and sustainable operational radar altimetry data exploitation. Recommendations provided are fundamental for optimizing further scientific and operational advances of oceanographic observations by altimetry, including requirements for spatial and temporal resolution of altimetric measurements, their accuracy and continuity. There are also new challenges and new openings mentioned in the paper that are particularly crucial for observations at higher latitudes, for coastal oceanography, for cryospheric studies and for hydrology. The paper starts with a general introduction followed by a section on Earth System Science including Ocean Dynamics, Sea Level, the Coastal Ocean, Hydrology, the Cryosphere and Polar Oceans and the ‘‘Green” Ocean, extending the frontier from biogeochemistry to marine ecology. Applications are described in a subsequent section, which covers Operational Oceanography, Weather, Hurricane Wave and Wind Forecasting, Climate projection. Instruments’ development and satellite missions’ evolutions are described in a fourth section. A fifth section covers the key observations that altimeters provide and their potential complements, from other Earth observation measurements to in situ data. Section 6 identifies the data and methods and provides some accuracy and resolution requirements for the wet tropospheric correction, the orbit and other geodetic requirements, the Mean Sea Surface, Geoid and Mean Dynamic Topography, Calibration and Validation, data accuracy, data access and handling (including the DUACS system). Section 7 brings a transversal view on scales, integration, artificial intelligence, and capacity building (education and training). Section 8 reviews the programmatic issues followed by a conclusion

Altimetry for the future: building on 25 years of progress

In 2018 we celebrated 25 years of development of radar altimetry, and the progress achieved by this methodology in the fields of global and coastal oceanography, hydrology, geodesy and cryospheric sciences. Many symbolic major events have celebrated these developments, e.g., in Venice, Italy, the 15th (2006) and 20th (2012) years of progress and more recently, in 2018, in Ponta Delgada, Portugal, 25 Years of Progress in Radar Altimetry. On this latter occasion it was decided to collect contributions of scientists, engineers and managers involved in the worldwide altimetry community to depict the state of altimetry and propose recommendations for the altimetry of the future. This paper summarizes contributions and recommendations that were collected and provides guidance for future mission design, research activities, and sustainable operational radar altimetry data exploitation. Recommendations provided are fundamental for optimizing further scientific and operational advances of oceanographic observations by altimetry, including requirements for spatial and temporal resolution of altimetric measurements, their accuracy and continuity. There are also new challenges and new openings mentioned in the paper that are particularly crucial for observations at higher latitudes, for coastal oceanography, for cryospheric studies and for hydrology. The paper starts with a general introduction followed by a section on Earth System Science including Ocean Dynamics, Sea Level, the Coastal Ocean, Hydrology, the Cryosphere and Polar Oceans and the “Green” Ocean, extending the frontier from biogeochemistry to marine ecology. Applications are described in a subsequent section, which covers Operational Oceanography, Weather, Hurricane Wave and Wind Forecasting, Climate projection. Instruments’ development and satellite missions’ evolutions are described in a fourth section. A fifth section covers the key observations that altimeters provide and their potential complements, from other Earth observation measurements to in situ data. Section 6 identifies the data and methods and provides some accuracy and resolution requirements for the wet tropospheric correction, the orbit and other geodetic requirements, the Mean Sea Surface, Geoid and Mean Dynamic Topography, Calibration and Validation, data accuracy, data access and handling (including the DUACS system). Section 7 brings a transversal view on scales, integration, artificial intelligence, and capacity building (education and training). Section 8 reviews the programmatic issues followed by a conclusion

Quantitative Unique Continuation and Eigenvalue Bounds for the Laplacian

I denne avhandlingen skal vi studere flere aspekter ved laplaceoperatoren, spesielt med hensyn på egenverdier og egenfunksjoner. En stor del av avhandlingen er dedikert til kvantitativ unik utvidelse ulikheter for harmoniske funksjoner og egenfunksjoner til laplaceoperatoren. For eksempel, bestemmer vi den best mulige vekstraten med hensyn på bølgetallet for tre-ball ulikheter for laplaceoperatoren på riemannske modell-mangfoldigheter. I tillegg skal vi vise egenverdi ulikheter for flere egenverdiproblemer (f.eks.\ dirichlet-, neumann-, steklovproblemet) der laplaceoperatoren er i stor grad tilstedeværende. Krumningen til den underliggende riemannske mangfoldigheten vil spille en sentral rolle fra start til slutt