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Let $��\subset \mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $��$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $��$ and $��$ on $\partial ��$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $��$. Write $��_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $��_L \in A_{\infty}(��)$, (ii) $L$ is $L^p(��)$-solvable for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies a Carleson measure estimate with respect to $��$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(��)$ for some (or for all) $q \in (0, \infty)$ for any null solution of $L$, and (v) $L$ is $\mathrm{BMO}(��)$-solvable. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $��_L$ with respect to $��$ in terms of local $L^2(��)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $��$-almost everywhere of the truncated conical square function for any bounded null solution of $L$., text overlap with arXiv:2101.06064

In this paper, we establish several different characterizations of the vanishing mean oscillation space associated with Neumann Laplacian $��_N$, written ${\rm VMO}_{��_N}(\mathbb{R}^n)$. We first describe it with the classical ${\rm VMO}(\mathbb{R}^n)$ and certain ${\rm VMO}$ on the half-spaces. Then we demonstrate that ${\rm VMO}_{��_N}(\mathbb{R}^n)$ is actually ${\rm BMO}_{��_N}(\mathbb{R}^n)$-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. Additionally, by means of the functional analysis, we obtain the duality between certain ${\rm VMO}$ and the corresponding Hardy spaces on the half-spaces. Finally, we present an useful approximation for ${\rm BMO}$ functions on the space of homogeneous type, which can be applied to our argument and otherwhere.

Since its inception six decades ago, astrobiology has diversified immensely to encompass several scientific questions including the origin and evolution of Terran life, the organic chemical composition of extraterrestrial objects, and the concept of habitability, among others. The detection of life beyond Earth forms the main goal of astrobiology, and a significant one for space exploration in general. This goal has galvanized and connected with other critical areas of investigation such as the analysis of meteorites and early Earth geological and biological systems, materials gathered by sample-return space missions, laboratory and computer simulations of extraterrestrial and early Earth environmental chemistry, astronomical remote sensing, and in-situ space exploration missions. Lately, scattered efforts are being undertaken towards the R&D of the novel and as-yet-spaceunproven 'life-detection' technologies capable of obtaining unambiguous evidence of extraterrestrial life, even if it is significantly different from Terran life. As the suite of space-proven payloads improves in breadth and sensitivity, this is an apt time to examine the progress and future of life-detection technologies.

Jupiter Trojan asteroids are located around the Sun-Jupiter Lagrange points (L4 or L5), and most of them are classified as D- or P-type, considered as volatile rich materials with rocks, water ices, and organics. However, the origin and evolution, composi- tion and physical conditions of these bodies still remain unknown. Jupiter Trojans are the missing links of ma- terials that originates from inner and outer solar system, and they are also intermediate size to undergo aqueous and thermal alteration between primitive smaller and evolved larger bodies. They are the key targets to un- derstand the evolution process and the chemical radial distribution in the solar system. An engineering mission to a Jupiter Trojan asteroid is being studied in Japan using a hybrid propulsion system of a large area solar-power sail (SPS) and a sophisticated ion engine [1]. After arriving there, the asteroid will be investigated for scientific purpose and for landing site selection by remote sensing. A lander will be deployed to land on the asteroid surface and conduct in situ bservations. A Jupiter Trojan multi-flyby mission (LUCY) [2] has been selected as the next Discovery class mission, which aims at understanding the variation and diversity of Jupiter Trojans, contrary to the SPS mission which will rendezvous and land on the asteroid and conduct in-depth measurements. The SPS mission is jointly studied between engineers and scientists both from Japan and Europe [3]. We will present here the key scientific objectives and the strawman payloads for the SPS mission.

A new mission to visit and explore a Jupiter Trojan asteroid is under study in Japan using a hybrid propulsion system of solar-powered sail (SPS) and sophisticated ion engine, and additionally using a scientific lander, which is jointly investigated between engineers and scientists from Japan and Europe [1]. We present here the key objectives and the strawman payloads of science experiments on the asteroid in this mission.