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The authors study algebras of singular integral operators on \\mathbb R^n and nilpotent Lie groups that arise when considering the composition of Calderón-Zygmund operators with different homogeneities, such as operators occuring in sub-elliptic problems and those arising in elliptic problems. These algebras are characterized in a number of different but equivalent ways: in terms of kernel estimates and cancellation conditions, in terms of estimates of the symbol, and in terms of decompositions into dyadic sums of dilates of bump functions. The resulting operators are pseudo-local and bounded on L^p for 1 \\lt p \\lt \\infty . While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators studied here fall outside this class, and reflect a multi-parameter structure.

We summarise the present status of the gravitational wave detectors on the Earth.Then, we discuss some of the most intriguing results obtained during the last data collection O3. We will focus on the statistical distributionof the population of these compact objects and we will discuss a couple of potential tests to be carried on the on the Bekenstein-Hawking thermodynamics using the gravitational wave signals.

We consider the Hodge Laplacian In this paper we address three main, related questions: Our first main result shows that the Next, we consider We then use this decomposition to prove a sharp Mihlin–Hörmander multiplier theorem for each condition of order Finally, we extend this multiplier theorem to the Dirac operator.

We review the present status of the Gravitational wave detectors on the Earth, focusing the attention on the present innovations and the longer term perspectives to improve their sensitivity. Then we conclude mentioning few potential searches of new Physics phenomena to be performed with these detectors and those of the third generation.

It is widely expected that in the coming quinquennium the first gravitational wave signal will be directly detected. The ground-based advanced LIGO and Virgo detectors are being upgraded to a sensitivity level such that we expect to be measure a significant binary merger rate. Gravitational waves events are likely to be accompanied by electromagnetic counterparts and neutrino emission carrying complementary information to those associated to the gravitational signals. If it becomes possible to measure all these forms of radiation in concert, we will end up an impressive increase in the comprehension of the whole phenomenon. In the following we summarize the scientific outcome of the interferometric detectors in the past configuration. Then we focus on some of the potentialities of the advanced detectors once used in the new context of the multimessenger astronomy.

Cooling down large test masses up to 200 kg, as foreseen for the Einstein Telescope, is a complex challenge combining cutting-edge technological achievements from different disciplines with the experience gained from both room-temperature and cryogenic-temperature detector development communities. We set up an apparatus designed to test cryogenic mechanical suspensions for the payload system. They should have high quality factors and enable sufficient heat extraction greater than 0.3 W. The facility is on a university campus where cryofluid servicing is not feasible. As a result, a system that incorporates conductive cooling technology was developed. The project has two main goals: validating crystalline suspensions in a realistic Einstein Telescope cryogenic payload and testing new solutions for radiative thermal shielding. No particular measures are planned for the vibration isolation system.

Core collapse supernovae are the most energetic explosions in the modern Universe and, because of their properties, they are considered a potential source of detectable gravitational waveforms for long time. The main obstacles to their detection are the weakness of the signal and its complexity, which cannot be modeled, making it almost impossible to apply matching filter techniques as the ones used for detecting compact binary coalescences. Although the first obstacle will probably be overcome by next-generation gravitational wave detectors, the second one can be overcome by adopting machine learning techniques. In this contribution, a novel method based on a classification procedure of the time-frequency images using a convolutional neural network will be described, showing the CCSN detection capability of the next-generation gravitational wave detectors, with a focus on the Einstein Telescope.

The search for direct detection of Gravitational Wave made a huge step forward in the years between 2015-2017. After the first detection signals from the coalescence of binary black hole systems, we had both the great success of the LISA pathfinder mission and the detection of the first event due to a neutron star - neutron star merger tagged as the birth of the multi messenger astronomy. A new era is now opened where the GW events are detected routinely by the triangular network of LIGO and Virgo that in the nearest future will be enlarged by including KAGRA, the fourth detector in Japan. Here we review the evolution of the existing detectors focusing our attention essentially on the middle and long-term evolution of those on the Earth.

The observation of supermassive black holes by the Event Horizon Telescope Collaboration and the detection of gravitational waves emitted during the merging phase of compact binary objects to stellar-mass black holes by the LIGO–Virgo–KAGRA collaboration constitute major achievements of modern science. Gravitational wave signals emitted by stellar-mass black holes are being used to test general relativity in an unprecedented way in the regime of strong gravitational fields, as well as to address other physics questions such as the formation of heavy elements or the Hawking Area Theorem. These discoveries require further research in order to answer critical questions about the population density and the formation processes of binary systems. The detection of supermassive black holes considerably extends the range of scientific investigation by making it possible to probe the structure of spacetime around the horizon of the central mass of our galaxy as well as other galaxies. The huge amount of information collected by the VLBI worldwide network will be used to investigate general relativity in a further range of physical conditions. These investigations hold the potential to pave the way for the detection of quantum-mechanical effects such as a possible graviton mass. In this paper we will review, in a cursory way, some of the results of both the LIGO–Virgo–KAGRA and the EHT collaborations.

For a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space S(K\\G/K) isomorphically onto the space S(ΣD), where ΣD is an embedded copy of the Gelfand spectrum in Rℓ, canonically associated to a generating system D of G-invariant differential operators on G/K, and S(ΣD) consists of restrictions to ΣD of Schwartz functions on Rℓ. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair (Mn,SOn) with n=3,4. The rather trivial case n=2 is included in previous work by the same authors.