Equivalence of norms on finite linear combinations of atoms

Let M be a space of homogeneous type and denote by F^\infty_{cont}(M) the space of finite linear combinations of continuous (1,\infty)-atoms. In this note we give a simple function theoretic proof of the equivalence on F^\infty_{cont}(M) of the H^1-norm and the norm defined in terms of finite linear combinations of atoms. The result holds also for the class of nondoubling metric measure spaces considered in previous works of A. Carbonaro and the authors.Comment: 10 pages, revised argumen

Estimates for functions of the Laplacian on manifolds with bounded geometry

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel.Comment: 19 page

Higher order Riesz transforms on noncompact symmetric spaces

In this note we prove various sharp boundedness results on suitable Hardy type spaces for Riesz transforms of arbitrary order on noncompact symmetric spaces of arbitrary rank.Comment: v2: the first version has been revised and splitted up in two papers, of which this new version is one par

A short proof of Stein's universal multiplier theorem

We give a short proof of Stein's universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods)

The Impossible Schools: Rural Classrooms in the Paintings of Italian Artists during the Second Half of the Nineteenth Century

Il presente lavoro si propone di studiare le modalit\ue0 di rappresentazione delle scuole rurali dell\u2019Italia Centrale e Meridionale nei dipinti di alcuni pittori attivi in quei contesti territoriali nella seconda met\ue0 dell\u2019Ottocento. La caratteristica fondamentale di questi dipinti \ue8 quella di rappresentare aule spoglie e vetuste, malamente illuminate e poco curate dal punto di vista igienico, all\u2019interno delle quali classi mai particolarmente numerose sono istruite da anziani maestri ancora ricorrenti al metodo individuale, gi\ue0 ampiamente superato all\u2019epoca dal metodo normale, oltre che ai cosiddetti \uabcastighi d\u2019obbrobrio\ubb, sconsigliati dalla moderna pedagogia. Le pessime condizioni materiali delle scuole rurali all\u2019indomani dell\u2019Unit\ue0 d\u2019Italia, particolarmente nelle zone rurali e montane e nelle regioni meridionali, sono gi\ue0 note alla storiografia. La fosca rappresentazione che di esse era fornita in questi dipinti, tuttavia, merita di essere approfondita. Sappiamo che in alcuni casi quelle rappresentate nei dipinti erano scuole coeve agli artisti, osservate con ogni probabilit\ue0 nelle campagne e quindi riprodotte su tela allo scopo di denunciarne lo stato di arretratezza; in altri invece, come per esempio nell\u2019opera di Cicconi, ad essere rappresentate erano le \u201cscuole dei vecchi tempi\u201d, che sottoponevano al pubblico biasimo gli antiquati sistemi scolastici in vigore negli stati pre-unitari, idealmente contrapposti a quelli pi\uf9 moderni adottati dal neonato stato unitario. Era d\u2019altronde un dato di fatto \u2013 risultante da numerose altre testimonianze \u2013 che il metodo individuale fosse ancora ampiamente in uso in numerose scuole del Mezzogiorno molti anni dopo la definitiva affermazione del metodo normale, impostosi nelle regioni settentrionali sottoposte alla dominazione asburgica gi\ue0 all\u2019inizio del secolo e quindi estesosi al resto della penisola attraverso la nascita di un vero movimento pedagogico. La principale accusa che si muoveva al metodo individuale era quella di annoiare gli alunni per la sua eccessiva lentezza, inducendoli ad atteggiamenti d\u2019insofferenza, insubordinazione e scherno e provocando la comminazione da parte dell\u2019insegnante di dolorose punizioni corporali o altri provvedimenti disciplinari. Non appare dunque casuale che burle, tumulti, intemperanze, castighi e penitenze abbondino in questi dipinti, il cui scopo \u2013 seppur per ragioni differenti \u2013 era quello di mettere in evidenza i tanti difetti di questo obsoleto metodo d\u2019insegnamento. Nel presente lavoro si cerca infine di comprendere per quale committenza venissero realizzati questi dipinti e quali ne fossero le reali motivazioni d\u2019acquisto, al fine di meglio comprendere le ragioni dell\u2019appeal esercitato da un soggetto dalle evidenti implicazioni politiche su un pubblico di facoltosi acquirenti, probabilmente interessati alla \uabpittura di interni\ubb di gusto realistico pi\uf9 per il suo \uabessere di moda\ubb che per l\u2019implicita denuncia delle precarie condizioni di vita in cui versavano delle classi popolari emergente dalla misera intimit\ue0 in essi raffigurata. Sono presi in considerazione circa trenta dipinti realizzati da Ferdinando Cicconi (1831-1886), Giuseppe Costantini (1844-1894), Francesco Bergamini (1850-post 1905), Vincenzo Loria (1849-1939) e Augusto Daini (1860-1920); si fa inoltre riferimento \u2013 per antitesi \u2013 ad alcuni dipinti di Gioacchino Toma (1836-1891), Demetrio Cosola (1851-1895), Giacomo Mantegazza (1853-1920) e Alessandro Battaglia (1870-1940), che \u2013 sempre nel contesto immediatamente post-unitario \u2013 rappresentavano le scuole elementari urbane dell\u2019Italiana settentrionale.This work aims to study the ways of representing rural schools of Central and Southern Italy in the paintings of some artists active in those regions in the second half of the nineteenth century. The main characteristic of these paintings is that of representing old and bare classrooms, badly lit and scarcely cleaned, where a small group of students is taught by an old schoolmaster who is still using the individual method. Actually, the individual method had been largely overcome by the normal method at the time, as well as the so-called \u201cshame punishments\u201d, which were not allowed by the modern pedagogy, but are nonetheless represented in these paintings. The poor material conditions of the rural schools in the aftermath of the Italian Unification (1861), particularly in country and mountain areas and in southern regions, have already been brought to light by the historiography. The gloomy representation of these classrooms in these paintings, however, has to be better analyzed. We know that in some cases these paintings actually represented the classrooms that the artists had observed in the countryside and then reproduced on canvas in order to denounce their backwardness; in other cases, however, as for example in Cicconi\u2019s paintings, the \uabschools of the old times\ubb were represented, which named and shamed the obsolete school methods used in use in the pre-unitary states, ideally opposed to the more modern ones adopted by the newborn Kingdom of Italy. It was indeed a fact \u2013 as testified by other sources \u2013 that the individual method was still widely in use in many small schools of the southern regions many years after the definitive success of the normal method, imposed at the beginning of the 19th century in the northern regions subjected to the Habsburg Empire and to the Italian Napoleonic Kingdom and then extended to the rest of Italy after the Unification. The main objections raised against the individual method were that it caused boredom in the pupils due to its excessive slowness and made the control of the pupils very difficult, since they ought to be busy with different tasks. Hence this way of teaching prompted pupils to acts of insubordination and mockeries and led the teacher to inflict physical punishments or other disciplinary measures. Thus, it is not a coincidence that in these paintings there are a lot of pranks, riots, excesses, punishments and mortifications, whose purpose \u2013 albeit for different reasons \u2013 was to highlight the many faults of this outdated teaching method, belonging to the old regime States. Finally, in this work we try to understand for which clients these paintings were made, and their real purchasing motivations, in order to better understand the reasons for the appeal of a subject with obvious political implications. About thirty paintings by Ferdinando Cicconi (1831-1886), Giuseppe Costantini (1844-1894), Francesco Bergamini (1850-post 1905), Vincenzo Loria (1849-1939) and Augusto Daini (1860-1920) are analyzed; reference is also made \u2013 by contrast \u2013 to some paintings by Gioacchino Toma (1836-1891), Demetrio Cosola (1851-1895), Giacomo Mantegazza (1853-1920) and Alessandro Battaglia (1870-1940), who \u2013 in the same period \u2013 represented the urban schools of Northern Italy

Revealing interference by continuous variable discordant states

In general, a pair of uncorrelated Gaussian states mixed in a beam splitter produces a correlated state at the output. However, when the inputs are identical Gaussian states the output state is equal to the input, and no correlations appear, as the interference had not taken place. On the other hand, since physical phenomena do have observable effects, and the beam splitter is there, a question arises on how to reveal the interference between the two beams. We prove theoretically and demonstrate experimentally that this is possible if at least one of the two beams is prepared in a discordant, i.e. Gaussian correlated, state with a third beam. We also apply the same technique to reveal the erasure of polarization information. Our experiments involves thermal states and the results show that Gaussian discordant states, even when they show a positive Glauber P-function, may be useful to achieve specific tasks.Comment: published versio