Authors:Valentina Casarino Pages: 1 - 17 Abstract: We give a survey of recent works concerning the mapping properties of joint harmonic projection operators, mapping the space of square integrable functions on complex and quaternionic spheres onto the eigenspaces of the Laplace-Beltrami operator and of a suitably defined subLaplacian. In particular, we discuss similarities and differences between the real, the complex and the quaternionic framework. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Marco Falconi Pages: 18 - 35 Abstract: We review some aspects of semiclassical analysis for systems whose phase space is of arbitrary (possibly infinite) dimension. An emphasis will be put on a general derivation of the so-called Wigner classical measures as the limit of states in a noncommutative algebra of quantum observables. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Davide Barbieri Pages: 36 - 52 Abstract: This is a joint work with E. Hernández, J. Parcet and V. Paternostro. We will discuss the structure of bases and frames of unitary orbits of discrete groups in invariant subspaces of separable Hilbert spaces. These invariant spaces can be characterized, by means of Fourier intertwining operators, as modules whose rings of coefficients are given by the group von Neumann algebra, endowed with an unbounded operator valued pairing which defines a noncommutative Hilbert structure. Frames and bases obtained by countable families of orbits have noncommutative counterparts in these Hilbert modules, given by countable families of operators satisfying generalized reproducing conditions. These results extend key notions of Fourier and wavelet analysis to general unitary actions of discrete groups, such as crystallographic transformations on the Euclidean plane or discrete Heisenberg groups. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Marco Mughetti Pages: 53 - 68 Abstract: We are concerned with the problem of the analytic hypoellipticity; precisely, we focus on the real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We discuss a model operator, P, (firstly appeared and studied in [3]) having a single symplectic stratum and prove that it is not analytic hypoelliptic. This yields a counterexample to the sufficient part of Treves conjecture; the necessary part is still an open problem. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Giovanni Molica Bisci, Raffaella Servadei Pages: 69 - 84 Abstract: In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Andrea Pinamonti, Gareth Speight Pages: 85 - 96 Abstract: We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Eugenio Vecchi Pages: 97 - 115 Abstract: The classical Steiner formula expresses the volume of the ∈-neighborhood Ω∈ of a bounded and regular domain Ω⊂Rn as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltàn Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildrick PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Alessia E. Kogoj Pages: 116 - 128 Abstract: For every bounded open set Ω in RN+1, we study the first boundary problem for a wide class of hypoelliptic evolution operators. The operators are assumed to be endowed with a well behaved global fundamental solution that allows us to construct a generalized solution in the sense of Perron-Wiener of the Dirichlet problem. Then, we give a criterion of regularity for boundary points in terms of the behavior, close to the point, of the fundamental solution of the involved operator. We deduce exterior conetype criteria for operators of Kolmogorov-Fokker-Planck-type, for the heat operators and more general evolution invariant operators on Lie groups. Our criteria extend and generalize the classical parabolic-cone condition for the classical heat operator due to Effros and Kazdan. The results presented are contained in [K16]. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Raffaella Giova, Antonia Passarelli di Napoli Pages: 129 - 146 Abstract: We give an overview on recent regularity results of local vectorial minimizers of under two main features: the energy density is uniformly convex with respect to the gradient variable only at infinity and it depends on the spatial variable through a possibly discontinuous coefficient. More precisely, the results that we present tell that a suitable weak differentiability property of the integrand as function of the spatial variable implies the higher differentiability and the higher integrability of the gradient of the local minimizers. We also discuss the regularity of the local solutions of nonlinear elliptic equations under a fractional Sobolev assumption. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Sunra Mosconi, Marco Squassina Pages: 147 - 164 Abstract: We overview some recent existence and regularity results in the theory of nonlocal nonlinear problems driven by the fractional p-Laplacian. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Davide Guidetti Pages: 165 - 174 Abstract: We illustrate some old and new results, concerning linear parabolic mixed problems in spaces of Hölder continuous functions: we begin with the classical Dirichlet and oblique derivative problems and continue with dynamic and Wentzell boundary conditions. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)

Authors:Angelo Favini, Gabriela Marinoschi Pages: 175 - 188 Abstract: A degenerate identification problem in Hilbert space is described, improving a previous paper [2]. An application to second order evolution equations of hyperbolic type is given. The abstract results are applied to concrete differential problems of interest in applied sciences. PubDate: 2017-02-10 Issue No:Vol. 7, No. 1 (2017)