517 research outputs found
Bernstein and Jackson theorems for the Heisenberg group
No full textWe describe on the Heisenberg group H_n a family of spaces M(h, X) of functions which play a role
analogous to the trigonometric polynomials in T" or the functions of exponential type in R". In
particular we prove that for the space M(h, X), Jackson's theorem holds in the classical form while
Bernstein's inequality hold in a modified form. We end the paper with a characterization of the
functions of the Lipschitz space Λ', by the behavior of their best approximations by functions in the
space M(h, X)
The Martin compactification of the Cartesian product of two hyperbolic spaces
No full textConsider H_m × H_n the Cartesian product of two hyperbolic spaces with dimensions m and n respectively. It carries the product Riemannian structure and corresponding Laplace-Beltrami operator Δ=Δ_m x Δ_n, the sum of the Laplace Beltrami operators on the two factors. It is well known that there exist positive functions h on H_m × H_n which satisfy Δh=λh if and only if λ≥λ_0, where λ_0=-((m-1)/2)^2-((n-1)/2)^2 is the bottom of the positive spectrum
Weak type (1,1) estimates for heat kernel maximal functions on Lie groups
No full textFor a Lie group G with left-invariant Haar measure and associated Lebesgue spaces L^p(G), we consider the heat kernels {p_t\}_{t>0} arising from a right-invariant Laplacian Δ on G: that is, u(t, .) = p_t * f solves the heat equation (∂/∂t - Δ)u = 0 with initial condition u(0, .) = f(.). We establish weak-type (1, 1) estimates for the maximal operator M (M f = \sup_{t>0} |p_t * f|) and for related Hardy-Littlewood maximal operators in a variety of contexts, namely for groups of polynomial growth and for a number of classes of Iwasawa AN groups. We also study the "local" maximal operator M_0 (M_0 f = \sup_{0<t<1} |p_t * f|) and related Hardy-Littlewood operators for all Lie groups
Lp-Lq estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I
No full textLet X = G/K be a symmetric space of noncompact type, L be the Laplace-Beltrami operator
on X, and b the bottom of its spectrum. In this paper we study the Lp-Lq mapping properties
of several families of operators naturally associated with L:
θ-heat semigroup, H_{t,θ) = exp(tL−θb),
complex powers of resolvent operator, H^α_θ=(L−θb)^{−α/2}, and
S^α_θ = (L−θb)^{−α/2} exp(i(L−θb)), where 0≤θ≤1, Reα≥ 0, closely related to the Cauchy problem for the Schrodinger operator on X. The
techniques mix harmonic analysis on semisimple Lie groups (Plancherel measure, c-function) and
functional analysis (interpolation, semigroup theory).
One of the contribution in this paper is to give precise estimates for the Lp-Lq operator norm of H_{t,θ) for large time t on all noncompact symmetric spaces. These estimates show that the interpolation and extrapolation methods of ultracontractivitywhich work well for small t or on groups of polynomial growth are not applicable here; in particular, log(|||H_{t,θ)|||_{p,q}) does not depend linearly on 1/p and 1/q
Illustrare la storia: il ruolo delle immagini nella storiografia milanese del Settecento. Qualche nota per Latuada e Giulini
No full textPercorso di analisi di come l'immagine della città di Milano si è evoluta nel corso del secolo XVIII, attraverso alcune guide cittadine e repertori di immagini a stampa; in particolare il saggio si sofferma sulla scelta dei monumenti effettuata da due storiografi locali, Serviliano Lattuada e Giorgio Giulini considerando anche il repertorio inciso posto a corredo dei loro test
Norms of characters and convergence of Fourier series on compact Lie groups
No full textThis is a short note in which we point out some estimates of the L^p norms of characters of compact connected semisimple Lie groups G
Lp-Lq estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III
No full textThis paper is the third of a series on semigroups of operator related to the Laplace Beltrami operator on a symmetric space of the non compact type. We consider the Poisson semigroup P_{τ,θ}, when θ=1 and τ is complex and Reτ>0. We remark that the shifted Laplace Beltrami operator -L+b, corresponding to the case θ=1, occurs naturally in geometry, as it is conformally invariant.
Our main theorem describes the behaviour of the Lp-Lq operator norm of P_{τ,1} for various possible values of p and q and for τ in various subsets of the right half of the complex plane. This description is nearly complete, but when p<2<q and |τ| is large but τ is nearly imaginary, our methods do not yield good estimates
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