Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

The resolvent of a singularly perturbed elliptic operator

Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?^{2(m ? n)}A + B + λ²ⁿI)u_? = f and (B + λ²ⁿI)u = f in a bounded domain Ω in R^k with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.     

On an estimate for Bloch and Doob norm and covering problem in Doob's class

For any fixed 0 < π ? 2π, let D(π) be the family of all holomorphic functions in the unit disk Δ which satisfy (i)f(0) = 0 and (ii) $\text{lim inf}{}_{\mathrm{z\; \to \; \pi ¦f(z)¦\; ?\; 1}}$, for all π lying on some arc A_f ? ?Δ with arclength $\text{¦A}{}_{\mathrm{f}}\text{¦ ? π}$. We show that for each 0 < ε < 1, there is a π₀ > 0 such that for any f?D(π) with π < π₀, the Bloch and Doob norm respectively satisfy

$\text{6f6}{}_{\mathrm{B}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|}{}^2\text{) > 2(1 ? ε)}\text{log}\text{1+}\text{cos}\text{(}\text{p}\text{2}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

$\text{6f6}{}_{\mathrm{D}}\text{=}\text{sup}\text{z?Δ}\text{|f′(z)| (1?|z|) > (1 ? ε)}\text{log}\text{1}\text{1?}\text{cos}\text{(}\text{p}\text{2}{}^{\mathrm{?1}}$

These two estimates do not hold with ε = 0.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

Polynomials which take Gaussian integer values at Gaussian integers

A factorial set for the Gaussian integers is a set G = {g₁, g₂ … g_n} of Gaussian integers such that $\text{G(z) =}\text{Π}{}_{\mathrm{k}}\text{(z ? g}{}_{\mathrm{k}}\text{)}\text{g}{}_{\mathrm{k}}$ takes Gaussian integer values at Gaussian integers. We characterize factorial sets and give a lower bound for $\text{max}{}_{\mathrm{<mtext>\parallel z\parallel 2=</mtext><mtext>n</mtext><mtext>\pi </mtext>}}\text{∥ G(z)∥}$. It is conjectured that there are infinitely many factorial sets. A Gaussian integer valued polynomial (GIP) is a polynomial with the title property. A bound similar to the above is given for max_∥z∥2=n ∥ G(z)∥ if G(z) is a GIP. There is a relation between factorial sets and testing for GIP's. We discuss this and close with some examples of factorial sets, and speculate on how to find more.     

Estimates on the first eigenvalue of an integral equation

We show how inequalities of the type $\text{∥F∥}{}_{\mathrm{p}}\text{? C(p, q) a}{}^{\mathrm{<mtext>1\; +\; (</mtext><mtext>1</mtext><mtext>p</mtext><mtext>)?\; (</mtext><mtext>1</mtext><mtext>q</mtext><mtext>)</mtext>}}\text{∥ F ′ ∥}{}_{\mathrm{q\prime }}$ when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝₀^ak(s, z)F(s) ds.     

The Hardy classes for functions in the class MV[α, k]

Suppose that f(z) = z + a₂z² + ··· + a_nz_n + ··· is regular in the unit disc $\text{D}\text{with}\text{[f(z) f′}\text{(z)}\text{z] ≠ 0}\text{in}\text{D}$, and further let α ? 0 and k ? 2. If $\text{∝}{}^{\mathrm{2\pi }}{}_{\mathrm{o}}\text{¦}\text{Re}\text{{(1 ? α)z[}\text{f′(z)}\text{f(z)}\text{] + α(1 + z[}\text{f″(z)}\text{f′(z)}\text{])}¦ dθ ? kπ}\text{for}\text{z ? D}$, then f(z) is said to belong to the class MV[α, k]. This class contains many of the special classes of regular and univalent functions. The authors determine the Hardy classes of which f(z), f′(z) and f″(z) belong and obtain growth estimates of a_n.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

On the convergence of solutions of functional differential equations with infinite delay

Let $\text{C(β), S1(β),}\text{and}\text{K(β, λ)}$ be the classes of univalent functions defined in $\text{E = {z: ¦z¦< 1}}$, which are convex of order β, starlike of order β and close-to-convex of order β type λ. Let $\text{f(z) = (}\text{1}\text{α}\text{)z}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>\alpha </mtext>}}\text{∝}{}^{\mathrm{z}}{}_0\text{z}{}^{\mathrm{<mtext>1</mtext><mtext>x?2</mtext>}}\text{F(z)dz, 0 ? α < 1}$. We discuss the properties of the function f when this function F belongs to the class K(β, λ) and its various subclasses.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

The spectra of composition operators on Hp

For H^p, 1 ? p < ∞, composition operators C_?, defined by $\text{C}{}_{\mathrm{?}}\text{(?) = ? ° ?}$ for $\text{? ? H}{}^{\mathrm{p}}$, ? analytic on $\text{D = {z ¦ ¦ z ¦ < 1}}$ are considered, and their spectra determined in the case where ? is analytic on an open region containing D?.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Equations du type de Monge-Ampère sur les variétés Riemanniennes compactes,III

Let (V_n, g) be a C^∞ compact Riemannian manifold without boundary. Given the following changes of metric: $\text{g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{= g +}\text{Hess}\text{? ±}\text{l}\text{α}{}^2\text{(▽ ? ? ▽?),}\text{g}\text{?}{}^{\mathrm{\pm }}\text{= ±?g + α}{}^2\text{Hess}\text{?}$, where a is a fixed constant, we study the corresponding Monge-Ampère equations (1)^±$\text{Log}\text{(¦g′}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P,▽?;?)}$, (2)^±$\text{Log}\text{(¦}\text{g}\text{?}{}_{\mathrm{?}}{}^{\mathrm{\pm }}\text{¦ ¦g¦}{}^{\mathrm{?1}}\text{) = F(P, ▽?; ?)}$. We first solve Eq. (2)^?, under some simple assumptions on F?C^∞. Then, using an appropriate change of functions that enables us to take advantage of the estimates just carried out for Eq. (2)^?, we extend to Eq.(1)^? all the results proved in our previous articles [5, 6] for the usual Monge-Ampère equation. Although equation (2)⁺ is not locally invertible, and does not even admit a solution for all $\text{F = λ? + ?, λ > 0, f ? C}{}^{\mathrm{\infty }}\text{(V}{}_{\mathrm{n}}\text{)}$, a similar change of functions leads to partial results about Eq. (1)⁺, via C² and C³ estimates for Eq. (2)⁺. Eventually we give some comments and errata of our previous article (P. Delanoë, J. Funct. Anal.41 (1981), 341–353).     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

A decomposition theorem for an analytic function

Let f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑_{k = 0}^∞c_kz^λ_k. Set and define the growth constants ?, λ, T, t by

$\text{?}\text{λ}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{}\text{log}\text{[Rr /(R?r)]}{}^{\mathrm{?1}}\text{log}{}^{\mathrm{+}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

, and if 0 < ? < ∞,

$\text{T}\text{t}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{[Rr /(R?r)]}{}^{\mathrm{??}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

. Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z).     

One-sided inequalities for ternary forms: II

For an indefinite quadratic form f(x₁, …, x_n) let P(f) denote the greatest lower bound of the positive values assumed by f for integers x₁, …, x_n. This paper investigates the values of $\text{P}{}^3\text{∥d∥}$ for nonzero ternary forms of signature ?1 and finds two new classes of forms with $\text{P}{}^3\text{∥d∥}\text{>}\text{1}\text{2}$.     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.