The Hardy Operator and the Gap Between L{infty} and BMO

We study boundedness and compactness properties of the Hardyintegral operator from a weighted Banach function space X(v) into L and BMO. We give a new simplecharacterization of compactness of T from X(v) into BMO. Weconstruct examples of spaces X(v) such that T(X(v)) is (a) boundedin L but not compact in BMO; (b) compact in BMO but not boundedin L; (c) bounded in BMO but neither bounded in L nor compactin BMO; (d) bounded in L, compact in BMO and yet not compactin L. In order to obtain the last of the counterexamples weconstruct a new weighted Banach function space.     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

Orbits in the Maximal Ideal Space of H{infty}

Let H be the Banach algebra of bounded analytic functions inthe open unit disc D. We can define the rotation in the maximalideal space M(H). For a point x in M(H)\D, an orbit O(x) isnot closed in M(H). It is proved that there exists a point xin M(H) such that x is not contained in the Shilov boundaryX and cl O(x), the closure of O(x), contains X, and there existsa point y in M(H)\(D X) such that cl O(y) X. The rotationpresents many problems concerning H. The purpose of this paperis to discuss these problems.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Embedding lp, {infty} into Lp, {infty}[0, 1] Complementably

We show that l^p, embeds into L^p, [0, 1] as a complemented sublattice.     

Asymptotic Interpolating Sequences in Uniform Algebras

T. Hosokawa, K. Izuchi and D. Zheng recently introduced theconcept of asymptotic interpolating sequences (of type 1) inthe unit disk for H(D). It is shown that these sequences coincidewith sequences that are interpolating for the algebra QA. Alsoa characterization is given of the interpolating sequences oftype 1 for H(D), and asymptotic interpolating sequences in thespectrum of H(D) are studied. The existence of asymptotic interpolatingsequences of type 1 for H() on arbitrary domains is verified.It is shown that any asymptotic interpolating sequence in auniform algebra eventually is interpolating.     

Smooth Approximation of Sobolev Functions on Planar Domains

We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

Majorations Uniformes de Normes D'Inverses Dans Les Algebres de Beurling

The Beurling algebras l¹(D,)(D=N,Z) that are semi-simple, withcompact Gelfand transform, are considered. The paper gives anecessary and sufficient condition (on ) such that l¹(D,) possessesa uniform quantitative version of Wiener's theorem in the sensethat there exists a function :]0,+[]0,+ such that, for everyinvertible element x in the unit ball of l¹(D,), we have ||x^–1||(r(x^–1)) r(x^–1) is the spectral radiusof x^–1.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences II

Let [ ] denote the integer part. Among other results in [3]we gave a complete solution to the following problem. PROBLEM. Given an increasing sequence a_n R⁺, n = 1, 2, ...,where a_n as n , are there infinitely many primes in the sequence[a_n] for almost all ?     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

On the Reflexivity of Operator Algebras with Isometric Functional Calculus

Every subalgebra of L(H) which is isometrically weak*-homeomorphicwith H() is reflexive, when is a finitely connected regionin the complex plane.     

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces H^p() = {f Hol(): |f|^p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for H^p(\E) if H^p(\E) Hol(), and a strongly removable singularity for H^p(\E) if H^p(\E)= H^p(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all H^p, but not strongly removablefor any H^p(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for H^p(\E)and q > p, then E is also weakly removable for H^q(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

The Full Muntz Theorem in C[0, 1] and L1[0, 1]

The main result of this paper is the establishment of the ‘fullMüntz Theorem’ in C[0, l]. This characterizes thesequences of distinct, positive real numbers for which span{l, x^₁, x^₂, ...} is dense in C[0,1]. The novelty of this result is the treatment of the mostdifficult case when inf_ii = 0 while sup_ii = . The paper settlesthe L and L₁ cases of the following. THEOREM (Full Müntz Theorem in L_p[0,1]). Let p [l, ].Suppose that is a sequence of distinct real numbers greater than –1/p. Then span{x^₀,x^₁, ...} is dense in L_p[0, 1] if and only if      

Weighted Bloch Spaces, Hp, and BMOA

It is determined which Bloch-type conditions on a function f _0<p<H^p ensure that f BMOA.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).