Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

Interpolation problems on the spectrum of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

We characterize those sequences (x _n ) in the spectrum of H ^∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = f _n (x) for every x ∈ P(x _n ), where P(x _n ) is the Gleason part associated with the point x _n and where (f _n ) is an arbitrary sequence of functions in the unit ball of H ^∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions, a result first proved by Gallardo and Gorkin.      

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Bounded <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-Calculus for Strip-Type Operators

We study the functional calculus for operators of strong strip type, including both a characterisation by means of weak bounded variation of the resolvent and some perturbation theory. The results are broadly similar in style to those for sectorial operators, but they are not corollaries and there are some surprising differences.     

On peak phenomena for non-commutative <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

A non-commutative extension of Amar and Lederer’s peak set result is given. As its simple applications it is shown that any non-commutative H ^∞-algebra H ^∞(M, τ) has unique predual, and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality. Dedicated to Professor Fumio Hiai on the occasion of his 60th birthday. Yoshimichi Ueda was supported in part by Grant-in-Aid for Young Scientists (B)17740096.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Solvability of the <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> Algebraic Riccati Equation in Banach Algebras

Let R be a commutative complex unital semisimple Banach algebra with the involution ·^*{\cdot ^\star} . Sufficient conditions are given for the existence of a stabilizing solution to the H ^∞ Riccati equation when the matricial data has entries from R. Applications to spatially distributed systems are discussed.     

<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-calculus for Products of Non-Commuting operators

It is shown that the product of two sectorial operators A and B admits a bounded H^∞-calculus on a Banach space X provided suitable commutator estimates and Kalton-Weis type assumptions on A and B are satisfied.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

<Emphasis Type="Italic">C</Emphasis><Superscript>∞</Superscript> regularity of solutions of the Levi equation in<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">2n+1</Emphasis></Superscript>

We study the regularity of the solutions of the Levi equation in ?²ⁿ⁺¹. It is a second order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every functionu∈C ². We show that the operator associated to the equation can be represented as a sum of squares of non linear vector fields. Then, by using a freezing method, we prove theC ^∞ regularity of the solutions.     

Estimates in corona theorems for some subalgebras of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

If n is a non-negative integer, then denote by ∂^-n H ^∞ the space of all complex-valued functions f defined on such that f, f ⁽¹⁾, f ⁽²⁾,...,f ⁽ⁿ⁾ belong to H ^∞, with the norm

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

Ultracontractivity and embedding into <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>

Given a self-adjoint semigroup e^−tA satisfying an ultracontractivity bound of the type , we find conditions on the sequence that imply that f is a bounded function. Sobolev’s classical embedding theorem says that, when A is the Laplace operator on , for some k > d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence and depends on the behavior of the ultracontractivity function m. Research of A. Bendikov was supported by the Polish Goverment Scientific Research Fund, Grant 1 PO3 A 03129. Research of T. Coulhon was partially supported by the European Commission (IHP Network “Harmonic Analysis and Related Problems” 2002–2006, Contract HPRN-CT-2001-00273-HARP). Research of L. Saloff-Coste was partially supported by NSF grant DMS-0102126.     

Solving the <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-control problem with reduced-dimensionality controllers in the presence of harmonic perturbations

We consider H -control in an additively perturbed system. Some components of the perturbation vector are assumed linear combinations of harmonic processes with unknown amplitudes and known frequencies. We solve the problem of output-feedback design of linear H -controllers. As an example, we consider the problem of suppressing the perturbations in an elastically connected two-body system.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 107–114, 2002.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

Toeplitz condition numbers as an <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> interpolation problem

The condition numbers CN(T) = ∥T∥ · ∥T⁻¹∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with ∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λ_i| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem. Bibliography: 2 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179.