Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ _∞ ^λ -spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ₂, we obtain a ℒ _∞ ^λ -spaceX with the RNP such that the projective tensor product containsc ₀ and hence fails the RNP. TakingE=L ¹, we obtain a ℒ _∞ ^λ -space failing the RNP but nevertheless not containingc ₀.     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

Certain asymptotic expansions for laguerre polynomials and Charlier polynomials

Here proposed are certain asympotic expansion formulas for L _n ^(∞-1) (λz) and C _n ^(∞) (λz) in which 0<w=0(λ) and C_n/(^w)(λz), z being a complex number. Also presented are certain estimates for the remainders (error bounds) of the asymptotic expansions within the regions D₁(-∞<R_ez<=1/2(ω/λ) and D₂(1/2(ω/λ)<=Re_z<∞), respectively. Supported by NSERC (Canada) and also by the National Natural Science Foundation of China.     

On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets

We consider natural Laplace operators on random recursive affine nested fractals based on the Sierpinski gasket and prove an analogue of Weyl’s classical result on their eigenvalue asymptotics. The eigenvalue counting function N(λ) is shown to be of order λ ^ds/2 as λ→∞ where we can explicitly compute the spectral dimension d _s . Moreover the limit N(λ) λ ^−ds/2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable. This random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal. Received: 22 January 1999 / Revised version: 2 September 1999 /?Published online: 30 March 2000     

Uniform algebras as Banach spaces

Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L^∞(μ) such that: 1)|F|= 1 a.e. relative to μ; 2) F _μ ε A¹; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L^∞(μ), L¹(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H^∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H^∞ to a very broad class of uniform algebras. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 80–89, 1976.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997     

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

The stable mapping class group and Q(ℂP∞+)

In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction ℤ×BΓ⁺ _∞ has an infinite loop space structure. This result and the tools developed in [BM] to analyse transfer maps, are used here to show the following splitting theorem. Let Σ^∞(ℂP ^∞ ₊)^∧ _p ≃E ₀∨...∨E _p-2 be the “Adams-splitting” of the p-completed suspension spectrum of ℂP ^∞ ₊. Then for some infinite loop space W _p ,?(ℤ×BΓ⁺ _∞)^∧ _p ≃Ω^∞(E ₀)×...×Ω^∞(E _p-3 )×W _p ?where Ω^∞ E _i denotes the infinite loop space associated to the spectrum E _i . The homology of Ω^∞ E _i is known, and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group. This splitting also detects all the Miller-Morita-Mumford classes. Our results suggest a homotopy theoretic refinement of the Mumford conjecture. The above p-adic splitting uses a certain infinite loop map?α_∞:ℤ×BΓ⁺ _∞?Ω^∞ℂP ^∞ _-1?that induces an isomorphims in rational cohomology precisely if the Mumford conjecture is true. We suggest that α_∞ might be a homotopy equivalence. Oblatum 2-VIII-1999 & 28-III-2001?Published online: 18 June 2001     

The projection constant of finite-dimensional spaces whose unconditional basis constant is 1

The relations of the projection constant λ(E) and the isomorphic distance d(E, 1 _n ^∞ ) of finite-dimensional spacesE whose unconditional basis constant is 1 are investigated. It turns out that both are proportional to the norm of a certain vector inE.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P ([0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

On some new generalized difference sequence spaces defined by a sequence of moduli

In this paper, we define the new generalized difference sequence spaces [V, λ, F, p, q]₀(Δ _v ^m ), [V, λ, F, p, q]₁(Δ _v ^m ) and [V, λ, F, p, q]_∞(Δ _v ^m ). We also study some inclusion relations between these spaces.     

On weighted spaces of holomorphic functions of several variables

We generalize the results of [11] and [12] for the unit ball $ \mathbb{B}_d $ \mathbb{B}_d of ℂ ^d . In particular, we show that under the weight condition (B) the weighted H ^∞-space on $ \mathbb{B}_d $ \mathbb{B}_d is isomorphic to ℓ^∞ and thus complemented in the corresponding weighted L ^∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂ ^d . In addition, we show that for the polydisc $ \mathbb{D}^d $ \mathbb{D}^d ^d , the weighted H ^∞-space is never isomorphic to ℓ^∞.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ ϕ(λ,μ)dE_λTdF_μ, where E_λ, F_μ are the spectral functions of tow self-adjoint operators A, B on a Hilbert space and T is a bounded operator. In principal, the case where A has finite spectrum is studied. Nonlinear estimates of ‖f(A)T-T f(B)‖ in terms of the norm of ‖AT-TB‖ for f∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 148–173. Translated by S. V. Kislyakov.