Operators Commuting with the Volterra Operator are not Weakly Supercyclic

We prove that any bounded linear operator on L _p [0, 1] for 1 ≤ p < ∞, commuting with the Volterra operator V, is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic type condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V.     

The negative discrete spectrum of the operator (-τ) l -αV inL 2(R d ) ford even and 2l≥dinL 2(R d ) ford even and 2l≥d

We study the asymptotic behaviour ofN(α)—the number of negative eigenvalues of the operator (-τ) ^l -αV inL ₂(R ^d ) for an evend and2l≥d. This is the only case where the previously known results were far from being complete. In order to describe our results we introduce an auxiliary ordinary differential operator (system) on the semiaxis. Depending on the spectral properties of this operator we can distinguish between three cases whereN(α) is of the Weyl-type,N(α) is of the Weyl-order but not the Weyl-type coefficient and finally whereN(α)=O(α^q) withq>d/2l.     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

Maximal operators and singular integrals with non-isotropic dilation on product domains

In this paper, the authors study the with non-isotropic dilation on product domains. LP-mapping properties of certain maximal operators As an application, the LP-boundedness of the corre- sponding nomisotropic multiple singular integral operator is also obtained. Here the integral kernel functions Ω belong to the spaces L（logL）a（E1 × E2） for some a 〉 0, which is optimal.     

Some estimates related to fractal measures and Laplacians on manifolds

Let Δ be the Laplace-Beltrami operator on ann dimensional completeC ^∞ manifoldM In this paper we establish an estimate ofe ^tΔ (dμ) valid for allt>0 wheredμ is a locally uniformly α dimensional measure onM 0≤α≤n The result is used to study the mapping properties of (I-tΔ)^-β considered as an operator fromL ^p (M dμ) toL ^p (M dx) wheredx is the Riemannian measure onM β>(n−α)/2p′ 1/p+1/p′=1 1≤p≤∞     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

A random walk for the solution sought: Remarks on the difference scheme approach to nonlinear semigroups and evolution operators

In [2], Crandall and Evans show existence of mild solution to an abstract Cauchy Problem: u′(t)+Au(t)∋f(t), 0≤t≤T, u(0)=x₀, where A is an accretive operator in a general Banach space X and f ε L¹(0,T;X). Their method involves proving convergence in the L^∞-norm of a sequence of step function approximations α_n(σ, τ) to the solution of a first order partial differential equation. We consider a more general Cauchy Problem and show a.e. existence of mild solution by proving convergence of the step functions α_n(σ, τ) in the L¹-norm. Fundamental to the proof is a nonhomogeneous random walk in the plane.     

On the negative discrete spectrum of a preiodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential

Let Ω ⊂R ^d be an unbounded domain, periodic along a chosen direction (a waveguide-type domain),P be a self-adjoint elliptic second order operator inL ₂(Ω) periodic along the same direction, andV be a real-valued decaying potential. We suppose that the bottom of the spectrum ofP is λ=0 and study the asymptotic behaviour of the number of negative eigenvalues of the opeatorP−aV as the parameter α tends to +∞. We show that typically the Weyl asymptotic law for this quantity is violated and find a substitute for this law.     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t)=Au(t)+f(t)(0 ≤t≤2π),where A is a closed operator in a Banach space X,0α1 and Dα is the fractional derivative in the sense of Weyl.Although this problem is not always well-posed in Lp(0,2π;X) or periodic continuous function spaces Cper([0,2π];X),we show by using the method of sum that it is well-posed in some subspaces of L p(0,2π;X) or C per([0,2π];X).     

On Linearly Isometric Extensions for 1-Lipschitz Mappings Between Unit Spheres of ALP-spaces （p 〉 2）

In this paper, we show that if Vo is a 1-Lipschitz mapping between unit spheres of two ALP-spaces with p 〉 2 and -Vo（S1（LP）） C Vo（S1（LP））, then V0 can be extended to a linear isometry defined on the whole space. If 1 〈 p 〈 2 and Vo is an ＂anti-l-Lipschitz＂ mapping, then Vo can also be linearly and isometrically extended.     

Similarity of operators associated with the Volterra relation

The “Volterra relation” is the commutation relation [S,V]⊂V ², where S is a not necessarily bounded operator, V is a bounded operator leaving D(S) invariant, and [⋅,⋅] is the Lie product. When S,V are so related, and in addition iS generates a bounded C ₀-group of operators and V has some general property, it is known that S+α V (α∈ℂ) is similar to S if and only if ℜ α=0 (cf. Theorem 11.17 in Kantorovitz, Spectral Theory of Banach Space Operators, Springer, Berlin, 1983). In particular, S−V is not similar to S. However, it is shown in this note that (without any restriction on V and on the group S(⋅) generated by iS), the perturbations (S−V)+P are similar to S for all P in the similarity sub-orbit {S(a)VS(−a);a∈ℝ} of V. When S is bounded, the above perturbations are similar to S for all P in the wider similarity sub-orbit {e ^aS Ve ^−aS ;a∈ℂ}.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Operator-semistable,operator semi-selfdecomposable probability measures and related nested classes on <Emphasis Type="Italic">p</Emphasis>-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).