Dirac system with potential lying in Besov spaces

We study the spectral properties of the Dirac operator L_P,U generated in the space (L₂[0, π])² by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y₁, y₂)^t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ_n}_n∈Z of the operator L_P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B_1,∞^θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L_P,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.     

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L ₀ + V be a Schrödinger type operator, where L ₀ is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ^{?t L} . Furthermore, the authors impose extra regularity assumptions on V to obtain the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].     

Uniform asymptotics of the eigenvalues and eigenfunctions of the Dirac system with an integrable potential

We consider the Dirac operator on a finite interval with a potential belonging to some set X completely bounded in the space L₁[0, π] and with strongly regular boundary conditions. We derive asymptotic formulas for the eigenvalues and eigenfunctions of the operator; moreover, the constants occurring in the estimates for the remainders depend on the boundary conditions and the set X alone.     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

On the properties of maps connected with inverse Sturm-Liouville problems

Let L _D be the Sturm-Liouville operator generated by the differential expression L _y = ?y″ + q(x)y on the finite interval [0, π] and by the Dirichlet boundary conditions. We assume that the potential q belongs to the Sobolev space W ₂ ^? [0, π] with some ? ≥ ?1. It is well known that one can uniquely recover the potential q from the spectrum and the norming constants of the operator L _D. In this paper, we construct special spaces of sequences ? ₂ ^θ in which the regularized spectral data {s _k } _?∞ ^∞ of the operator L _D are placed. We prove the following main theorem: the map F _q = {s _k } from W ₂ ^? to ? ₂ ^θ is weakly nonlinear (i.e., it is a compact perturbation of a linear map). A similar result is obtained for the operator L _DN generated by the same differential expression and the Dirichlet-Neumann boundary conditions. These results serve as a basis for solving the problem of uniform stability of recovering a potential. Note that this problem has not been considered in the literature. The uniform stability results are formulated here, but their proof will be presented elsewhere.     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators ?_P,U and ?_0,U with potential P summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of P ∈ L_?[0, π], ? ∈ (1,∞], equiconvergence holds for every function f ∈ L_μ[0, π], μ ∈ [1,∞], in the norm of the space L_ν[0, π], ν ∈ [1,∞], if the indices ?, μ, and ν satisfy the inequality 1/? + 1/μ ? 1/ν ≤ 1 (except for the case when ? = ν = ∞ and μ = 1). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval [0, π] is proved for an arbitrary function f ∈ L₂[0, π].     

An elementary proof of the <Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript> bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A _p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L ¹ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A ² bounds in that setting.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Traces of Higher Negative Orders for a String with a Singular Weight

We study the linear operator pencil A(λ) = L?λV, λ ∈ ?, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ₂ ^?1 [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λ_n, n ∈ ?, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L^?1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.     

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.     

〈2, 1〉-Compact operators

We consider the class of the continuous L _2,1 linear operators in L ₂ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L ₂ into a compact subset of L ₁. We prove that a functional equation with an operator from L _2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L _2,1 and VTV ^?1 ∈ L _2,1 for all unitary operators V in L ₂ then T = α1 + C, where α is a scalar, 1 is the identity operator in L ₂, and C is a compact operator in L ₂.     

Regular growth of various characteristics of entire functions of order zero

We study the relationship between the strongly regular growth of an entire function f of order zero, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of the logarithm of f, and the regular growth of the logarithm of the modulus and the argument of f in the L^p[0, 2π]-metric, p ≥ 1.     

Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = Δ_E² + q to be separated in the space L² (M) on a complete Riemannian manifold (M,g) with metric g, where Δ_E is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L² (M) if for all u ∈ L² (M) such that Au ∈ L² (M) we have Δ_E²u ∈ L² (M) and qu ∈ L² (M).     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

Regularized traces of singular differential operators with canonical boundary conditions

A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L ₂[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k ₁ < k ₂ < ... < k _m ≤ 2m ? 1 and {k _s } _s=1 ^m ∪ {2m ? 1 ? k _s } _s=1 ^m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L ₂[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated.     

Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

In this paper we prove that for an arbitrary pair {T ₁, T ₀} of contractions on Hilbert space with trace class difference, there exists a function ξ in L ¹(T) (called a spectral shift function for the pair {T ₁, T ₀}) such that the trace formula trace(f(T ₁) ? f(T ₀)) = ∫_T f′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.