Weak-Type Boundedness of the Hardy–Littlewood Maximal Operator on Weighted Lorentz Spaces

The main goal of this paper is to provide a characterization of the weak-type boundedness of the Hardy–Littlewood maximal operator, M, on weighted Lorentz spaces \(\Lambda ^p_u(w)\), whenever \(p>1\). This solves a problem left open in (Carro et al., Mem Am Math Soc. 2007). Moreover, with this result, we complete the program of unifying the study of the boundedness of M on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph.     

On the Operator Semirings of a Γ-Semiring

In this paper we introduce the notion of operator semirings of a -semiring to study -semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a -semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various -semirings.AMS Subject Classification (2000): 16Y60, 16Y99     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

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≤∞.      

L^p（R^n） Boundedness for the Maximal Operator of Multilinear Singular Integrals with Calderon-Zygmund Kernels

In this paper, the authors consider a class of maximal multilinear singular integral operators and maximal multilinear oscillatory singular integral operators with standard Calderón–Zygmund kernels, and obtain their boundedness on L ^p (ℝ ⁿ ) for 1 < p < ∞. Research supported by Professor Xu Yuesheng’s Research Grant in the program of "One hundred Distinguished Young Scientists" of the Chinese Academy of Sciences     

Large Deviation of a Capacity Related to the Ornstein-Uhlenbeck Operator

We discuss the large deviation estimates of a capacity related to the Ornstein-Uhlenbeck operator and present a refinement of Strassen's form of the law of the iterated logarithm for Brownian metion from the viewpoint of the Dirichlet space theory.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Stability of the Magnetic Schrödinger Operator in a Waveguide

Abstract

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.     

Boundedness of the Maximal Operator on Lipα（R^n）（0〈α〈1）

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＡｌｏｃａｌｌｙｉｎｔｅｇｒａｂｌｅｆｕｎｃｔｉｏｎｆ（ｘ）ｂｅｌｏｎｇｓｔｏＬｉｐα（Ｒｎ）,ｉｆｔｈｅｒｅｉｓａｃｏｎｓｔａｎｔＣ,ｓｕｃｈｔｈａｔｆｏｒｅｖｅｒｙｘ,ｙ∈Ｒｎ｜ｆ（ｘ）－ｆ（ｙ）｜≤Ｃ｜ｘ－ｙ｜α　　ＴｈｅｓｍａｌｌｅｓｔｃｏｎｓｔａｎｔＣｓａｔｉｓｆｉｅｓａｂｏｖｅｉｓｃａｌｌｅｄＬｉｐｓｃｈｉｔｚｎｏｒｍｏｆｆａｎｄｉｓｄｅｎｏｔｅｄｂｙｙｆｙ∧α．Ｂｙ［１］,ｆ∈Ｌｉｐα（Ｒｎ）ｅｑｕｉｖａｌｅｎｔｔｏｆ∈εα,２,ｗｈｅｒｅεα,２＝…     

The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), where K is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of H(K) for all sufficiently small values of the zero-range attractive potentials is established.The asymptotics is found for the number of eigenvalues N(0,z) lying below . Moreover, for all sufficiently small nonzero values of the three-particle quasimomentum K, the finiteness of the number of eigenvalues below the essential spectrum of H(K) is established and the asymptotics of the number N(K,0) of eigenvalues of H(K) below zero is given.     

The Degree Theory of a New Operator

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｔｈｅｄｅｇｒｅｅｔｈｅｏｒｙｆｏｒａｋ－ｓｅｔ－ｃｏｎｔｒａｃｔｏｐｅｒａｔｏｒｈａｓｍａｎｙａｐｐｌｉｃａｔｉｏｎｓｏｎｔｈｅｅｘｉｓｔｅｎｃｅｏｆｔｈｅｓｏｌｕｔｉｏｎｓｏｆｓｏｍｅｅｑｕａｔｉｏｎｓ（ｓｅｅ［１］，［２］，［３］）．Ｈｏｗｅｖｅｒ，ｓｏｍｅｏｐｅｒａ－ｔｏｒｓａｒｅｎｏｔｋ－ｓｅｔ－ｃｏｎｔｒａｃｔｏｐｅｒａｔｏｒｓ．Ｉｔｉｓａｎｅｗｐｒｏｂｌｅｍｔｈａｔｗｅｅｓｔａｂｌｉｓｈａｄｅｇｒｅｅｔｈｅｏｒｙｆｏ…     

Some Properties of Eigenfunctions of the Schrödinger Operator in a Magnetic Field

We study the behavior of eigenfunctions corresponding to a positive point spectrum of the Schrödinger operator with magnetic and electric potentials.     

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new.     

On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.     

Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

Maximal invariant subspaces for a class of operators

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim?M ? TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim?M ? zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Solution of the Operator Equation AT-TB=S

Let H, K be two Hilbert spaces over complex field A. Let B(HI, K) denote the set of all bounded linear operators from H to K. If H=K, we write B(H) instead of B(H, K). Consider the operator defined by the equation