Maslov’s canonical operator in arbitrary coordinates on the Lagrangian manifold

We present the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold. The construction does not require any additional choice of the phase function.     

New formulas for Maslov’s canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics

We suggest a new representation of Maslov’s canonical operator in a neighborhood of caustics using a special class of coordinate systems (eikonal coordinates) on Lagrangian manifolds. We present the results in the two-dimensional case and illustrate them with examples.     

Maslov’s canonical operator,Hörmander’s formula,and localization of the Berry-Balazs solution in the theory of wave beams

For three-dimensional Schrödinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry-Balazs solutions) and a Bessel function and known as Airy-Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov’s canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hörmander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov’s canonical operator to “move” the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions.     

Maurey’s factorization theory for operator spaces

We prove an operator space version of Maurey’s theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically absolutely q-summing for q > p. Our results imply in particular that every completely bounded map from B(H) with values in Pisier’s operator space OH is completely p-summing for p > 2. This fails for p = 2. As applications, we obtain eigenvalue estimates for translation invariant maps defined on the von Neumann algebra V N(G) associated with a discrete group G. We also develop a notion of cotype which is compatible with factorization results on noncommutative L _p spaces.     

On the representation of localized functions in ℝ2 by the Maslov canonical operator

We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being the Maslov canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.     

Egoroff’s theorem in measurable operator spaces associated with a Von Neumann algebra

In this article, some kinds of convergence about the τ–measurable operators affiliated with a von Neumann algebra are defined. Moreover, the relationships among these kinds of convergence are considered.     

Witt’s theorem in abstract geometric algebra

In an earlier paper of the author, a version of the Witt’s theorem was obtained within a specific subcategory of the category of A{mathcal A}-modules: the full subcategory of convenient A{mathcal A}-modules. A further investigation yields two more versions of the Witt’s theorem by revising the notion of convenient A{mathcal A}-modules. For the first version, the A{mathcal A}-bilinear form involved is either symmetric or antisymmetric, and the two isometric free sub-A{mathcal A}-modules, the isometry between which may extend to an isometry of the non-isotropic convenient A{mathcal A}-module concerned onto itself, are assumed pre-hyperbolic. On the other hand, for the second version, the A{mathcal A}-bilinear form defined on the non-isotropic convenient A{mathcal A}-module involved is set to be symmetric, and the two isometric free sub-A{mathcal A}-modules, whose orthogonals are to be proved isometric, are assumed strongly non-isotropic and disjoint.     

Jackson’s theorem in Q_p spaces

A Jackson type inequality in Q p spaces is established, i.e., for any f (z) = Σ∞ j=0 ajzj ∈ Qp , 0≤p ∞, a 1, and k-1 ∈ N,where ω(1/k, f, Q p ) is the modulus of continuity in Q p spaces and C(a) is an absolute constant depending only on the parameter a.     

Moreau’s decomposition in Banach spaces

Moreau’s decomposition is a powerful nonlinear hilbertian analysis tool that has been used in various areas of optimization and applied mathematics. In this paper, it is extended to reflexive Banach spaces and in the context of generalized proximity measures. This extension unifies and significantly improves upon existing results.     

On Stein?s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein?s method to “infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein?s identity for abstract Wiener measures and solve the corresponding Stein?s equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg-Lévy type limit theorem. In addition, an analogous of Berry-Esséen type estimate for abstract Wiener measures will be obtained.     

Meir–Keeler-type conditions in abstract metric spaces

Various Meir–Keeler-type conditions for mappings acting in abstract metric spaces are presented and their connections are discussed. Results about associated symmetric spaces, obtained in [S. Radenovi?, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (2011), 38–50] are used to show that the regularity condition for the underlying cone can be dropped in some fixed point results that have appeared recently.     

Wolff’s inequality in multi-parameter Morrey spaces

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L ^p (R ^d ) and multi-parameter Morrey spaces ${L^p_\lambda (R^d)}$ , where ${R^d=R^{n_1} \times R^{n_2} \times \cdots \times R^{n_k},\, \lambda = (\lambda _1,\ldots ,\lambda _k})$ and 0?<?λ _i ≤ n _i , 1?≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.     

Doob’s optional sampling theorem in Riesz spaces

The notions of stopping times and stopped processes for continuous stochastic processes are defined and studied in the framework of Riesz spaces. This leads to a formulation and proof of Doob’s optional sampling theorem.     

On Dekster’s angle measure in Minkowski spaces

Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between and 8.     

Area operator on Bergman spaces

We characterize the non-negative measuresμon the unit disk D for which the area operator Aμis bounded from Bergman space Aαp to Lq ((?)D).     

Stochastic integrals and BDG’s inequalities in Orlicz-type spaces

In this paper we extend an inequality of Lenglart et al. (1980, Lemma 1.1) to general continuous adapted stochastic processes with values in topological spaces. Using this inequality we prove Burkholder–Davies–Gundy’s inequality for stochastic integrals in Orlicz-type spaces (a class of quasi-Banach spaces) with respect to cylindrical Brownian motions. As an application, we show the well-posedness of stochastic heat equations in Orlicz spaces.     

Strong convergence of Browder’s and Halpern’s type iterations in Hilbert spaces

We use results from theory of nonexpansive mappings to unify and deduce the recent results of Lin and Takahashi (Positivity 16:429–453, 2012) and of Takahashi (J Optim Theory Appl 157:781–802, 2013).     

Blowup solutions of Grushin’s operator

In this note, we consider the blowup phenomenon of Grushin’s operator. By using the knowledge of probability, we first get an expression of heat kernel of Grushin’s operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solution will blow up in finite time.     

Special values of canonical Green’s functions

We give a precise formula for the value of the canonical Green’s function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the ‘energy’ of the Weierstrass points in terms of a spectral invariant recently introduced by N. Kawazumi and S. Zhang. It follows that the energy is strictly larger than log 2. Our results generalize known formulas for elliptic curves.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.