Selfadjoint extensions of the operator of the Dirichlet problem in a 3-dimensional region with an edge

Taking various viewpoints, we study the selfadjoint extensions $ \mathcal{A} $ \mathcal{A} of the operator A of the Dirichlet problem in a 3-dimensional region Ω with an edge Γ. We identify the infinite dimensional nullspace def A with the Sobolev space H ^−ϰ(Γ) on Γ with variable smoothness exponent −ϰ ∈ (−1, 0); while the selfadjoint extensions, with selfadjoint operators $ \mathcal{T} $ \mathcal{T} on the subspaces of H ^−ϰ(Γ). To the boundary value problem in the region with a “smoothed” edge we associate a concrete extension, which yields a more precise approximate solution to the singularly perturbed problem.     

Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) of Nikol’skii-Besov and Lizorkin-Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $ \mathcal{W}_m^\mathbb{I} $ \mathcal{W}_m^\mathbb{I} of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in B _pq ^sm ($ \mathbb{I} $ \mathbb{I} ) and L _pq ^sm ($ \mathbb{I} $ \mathbb{I} ^k ) by special partial sums of these series in the metric of L _r ($ \mathbb{I} $ \mathbb{I} ^k ) for a number of relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ℝ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ℕ ⁿ , k = m ₁ +... + m _n , and $ \mathbb{I} $ \mathbb{I} = ℝ or $ \mathbb{T} $ \mathbb{T} ). In the periodic case, we study the Fourier widths of these function classes.     

Representation of a solution of the dirichlet problem in a planar cusp domain as a logarithmic single-layer potential

The problem of representing the solution of the Dirichlet problem for the Laplace equation as a single-layer potential V _ϱ with unknown density ϱ is known to lead to the equation V _ϱ = f for density ϱ, where f is the Dirichlet boundary data. Let Γ be the boundary of a bounded planar domain with an outward or inward peak and T(Γ) be the space of the traces on Γ of functions with finite Dirichlet integral over R ². It is shown that the operator $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) is continuous, and the operator $ \varrho \to V\varrho - \overline {V\varrho } $ \varrho \to V\varrho - \overline {V\varrho } (where $ \bar u $ \bar u denotes u averaged over Γ) can be uniquely extended to the isomorphism      

Trace class Toeplitz operators with unbounded symbols on weighted Bergman spaces

In this note we construct a function φ in L2（Bn,dμ） which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on weighted Bergman space Lα2（Bn,dμ） for several complex variables.     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

Spaces with fibered approximation property in dimension <Emphasis Type="Italic">n</Emphasis>

A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M there exists a map g′: $ \mathbb{I} $ \mathbb{I} ^m × $ \mathbb{I} $ \mathbb{I} ⁿ → M such that g′ is ɛ-homotopic to g and dim g′ ({z} × $ \mathbb{I} $ \mathbb{I} ⁿ ) ≤ n for all z ∈ $ \mathbb{I} $ \mathbb{I} ^m . The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].     

Two operators related to Marcinkiewicz integrals

In this paper, the authors consider the boundedness of generalized higher commutator of Marcinkiewicz integral μΩ^b, multilinear Marcinkiewicz integral μΩ^A and its variation μΩ^A on Herz-type Hardy spaces, here Ω is homogeneous of degree zero and satisfies a class of L^s-Dini condition. And as a special case, they also get the boundedness of commutators of Marcinkiewicz integrals on Herz-type Hardy 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^∞.     

Isometric immersions into warped product spaces

This paper concerns the submanifold geometry in the ambient space of warped productmanifolds F^n×σ R, this is a large family of manifolds including the usual space forms R^m, S^m and H^m. We give the fundamental theorem for isometric immersions of hypersurfaces into warped product space R^n×σ R, which extends this kind of results from the space forms and several spaces recently considered by Daniel to the cases of infinitely many ambient spaces.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

On the smoothness of solutions of a class of regular equations in a strip

The paper considers a class of regular, hypoelliptic in x ₁, two-dimensional operators P(D) = P(D ₁,D ₂) in rather wide strip Ω _H = {x = (x ₁; x ₂) ∈ $ \mathbb{E} $ \mathbb{E} ², |x ₁| < H, x ₂ ∈ $ \mathbb{E} $ \mathbb{E} ¹}. It is proved the infinite differentiability in Ω _H of those generalized solutions of the equation P(D) _u = 0, for which D ₂ ^j u ∈ L ₂(Ω _H ), j = 0, …, ord _x2 P.     

Σ-definability of uncountable models of <Emphasis Type="Italic">c</Emphasis>-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ$ \mathbb{H} $ \mathbb{H} ($ \mathbb{L} $ \mathbb{L} ), where $ \mathbb{L} $ \mathbb{L} is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.     

The Bergman kernel function

In this note, we point out that a large family of n×n matrix valued kernel functions defined on the unit disc $ \mathbb{D} \subseteq \mathbb{C} $ \mathbb{D} \subseteq \mathbb{C} , which were constructed recently in [9], behave like the familiar Bergman kernel function on $ \mathbb{D} $ \mathbb{D} in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on $ \mathbb{D} $ \mathbb{D} can be answered using this likeness.     

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Explicit sum-product theorems for large subsets of $$ \mathbb{F} $$p

In this note, we use ‘classical’ methods to obtain sum-product theorems for subsets A⊂$ \mathbb{F} $ \mathbb{F} _p .     

Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces

In this paper,we obtain the boundedness of the parabolic singular integral operator T with kernel in L(logL)1/γ(Sn-1) on Triebel-Lizorkin spaces.Moreover,we prove the boundedness of a class of Marcinkiewicz integrals μΩ,q(f) from ∥f∥ F˙p0,q(Rn) into Lp(Rn).     

Constant angle surfaces in the Heisenberg group

In this article we extend the notion of constant angle surfaces in $ \mathbb{S}^2 $ \mathbb{S}^2 × ℝ and ℍ² × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give a complete local classification in the Heisenberg group.     

On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.