A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product \({\phi}\) on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of \({M_\phi}\) are orthogonal. When the order n of \({\phi}\) is 2 or 3, we show that \({M_\phi}\) is reducible on D if and only if \({\phi}\) is equivalent to \({z^n}\). When the order of \({\phi}\) is 4, we determine the reducing subspaces for \({M_\phi}\), and we see that in this case \({M_\phi}\) can be reducible on D when \({\phi}\) is not equivalent to \({z^4}\). The same phenomenon happens when the order n of \({\phi}\) is not a prime number. Furthermore, we show that \({M_\phi}\) is unitarily equivalent to \({M_{z^n} (n > 1)}\) on D if and only if \({\phi = az^n}\) for some unimodular constant a.     

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).     

{\mathbb{Z}_2}-bordism and the Borsuk–Ulam Theorem

The purpose of this work is to classify, for given integers \({m,\, n\geq 1}\), the bordism class of a closed smooth \({m}\)-manifold \({X^m}\) with a free smooth involution \({\tau}\) with respect to the validity of the Borsuk–Ulam property that for every continuous map \({\phi : X^m \to \mathbb{R}^n}\) there exists a point \({x\in X^m}\) such that \({\phi (x)=\phi (\tau (x))}\). We will classify a given free \({\mathbb{Z}_2}\)-bordism class \({\alpha}\) according to the three possible cases that (a) all representatives \({(X^m, \tau)}\) of \({\alpha}\) satisfy the Borsuk–Ulam property; (b) there are representatives \({({X_{1}^{m}}, \tau_1)}\) and \({({X_{2}^{m}}, \tau_2)}\) of \({\alpha}\) such that \({({X_{1}^{m}}, \tau_1)}\) satisfies the Borsuk–Ulam property but \({({X_{2}^{m}}, \tau_2)}\) does not; (c) no representative \({(X^m, \tau)}\) of \({\alpha}\) satisfies the Borsuk–Ulam property.     

Symmetry for a general class of overdetermined elliptic problems

Let \({\Omega}\) a bounded domain in \({\mathbb{R} ^N }\), and let \({u\in C^1 (\overline{\Omega})}\) a weak solution of the following overdetermined BVP: \({-\nabla (g(|\nabla u|)|\nabla u|^{-1}\nabla u)=f(|x|,u)}\), \({ u > 0 }\) in \({\Omega }\) and \({u=0, \ |\nabla u(x)|=\lambda (|x|)}\) on \({\partial \Omega }\), where \({g\in C([0,+\infty)\cap C^1 ((0,+\infty ) ) }\) with \({g(0)=0}\), \({g'(t) > 0}\) for \({t > 0}\), \({f\in C([0,+\infty ) \times [0, +\infty ) )}\), f is nonincreasing in \({|x|}\), \({\lambda \in C([0, +\infty )) }\) and \({\lambda }\) is positive and nondecreasing. We show that \({\Omega }\) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.     

The exact asymptotic behavior of boundary blow-up solutions to infinity Laplacian equations

In this paper, we study the asymptotic behavior of viscosity solutions to boundary blow-up elliptic problem \({\Delta_{\infty}u=b(x)f(u),\, x\in\Omega,\,u|_{\partial\Omega}=+\infty,}\) where \({\Omega}\) is a bounded domain with C²-boundary in \({\mathbb{R}^{N}}\), \({b\in \rm C(\bar{\Omega})}\) is positive in \({\Omega}\), which may be vanishing on the boundary, \({f\in C^{1}([0, \infty))}\) is regularly varying or is rapidly varying at infinity.     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

An inverse spectral theorem for Kreĭn strings with a negative eigenvalue

A string is a pair \({(L, \mathfrak{m})}\) where \({L \in[0, \infty]}\) and \({\mathfrak{m}}\) is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \({\mathfrak{m}}\) as its mass density. To each string a differential operator acting in the space \({L^2(\mathfrak{m})}\) is associated. Namely, the Kre?n–Feller differential operator \({-D_{\mathfrak{m}}D_x}\) ; its eigenvalue equation can be written, e.g., as

$$f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.$$

A positive Borel measure τ on \({\mathbb R}\) is called a (canonical) spectral measure of the string \({\textsc S[L, \mathfrak{m}]}\) , if there exists an appropriately normalized Fourier transform of \({L^2(\mathfrak{m})}\) onto L ²(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) \({\int_{\mathbb R} \frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) . (2) Either \({{\rm supp} \tau \subseteq [0, \infty)}\) , or τ is discrete and has exactly one point mass in (?∞, 0). It is a deep result, going back to Kre?n in the 1950’s, that each measure with \({\int_{\mathbb R}\frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) and \({{\rm supp} \tau \subseteq [0, \infty)}\) is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with \({{\rm supp} \tau \not\subseteq [0, \infty)}\) is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.

     

Inequalities between Dirichlet and Neumann eigenvalues of vibrating strings

The Dirichlet eigenvalues \({\{\lambda_{n}\}_{n=1}^{\infty}}\) and Neumann eigenvalues \({\{\mu_{n}\}_{n=1}^{\infty}}\) of the string equation \({\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}\) are considered. It is known that \({ \mu_{n} < \lambda_{n} < \mu_{n+2}}\) for all n. The purpose of this paper is to provide conditions on the mass density \({\rho(x)}\) under which \({\lambda_{n} < \mu_{n+1}}\) or \({\mu_{n+1} < \lambda_{n}.}\)     

On countably {\mu}-paracompact spaces

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F_i} of non-empty closed sets with \({\bigcap_{i=1}^{\infty} F_{i} = \emptyset}\) there exists a sequence {G_i} of open sets such that \({\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}\) and \({F_{i} \subset G_{i}}\) for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a \({\mu}\)-normal generalized topological space satisfying the analogue of A which is not even countably \({\mu}\)-metacompact. Then we study the relationships between countably \({\mu}\)-paracompactness, countably \({\mu}\)-metacompactness and the condition corresponding to condition A in generalized topological spaces.     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.     

On certain functional equations related to Jordan triple $${(\theta, \phi)}$$-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let \({\theta}\) and \({\phi}\) be automorphisms of R. Suppose \({T:R\rightarrow R}\) is an additive mapping satisfying the relation \({T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}\), for all pairs \({x,y\in R}\). In this case T is of the form \({2T(x)=q\theta (x)+\phi (x)q}\), for all \({x\in R}\) and some fixed element \({q\in Q_{s}}\).