On the Riccati equations

In this article we give, in terms of so-called Berezin symbols, some necessary conditions for the solvability of the Riccati equation XAX+XB-CX-D=0XAX+XB-CX-D=0 on the set T{\cal T} of all Toeplitz operators on the Hardy space H²(\Bbb D)H^2({\Bbb D}) .     

Quantization of Pseudo-differential Operators on the Torus

Pseudo-differential and Fourier series operators on the torus \mathbbTⁿ=(\BbbR/2p\BbbZ)ⁿ{{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n} are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L ² under certain conditions on their phases and amplitudes.     

The Boundedness of Monge-Ampère Singular Integral Operators

We first define molecules for Hardy spaces H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}) associated with a family F\mathcal{F} of sections which is closely related to the Monge-Ampère equation and prove their molecular characters. As an application, we show that Monge-Ampère singular operators are bounded on H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}).     

An eigenform of weight one expressed by meromorphic η-quotients

In this paper, we construct an eigenform of weight one by using meromorphic η-quotients. It is the modular form associated to an irreducible representation r: Gal(L/\mathbb Q)? GL₂(\mathbb C)\rho : {\rm Gal}(L/\mathbb {Q})\rightarrow {\rm GL}_{2}(\mathbb {C}) with L=\mathbb Q(?{-3},³?{3})L=\mathbb {Q}(\sqrt{-3},\sqrt[3]{3}).     

Isomorphic classification of the spaces $H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R)$

We give a representation of the spaces H^k_loc(\Bbb R)?L²(\Bbb R)H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R) as Köthe sequence spaces and use some recent topological invariants to investigate their isomorphic classification.     

Two-dimensional complex tori with multiplication by $\sqrt {d}$

We give an elementary argument for the well known fact that the endomorphism algebra End(A)?\Bbb Q {\rm {End}}(A)\otimes {\Bbb Q } of a simple complex abelian surface A can neither be an imaginary quadratic field nor a definite quaternion algebra. Another consequence of our argument is that a two-dimensional complex torus T with \Bbb Q (?d)\hookrightarrow End_{\Bbb Q} (T){\Bbb Q }(\sqrt {d})\hookrightarrow {\rm{End_{{\Bbb Q }}}}(T) where \Bbb Q (?d){\Bbb Q }(\sqrt {d}) is real quadratic, is algebraic.     

Composition Operators and Endomorphisms

If b is an inner function, then composition with b induces an endomorphism, β, of L^￥(\mathbbT){L^\infty({\mathbb{T}})} that leaves H^￥(\mathbbT){H^\infty({\mathbb{T}})} invariant. We investigate the structure of the endomorphisms of B(L²(\mathbbT)){B(L^2({\mathbb{T}}))} and B(H²(\mathbbT)){B(H^2({\mathbb{T}}))} that implement β through the representations of L^￥(\mathbbT){L^\infty({\mathbb{T}})} and H^￥(\mathbbT){H^\infty({\mathbb{T}})} in terms of multiplication operators on L²(\mathbbT){L^2({\mathbb{T}})} and H²(\mathbbT){H^2({\mathbb{T}})} . Our analysis, which is based on work of Rochberg and McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert C*-modules.     

Fourier Series with the Continuous Primitive Integral

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||_\mathbbT=sup_{|I| ￡ 2p}|ò_I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ¹ Fourier series continue to hold for this larger space, with the L ¹ norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? A_c(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||_￥ ￡ ||f||_\mathbbT ||g||_BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L¹(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||_\mathbbT ￡ ||f||_{\mathbb T} ||g||₁\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _n be the Dirichlet kernel and let f ? L¹(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||D_n*f-f||_\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.     

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(?), using “local smoothing” estimates. L ²(?) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in $L^{2}(\mathbb{T})In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L²(\mathbbT)L^{2}(\mathbb{T}). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L ²(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in L²(\mathbbT)L^{2}(\mathbb{T}).     

A Seidel-Walsh theorem with linear differential operators

Assume that {S_n}₁^￥ \{S_n\}_1^\infty is a sequence of automorphisms of the open unit disk \Bbb D{\Bbb D} and that {T_n}₁^￥\{T_n\}_1^\infty is a sequence of linear differential operators with constant coefficients, both of them satisfying suitable conditions. We prove that for certain spaces X of holomorphic functions in the open unit disk, the set of functions f ? Xf \in X such that {(T_n f) °S_n:  n ? \Bbb N}\{(T_n\,f) \circ S_n: \, n \in {\Bbb N}\} is dense in H(\Bbb D)H({\Bbb D}) is residual in X. This extends the Seidel-Walsh theorem together with some subsequent results.     

Anisotropic Local Hardy Spaces

In this paper we introduce and study the anisotropic local Hardy spaces h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}) 0<p≤1, associated with the expansive matrix A. We obtain an atomic characterization of the distributions in h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}). Also we describe the dual spaces of our local Hardy anisotropic spaces as anisotropic Campanato type spaces.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

Connectedness of Spectra of Toeplitz Operators on Hardy Spaces with Muckenhoupt Weights Over Carleson Curves

Harold Widom proved in 1966 that the spectrum of a Toeplitz operator T(a) acting on the Hardy space H^p(\mathbbT)H^p({\mathbb{T}}) over the unit circle \mathbbT{\mathbb{T}} is a connected subset of the complex plane for every bounded measurable symbol a and 1 < p < ∞. In 1972, Ronald Douglas established the connectedness of the essential spectrum of T(a) on H²(\mathbbT)H^2({\mathbb{T}}). We show that, as was suspected, these results remain valid in the setting of Hardy spaces H^p(Γ,w), 1 < p < ∞, with general Muckenhoupt weights w over arbitrary Carleson curves Γ.     

On the Second Cohomology of K?hler Groups

Carlson and Toledo conjectured that if an infinite group Γ is the fundamental group of a compact K?hler manifold, then virtually H²(G, \mathbb R) 1 0{H^{2}(\Gamma, {\mathbb R}) \ne 0} . We assume that Γ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ( \mathbbC{\mathbb{C}} -VHS) on the K?hler manifold. We prove the conjecture under some assumption on the \mathbbC{\mathbb{C}} -VHS. We also study some related geometric/topological properties of period domains associated to such a \mathbbC{\mathbb{C}} -VHS.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

On the Translates of Powers of a Continuous Periodic Function

We characterize the set of real-valued, 2π -periodic, continuous functions f for which the translation invariant subspace V(f) generated by f ⁿ , n≥0, is dense in C(\mathbbT)C(\mathbb{T}). In particular, it follows that if f takes a given value at only one point then V(f) is dense in C(\mathbbT)C(\mathbb{T}).