Sufficient Criteria for \varepsilon Starlike Mappings in a Complex Banach Space

Several sufficient conditions for $\varepsilon $ starlike mappings on the unit ball $B$ in a complex Banach space are provided. From these, we may construct many concrete $\varepsilon $ starlike mappings on $B$ . Furthermore, several growth results associated with these sufficient conditions are also provided.     

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

Sharp regularity of linearization for C^{1,1} hyperbolic diffeomorphisms

$C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization. For $C^{1,1}$ hyperbolic diffeomorphisms on Banach spaces $C^1$ linearization was proved under a gap condition together with a band condition of the spectrum. In this paper, the result of $C^1$ linearization in Banach spaces is strengthened to $C^{1,\beta }$ linearization with a constant $\beta >0$ under a weaker band condition by a decomposition with invariant foliations. The weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum. Furthermore, we give an estimate for the exponent $\beta $ and prove that the estimate is sharp in the planar case.     

Universal geometric cluster algebras

We consider, for each exchange matrix $B$ , a category of geometric cluster algebras over $B$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $R$ , usually $\mathbb {Z},\,\mathbb {Q}$ , or $\mathbb {R}$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $B$ with universal geometric coefficients, or the universal geometric cluster algebra over $B$ . Constructing universal geometric coefficients is equivalent to finding an $R$ -basis for $B$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan ${\mathcal {F}}_B$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between ${\mathcal {F}}_B$ and $\mathbf{g}$ -vectors. We construct universal geometric coefficients in rank $2$ and in finite type and discuss the construction in affine type.     

Rational matrix pseudodifferential operators

The skewfield $\mathcal{K }(\partial )$ of rational pseudodifferential operators over a differential field $\mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $\mathcal{K }[\partial ]$ . In our previous paper, we showed that any $H\in \mathcal{K }(\partial )$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in \mathcal{K }[\partial ],\,B\ne 0$ , and any common right divisor of $A$ and $B$ is a non-zero element of $\mathcal{K }$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $\mathcal{K }[\partial ]$ . In the present paper, we study the ring $M_n(\mathcal{K }(\partial ))$ of $n\times n$ matrices over the skewfield $\mathcal{K }(\partial )$ . We show that similarly, any $H\in M_n(\mathcal{K }(\partial ))$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in M_n(\mathcal{K }[\partial ]),\,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(\mathcal{K }[\partial ])$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(\mathcal{K } [\partial ])$ . We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .     

True Poly-Bergman and Poly-Bergman Kernels for the Complement of a Closed Disk

Let $k$ and $j$ be positive integers. We prove that the action of the two-dimensional singular integral operators $(S_\Omega )^{j-1}$ and $(S_\Omega ^*)^{j-1}$ on a Hilbert base for the Bergman space $\mathcal{A }^2(\Omega )$ and anti-Bergman space $\mathcal{A }^2_{-1}(\Omega ),$ respectively, gives Hilbert bases $\{ \psi _{\pm j , k } \}_{ k }$ for the true poly-Bergman spaces $\mathcal{A }_{(\pm j)}^2(\Omega ),$ where $S_\Omega $ denotes the compression of the Beurling transform to the Lebesgue space $L^2(\Omega , dA).$ The functions $\psi _{\pm j,k}$ will be explicitly represented in terms of the $(2,1)$ -hypergeometric polynomials as well as by formulas of Rodrigues type. We prove explicit representations for the true poly-Bergman kernels and more transparent representations for the poly-Bergman kernels of $\Omega $ . We establish Rodrigues type formulas for the poly-Bergman kernels of $\mathbb{D }$ .     

Asymptotically Spirallike Mappings in Reflexive Complex Banach Spaces

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces $X$ , and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in $\mathbb{C }^{n}$ which have parametric representation in the sense of Loewner theory. The authors introduced the notions of $A$ -asymptotic spirallikeness and $A$ -parametric representation on the Euclidean unit ball of $\mathbb{C }^{n}$ , where $A\in L(\mathbb{C }^{n})$ with $m(A)>0$ . They showed that these notions are equivalent whenever $k_+(A)<2m(A)$ . In this paper we prove that if $k_+(A)<2m(A)$ and $f\in S(B)$ has $A$ -parametric representation, then $f$ is also $A$ -asymptotically spirallike on the unit ball $B$ of $X$ . For the converse, we need the additional assumption that $f$ is a smooth $A$ -asymptotically spirallike mapping, except in the finite-dimensional case $X=\mathbb{C }^{n}$ with an arbitrary norm. The notion of asymptotic spirallikeness involves differential equations and may be regarded as giving a geometric characterization of certain domains in $X$ . That is one of the motivations for considering this notion in the case of reflexive complex Banach spaces.     

\mathfrak D -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C }^{m+2})$ with $\mathfrak D $ -parallel normal Jacobi operator ${\bar{R}}_N$ and $\mathfrak D $ -parallel structure Jacobi operator $R_{\xi }$ if the distribution $\mathfrak D $ or $\mathfrak D ^{\bot }$ component of the Reeb vector field is invariant by the shape operator, respectively.     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

A non-type (D) operator in c_0

Previous examples of non-type (D) maximal monotone operators were restricted to $\ell ^1$ , $L^1$ , and Banach spaces containing isometric copies of these spaces. This fact led to the conjecture that non-type (D) operators were restricted to this class of Banach spaces. We present a linear non-type (D) operator in $c_0$ .     

Weakly B-orthodox semigroups

We introduce the notions of a band category and of a weakly orthodox category over a band. Our focus is to describe a class of weakly $B$ -orthodox semigroups, where $B$ denotes a band of idempotents. In particular, we investigate orthodox semigroups, by using orthodox groupoids. Weakly $B$ -orthodox semigroups are analogues of orthodox semigroups, where the relations $\widetilde{\mathcal {R}}_B$ and $\widetilde{\mathcal {L}}_B$ play the role that ${\mathcal {R}}$ and $\mathcal {L}$ take in the regular case. We show that the category of weakly $B$ -orthodox semigroups and admissible morphisms is equivalent to the category of weakly orthodox categories over bands and orthodox functors. The same class of weakly $B$ -orthodox semigroups was studied in an earlier article by Gould and the author using generalised categories. Our approach here is more akin to that of Nambooripad. The significant difference in strategy is that it is more convenient to consider categories equipped with pre-orders, rather than with partial orders.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(\mathbb{R},w)}$ , where ${p\in(1,\infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R}\cup\{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.     

Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions

In this paper we consider the space ${{{BMO}_o(\mathbb{R}, X)}}$ of bounded mean oscillations and odd functions on ${{\mathbb{R}}}$ taking values in a UMD Banach space X. The functions in ${{{BMO}_o(\mathbb{R}, X)}}$ are characterized by Carleson type conditions involving Bessel convolutions and γ-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain γ-radonifying Carleson inequalities for Bessel–Poisson integrals of ${{{BMO}_o(\mathbb{R}, X)}}$ functions hold.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^{p}(\mathbb{R}, w)}$ , where ${p \in (1, \infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{A}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R} \cup \{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.     

Decomposable and indecomposable algebras of degree 8 and exponent 2

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let $B$ be a biquaternion algebra over $F(\sqrt{a})$ with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether $B$ has a descent to $F$ . This invariant is used to give examples of indecomposable algebras of degree $8$ and exponent 2 over a field of 2-cohomological dimension 3 and over a field $\mathbb M(t)$ where the $u$ -invariant of $\mathbb M$ is $8$ and $t$ is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.     

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters \alpha =\beta =-1

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $\alpha =\beta =-1$ in contrast to the classical case when $\alpha ,\beta >-1$ . For fixed $k\ge 0$ and appropriate values of the spectral parameter $\lambda ,$ the equation $\ell [y]=\lambda y$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$ These Jacobi polynomial solutions of degree $\ge 2$ form a complete orthogonal set in the Hilbert space $L^{2}((-1,1);(1-x^{2})^{-1})$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $\ge 0$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $T$ , generated by $\ell [\cdot ],$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $T$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.     

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N _X and N _Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L ^r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.