Besov-Morrey spaces: Function space theory and applications to non-linear PDE
Anna
L.
Mazzucato
1297-1364
Abstract: This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.
Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II
Martin
Dindos
1365-1399
Abstract: Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem $Lu-F(x,u)=0$ in $\Omega$. We improve our previous results by studying more general nonlinear terms $F(x,u)$ with polynomial (and in some cases exponential) growth in the variable $u$. We also study the case of nonnegative solutions.
On one-dimensional self-similar tilings and $pq$-tiles
Ka-Sing
Lau;
Hui
Rao
1401-1414
Abstract: Let $b \geq 2$ be an integer base, $\mathcal{D} = \{ 0, d_1, \cdots , d_{b-1}\} \subset \mathbb{Z}$ a digit set and $T = T(b, \mathcal{D})$the set of radix expansions. It is well known that if $T$ has nonvoid interior, then $T$ can tile $\mathbb{R}$ with some translation set $\mathcal{J}$ ($T$ is called a tile and $\mathcal{D}$ a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of $\mathcal{J}$; (ii) for a given $b$, characterize $\mathcal{D}$ so that $T$ is a tile. We show that for a given pair $(b,\mathcal{D})$, there is a unique self-replicating translation set $\mathcal{J} \subset \mathbb{Z}$, and it has period $b^m$ for some $m \in \mathbb{N}$. This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for $b = pq$ when $p,q$ are distinct primes. The only other known characterization is for $b = p^l$, due to Lagarias and Wang. The proof for the $pq$ case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.
On the capacity of sets of divergence associated with the spherical partial integral operator
Emmanuel
Montini
1415-1441
Abstract: In this article, we study the pointwise convergence of the spherical partial integral operator $S_Rf(x)=\int_{B(0,R)} \hat{f} (y) e^{2\pi ix\cdot y}dy$ when it is applied to functions with a certain amount of smoothness. In particular, for $f\in \mathcal{L}_{\alpha}^p(\mathbb{R} ^n)$, $\tfrac{n-1}{2} <\alpha\leq\tfrac{n}{p}$, $2\leq p<\tfrac{2n}{n-1}$, we prove that $S_Rf(x)\to G_{\alpha} *g(x)$ $C_{\alpha,p}$-quasieverywhere on $\mathbb{R} ^n$, where $g\in L^p({\mathbb{R} }^n )$ is such that $f=G_{\alpha}*g$ almost everywhere. A weaker version of this result in the range $0<\alpha\leq\tfrac{n-1}{2}$ as well as some related localisation principles are also obtained. For $1\leq p<2-\tfrac{1}{n}$ and $0\leq\alpha <\tfrac{(2-p)n-1}{2p}$, we construct a function $f\in\mathcal{L}_\alpha^p(\mathbb{R} ^n)$ such that $S_Rf(x)$ diverges everywhere.
Square-integrability modulo a subgroup
G.
Cassinelli;
E.
De Vito
1443-1465
Abstract: We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of square-integrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.
Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras
E.
Kaniuth;
A.
T.
Lau;
G.
Schlichting
1467-1490
Abstract: Let $G$ be a locally compact group and let $A(G)$ and $B(G)$ be the Fourier algebra and the Fourier-Stieltjes algebra of $G$, respectively. For any unitary representation $\pi$ of $G$, let $B_\pi(G)$ denote the $w^\ast$-closed linear subspace of $B(G)$ generated by all coefficient functions of $\pi$, and $B_\pi^0(G)$ the closure of $B_\pi(G) \cap A_c(G)$, where $A_c(G)$ consists of all functions in $A(G)$ with compact support. In this paper we present descriptions of $B_\pi^0(G)$ and its orthogonal complement $B_\pi^s(G)$ in $B_\pi(G)$, generalizing a recent result of T. Miao. We show that for some classes of locally compact groups $G$, there is a dichotomy in the sense that for arbitrary $\pi$, either $B_\pi^0(G) = \{0\}$ or $B_\pi^0(G) = A(G)$. We also characterize functions in ${\mathcal B}_\pi^0(G) = A_c(G) + B_\pi^0(G)$and study the question of whether ${\mathcal B}_\pi^0(G) = A(G)$ implies that $\pi$ weakly contains the regular representation.
Some two-step and three-step nilpotent Lie groups with small automorphism groups
S.
G.
Dani
1491-1503
Abstract: We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are ``small'' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms.
Quadratic iterations to ${\pi}$ associated with elliptic functions to the cubic and septic base
Heng
Huat
Chan;
Kok
Seng
Chua;
Patrick
Solé
1505-1520
Abstract: In this paper, properties of the functions $A_d(q)$, $B_d(q)$ and $C_d(q)$ are derived. Specializing at $d=1$ and $2$, we construct two new quadratic iterations to $\pi$. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series $_2F_1(1/3,1/6;1;z)$are also derived.
Higher Weierstrass points on $X_{0}(p)$
Scott
Ahlgren;
Matthew
Papanikolas
1521-1535
Abstract: We study the arithmetic properties of higher Weierstrass points on modular curves $X_{0}(p)$ for primes $p$. In particular, for $r\in \{2, 3, 4, 5\}$, we obtain a relationship between the reductions modulo $p$ of the collection of $r$-Weierstrass points on $X_{0}(p)$ and the supersingular locus in characteristic $p$.
On extendability of group actions on compact Riemann surfaces
Emilio
Bujalance;
F.
J.
Cirre;
Marston
Conder
1537-1557
Abstract: The question of whether a given group $G$ which acts faithfully on a compact Riemann surface $X$ of genus $g\ge 2$ is the full group of automorphisms of $X$ (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group $G$ in terms of a concrete partial presentation for $G$associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where $G$ is a non-cyclic abelian group.
Local geometry of singular real analytic surfaces
Daniel
Grieser
1559-1577
Abstract: Let $V\subset\mathbb{R} ^N$ be a compact real analytic surface with isolated singularities, and assume its smooth part $V_0$ is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $\mathbb{R} ^N$. We prove: 1. Each point of $V$ has a neighborhood which is quasi-isometric (naturally and ``almost isometrically'') to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expansion, for any $p\in V$, of the length of $V\cap\{q:{\rm dist\,}(q,p)=r\}$ as $r\to0$. 3. A Gauss-Bonnet Theorem, saying that each singular point contributes $1-l/(2\pi)$, where $l$ is the coefficient of the linear term in the expansion of (2). 4. The $L^2$ Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on $V_0$, an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic. As a central tool we use resolution of singularities.
Approximation of plurisubharmonic functions by multipole Green functions
Evgeny
A.
Poletsky
1579-1591
Abstract: For a strongly hyperconvex domain $D\subset{{\mathbb{C}}}^n$ we prove that multipole pluricomplex Green functions are dense in the cone in $L^1(D)$ of negative plurisubharmonic functions with zero boundary values.
Monomial bases for $q$-Schur algebras
Jie
Du;
Brian
Parshall
1593-1620
Abstract: Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of $\mathfrak{gl}_n$ and its associated monomial basis, we investigate $q$-Schur algebras $\mathbf{S}_q(n,r)$ as ``little quantum groups". We give a presentation for $\mathbf{S}_q(n,r)$ and obtain a new basis for the integral $q$-Schur algebra $S_q(n,r)$, which consists of certain monomials in the original generators. Finally, when $n\geqslant r$, we interpret the Hecke algebra part of the monomial basis for $S_q(n,r)$ in terms of Kazhdan-Lusztig basis elements.
Logmodularity and isometries of operator algebras
David
P.
Blecher;
Louis
E.
Labuschagne
1621-1646
Abstract: We generalize some facts about function algebras to operator algebras, using the ``noncommutative Shilov boundary'' or ``$C^*$-envelope'' first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided.
The $D$--module structure of $R[F]$--modules
Manuel
Blickle
1647-1668
Abstract: Let $R$ be a regular ring, essentially of finite type over a perfect field $k$. An $R$-module $\mathcal{M}$ is called a unit $R[F]$-module if it comes equipped with an isomorphism $F^{e*} \mathcal{M} \xrightarrow{ \ }\mathcal{M}$, where $F$ denotes the Frobenius map on $\operatorname{Spec}R$, and $F^{e*}$ is the associated pullback functor. It is well known that $\mathcal{M}$ then carries a natural $D_R$-module structure. In this paper we investigate the relation between the unit $R[F]$-structure and the induced $D_R$-structure on $\mathcal{M}$. In particular, it is shown that if $k$ is algebraically closed and $\mathcal{M}$ is a simple finitely generated unit $R[F]$-module, then it is also simple as a $D_R$-module. An example showing the necessity of $k$ being algebraically closed is also given.
Seiberg-Witten invariants, orbifolds, and circle actions
Scott
Jeremy
Baldridge
1669-1697
Abstract: The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point-free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include the fact that $b_+ {>} 1$ $4$-manifolds with fixed-point-free circle actions are simple type and a new proof of the equality $\mathcal{SW}_{Y^3\times S^1} = \mathcal{SW}_{Y^3}$. An infinite number of $4$-manifolds with $b_+=1$ whose Seiberg-Witten invariants are still diffeomorphism invariants is constructed and studied.
Couples contacto-symplectiques
Gianluca
Bande
1699-1711
Abstract: We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold $M$ is a pair $\left( \alpha ,\eta \right)$ where $\alpha$ is a Pfaffian form of constant class $2k+1$ and $\eta$ a $2$-form of constant class$2h$ such that $\alpha \wedge d\alpha ^{k}\wedge \eta ^{h}$ is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb's problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.
Projectively flat Finsler metrics of constant flag curvature
Zhongmin
Shen
1713-1728
Abstract: Finsler metrics on an open subset in ${R}^n$ with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.