# Indiana University Mathematics Journal

Published by Indiana University Mathematics Journal

Print ISSN: 0022-2518

Published by Indiana University Mathematics Journal

Print ISSN: 0022-2518

Publications

This paper characterizes, in terms of thinness, compact sets K in a suitable harmonic space Ω \Omega which have the following property: functions which are harmonic (resp. continuous and superharmonic) on a neighbourhood of K can be uniformly approximated on K by functions which are harmonic (resp. continuous and superharmonic) on Ω \Omega . The corresponding problems of approximating functions which are continuous on K and harmonic (resp. superharmonic) on the interior K ˚ \mathring {K} are also solved.

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The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [Ka]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized $h$-vector. In this paper we strengthen Karu's theorem by introducing a canonical bilinear form $(\cdot ,\cdot)_{\Phi}$ on the intersection cohomology $IH(\Phi)$ of a complete fan $\Phi$ and proving the Hodge-Riemann bilinear relations for $(\cdot ,\cdot)_{\Phi}$.

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This is an addendum to the article of Qi Zhong cited above [arXiv:0705.2000].
It outlines how to apply the main result of that article to calculate the
asymptotics of the expected energy of zeros of random polynomials on the
Riemann sphere $S^2$ with respect to the log chordal distance $\log [z, w]$.
The cited article did not calculate the asymptotic energy this way, but by an
ad-hoc method, and the calculation contained some errors. The correct
calculation here agrees (up to the stipulted remainder) with that of Armentano-
Beltran-Shub.

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One dimensional Dirac operators $$ L_{bc}(v) \, y = i \begin{pmatrix} 1 & 0 0 & -1 \end{pmatrix} \frac{dy}{dx} + v(x) y, \quad y = \begin{pmatrix} y_1 y_2 \end{pmatrix}, \quad x\in[0,\pi],$$ considered with $L^2$-potentials $ v(x) = \begin{pmatrix} 0 & P(x) Q(x) & 0 \end{pmatrix} $ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ it is shown that every eigenvalue of the free operator $L^0_{bc}$ is simple and has the form $\lambda_{k,\alpha}^0 = k + \tau_\alpha $ where $ \; \alpha \in \{1,2\}, \; k \in 2 \mathbb{Z} $ and $\tau_\alpha =\tau_\alpha (bc);$ if $|k|>N(v, bc) $ each of the discs $D_k^\alpha = \{z: \; |z-\lambda_{k,\alpha}^0| <\rho =\rho (bc) \} , $ $\alpha \in \{1,2\}, $ contains exactly one simple eigenvalue $\lambda_{k,\alpha} $ of $L_{bc} (v) $ and $(\lambda_{k,\alpha} -\lambda_{k,\alpha}^0)_{k\in 2\mathbb{Z}} $ is an $\ell^2 $-sequence. Moreover, it is proven that the root projections $ P_{n,\alpha} = \frac{1}{2\pi i} \int_{\partial D^\alpha_n} (z-L_{bc} (v))^{-1} dz $ satisfy the Bari--Markus condition $$\sum_{|n| > N} \|P_{n,\alpha} - P_{n,\alpha}^0\|^2 < \infty, \quad n \in 2\mathbb{Z}, $$ where $P_n^0 $ are the root projections of the free operator $L^0_{bc}.$ Hence, for strictly regular $bc,$ there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular $bc$ -- then in general there is no Riesz basis consisting of root functions but we prove that the corresponding system of two-dimensional root projections is a Riesz basis of projections.

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We show that any dimension 6 nearly K\"ahler (or nearly para-K\"ahler)
geometry arises as a projective manifold equipped with a $\mathrm{G}_2^{(*)}$
holonomy reduction. In the converse direction we show that if a projective
manifold is equipped with a parallel 7-dimensional cross product on its
standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler
manifold, if the cross product is definite; otherwise, if the cross product has
the other algebraic type, the manifold is in general stratified with nearly
K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which
canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a
projective infinity for the pseudo-Riemannian geometry elsewhere on the
manifold, and we establish how the Cartan distribution can be understood
explicitly, and also in terms of conformal geometry, as a limit of the ambient
nearly (para-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is
seen to arise as such a limit, because we can solve the geometric Dirichlet
problem of building a collar structure equipped with the required
holonomy-reduced projective structure.
Our approach is to use Cartan/tractor theory to understand all structures as
arising from a curved version of the algebra of imaginary (split) octonions as
a flat structure over its projectivization. The perspective is used to
establish results concerning the projective compactification of nearly
(para-)K\"ahler manifolds.

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Let $P_\gamma$ be the orthogonal projection from the space $L ^2
(\mathbb{B}_n, dv_\gamma)$ to the standard weighted Bergman space $L_a ^2
(\mathbb{B}_n, dv_\gamma)$. In this paper, we characterize the Schatten $p$
class membership of the commutator $[M_f, P_\gamma]$ when $\frac{2n}{n + 1 +
\gamma} < p < \infty$. In particular, if $\frac{2n}{n + 1 + \gamma} < p <
\infty$, then we show that $[M_f, P_\gamma]$ is in the Schatten $p$ class if
and only if the mean oscillation MO${}_\gamma (f)$ is in $ L^p(\mathbb{B}_n,
d\zeta)$ where $d\zeta$ is the M\"{o}bius invariant measure on $\mathbb{B}_n.$
This answers a question recently raised by K. Zhu.

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This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $\theta$ through $\mathcal{R} \Lambda^{-1} P(\Lambda) \theta$ where $\mathcal{R}$ denotes a Riesz transform, $\Lambda=(-\Delta)^{1/2}$ and $P(\Lambda)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(\Lambda)=I$ while the surface quasi-geostrophic (SQG) equation to $P(\Lambda) =\Lambda$. We obtain the global regularity for a class of equations for which $P(\Lambda)$ and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with $P(\Lambda)= (\log(I-\Delta))^\gamma$ for any $\gamma>0$ are globally regular.

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We prove that entire bounded monotone solutions to a certain class of fully nonlinear equations in 2D are one-dimensional. Our result also gives a new (non-variational) proof of the well known De Giorgi's conjecture.

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We characterize the possible behaviors at infinity of weak solutions to the
2D Euler equations in the full plane having bounded velocity and bounded
vorticity. We show that any such solution can be put in the form obtained by
Ph. Serfati in 1995 after a suitable change of reference frame. Our results
build on those of a recent paper of the author's, joint with Ambrose, Lopes
Filho, and Nussenzveig Lopes.

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We obtain a complete classification of the locally finite algebras and the operator algebras, given as algebraic inductive limits and Banach algebraic inductive limits respectively, of direct systems: A_1 contained in A_2 contained in A_3 and so on. Here the A_k are 2n-cycle algebras, where n is at least 3 and the inclusions are of rigid type. The complete isomorphism invariant is essentially the triple (K_0(A), H_1(A), Sigma(A)) where K_0(A) is viewed as a scaled ordered group, H_1(A) is a partial isometry homology group and Sigma(A), contained in the direct sum of K_0(A) and H_1(A), is the 2n-cycle joint scale.

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Consider steady-state weak solutions to the incompressible Navier-Stokes
equations in six spatial dimensions. We prove that the 2D Hausdorff measure of
the set of singular points is equal to zero. This problem was mentioned in 1988
by Struwe [24], during his study of the five dimensional case.

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For any Calderon-Zygmund operator T and any BMO function b we prove the following quadratic estimate: parallel to[b, T]parallel to(LP(w)) <= c parallel to b parallel to(BMO)(PP')(2)[w](A1)(2), 1 < p < infinity, w is an element of A(1), with constant c = c (n, T) being the estimate optimal on p and the exponent of the weight constant. As an endpoint estimate we prove w({x is an element of R-n : vertical bar[b, T]f(x)vertical bar > lambda}) <= c Phi([w]A(1))(2) integral(Rn) Phi(vertical bar f(x)vertical bar/lambda)w(x) dx, where Phi(t) = t(1 + log(+) t) and constant c = c(n,T, parallel to b parallel to(BMO)).

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Let T be an arbitrary L^2 bounded Calderon--Zygmund operator, and T_# its
maximal truncated version. Then T_# satisfies the following bound for all
1<p<\infty and all weights w\in A_p: \|T_# \|_{L^p(w)} << [w]_{A_p}^{1/p}
{[w]_{A_infty}^{1/p'}+[w^{1-p'}]_{A_infty}^{1/p}}.

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In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resulting linear system is not self-adjoint, of order 5, with coefficients depending on x and on physical parameters $\alpha, \beta$. We calculate Evans function associated with this linear system, using rigorous constructions of decreasing at $\pm \infty$ solutions of systems of ODE. We prove that for $\alpha$ small, there is no bounded solution of the linearized system.

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We introduce and systematically study the notion of Rokhlin dimension for
compact group actions on $C^*$-algebras. This notion generalizes the one
introduced by Hirshberg, Winter and Zacharias for finite groups, and contains
the Rokhlin property as the zero dimensional case. We show, by means of an
example, that commuting towers cannot always be arranged, even in the absence
of $K$-theoretic obstructions. For a compact Lie group action on a compact
Hausdorff space, freeness is equivalent to finite Rokhlin dimension of the
induced action. Building on work of Hirshberg and Phillips, we compare this
notion to other existing definitions of noncommutative freeness for compact
group actions. We obtain further $K$-theoretic obstructions to having an action
of a non-finite compact Lie group with finite Rokhlin dimension with commuting
towers, and use them to confirm a conjecture of Phillips. Finally, we study the
structure of the crossed product and fixed point algebra by such actions in
relation to their nuclear dimension, decomposition rank, and absorption of the
Jiang-Su algebra.

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We prove that if a finite group $G$ has a representation with fixity $f$,
then it acts freely and homologically trivially on a finite CW-complex homotopy
equivalent to a product of $f+1$ spheres. This shows, in particular, that every
finite group acts freely and homologically trivially on some finite CW-complex
homotopy equivalent to a product of spheres.

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We show that the category A(G) of actions of a locally compact group G on C*-algebras (with equivariant nondegenerate *-homomorphisms into multiplier algebras) is equivalent, via a full-crossed-product functor, to a comma category of maximal coactions of G under the comultiplication (C*(G),delta_G); and also that A(G) is equivalent, via a reduced-crossed-product functor, to a comma category of normal coactions under the comultiplication. This extends classical Landstad duality to a category equivalence, and allows us to identify those C*-algebras which are isomorphic to crossed products by G as precisely those which form part of an object in the appropriate comma category.

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We prove that Ahlfors 2-regular quasisymmetric images of the Euclidean plane
are bi-Lipschitz images of the plane if and only if they are uniformly
bi-Lipschitz homogeneous with respect to a group. We also prove that certain
geodesic spaces are bi-Lipschitz images of Carnot groups if they are inversion
invariant bi-Lipschitz homogeneous with respect to a group.

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Given a C*-dynamical system (A,G,\alpha), we say that A is a weakly proper
(X\rtimes G)-algebra if there exists a proper G-space X together with a
nondegenerate G-equivariant *-homomorphism \phi:C_0(X)->M(A). Weakly proper
G-algebras form a large subclass of the class of proper G-algebras in the sense
of Rieffel. In this paper we show that weakly proper (X\rtimes G)-algebras
allow the construction of full fixed-point algebras A^G corresponding to the
full crossed product A\rtimes_{\alpha}G, thus solving, in this setting, a
problem stated by Rieffel in his 1988's original article on proper actions. As
an application we obtain a general Landstad duality result for arbitrary
coactions together with a new and functorial construction of maximalizations of
coactions.
The same methods also allow the construction of exotic generalized
fixed-point algebras associated to crossed-product norms lying between the
reduced and universal ones. Using these, we give complete answers to some
questions on duality theory for exotic crossed products recently raised by
Kaliszewski, Landstad and Quigg.

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We consider positive solutions of the stationary Gierer-Meinhardt system d1Δ - u + up/vq + σ = 0 in Ω, d2Δv - + ur/vs = 0 in Ω, ∂u/∂ν = ∂v/∂ν = 0 on ∂Ω where Δ is the Laplace operator, Ω is a bounded smooth domain in ℝn, n ≥ 1, and ν is the unit outer normal to ∂Ω. Under suitable conditions on the exponents p, q, r, and s, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both σ > 0 and σ = 0 cases. Indiana University Mathematics Journal

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We study curvature-adapted submanifolds of general symmetric spaces. We
generalize Cartan's theorem for isoparametric hypersurfaces of spheres and
Wang's classification of isoparametric Hopf hypersurfaces in complex projective
spaces to any compact symmetric space. Our second objective is to investigate
such hypersurfaces in some specific symmetric spaces. Various classification
results in the Cayley projective and hyperbolic planes and in complex two-plane
Grassmannians are obtained under some additional assumptions.

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Integral representations are obtained of positive additive functionals on
finite products of the space of continuous functions (or of bounded Borel
functions) on a compact Hausdorff space. These are shown to yield
characterizations of the dual mixed volume, the fundamental concept in the dual
Brunn-Minkowski theory. The characterizations are shown to be best possible in
the sense that none of the assumptions can be omitted. The results obtained are
in the spirit of a similar characterization of the mixed volume in the
classical Brunn-Minkowski theory, obtained recently by Milman and Schneider,
but the methods employed are completely different.

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With the help of the representation of SL(2,Z) on the rank two free module
over the integer adeles, we define the transition operator of a Markov chain.
The real component of its spectrum exhibits a gap, whereas the non-real
component forms a circle of radius 1/\sqrt{2}.

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We rigorously derive multi-pulse interaction laws for the semi-strong
interactions in a family of singularly-perturbed and weakly-damped
reaction-diffusion systems in one space dimension. Most significantly, we show
the existence of a manifold of quasi-steady N-pulse solutions and identify a
"normal-hyperbolicity" condition which balances the asymptotic weakness of the
linear damping against the algebraic evolution rate of the multi-pulses. Our
main result is the adiabatic stability of the manifolds subject to this normal
hyperbolicity condition. More specifically, the spectrum of the linearization
about a fixed N-pulse configuration contains essential spectrum that is
asymptotically close to the origin as well as semi-strong eigenvalues which
move at leading order as the pulse positions evolve. We characterize the
semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix,
and rigorously bound the error between the N-pulse manifold and the evolution
of the full system, in a polynomially weighted space, so long as the
semi-strong spectrum remains strictly in the left-half complex plane, and the
essential spectrum is not too close to the origin.

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This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operator. In this paper we extend results of \cite{AP2} and \cite{AP3} to the case of unbounded self-adjoint operators $A$. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of \cite{AP2}. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers \cite{AP2} and \cite{AP3} were announced in \cite{AP1}. Comment: 34 pages

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Given a self-adjoint involution J on a Hilbert space H, we consider a
J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint
operator commuting with J and V a bounded J-self-adjoint operator
anti-commuting with J. We establish optimal estimates on the position of the
spectrum of L with respect to the spectrum of A and we obtain norm bounds on
the operator angles between maximal uniformly definite reducing subspaces of
the unperturbed operator A and the perturbed operator L. All the bounds are
given in terms of the norm of V and the distances between pairs of disjoint
spectral sets associated with the operator L and/or the operator A. As an
example, the quantum harmonic oscillator under a PT-symmetric perturbation is
discussed. The sharp norm bounds obtained for the operator angles generalize
the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint
perturbations.

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In the present paper, we derive an adjunction formula for the
Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a
complex manifold M. This adjunction formula is used to study the problem of
extending L2-cohomology classes of dbar-closed forms from the singular
hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel
extension theorem. We do that by showing that our formulation of the
L2-extension problem is invariant under bimeromorphic modifications, so that we
can reduce the problem to the smooth case by use of an embedded resolution of V
in M. The smooth case has recently been studied by Berndtsson.

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We study families of possibly overlapping self-affine sets. Our main example is a family that can be considered the self-affine version of Bernoulli convolutions and was studied, in the non-overlapping case, by F.Przytycki and M.Urbanski. We extend their results to the overlapping region and also consider some extensions and generalizations.

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We investigate the dimension theory of inhomogeneous self-affine carpets.
Through the work of Olsen, Snigireva and Fraser, the dimension theory of
inhomogeneous self-similar sets is now relatively well-understood, however,
almost no progress has been made concerning more general non-conformal
inhomogeneous attractors. If a dimension is countably stable, then the results
are immediate and so we focus on the upper and lower box dimensions and compute
these explicitly for large classes of inhomogeneous self-affine carpets.
Interestingly, we find that the `expected formula' for the upper box dimension
can fail in the self-affine setting and we thus reveal new phenomena, not
occurring in the simpler self-similar case.

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We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

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An iterated function system $\Phi$ consisting of contractive similarity mappings has a unique attractor $F \subseteq \mathbb{R}^d$ which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling $\mathcal{T}$ of the convex hull of the attractor. More precisely, it tiles the complement of the attractor within its convex hull. These tiles form a collection of sets whose geometry is typically much simpler than that of $F$, yet retains key information about both $F$ and $\Phi$. In particular, the tiles encode all the scaling data of $\Phi$. We give the construction, along with some examples and applications. The tiling $\mathcal{T}$ is the foundation for the higher-dimensional extension of the theory of \emph{complex dimensions} which was developed for the case $d=1$ in ``Fractal Geometry, Complex Dimensions, and Zeros of Zeta Functions,'' by Michel L. Lapidus and Machiel van Frankenhuijsen. Comment: 16 pages, 8 figures, referee comments incorporated, new counterexample

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In this paper, we prove that, if functions (concave) $\phi$ and (convex)
$\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is
monotone increasing, while the $L_{\psi}$ affine surface area is monotone
decreasing under the Steiner symmetrization. Consequently, we can prove related
affine isoperimetric inequalities, under certain conditions on $\phi$ and
$\psi$, without assuming that the convex body involved has centroid (or the
Santal\'{o} point) at the origin.

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We prove a refined Agler decomposition for bounded analytic functions on the bidisk and show how it can be used to reprove an interesting result of Guo et al. related to extending holomorphic functions without increasing their norm. In addition, we give a new treatment of Heath and Suffridge's characterization of holomorphic retracts on the polydisk. Comment: 11 pages

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In this paper we consider a stationary Schroedinger operator in the plane, in presence of a magnetic field of Aharonov-Bohm type with semi-integer circulation. We analyze the nodal regions for a class of solutions such that the nodal set consists of regular arcs, connecting the singular points with the boundary. In case of one magnetic pole, which is free to move in the domain, the nodal lines may cluster dissecting the domain in three parts. Our main result states that the magnetic energy is critical (with respect to the magnetic pole) if and only if such a configuration occurs. Moreover the nodal regions form a minimal 3-partition of the domain (with respect to the real energy associated to the equation), the configuration is unique and depends continuously on the data. The analysis performed is related to the notion of spectral minimal partition introduced in [20]. As it concerns eigenfunctions, we similarly show that critical points of the Rayleigh quotient correspond to multiple clustering of the nodal lines. Comment: 32 pages

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We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is now a general theorem for a class of Haar shift operators that can be used instead to deduce similar results. We invoke this general theorem to obtain the corresponding result for the Riesz transforms and the Beurling-Ahlfors operator. We can then extrapolate to Lp(w), and the results are sharp for 1 < p < 1.

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The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting. Comment: 14 pages

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We confirm the AJ conjecture [Ga04] that relates the A-polynomial and the
colored Jones polynomials for those hyperbolic knots satisfying certain
conditions. In particular, we show that the conjecture holds true for some
classes of two-bridge knots and pretzel knots. This extends the result of the
first author in [Le06] where he established the AJ conjecture for a large class
of two-bridge knots, including all twist knots. Along the way, we explicitly
calculate the universal character ring of the knot group of the
(-2,3,2n+1)-pretzel knot and show that it is reduced for all integers n.

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Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth
valuations} on a smooth manifold $M$, and shown that it admits a natural
commutative multiplication. Although Alesker's original construction is highly
technical, from a moral perspective this product is simply an artifact of the
operation of intersection of two sets. Subsequently Alesker and Bernig gave an
expression for the product in terms of differential forms. We show how the
Alesker-Bernig formula arises naturally from the intersection interpretation,
and apply this insight to give a new formula for the product of a general
valuation with a valuation that is expressed in terms of intersections with a
sufficiently rich family of smooth polyhedra.

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We generalize the classical study of Alexander polynomials of smooth or PL
locally-flat knots to PL knots that are not necessarily locally-flat. We
introduce three families of generalized Alexander polynomials and study their
properties. For knots with point singularities, we obtain a classification of
these polynomials that is complete except for one special low-dimensional case.
This classification extends existing classifications for PL locally-flat knots.
For knots with higher-dimensional singularities, we further extend the
necessary conditions on the invariants. We also construct several varieties of
singular knots to demonstrate realizability of certain families of polynomials
as generalized Alexander polynomials. These constructions, of independent
interest, generalize known knot constructions such as frame spinning and twist
spinning.

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We prove duality theorems for twisted Reidemeister torsions and twisted
Alexander polynomials generalizing the results of Turaev. As a corollary we
determine the parity of the degrees of twisted Alexander polynomials of
3-manifolds in many cases.

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It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on the cone of lower semicontinuous traces of a stable C*-algebra that arise from positive elements of the algebra.

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Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C* -algebra script O signX which includes, for particular choices of X, crossed products of A by ℤ, the Cuntz algebras script O signn, and the Cuntz-Krieger algebras script O signB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for script O sign∞.

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For $\MvN$ a separable, purely infinite von Neumann algebra with almost periodic weight $\phi$, a decomposition of $\MvN$ as a crossed product of a semifinite von Neumann algebra by a trace--scaling action of a countable abelian group is given. Then Takasaki's continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki's notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one--parameter trace--scaling action of $\Real_+$ on the injective II$_\infty$ factor. Finally, another model of the same action, due to work of Aubert and explained by Jones, is described.

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We apply the One-Sided Action Theorem from the first paper in this series to
prove that Rieffel's Morita equivalence between the reduced crossed product by
a proper saturated action and the generalized fixed-point algebra is a quotient
of a Morita equivalence between the full crossed product and a "universal"
fixed-point algebra. We give several applications, to Fell bundles over groups,
reduced crossed products as fixed-point algebras, and C*-bundles.

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We compute all dynamical degrees of monomial maps by interpreting them as
mixed volumes of polytopes. By exploiting further the isomorphism between the
polytope algebra of P. McMullen and the universal cohomology of complete toric
varieties, we construct invariant positive cohomology classes when the
dynamical degrees have no resonance.

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Let $k$ be a field. We characterize the group schemes $G$ over $k$, not
necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly
generated. We also describe the algebraic stacks that have finite cohomological
dimension in terms of their stabilizer groups.

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We introduce algebraic dynamical systems, which consist of an action of a
right LCM semigroup by injective endomorphisms of a group. To each algebraic
dynamical system we associate a C*-algebra and describe it as a semigroup
C*-algebra. As part of our analysis of these C*-algebras we prove results for
right LCM semigroups. More precisely we discuss functoriality of the full
semigroup C*-algebra and compute its K-theory for a large class of semigroups.
We introduce the notion of a Nica-Toeplitz algebra of a product system over a
right LCM semigroup, and show that it provides a useful alternative to study
algebraic dynamical systems.

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We associate to any compact semi-algebraic set $X \subset \mathbb{R}^n$ a
chain complex of currents $S_\ast (X)$ generated by integration along
semi-algebraic submanifolds, and we analyze the corresponding homology groups.
In particular, we show that these homology groups satisfy the
Eilenberg-Steenrod axioms and further, that they are isomorphic to both the
ordinary singular homology groups of $X$ and to the homology groups generated
by the integral currents supported on $X$. Using this result and a certain
neighborhood of $X$, we are able to prove homological mass minimization for
integral currents supported on $X$, and verify that cycles of $X$ of
sufficiently small mass bound.

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Let $V$ be a given time-dependent Navier-Stokes flow of an incompressible
viscous fluid in the whole space ($n=3,4$). Assume such $V$ to be small in
$L^\infty(0,\infty; L^{n,\infty})$, where $L^{n,\infty}$ denotes the weak-$L^n$
space. The energy stability of this basic flow $V$ with respect to any initial
disturbance in $L^2_\sigma$ has been established by Karch, Pilarczyk and
Schonbek. In this paper we study, under reasonable conditions, the algebraic
rates of energy decay of disturbances as $t\to\infty$.

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A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions about the existence of biholomorphic maps. Conversely, we show that the algebraic-geometric viewpoint leads to interesting questions in classical analysis.

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