We give an explicit upper bound on the number of non-constant holomorphic maps from a quasi-projective manifold into a complex hyperbolic manifold of finite volume. This gives an effective version of the results of Sunada and Noguchi.

A formula for the interior $\varepsilon$-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of $\varepsilon$ is shown to match quite closely with earlier predictions of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analog of the standard Cantor staircase function and reflects the self-similarity of the Koch curve. As a consequence, the possible complex dimensions of the Koch snowflake are computed explicitly.

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains.

We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form

$$-u''(t)=g(t)f(t,u(t)),\quad \text{for almost every } t \in (0,1),$$

where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space $C[0,1]$, of the form

$$u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds$$

where $\alpha[u]$, $\beta[u]$ are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive $u$. This means we actually cover many more differential equations than the simple equation written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well-known $m$-point boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.

We study concentration phenomena for the system

\[\varepsilon^2 \Delta v - v - \delta \phi v + \gamma v^{p} =0,\quad \Delta\phi+ \delta v^2=0 \]

in the unit ball $B_1$ of $\mathbb{R}^3$ with Dirichlet boundary conditions. Here $\varepsilon,\ \delta,\ \gamma >0$ and $p>1$. We prove the existence of positive radial solutions $(v_{\varepsilon}, \phi_{\varepsilon})$ such that $v_{\varepsilon}$ concentrates at a distance $({\varepsilon}/{2}) |{\rm log}\, {\varepsilon} |$ away from the boundary $\partial B_1$ as the parameter $\varepsilon$ tends to 0. The approach is based on a combination of Lyapunov–Schmidt reduction procedure together with a variational method.

Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

SOLUTIONS OF $p$-SUBLINEAR $p$-LAPLACIAN EQUATION VIA MORSE THEORY (J. London Math. Soc. (2) 72 (2005) 632–644)

YUXIA GUO AND JIAQUAN LIU

Department of Mathematics, Tsinghua University, Beijing 100084, PR China

LMAM, School of Mathematical Science, Beijing University, Beijing 100871, PR China

35J65

49K20

35J40

The above-mentioned paper was influenced by the work of V. Moroz [1] and Z. Q. Wang [2], which had been inadvertently omitted from the bibliography.

This paper deals with the lack of compactness in the nonlinear elliptic problem $-\Delta u+u=|u|^{p-2}u$ in $\Omega,\ u>0$ in $\Omega,\ u=0$ on $\partial \Omega$, when $\Omega$ is un unbounded domain in $\mathbb{R}^n$ and $2<p<2n/(n-2)$.

In particular, a domain $\widetilde\Omega$ is provided where non-converging Palais–Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on $\widetilde\Omega$.

In 1868 Zolotarev determined the polynomial which deviates least from zero with respect to the maximum norm on $[-1,1]$ among all polynomials of the form $x^n + \sigma x^{n-1} + a_{n-2}x^{n-2} + \ldots +a_1x + a_0$, where $\sigma \in {\mathbb R}$ is given. The polynomial was given explicitly in terms of elliptic functions by Zolotarev. It is now called the Zolotarev polynomial. Zolotarev also gave an explicit expression for the minimum deviation. In the sequel attempts have been made to replace the elliptic functions and to express the Zolotarev polynomial and the minimum deviation in terms of elementary functions, at least asymptotically. In 1913 Bernstein succeeded in finding an asymptotic formula for the minimum deviation, which has been improved several times since then. Here we give the first asymptotic representation of the Zolotarev polynomials. For the asymptotic representation we use the rational functions introduced by Bernstein. Furthermore, we obtain asymptotic representations of minimal polynomials with interpolation constraints which are of interest in the theory of the iterative solution of inconsistent linear systems of equations.

That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A–B (Aharonov–Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A–B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A–B effect appears strongly when the centers are on an even line.

Consider a parabolic $N\times N$-system of order $m$ on $\mathbb{R}^n$ with top-order coefficients $a_\alpha \in \mathrm{VMO} \cap L^\infty$. Let $1<p,q < \infty$ and let $\omega$ be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution $u$ satisfying

$$\|u'\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} + \|\mathcal{A} u\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)} \le C \|f\|_{L^q(J;L^p_\omega(\mathbb{R}^n)^N)},$$

where $\mathcal{A} u = \sum_{|\alpha| \le m}a_\alpha D^\alpha u$ and $J=[0,\infty)$. In particular, choosing $\omega =1$, the realization of $\mathcal{A}$ in $L^p({\mathbb{R}}^n)^N$ has maximal $L^p-L^q$ regularity.

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

We introduce a numerical radius operator space $(X, \mathcal{W}_n)$. The conditions to be a numerical radius operator space are weaker than Ruan's axiom for an operator space $(X, \mathcal{O}_n)$. Let $w(\cdot)$ be the numerical radius on $\mathbb{B}(\mathcal{H})$. It is shown that, if $X$ admits a norm $\mathcal{W}_n(\cdot)$ on the matrix space $\mathbb{M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{W}_n)$ into $(\mathbb{B}(\mathcal{H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{O}_n)$ and the numerical radius operator space $(X, \mathcal{W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.

We study the following question: given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $\Omega$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and ${\mathbb Z}^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $\Omega$ of two types: (a) packing properties of $\Omega$ and (b) spectral properties of $\Omega$. Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_{\Omega}f$ over possibly dispersed sets solely in dependence of the given measure $m:=|\Omega|$ of $\Omega$. In this respect we show that in ${\mathbb R}$ and ${\mathbb Z}$ the integral is maximal for intervals.

We show that, if $A$ is a separable simple unital $C^{*}$-algebra that absorbs the Jiang–Su algebra ${{\mathcal Z}}$ tensorially and that has real rank zero and finite decomposition rank, then $A$ is tracially approximately finite-dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of $C^{*}$-algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered $K$-theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.

THE BAIRE METHOD FOR THE PRESCRIBED SINGULAR VALUES PROBLEM (J. London Math. Soc. (2) 70 (2004) 719–734)

F. S. DE BLASI AND G. PIANIGIANI

Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via delle Ricerca Scientifica 1, 00133 Roma, Italy

Dipartimento di Matematica per le Decisioni, Università di Firenze, Via Lombroso 6/17, 50134 Firenze, Italy

35F30 (primary)

54E52 (secondary)

In the proof of Theorem 5.1 of the above-mentioned paper, the proof of Claim 1 is not correct (by a misuse of Mazur's theorem). With the same notation as in that paper, we here present the correct proof.

We give necessary and sufficient conditions in order that inequalities of the type

\[ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \quad f \in L^p(d\sigma), \]

hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x,y) f(y)\,d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ on $\mathbb{R}^n$, in the case where $p>q>0$ and $p>1$.

An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y)= \sum_{Q\in{\mathcal D}} K(Q)\chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\mathbb{R}^n$, and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$.

The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k \star f$ with positive radially decreasing kernel $k(|x-y|)$, the trace inequality

\[\| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \quad f \in L^p(dx),\]

holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$, where $s = q(p-1)/(p-q)$. Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{1/(p-1)} \mu(B(x,r))^{1/(p-1)} r^{n-1} \, dr$, and $\bar{k}(r)=(1/r^n)\int_0^r k(t) t^{n-1} \, dt$. Analogous inequalities for $1\le q < p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$.

In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice $\varGamma\subseteq\mathbb{Z}^n$ as a translation tiling set of $\mathbb{R}^n$. In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile $K$ and to determine all of the lattice tilings of $\mathbb^n$ by $K$. Moreover, we also propose and determine algorithmically another type of translation tiling of $\mathbb{R}^n$ by $K$, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set $K\cap (K+j),\ j\in\mathbb{Z}^n$, is strictly positive.

We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + \lambda_i Y x\}_{i=1}^m$, where $d_i\in {\mathbb R}$ and $\lambda_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of independent and identically distributed errors $y_1,y_2,\dotsc$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $\chi = {\mathbb E}[\log(\lambda Y)]$ be the Lyapunov exponent. Assuming that $\chi < 0$, we obtain a family of conditional measures $\nu_{\bf y}$ on the line, parametrized by ${\bf y} = (y_1,y_2,\dotsc)$, the sequence of errors. Our main result is that if $h > |\chi|$, then $\nu_{\bf y}$ is absolutely continuous with respect to the Lebesgue measure for almost every ${\bf y}$. We also prove that if $h < |\chi|$, then the measure $\nu_{\bf y}$ is singular and has dimension $h/|\chi|$ for almost every ${\bf y}$. These results are applied to a randomly perturbed iterated function system suggested by Sinai, and to a class of random sets considered by Arratia, motivated by probabilistic number theory.

We prove a general Borg-type result for reflectionless unitary CMV operators $U$ associated with orthogonal polynomials on the unit circle. The spectrum of $U$ is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle.

In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator $U$.