Applications of weighted estimates of Calderon’s commutators

The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition |V(z) ? V(??)| ?? w|z ? ??| and F is an entire function, then we put $$P_F^* f(z) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int\limits_{\left| {\zeta - z} \right| > \varepsilon } {F\left( {\frac{{V(\zeta ) - V(z)}} {{\zeta - z}}} \right)\frac{{f(\zeta )}} {{\left( {\zeta - z} \right)^2 }}d\sigma (\zeta )} } \right|.$$ It is shown that if the weight function ?? is a Muckenhoupt A _p weight for 1 < p < ??, then $$\left\| {P_F^* f} \right\|_{p,\omega } \leqslant C(F,w,p)\left\| f \right\|_{p,\omega } .$$ .     

Strengthening Kazhdan’s property (T) by Bochner methods

In this paper, we propose a property which is a natural generalization of Kazhdan??s property (T) and prove that many, but not all, groups with property (T) also have this property. Let ?? be a finitely generated group. One definition of ?? having property (T) is that ${H^{1}(\Gamma, \pi, {\mathcal{H}}) = 0}$ where the coefficient module ${{\mathcal{H}}}$ is a Hilbert space and ?? is a unitary representation of ?? on ${{\mathcal{H}}}$ . Here we allow more general coefficients and say that ?? has property ${F \otimes {H}}$ if ${H^{1}(\Gamma, \pi_{1}{\otimes}\pi_{2}, F{\otimes} {\mathcal{H}}) = 0}$ if (F, ?? ₁) is any representation with dim(F) <??? and ${({\mathcal{H}}, \pi_{2})}$ is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property ${F \otimes {H}}$ if and only if it has property (T). The proof hinges on an extension of a Bochner-type formula due to Matsushima?CMurakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenb?ck formula??s for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products (Fisher and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1?C19, 2006). Some further applications of property ${F\otimes {H}}$ in the context of group actions will be given in Fisher and Hitchman (in preparation).     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

Uniqueness properties of degenerate elliptic operators

Let ?? be an open subset of R ^d and ${ K=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j+\sum^d_{i=1}c_i\partial_i+c_0}$ a second-order partial differential operator with real-valued coefficients ${c_{ij}=c_{ji}\in W^{1,\infty}_{\rm loc}(\Omega),c_i,c_0\in L_{\infty,{\rm loc}}(\Omega)}$ satisfying the strict ellipticity condition ${C=(c_{ij}) >0 }$ . Further let ${H=-\sum^d_{i,j=1} \partial_i\,c_{ij}\,\partial_j}$ denote the principal part of K. Assuming an accretivity condition ${C\geq \kappa (c\otimes c^{\,T})}$ with ${\kappa >0 }$ , an invariance condition ${(1\!\!1_\Omega, K\varphi)=0}$ and a growth condition which allows ${\|C(x)\|\sim |x|^2\log |x|}$ as |x| ?? ?? we prove that K is L ₁-unique if and only if H is L ₁-unique or Markov unique.     

Integrability of vector-valued lacunary series with applications to function spaces

Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 <  r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) =  x y ^p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$      

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.     

Topological transitivity for a class of monotonic mod one transformations

Suppose that f : [0, 1] ?? [0, 2] is a continuous strictly increasing piecewise differentiable function, and define T _f x :=?f(x) (mod 1). Let ${\beta \geq \sqrt[3]{2}}$ . It is proved that T _f is topologically transitive if inf f???????? and ${f(0)\geq\frac{1}{\beta+1}}$ . Counterexamples are provided if the assumptions are not satisfied. For ${\sqrt[3]{2}\leq\beta < \sqrt{2}}$ and 0????????? 2 ? ?? it is shown that ??x?+??? (mod 1) is topologically transitive if and only if ${\alpha < \frac{1}{\beta^2+\beta}}$ or ${\alpha >2 -\beta-\frac{1}{\beta^2+\beta}}$ .     

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).     

F-signature exists

Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let $R^{1/p^{e}}$ be the ring of p ^e -th roots of elements of R for e???, and let a _e denote the maximal rank of a free R-module appearing in a direct sum decomposition of $R^{1/p^{e}}$ . We show the existence of the limit $s(R) := \lim_{e \to\infty} \frac{a_{e}}{p^{ed}}$ , called the F-signature of R. This invariant??which can be extended to all local rings in prime characteristic??was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391?C404, 2002) and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of arbitrary finite quotient singularities.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

Sharp Estimates for Integral Means for Three Classes of Domains

In this paper, the following sharp estimate is proved: $$\int_{0}^{2{\pi }} {\left| {F\prime \left( {e^{i\theta } } \right)} \right|^p d\theta \leqslant \sqrt {\pi } 2^{1 + p} \frac{{\gamma \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}} {{\gamma \left( {1 + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} ,\quad p > - 1,$$ where F is the conformal mapping of the domain $D^ - = \left\{ {\zeta :\left| \zeta \right| > 1} \right\}$ onto the exterior of a convex curve, with $F\prime \left( \infty \right) = 1$ . For p=1, this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D ^-).     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

Eine Erweiterung des Riemann-Lebesgue-Lemmas

It is proved that iff∈L ₁(?),f'∈L ₁(?) and ∫∣x∣ ⁱ ∣f(x)∣dx<∞ fori=1, ...,k?1 and ifA=(a _ij ) is a (k×k)-matrix with non-vanishing determinant, for $$\tilde f_A (\zeta ): = \smallint \exp (i\zeta _1 \sum\limits_{j = 1}^k {a_{1j} x^j } + ... + i\zeta _k \sum\limits_{j = 1}^k {a_{kj} x^j } )f(x)dx$$ the following relation holds: $$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$ .     

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation $$x^2 u''=axu'+bu-c \bigl(u'-1\bigr)^2, \quad x\in(0,x_0), $$ with $a\in\mathbb{R}, b\in\mathbb{R}$ , c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b<0 then no continuous solutions exist, whereas if a+b>0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x ₀=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.     

On the mean square value of Hurwitz zeta function

R Balasubramanian has shown that $$\mathop \smallint \limits_1^{\rm T} |\zeta (\tfrac{1}{2} + it)|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by $$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\| {2a} \right\|}}} \right)$$ for 0 <a < 1.     

A characterization of the Wishart exponential families by an invariance property

E is the space of real symmetric (d, d) matrices, andS and \(\bar S\) are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on \(\bar S\) defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μ_p is a suitable measure on \(\bar S\) . LetGL(?^d) be the subset ofE of invertible matrices. Fora inGL(?^d), define the automorphismg _a ofE byg _a(x)=^t axa, where ^t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(?^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx??x. (Theorem).     

Statistical limit of Lebesgue measurable functions at ∞ with applications in Fourier Analysis and Summability

This is basically a survey paper on recent results indicated in the title. A function s: [a, ∞) → ?, measurable in Lebesgue’s sense, where a ≥ 0, is said to have statistical limit ? at ∞ if for every ? > 0, $\mathop {\lim }\limits_{b \to \infty } (b - a)^{ - 1} |\{ \nu \in (a,b):|s(\nu ) - \ell | > \varepsilon \} | = 0$ . We briefly summarize the main properties of this new concept of statistical limit at ∞. Then we demonstrate its applicability in Fourier Analysis. For example, the classical inversion formula involving the Fourier transform $\hat s$ of a function s ∈ L ¹(?) remains valid even in the general case when $\hat s\not \in L^1 (\mathbb{R})$ . We also present Tauberian conditions, under which the ordinary limit of a function s ∈ L _loc ¹ [1,∞) follows from the existence of the statistical limit of its logarithmic mean at ∞.