Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain : (1)∂_tu=Au−f(u)+g, u|_∂Ω=0. Here u=(u¹,…,u^k) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where a_ij^l(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor of the associated dynamical system and that the attractors  , 0<<₀1, converge to the attractor of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized and the homogenized attractors in terms of the parameter .     

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].     

Two-parameter estimates for joint spectral projections on complex spheres

We prove sharp two-parameter estimates for the L ^p -L ² norm, 1 ≤ p ≤ 2, of the joint spectral projectors associated to the Laplace–Beltrami operator and to the Kohn Laplacian on the unit sphere S ^2n-1 in . Then, by using the notion of contraction of Lie groups, we deduce the estimates recently obtained by H. Koch and F. Ricci for joint spectral projections on the reduced Heisenberg group h ¹.      

Analytic regularity for an operator with Treves curves

We study a partial differential operator with analytic coefficients, which is of the form “sum of squares”. is hypoelliptic on any open subset of , yet possesses the following properties: (1) is not analytic hypoelliptic on any open subset of that contains 0. (2) If u is any distribution defined near with the property that is analytic near 0, then u must be analytic near 0. (3) The point 0 lies on the projection of an infinite number of Treves curves (bicharacteristics).These results are consistent with the Treves conjectures. However, it follows that the analog of Treves conjecture, in the sense of germs, is false.As far as we know, is the first example of a “sum of squares” operator which is not analytic hypoelliptic in the usual sense, yet is analytic hypoelliptic in the sense of germs.     

First-order differentiability of the flow of a system withL p controls

This paper considers the study of the regularity of the flow of a nonautonomous nonlinear control process when the set of control maps is endowed with theL ^p -topology. Roughly speaking, it is proved that, if the norm of the mapf(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm ofu not faster thanu ^p ,p>1, then the flow isC ¹ in theL ^p -topology. This property implies that, if the control maps are bounded, then the flow is differentiable in anyL ^p ,p>1. Moreover, it is proved that the only systems for which the flow is differentiable inL ¹ are the affine ones.This research was supported by a grant from Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica, Rome, Italy.     

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

In this paper we study the nonlocal p-Laplacian type diffusion equation,

If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation u_t=div(|u|^p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L^∞(0,T;L^p(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data

In this paper we investigate vector-valued parabolic initial boundary value problems , subject to general boundary conditions in domains G in with compact C ^2m -boundary. The top-order coefficients of are assumed to be continuous. We characterize optimal L ^p -L ^q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on and the Lopatinskii–Shapiro condition on are necessary for these L ^p -L ^q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.      

Discrete Fourier–Neumann series

Let J_μ denote the Bessel function of order μ. The systemwith n=0,1,…,α>−1, and where p_s denotes the sth positive zero of J_α(ax), is orthonormal in . In this paper, we study the mean convergence of the Fourier series with respect to this system. Also, we describe the space in which the span of the system is dense.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

Classification and Liouville type theorems for <Emphasis Type="Italic">p</Emphasis>-harmonic morphisms

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the L ^q -energy is finite, then is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).      

L r Estimates for Degenerate Elliptic Problems

We prove L ^r estimates for the Dirichlet problem –div(a(x,u,Du))=f with f in L ^q for 1q+, where the operator satisfies (|s|)|| ^p a(x,s,), with p>1. These estimates are obtained without symmetrization and are sharp in some cases.     

Approximation in Lp( ) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function

The problem of approximating smooth L_p-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout ^d, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of ^d. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ= ^d, apply specifically to the examples of the Gauss kernel and the generalized multiquadric.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

Some rigidity results for non-commutative Bernoulli shifts

We introduce the outer conjugacy invariants , for cocycle actions σ of discrete groups G on type II₁ factors N, as the set of real numbers t>0 for which the amplification σ^t of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly and the fundamental group of σ, , in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and σ is an action of G on the hyperfinite II₁ factor by Connes–Størmer Bernoulli shifts of weights {t_i}_i. Thus, and coincide with the multiplicative subgroup S of generated by the ratios {t_i/t_j}_i,j, while if S={1} (i.e. when all weights are equal), and otherwise. In fact, we calculate all the “1-cohomology picture” of σ^t,t>0, and classify the actions (σ,G) in terms of their weights {t_i}_i. In particular, we show that any 1-cocycle for (σ,G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on to the algebra pNp, for p a projection in N, p≠0,1, cannot be perturbed to a genuine action.     

On the behavio r of complete g raphs in $${\mathbb{R}\sp{n+1}}$$ with Lp-finite r-cu rvatu re

We study the behavior of complete graphs in with L ^p -finite r-curvature, that is, whose length of the r-th Newton transformation |P _r | is in L ^p , for some p ≥ 1. Moreover, we use a monotonicity formulae to establish an L ^p -lower bound for |P _r | in balls. As application, we prove some new Bernstein-type results.