Empirical multifractal moment measures of self-similar measure for q < 0*

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L ^q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L ^q - spectrum for q < 0 is generally considered significantly more difficult since the L ^q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L ^q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.     

Multifractal formalism for self-affine measures with overlaps

We prove that for a class of self-affine measures defined by an expanding matrix whose eigenvalues have the same modulus, the L^q-spectrum τ(q) is differentiable for all q > 0. Furthermore, we prove that the multifractal formalism holds in the region corresponding to q > 0. Received: 4 August 2008, Revised: 19 January 2009     

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

Very recently bounds for the L ^q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L ^q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L ^q spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bounds for the Commutator of Parabolic Singular Integral with Rough Kernels

In this paper, the authors give the L ^p (1 < p < ∞ ) boundedness of the k-th order commutator of parabolic singular integral with the kernel function Ω ∈ L(log⁺ L) ^k + 1(S ^n − 1). The result in this paper is an extension of some known results. The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

Atomic hardy-type spaces between <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> on metric spaces with non-doubling measures

Let ( Y,d,dl )\left( {\mathcal{Y},d,d\lambda } \right) be (ℝ ⁿ , |·|, μ), where |·| is the Euclidean distance, μ is a nonnegative Radon measure on ℝ ⁿ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ ⁿ , |·|, d _λ ), or the space (S, d, ρ), where S ≡ ℝ ⁿ ⋉ ℝ⁺ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces { X_s ( Y ) }_{0 < s \leqslant ￥}\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty } and the BMO-type spaces { BMO( Y, s ) }_{0 < s \leqslant ￥}\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }. Let H ¹ ( Y )\left( \mathcal{Y} \right) be the known atomic Hardy space and L ₀¹ ( Y )\left( \mathcal{Y} \right) the subspace of f ∈ L ¹ ( Y )\left( \mathcal{Y} \right) with integral 0. The authors prove that the dual space of X _s ( Y )\left( \mathcal{Y} \right) is BMO( Y,s )BMO\left( {\mathcal{Y},s} \right) when s ∈ (0,∞), X _s ( Y )\left( \mathcal{Y} \right) = H ¹ ( Y )\left( \mathcal{Y} \right) when s ∈ (0, 1], and X _∞ ( Y )\left( \mathcal{Y} \right) = L ₀¹ ( Y )\left( \mathcal{Y} \right) (or L ¹ ( Y )\left( \mathcal{Y} \right)). As applications, the authors show that if T is a linear operator bounded from H ¹ ( Y )\left( \mathcal{Y} \right) to L ¹ ( Y )\left( \mathcal{Y} \right) and from L ¹ ( Y )\left( \mathcal{Y} \right) to L ^1,∞ ( Y )\left( \mathcal{Y} \right), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from X _r ( Y )\left( \mathcal{Y} \right) to the Lorentz space L ^1,s ( Y )\left( \mathcal{Y} \right), which applies to the Calderón-Zygmund operator on (ℝ ⁿ , |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ ⁿ , |·|, d _γ ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.     

Eigensystem of an <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-perturbed harmonic oscillator is an unconditional basis

For any complex valued L ^p -function b(x), 2 ≤ p < ∞, or L ^∞-function with the norm ‖b↾L ^∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d ²/dx ² + x ² + b(x) in L ²(ℝ¹) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L ²(ℝ).     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

A weak invertibility criterion in the weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces of holomorphic functions in the ball

We obtain a necessary and sufficient condition on a weight function for every nowhere vanishing holomorphic function in the unit ball in the weighted L ^p -space to be weakly invertible in the corresponding L ^q -space for all q < p.     

<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)     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

Optimal Domains for <Emphasis Type="Italic">L</Emphasis><Superscript>0</Superscript>-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L⁰([0, 1]), where E is an F–function space and L⁰([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions. Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).     

Extrapolation results for <Emphasis Type="Italic">q</Emphasis> < 1

Given a sublinear operator T such that is bounded, it can be shown that is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : L ^q (μ) → X is bounded, with constant C/(1−q), for every and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped. This work has been partially supported by MTM2004-02299 and by 2005SGR00556.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Functional Inequalities and Weighted Porous Media Equations

Using measure-capacity inequalities we study new functional inequalities, namely L ^q -Poincaré inequalities

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).