A short note on Lq theory for Stokes problem with a pressure-dependent viscosity

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u, namely, it is represented by a stress tensor T (Du, p):= v(p, |D|²)D which satisfies r-growth condition with r ∈ (1, 2]. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998).     

A Remark on the Boundedness of Calderón–Zygmund Operators in Nony–homogeneous Spaces

Let μ be a Radon measure on Rd which may be non–doubling. The only condition satisfied by μ is that μ(B(x, r)) ≤ Cr ⁿ for all x ∈ ? ^d , r > 0 and some fixed 0 < n ≤ d. In this paper, the authors prove that the boundedness from H ¹(μ) into L ¹,^∞(μ) of a singular integral operator T with Calderón–Zygmund kernel of Hörmander type implies its L ²(μ)–boundedness.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

Weighted weak type (1,1) estimate for the Christ-Journé type commutator

Let K be the Calderón-Zygmund convolution kernel on R~d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R~d)by T_af(x)=p.v.∫_(R~d)K(x-y)m_x,y~a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) boundedness for Calderón commutator with rough variable kernel

For b ∈ Lip(?ⁿ); the Calderón commutator with variable kernel is defined by

$$[b,{T_1}]f(x) = p.v.\int_{{R n}} {\frac{{\Omega (x,x - y)}}{{|x - y{| {n + 1}}}}(b(x) - b(y))f(y)dy} $$

In this paper, we establish the L²(?ⁿ) boundedness for [b, T₁] with Ω(x, z′) ∈ L^∞(?ⁿ)-L^q(S^n-1) (q > 2(n-1)=n) satisfying certain cancellation conditions. Moreover, the exponent q > 2(n-1)/n is optimal. Our main result improves a previous result of Calderón.

     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.     

Convexity Inequalities for Positive Operators

We prove pointwise convexity (Jensen-type) inequalities of the form Open image in new window where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Hölder inequality T(fg) ≤ (Tf ^p )^{1/ p} (Tf ^q )^{1/ q} for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a ``real'' proof of the Riesz-Thorin theorem for positive operators.     

On the zeros of the Davenport–Heilbronn function

Let N ₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N ₀(T) ? T (ln T)^1/2+1/16?ε, where ε is an arbitrarily small positive number.     

Essential spectrum of difference operators on periodic metric spaces

The paper deals with the study of Fredholm property and essential spectrum of general difference (or band) operators acting on the spaces l ^p (X) on a discrete metric space X periodic with respect to the action of a finitely generated discrete group. The Schrödinger operator on a combinatorial periodic graph is a prominent example of a band operator of this kind.     

A modulation invariant Carleson embedding theorem outside local <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

Yen Do and Christoph Thiele developed a theory of Carleson embeddings in outer L_p spaces for the wave packet transform

$${F_\phi }(f)(u,t,\eta ) = \int {f(x){e^{i\eta (u - x)}}\phi \left( {\frac{{u - x}}{t}} \right)} \frac{{dx}}{t},(u,t,\eta ) \in R \times (0,\infty ) \times R$$

of functions f ∈ L_p(R) in the range 2 ≤ p ≤ ∞, referred to as local L². In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of L^p estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation.

     

Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so–called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self–adjoint integral operator in a suitable L ₂(μ)–space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion X _H with Hurst parameter H∈(0,1), indexed by a self–similar set T?? ^N of Hausdorff dimension D. This rate turns out to be of order n ^?H/D (log?n)^1/2. In the case T=[0,1] ^N we present a concrete wavelet representation of X _H leading to an approximation of X _H with the optimal rate n ^?H/N (log?n)^1/2.     

Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Monge–Ampère equations in big cohomology classes

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L ^1+ε -density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

To the Spectral Theory of One-Dimensional Matrix Dirac Operators with Point Matrix Interactions

We investigate one-dimensional (2p × 2p)-matrix Dirac operators D_X,α and D_X,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators D_X,α and D_X,β (self-adjointness, discreteness of the spectrum, etc.) are identical to the corresponding properties of some Jacobi matrices B_X,α and B_X,β with (p × p)-matrix entries. The relationship found is used to describe these properties as well as conditions of continuity and absolute continuity of the spectra of the operators D_X,α and D_X,β. Also the non-relativistic limit at the velocity of light c → ∞ is studied.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices

Let X ⁽ⁿ⁾=(X _ij ) be a p×n data matrix, where the n columns form a random sample of size n from a certain p-dimensional distribution. Let R ⁽ⁿ⁾=(ρ _ij ) be the p×p sample correlation coefficient matrix of X ⁽ⁿ⁾, and \(S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}\) be the sample covariance matrix of X ⁽ⁿ⁾, where \(\bar{X}\) is the mean vector of the n observations. Assuming that X _ij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R ⁽ⁿ⁾ converges almost surely to the limit \((1-\sqrt{c}\,)^{2}\) as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S ⁽ⁿ⁾ converges almost surely to \((1-\sqrt{c}\,)^{2}\).     

Bloom’s inequality: commutators in a two-weight setting

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L^p spaces, where both weights are in A_p. The characterization is in terms of a novel BMO condition. We give a ‘modern’ proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom’s result to all Calderón–Zygmund operators and dimensions.