Maximal Marcinkiewicz multipliers

Let \(\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}\) be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in \(\mathbb{R}^{n}\). We show that the L ^p norm, 1<p<∞, of the related maximal operator

$$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$

is at most C(log(N+2)) ^n/2. We show that this bound is sharp.

     

On a class of weighted inequalities containing quasilinear operators

A characterization of weighted L^p–L^r inequalities on a half-axis is obtained for positive quasilinear operators with Oinarov kernels.     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

Necessary and Sufficient Conditions for the Solvability of the Neumann and the Regularity Problems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of Some Schrödinger Equations on Lipschitz Domains

Let n ≥? 3and Ω be a bounded Lipschitz domain in \(\mathbb {R}^{n}\). Assume that the non-negative potential V belongs to the reverse Hölder class \(RH_{n}(\mathbb {R}^{n})\) and p ∈ (2, ∞). In this article, two necessary and sufficient conditions for the unique solvability of the Neumann and the Regularity problems of the Schrödinger equation ? Δu + V u =? 0 in Ω with boundary data in L ^p , in terms of a weak reverse Hölder inequality with exponent p and the unique solvability of the Neumann and the Regularity problems with boundary data in some weighted L ² space, are established. As applications, for any p ∈ (1, ∞), the unique solvability of the Regularity problem for the Schrödinger equation ? Δu + V u =?0 in the bounded (semi-)convex domain Ω with boundary data in L ^p is obtained.     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

Norm convergence of multiple ergodic averages on amenable groups

We consider abstract non-negative self-adjoint operators on L²(X) which satisfy the finite-speed propagation property for the corresponding wave equation. For such operators, we introduce a restriction type condition, which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next, we show that in the considered abstract setting, our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on Rⁿ, the harmonic oscillator, elliptic operators on compact manifolds, and Schr¨odinger operators on asymptotically conic manifolds.     

The best extension of algebraic polynomials from the unit circle

We study the class \(\mathfrak{P}_n \) of algebraic polynomials P _n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ₁ centered at the origin is at most 1: \(\left\| {P_n } \right\|_{C(\Gamma _1 )} \) ≤ 1. The extension of polynomials from the class \(\mathfrak{P}_n \) to the plane with the least uniform norm on the concentric circle Γ _r of radius r is investigated. It is proved that the values θ _n (r) of the best extension of the class \(\mathfrak{P}_n \) satisfy the equalities θ _n (r) = r ⁿ for r > 1 and θ _n (r) = r ^n?1 for 0 < r < 1.     

Dirac system with potential lying in Besov spaces

We study the spectral properties of the Dirac operator L_P,U generated in the space (L₂[0, π])² by the differential expression By′ + P(x)y and by Birkhoff regular boundary conditions U, where y = (y₁, y₂)^t, \(B = \left( {\begin{array}{*{20}{c}} { - i}&0 \\ 0&i \end{array}} \right)\), and the entries of the matrix P are complexvalued Lebesgue measurable functions on [0, π]. We also study the asymptotic properties of the eigenvalues {λ_n}_n∈Z of the operator L_P,U as n → ∞ depending on the “smoothness” degree of the potential P; i.e., we consider the scale of Besov spaces B_1,∞^θ, θ ∈ (0, 1). In the case of strongly regular boundary conditions, we study the asymptotic behavior of the system of normalized eigenfunctions of the operator L_P,U, and in the case of regular but not strongly regular boundary conditions, we find the asymptotics of two-dimensional spectral projections.     

Geometric and probabilistic analysis of convex bodies with unconditional structures,and associated spaces of operators

Let \({{\|\cdot\|}}\) be a norm on \({\mathbb{R}^n}\) and \({\|.\|_*}\) be the dual norm. If \({\|\cdot\|}\) has a normalized 1-symmetric basis \({\{e_i\}_{i=1}^n}\) then the following inequalities hold: for all \({x,y\in \mathbb{R}^n}\), \({\|x\|\cdot\|y\|_*\le \max(\|x\|_1\cdot\|y\|_\infty,\|x\|_\infty\cdot\|y\|_1)}\) and if the basis is only 1-unconditional and normalized then for all \({x \in \mathbb{R}^n}\) , \({\|x\|+\|x\|_{*}\leq \|x\|_1+\|x\|_\infty}\) . We consider other geometric generalizations and apply these results to get, as a special case, estimates on best random embeddings of k-dimensional Hilbert spaces in the spaces of nuclear operators \({{\mathcal N}(K,K)}\) of dimension n ², for all k = [λn ²] and 0 < λ < 1. We obtain universal upper bounds independent on the 1-symmetric norm \({\|.\|}\) for the products of pth moments

$\left( {\mathbb{E}} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|^p\cdot\, \mathbb {E} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|_*^p\right)^{1/p}$

for independent random variables {f _i (ω)}, and 1 ≤ p < ∞.

     

Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato

In this paper, we show that, for doubling manifolds satisfiying the scaled Poincaré inequalities and \(p\in (2,\infty )\), the boundedness of the Riesz transform dΔ^?1/2 on L ^p , is essentially equivalent to the fact that \(H_{1,d}^{p}\) is equal the L ^p closure of the set of L ^p exact harmonic 1-forms. Here, \(H_{1,d}^{p}\) is a Hardy space of exact 1 ?forms, naturally associated with the Riesz transform, as defined by Auscher, McIntosh and Russ.     

Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

We study the sharp Nikol’skii inequality between the uniform norm and the L _q norm of algebraic polynomials of a given (total) degree n ≥ 1 on the unit sphere \(\mathbb{S}^{m - 1} \) of the Euclidean space ? ^m for 1 ≤ q < ∞. We prove that the polynomial ? _n in one variable with unit leading coefficient that deviates least from zero in the space L _q ^ψ (?1, 1) of functions f such that |f| ^q is summable over (?1, 1) with the Jacobi weight ψ(t) = (1 - t)^α(1 + t)^β, α = (m - 1)/2, β = (m - 3)/2 as a zonal polynomial in one variable t = ξ _m , where x = (ξ ₁, ξ ₂, …, ξ _m ) ∈ \(\mathbb{S}^{m - 1} \), is (in a certain sense, unique) extremal polynomial in the Nikol’skii inequality on the sphere \(\mathbb{S}^{m - 1} \). The corresponding one-dimensional inequalities for algebraic polynomials on a closed interval are discussed.     

<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.

     

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of

$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$

with f subject to the general structural conditions

$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$

where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ L ^t (Ω) for some t > n/p ^?, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L ^∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L ^∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.

     

An elementary proof of the <Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript> bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A _p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L ¹ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A ² bounds in that setting.     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces \({\mathcal{M}^{p}}\) and Stepanoff spaces \({\mathcal{S}^p}\), 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (\({M^{p}_{0}}\)). We construct a function f that belongs to these spaces and has the property that all approximate identities \({\phi_\varepsilon * f}\) converge to f pointwise but they never converge in norm.     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

Conditional exponential stability of a differential system with linear dichotomous Coppel-Conti approximation

We prove the conditional exponential stability of the zero solution of the nonlinear differential system

$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$

with L ^p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X _A (t), X _A (0) = E, satisfies the condition

$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$

where P ₁ and P ₂ are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R ⁿ that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.

     

The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval

Let P _n (α) be the set of algebraic polynomials p _n of order n with real coefficients and zero weighted mean value with ultraspherical weight \(\phi ^{(\alpha )} (t) = (1 - t^2 )^\alpha \) on the interval \([ - 1,1]:\int_{ - 1}^1 {\phi ^{(\alpha )} (t)p_n (t)dx = 0} \). We study the problem on the smallest value µ _n = inf{m(p _n ): p _n ∈ P _n (α)} of the weighted measure \(m(p_n ) = \int_{\chi (p_n )} {\phi ^{(\alpha )} (t)dt} \) of the set where p _n is nonnegative. The order of µ _n with respect to n is found: it is proved that \(\mu _n (\alpha ) \asymp n^{ - 2(\alpha + 1)} \) as n→∞.