Zygmund Inequalities Research Articles

Global Calderón–Zygmund inequalities on complete Riemannian manifolds

Global Calderón–Zygmund inequalities on complete Riemannian manifolds

$L^{p}$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds

In this paper, we investigate the validity of first and second order L^{p} estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^{p} estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain L^{p} estimates for the second order Riesz transform (or, equivalently, the validity of L^{p} Calderón–Zygmund inequalities) on the whole scale 1<p<+\infty by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1<p \leq 2 , analogous L^{p} bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.

Vector-valued Kahane–Salem–Zygmund inequalities with asymptotically bounded constants

We establish new vector-valued Kahane–Salem–Zygmund inequalities, with asymptotically bounded constants, on spaces with unconditional Schauder basis.

Mapping properties of operator-valued Bergman projections

In this paper, we study the boundedness theory for Bergman projection in the operator-valued setting. More precisely, let D \mathbb {D} be the open unit disk in the complex plane C \mathbb {C} and M \mathcal {M} be a semifinite von Neumann algebra. We prove that ‖ P ( f ) ‖ L 1 , ∞ ( N ) ≤ C ‖ f ‖ L 1 ( N ) , \begin{equation*} \|P(f)\|_{L_{1,\infty }(\mathcal {N})}\leq C \|f\|_{L_1(\mathcal {N})}, \end{equation*} where N = L ∞ ( D ) ⊗ ¯ M \mathcal {N}=L_{\infty }(\mathbb {D})\bar {\otimes }\mathcal {M} and P P denotes the Bergman projection. Consequently, P P is bounded on L p ( N ) L_{p}(\mathcal {N}) with 1 > p > ∞ 1>p>\infty . As applications, we also obtain Kolmogorov and Zygmund inequalities for the Bergman projection P P .

Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

The main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages sup _{1le jle N}Big |sum _{i=1}^K a_i(j) gamma _iBig |,, where (a_i(j)) in ell _infty ^N, , 1 le i le K and the (gamma _i) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.

Approximation and Localized Polynomial Frame on Double Hyperbolic and Conic Domains

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

On the quadrature exactness in hyperinterpolation

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to \(n+k\) with \(0<k\le n\). Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the \(L^2\) norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as \(n\rightarrow \infty \) if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs.

VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

AbstractWe prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$ . Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$ -multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.

Salem–Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

In this manuscript we give an extension of the classic Salem–Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width O(n-2(logn)-1/2-γ),γ>1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ ext {O}( n^{-2}(\\log n)^{-1/2-\\gamma }), \\quad \\gamma >1/2\\end{aligned}$$\\end{document}around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least n^{-rho }, rho in (0,1/4) with probability 1-text {O}( n^{-2rho }).

Constants of the Kahane–Salem–Zygmund inequality asymptotically bounded by 1

The Kahane–Salem–Zygmund inequality for multilinear forms in ℓ∞ spaces claims that, for all positive integers m,n1,...,nm, there exists an m-linear form A:ℓ∞n1×⋯×ℓ∞nm⟶K (K=R or C) of the typeA(z(1),...,z(m))=∑j1=1n1⋯∑jm=1nm±zj1(1)⋯zjm(m), satisfying‖A‖≤Cmmax⁡{n11/2,…,nm1/2}∏j=1mnj1/2, forCm≤κmlog⁡mm! and a certain κ>0. Our main result shows that given any ϵ>0 and any positive integer m, there exists a positive integer N such thatCm<1+ϵ, when we consider n1,...,nm>N. In addition, while the original proof of the Kahane–Salem–Zygmund relies on highly non-deterministic arguments, our approach is constructive. We also provide the same asymptotic bound (which is shown to be optimal in some cases) for the constant of a related non-deterministic inequality proved by G. Bennett in 1977. Applications to Berlekamp's switching game are given.

Density and non-density of [formula omitted] on complete manifolds with curvature bounds

We investigate the density of compactly supported smooth functions in the Sobolev space Wk,p on complete Riemannian manifolds. In the first part of the paper, we extend to the full range p∈[1,2] the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when k=2) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order k−3 (when k>2). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every n≥2 and p>2 we construct a complete n-dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in Wk,p does not hold for any k≥2. We also deduce the existence of a counterexample to the validity of the Calderón–Zygmund inequality for p>2 when Sec≥0, and in the compact setting we show the impossibility to build a Calderón–Zygmund theory for p>2 with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.

The $$L^p$$-Calderón–Zygmund inequality on non-compact manifolds of positive curvature

We construct, for $$p>n$$ , a concrete example of a complete non-compact n-dimensional Riemannian manifold of positive sectional curvature which does not support any $$L^p$$ -Calderón–Zygmund inequality: $$\begin{aligned}\Vert {{\,\mathrm{Hess}\,}}\varphi \Vert _{L^p}\le C(\Vert \varphi \Vert _{L^p}+\Vert \Delta \varphi \Vert _{L^p}), \qquad \forall \,\varphi \in C^{\infty }_c(M). \end{aligned}$$ The proof proceeds by local deformations of an initial metric which (locally) Gromov–Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by De Philippis and Núñez–Zimbron dealing with the case of compact manifolds. As a straightforward consequence, we obtain that the $$L^p$$ -gradient estimates and the $$L^p$$ -Calderón–Zygmund inequalities are generally not equivalent, thus answering an open question in the literature. Finally, our example gives also a contribution to the study of the (non-)equivalence of different definitions of Sobolev spaces on manifolds.

On Marcinkiewicz–Zygmund Inequalities and $$A_p$$-Weights for $$L$$-Shape Arcs

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$.

Sampling, Marcinkiewicz–Zygmund inequalities, approximation, and quadrature rules

Given a sequence of Marcinkiewicz-Zygmund inequalities in L2, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.

On unimodular multilinear forms with small norms on sequence spaces

The Kahane–Salem–Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries 1 and −1 generating unimodular m-linear forms Am,n:ℓp1n×⋯×ℓpmn⟶R (or C) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of Am,n when {p1,…,pm}⊂[2,∞] and when {p1,…,pm}⊂[1,2) are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: {p1,…,pm} intercepts both [2,∞] and [1,2). In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

AbstractWe consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

Zygmund inequality of the conjugate function on Morrey-Zygmund spaces

Abstract We generalize the Zygmund inequality for the conjugate function to the Morrey type spaces defined on the unit circle T. We obtain this extended Zygmund inequality by introducing the Morrey-Zygmund space on T.

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing $$\mathsf {L}^{p}$$ -bounds on the second fundamental form of an isometric immersion from $$\mathsf {L}^{p}$$ -bounds on its mean curvature vector field, we prove a nonlinear Calderon–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

Asymptotically honest confidence regions for high dimensional parameters by the desparsified conservative Lasso

In this paper we consider the conservative Lasso which we argue penalizes more correctly than the Lasso and show how it may be desparsified in the sense of van de Geer et al. (2014) in order to construct asymptotically honest (uniform) confidence bands. In particular, we develop an oracle inequality for the conservative Lasso only assuming the existence of a certain number of moments. This is done by means of the Marcinkiewicz–Zygmund inequality. We allow for heteroskedastic non-subgaussian error terms and covariates. Next, we desparsify the conservative Lasso estimator and derive the asymptotic distribution of tests involving an increasing number of parameters. Our simulations reveal that the desparsified conservative Lasso estimates the parameters more precisely than the desparsified Lasso, has better size properties and produces confidence bands with superior coverage rates.