Maximal Function Estimates Research Articles

Weighted norm inequalities for Toeplitz type operators associated to generalized Calderón-Zygmund operators.

Let T_1 be a generalized Calderón–Zygmund operator or pm I ( the identity operator), let T_2 and T_3 be the linear operators, and let T_3=pm I. Denote the Toeplitz type operator by \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T^b=T_1M^bI_\\alpha T_2+T_3I_\\alpha M^b T_4, \\end{aligned}$$\\end{document}Tb=T1MbIαT2+T3IαMbT4,where M^bf=bf, and I_alpha is fractional integral operator. In this paper, we establish the sharp maximal function estimates for T^b when b belongs to weighted Lipschitz function space, and the weighted norm inequalities of T^b on weighted Lebesgue space are obtained.

Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.

Uniform regularity estimates in homogenization theory of elliptic system with lower order terms

In this paper, we extend the uniform regularity estimates obtained by M. Avellaneda and F. Lin in [3,6] to the more general second order elliptic systems in divergence form {Lε,ε>0}, with rapidly oscillating periodic coefficients. We establish not only sharp W1,p estimates, Hölder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichlet problem on a bounded C1,η domain, but also a sharp O(ε) convergence rate in H01(Ω) by virtue of the Dirichlet correctors. Moreover, we define the Green's matrix associated with Lε and obtain its decay estimates. We remark that the well known compactness methods are not employed here, instead we construct the transformations (1.11) to make full use of the results in [3,6].

Singular integrals, maximal functions and Fourier restriction to spheres: The disk multiplier revisited

Several estimates for singular integrals, maximal functions and spherical summation operator are given in the spaces LradpLang2(Rn), n≥2.

Spherically Averaged Maximal Function and Scattering for the Two-Dimensional Cubic Derivative Schrödinger Equation

We prove scattering for the 2D cubic derivative Schr\"odinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate for the 2D Schr\"odinger equation. We also prove a global well-posedness result for the 2D Schr\"odinger map in the critical Besov space with one degree angular regularity. The key ingredients for the latter results are the spherically averaged maximal function estimate, null form structure observed in \cite{Bej}, as well as the generalised spherically averaged Strichartz estimates obtained in \cite{Guo2} in order to exploit the null form structure.

Two-Weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces

Abstract We derive criteria governing two-weight estimates for multilinear fractional integrals and appropriate maximal functions. The two and one weight problems for multi(sub)linear strong fractional maximal operators are also studied; in particular, we derive necessary and sufficient conditions guaranteeing the trace type inequality for this operator. We also establish the Fefferman-Stein type inequality, and obtain one-weight criteria when a weight function is of product type. As a consequence, appropriate results for multilinear Riesz potential operator with product kernels follow.

$$M^k$$ M k -type sharp estimates and boundedness on Morrey space for Toeplitz type operators associated to fractional integral and singular integral operator with general kernel

In this paper, we prove the \(M^k\)-type sharp maximal function estimates for the Toeplitz type operators associated to the fractional integral and singular integral operator with general kernel. As an application, we obtain the weighted boundedness of the operators on the Morrey space.

A posteriorierror estimates for elliptic problems with Dirac measure terms in weighted spaces

In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation operators, without need of modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.

The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $$t$$ -independent complex bounded measurable coefficients ( $$t$$ being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $$L^{p'}$$ , subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $$L^p$$ . Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $$t$$ -independent real (possibly non-symmetric) coefficients there exists a $$p>1$$ such that the Regularity problem is well-posed in $$L^p$$ .

Weighted boundedness for Toeplitz type operator associated to general integral operators

In this paper, we establish the weighted sharp maximal function estimates for the Toeplitz type operators associated to some integral operators and the weighted Lipschitz and BMO functions. As an application, we obtain the boundedness of the Toeplitz type operators on weighted Lebesgue and Morrey spaces. The operator includes Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.

Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

We consider divergence form elliptic operators L = − d i v A ( x ) ∇ L= {-}\mathrm {div}\, A(x) \nabla , defined in the half space R + n + 1 \mathbb {R}^{n+1}_+ , n ≥ 2 n\geq 2 , where the coefficient matrix A ( x ) A(x) is bounded, measurable, uniformly elliptic, t t -independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation L u = 0 Lu=0 , and we then combine these estimates with the method of “ ϵ \epsilon -approximability” to show that L L -harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A ∞ A_\infty with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in L p L^p , for some p > ∞ p>\infty ). Previously, these results had been known only in the case n = 1 n=1 .

Sharp function estimates and boundedness for commutators associated with general singular integral operator

In this paper, we establish the sharp maximal function estimates for the commutator associated with the singular integral operator with general kernel. As an application, we obtain the boundedness of the commutator on weighted Lebesgue, Morrey, and Triebel-Lizorkin spaces.

Weighted boundedness for Toeplitz type operators related to strongly singular integral operators

In this paper, we show the sharp maximal function estimates for the Toeplitz type operators related to the strongly singular integral operators. As an application, we obtain the boundedness of the operators on weighted Lebesgue and Triebel-Lizorkin spaces.

$M^2$-type sharp estimates and weighted boundedness for commutators related to singular integral operators satisfying a variant of Hörmander's condition

In this paper, we prove the $M^k$-type sharp maximal function estimates for the commutators related to some singular integral operators satisfying a variant of Hörmander's condition. As an application, we obtain the weighted boundedness of the commutators on Lebesgue and Morrey spaces.

M 2 -Type sharp estimates and boundedness on a Morrey space for Toeplitz-type operators associated to singular integral operators satisfying a variant of Hörmander’s condition

In this paper, we prove the -type sharp maximal function estimates for the Toeplitz-type operators associated to certain singular integral operators satisfying a variant of Hormander’s condition. As an application, we obtain the weighted boundedness of the operators on the Lebesgue and Morrey spaces. MSC:42B20, 42B25.

Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data

We show that there exists sc>0 such that the cubic (quartic) non-elliptic derivative Schrödinger equations with small data in modulation spaces M2,1s(Rn) for n≥3 (n=2) are globally well-posed if s≥sc, and ill-posed if s<sc. In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schrödinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space Lx12Lx2,t∞. By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger–Segal algebra or in weighted Sobolev spaces are sufficiently small.

Square function and maximal function estimates for operators beyond divergence form equations

We prove square function estimates in L 2 for general operators of the form B 1 D 1 + D 2 B 2, where D i are partially elliptic constant coefficient homogeneous first-order self-adjoint differential operators with orthogonal ranges, and B i are bounded accretive multiplication operators, extending earlier estimates from the Kato square root problem to a wider class of operators. The main novelty is that B 1 and B 2 are not assumed to be related in any way. We show how these operators appear naturally from exterior differential systems with boundary data in L 2. We also prove non-tangential maximal function estimates, where our proof needs only off-diagonal decay of resolvents in L 2, unlike earlier proofs which relied on interpolation and L p estimates.

Homogenization of elliptic systems with Neumann boundary conditions

For a family of second-order elliptic systems with rapidly oscillating periodic coefficients in a $C^{1,\alpha }$ domain, we establish uniform $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates on solutions with Neumann boundary conditions.

Weighted boundedness for Toeplitz type operators related to Riesz transforms of the Schrödinger operator

This paper addresses the issue of sharp maximal function estimates of Toeplitz type operators. As an application, the boundedness of the operator on weighted Lebesgue and Triebel–Lizorkin spaces is established.

On joint estimates for maximal functions and singular integrals on weighted spaces

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for A p A_p weights with 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty , through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding L p L^p spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele.