Maximal Function Estimates Research Articles

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some Lp(ℝn) to Lq(ℝn) when the m–th order derivatives of A belong to Lr(ℝn) for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m–th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are (Lp, Lp)–type operators for 1 < p < ∞.

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Let Ω be a bounded Lipschitz domain in ℝ n , n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ L p (∂Ω) on ∂Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ϵ, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(∇u)*‖ p ≤ C‖f‖ p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

Maximal function estimates with applications to a modified Kadomstev-Petviashvili equation

In this paper we give optimal (up to endpoint) maximal function type estimates for the solution of the linear initial value problem associated with the Kadomstev-Petviashvili I equation. These estimates enable us to obtain a well-posedness result for a modified Kadomstev-Petviashvili I equation.

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K ∗ in L p 0(∂ Ω) is less than 1 2 , whenever Ω is a bounded convex domain and 1< p⩽2.

Maximal theorems of Menchoff–Rademacher type in non-commutative Lq-spaces

Estimates for maximal functions provide the fundamental tool for solving problems on pointwise convergence. This applies in particular for the Menchoff–Rademacher theorem on orthogonal series in L 2[0,1] and for results due independently to Bennett and Maurey–Nahoum on unconditionally convergent series in L 1[0,1]. We prove corresponding maximal inequalities in non-commutative L q -spaces over a semifinite von Neumann algebra. The appropriate formulation for non-commutative maximal functions originates in Pisier's recent work on non-commutative vector valued L q -spaces.

Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], n ≥ 3. Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a second order elliptic system with constant coefficients satisfying the Legendre-Hadamard condition. We consider the Dirichlet problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 0 in Ω, u = f on ∂Ω with boundary data f in the Morrey space L 2,λ (∂Ω). Assume that 0 ≤ λ &lt; 2 + ε for n ≥ 4 where ε &gt; 0 depends on Ω, and 0 ≤ λ ≤ 2 for n = 3. We obtain existence and uniqueness results with nontangential maximal function estimate ║( u )*║ L 2,λ(∂Ω) ≤ C ║ f ║ L 2,λ(∂Ω) . If [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] satisfies the strong elliptic condition and 0 ≤ λ &lt; min ( n -1, 2+ε), we show that the Neumann type problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] = 0 in Ω, ∂ u / ∂ν = g ∈ H 2,λ (∂Ω) on ∂Ω, ║(∇ u )*║ H 2,λ (∂Ω) &lt; ∞ has a unique solution. Here H 2,λ (∂Ω) is an atomic space with the property ( H 2,λ (∂Ω))* = L 2,λ (∂Ω). The invertibility of layer potentials on L 2,λ (∂Ω) and H 2,λ (∂Ω) is also obtained. Finally we study the Dirichlet problem for the biharmonic equation. We establish a similar estimate in L 2,λ for the biharmonic equation, in which case the range 0 ≤ λ &lt; 2 + ε is sharp for n = 4 or 5.

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor g j k d x j ⊗ d x k g_{jk} dx_j\otimes dx_k has low regularity. Under the assumption that \[ | g j k ( x ) − g j k ( y ) | ≤ C ω ( | x − y | ) , |g_{jk}(x)-g_{jk}(y)|\leq C\,\omega (|x-y|), \] where the modulus of continuity ω \omega satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with L p L^p boundary data, for sharp ranges of p p ’s and with optimal nontangential maximal function estimates.

Weighted estimates for fractional maximal functions related to spherical means

We prove weighted Lp-Lq estimates for the maximal operators ℳα, given by , where μt denotes the normalised surface measure on the sphere of centre 0 and radius t in Rd. The techniques used involve interpolation and the Mellin transform. To do this, we also prove weighted Lp-Lq estimates for the operators of convolution with the kernels |·|−α−iη.

Tangential boundary behavior of functions of several variables

In this paper we study the asymptotic behavior of functions defined on domains of a multidimensional real or complex space when the point tends to the boundary in the approach region with different orders of tangency. The main results are related to the boundary behavior of functions from Hardy-Sobolev spaces in a multidimensional complex ball and of solutions to elliptic boundary-value problems in a Lipschitz domain of a real Euclidean space. The methods used are based on two-weighted estimates for tangential maximal functions in an abstract ball. The boundary of this ball is a space equipped with measure and quasimetric.

Concerning Nikodym-type sets in 3-dimensional curved spaces

We investigate maximal functions involving averages over geo- desics in three-dimensional Riemannian manifolds. We first show that one can easily extend the Euclidean results of Bourgain and Wolff if one assumes constant curvature. These results need not hold if this assumption is dropped. Nonetheless, we formulate a generic geometric condition which allows favorable estimates. Curiously, this condition ensures that one is in some sense as far as possible from the constant curvature case. Assuming this, one can prove dimensional estimates for Nikodym-type sets which are essentially optimal. Optimal estimates for the related maximal functions are still open though.

Maximal function estimates of solutions to general dispersive partial differential equations

Let u ( x , t ) = ( S Ω f ) ( x , t ) u(x,t)=(S_\Omega f)(x,t) be the solution of the general dispersive initial value problem: \[ ∂ t u − i Ω ( D ) u = 0 , u ( x , 0 ) = f ( x ) , ( x , t ) ∈ R n × R \partial _tu-i\Omega (D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb {R}^n \times \mathbb {R} \] and S Ω ∗ ∗ f S^{**}_\Omega f the global maximal operator of S Ω f S_\Omega f . Sharp weighted L p L^p -esimates for S Ω ∗ ∗ f S^{**}_\Omega f with f ∈ H s ( R n ) f\in H_s(\mathbb {R}^n) are given for general phase functions Ω \Omega .

A geometric proof of the circular maximal theorem

is bounded on L(R) for p > 2. Simple examples show that this fails for p = 2. In this note we derive Bourgain’s result by geometric and combinatorial methods. In particular, we do not use the Fourier transform in any way. Our proof is based on a combinatorial argument from [6], which in turn uses Marstrand’s three circle lemma [7], and a lemma involving two circles that seems to originate in [10]. The three circle lemma was used in [7] to prove the following result, which is a simple consequence of Bourgain’s theorem: Suppose a planar set E has the property: for every point in the plane, E contains some circle with that point as center. Then E has positive measure. Thus we show here that Marstrand’s lemma, in combination with other ideas, does indeed allow one to establish the stronger maximal function estimate. Furthermore, we demonstrate in section 4 how to obtain the entire known range of L → L estimates for the circular maximal function (which is optimal possibly up to endpoints), see [9] and [11], by using the methods from sections 2 and 3. One — perhaps significant — distinction from the techniques developed in [1], [8], [11], and [12], which involve the Fourier transform, is the fact that the methods presented here do not seem to yield estimates for the global maximal function Mf(x) = sup 0<t<∞ ???

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations V ( x ) V(x) satisfying | V ( x ) | ≤ C ( 1 + | x | ) − α , |V(x)|\leq C(1+|x|)^{-\alpha }, α &gt; 1 2 . \alpha &gt;\frac {1}{2}. This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

Harmonic Analysis and Ultracontractivity

Let ${({T_t})_{t > 0}}$ be a symmetric contraction semigroup on the spaces ${L^p}(M)\;(1 \leq p \leq \infty )$, and let the functions $\phi$ and $\psi$ be "regularly related". We show that ${({T_t})_{t > 0}}$ is $\phi$-ultracontractive, i.e., that ${({T_t})_{t > 0}}$ satisfies the condition ${\left \| {{T_t}f} \right \|_\infty } \leq C\phi {(t)^{ - 1}}{\left \| f \right \|_1}$ for all $f$ in ${L^1}(M)$ and all $t$ in ${{\mathbf {R}}^ + }$, if and only if the infinitesimal generator $\mathcal {G}$ has Sobolev embedding properties, namely, ${\left \| {\psi {{(\mathcal {G})}^{ - \alpha }}f} \right \|_q} \leq C{\left \| f \right \|_p}$ for all $f$ in ${L^p}(M)$, whenever $1 < p < q < \infty$ and $\alpha = 1/p - 1/q$ . We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on $m$ for $m(\mathcal {G})$ to map ${L^p}(M)$ to ${L^q}(M)$, and for the example where there exists $\mu$ in ${{\mathbf {R}}^ + }$ such that $\phi (t) = {t^\mu }$ for all $t$ in ${{\mathbf {R}}^ + }$ , we give conditions which ensure that the maximal function ${\sup _{t > 0}}|{t^\alpha }{T_t}f( \bullet )|$ is bounded.

Some integral and maximal operators related to starlike sets

We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product sp

The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type

We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in C n {C^n} contains an open set in C n {C^n} which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.

The Hull of Holomorphy of a Nonisotropic Ball in a Real Hypersurface of Finite Type

We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in ${C^n}$ contains an open set in ${C^n}$ which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.

Estimates for maximal functions along hypersurfaces

Estimates for maximal functions along hypersurfaces

Strong Lp estimates for maximal functions with respect to singular measures; with applications to exceptional sets

Strong Lp estimates for maximal functions with respect to singular measures; with applications to exceptional sets

𝐿^{𝑝} estimates for maximal functions and Hilbert transforms along flat convex curves in 𝐑²

𝐿^{𝑝} estimates for maximal functions and Hilbert transforms along flat convex curves in 𝐑²