Non-tangential Maximal Function Research Articles

A CHARACTERIZATION OF HARDY SPACE ON STRONGLY LIPSCHITZ DOMAINS OF ℝn BY LITTLEWOOD–PALEY–STEIN FUNCTION

Let Ω be a strongly Lipschitz domain of ℝn and define Hardy spaces on Ω by non-tangential maximal function. In this paper, we will give a characterization of the Hardy spaces on Ω by Littlwood–Paley–Stein function associated to L, where L is an elliptic second-order divergence operator. In order to get our result, we also consider the Lusin area integral characterization of the Hardy spaces on Ω.

Boundary value problems for the Laplacian in convex and semiconvex domains

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in R n , when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel–Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).

Mixed boundary-value problems for Maxwell’s equations

We study the Maxwell system with mixed boundary conditions in a Lipschitz domain Ω in ℝ 3 . It is assumed that two disjoint, relatively open subsets ∑ e , ∑ h of ∂Ω such that ∑ e ∩ ∑ h = ∂∑ e = ∂∑ h have been fixed, and one prescribes the tangential components of the electric and magnetic fields on ∑ e and ∑ h , respectively. Under suitable geometric assumptions on ∂Ω, ∑ e and ∑ h , we prove that this boundary value problem is well-posed when L P -estimates for the nontangential maximal function are sought, with p near 2. A higher-dimensional version of this result is established as well, in the language of differential forms. This extends earlier work by R. Brown and by the author and collaborators.

Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n .F or α ∈ (0, ∞ )d enote byH p(X), H p d(X), and H ∗ ,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X )w henp ∈ (1, ∞) and with each other when p ∈ (n/(n +1 ) ,1). An atomic characterization for H ∗ ,p (X )w ithp ∈ (n/(n +1 ),1) is also established; moreover, in the range p ∈ (n/(n +1 ), 1), it is proved that the spaceH ∗ ,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X )t o some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩(1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

W 1,p (R n )-boundedness for maximal functions

In this paper, we get W1,p(Rn)-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results.

The mixed problem for the Lamé system in a class of Lipschitz domains

We consider the mixed problem for the Lamé system { L u = 0 in Ω , u | D = f D on D , ∂ u ∂ ρ = f N on N , ( ∇ u ) ∗ ∈ L p ( ∂ Ω ) in the class of bounded Lipschitz creased domains. Here D and N partition ∂ Ω and ∂ / ∂ ρ stands for the traction operator. We suppose the Dirichlet data f D has one derivative in L p ( D ) and the traction data f N is in L p ( N ) . For p in a small interval containing 2, we find a unique solution to the mixed problem subject to the condition that the non-tangential maximal function of the gradient of the solution is in L p ( ∂ Ω ) .

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H (X), H d (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderon reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ∗,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

On the energy of inviscid singular flows

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space B 3 , ∞ 1 / 3 . When the singular set of the solution is (or belongs to) a smooth manifold, we derive various L p -space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of u is conserved provided the one-sided non-tangential limits to the surface exist and the non-tangential maximal function is L 3 integrable, while the maximal function of the pressure is L 3 / 2 integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.

The regularity and Neumann problem for non-symmetric elliptic operators

We establish optimal $L^p$ bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the $L^p$-norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data.

On a Littlewood-Paley type inequality

The following is proved: If u is a function harmonic in the unit ball B C R N and if 0 < p < 1, then the inequality ∫ ∂B u*(y) p dσ ≤ C p,N (|u(0)| p +∫ B (1-|x| p-1 |∇u(x)| p dV(x)) holds, where u* is the nontangential maximal function of u. This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

We prove distributional inequalities that imply the comparability of theLpnorms of the multiplicative square function ofuand the nontangential maximal function oflogu, whereuis a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case whenuis a harmonic function is considered.

Vector-valued Hardy spaces in non-smooth domains

We characterize the Radon–Nikodým property of a Banach space X in terms of the existence of non-tangential limits of X -valued harmonic functions u defined in a domain D ⊂ R n , n > 2 , with Lipschitz boundary and belonging to maximal Hardy spaces. This extends the same result previously known for the unit disk of C . We also prove an atomic decomposition of the Borel X -valued measures in ∂ D that arise as boundary limits of X -valued harmonic functions whose non-tangential maximal function is integrable with respect to harmonic measure of ∂ D.

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.

Multiplicative Square Functions

We study regularity properties of a positive measure in the euclidean space in terms of two square functions which are the multiplicative analogues of the usual martingale square function and of the Lusin area function of a harmonic function. The size of these square functions is related to the rate at which the measure doubles at small scales and determines several regularity properties of the measure. We consider the non-tangential maximal function of the logarithm of the densities of the measure in the dyadic setting, and of the logarithm of the harmonic extension of the measure, in the continuous setting. We relate the size of these maximal functions to the size of the corresponding square functions. Fatou type results, $L^p$ estimates and versions of the Law of the Iterated Logarithm are proved. As applications we introduce a hyperbolic version of the Lusin Area function of an analytic mapping from the unit disc into itself, and use it to characterize inner functions. Another application to the theory of quasiconformal mappings is given showing that our methods can also be applied to prove a result by Din'kyn's on the smoothness of quasiconformal mappings of the disc.

On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains

AbstractWe consider the Neumann problem for the Schrödinger equations –Δu + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class ℬn, in a connected Lipschitz domain Ω Rn. Given boundary data g in Hp or Lp for 1 – ɛ &lt; p ≤ 2, where 0 &lt; ɛ &lt; , it is shown that there is a unique solution, u, that solves the Neumann problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂Ω). Moreover, the uniform estimates are found.

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.

On the atomic decomposition for Hardy spaces on Lipschitz domains of [formula omitted

Let Ω be a special Lipschitz domain on R n , and L be a second-order elliptic self-adjoint operator in divergence form L=−div( A∇) on Lipschitz domain Ω subject to Neumann boundary condition. In this paper, we give a simple proof of the atomic decomposition for Hardy spaces H N p(Ω) of Ω for a range of p, by means of nontangential maximal function associated with the Poisson semigroup of L.

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.

Hardy spaces and divergence operators on strongly Lipschitz domains of [formula omitted

Let Ω be a strongly Lipschitz domain of R n . Consider an elliptic second-order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with H 1( R n) if Ω= R n , with H r 1(Ω) under the Dirichlet boundary condition, and with H z 1(Ω) under the Neumann boundary condition.

Anisotropic Hardy spaces and wavelets

In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky. Given a dilation A, that is an n × n matrix all of whose eigenvalues λ satisfy |λ| > 1, define the radial maximal function M φf(x) := sup k∈Z |(f ∗ φk)(x)|, where φk(x) = | detA|φ(Ax). Here φ is any test function in the Schwartz class with ∫ φ 6= 0. For 0 < p < ∞ we introduce the corresponding anisotropic Hardy space H A as a space of tempered distributions f such that M φf belongs to L (R). Anisotropic Hardy spaces enjoy the basic properties of the classical Hardy spaces. For example, it turns out that this definition does not depend on the choice of the test function φ as long as ∫ φ 6= 0. These spaces can be equivalently introduced in terms of grand, tangential, or nontangential maximal functions. We prove the Calderon-Zygmund decomposition which enables us to show the atomic decomposition of H A. As a consequence of atomic decomposition we obtain the description of the dual to H A in terms of Campanato spaces. We provide a description of the natural class of operators acting on H A, i.e., Calderon-Zygmund singular integral operators. We also give a full classification of dilations generating the same space H A in terms of spectral properties of A. In the second part of this paper we show that for every dilation A preserving some lattice and satisfying a particular expansiveness property there is a multiwavelet in the Schwartz class. We also show that for a large class of dilations (lacking this property) all multiwavelets must be combined minimally supported in frequency, and thus far from being regular. We show that r-regular (tight frame) multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy space H A. We also describe the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space. 2000 Mathematics Subject Classification. Primary 42B30, 42C40; Secondary 42B20, 42B25.