Non-tangential Maximal Function Research Articles

The Dirichlet problem for higher order equations in composition form

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB⁎∇(adivA∇u)=0, where A and B are elliptic matrices with complex-valued bounded measurable coefficients and a is an accretive function. Elliptic operators of this type naturally arise, for instance, via a pull-back of the bilaplacian Δ2 from a Lipschitz domain to the upper half-space. More generally, this form is preserved under a Lipschitz change of variables, contrary to the case of divergence-form fourth-order differential equations. We establish well-posedness of the Dirichlet problem for the equation Lu=0, with boundary data in L2, and with optimal estimates in terms of nontangential maximal functions and square functions.

The mixed problem for the Lamé system in two dimensions

We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R2. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space Lp(N) and Dirichlet data from a Sobolev space W1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p0>1 so that for 1<p<p0, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in Lp(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy–Sobolev spaces with p in (p1,1] for some p1<1.

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.

Real-variable characterizations of Orlicz–Hardy spaces on strongly Lipschitz domains of $\mathbb{R}^n$

Let \Omega be a strongly Lipschitz domain of \mathbb{R}^n , whose complement in \mathbb{R}^n is unbounded. Let L be a second order divergence form elliptic operator on L^2 (\Omega) with the Dirichlet boundary condition, and the heat semigroup generated by L having the Gaussian property (G_{\mathrm{diam}(\Omega)}) with the regularity of its kernels measured by \mu\in(0,1] , where \mathrm{diam}(\Omega) denotes the diameter of \Omega . Let \Phi be a continuous, strictly increasing, subadditive and positive function on (0,\infty) of upper type 1 and of strictly critical lower type p_{\Phi}\in(n/(n+\mu),1] . In this paper, the authors introduce the Orlicz–Hardy space H_{\Phi,\,r}(\Omega) by restricting arbitrary elements of the Orlicz–Hardy space H_{\Phi}(\mathbb{R}^n) to \Omega and establish its atomic decomposition by means of the Lusin area function associated with \{e^{-tL}\}_{t\ge0} . Applying this, the authors obtain two equivalent characterizations of H_{\Phi,\,r}(\Omega) in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by L .

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$. Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension $3$ or higher. The existence of a proof {\em a priori} to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.

The mixed problem in Lipschitz domains with general decompositions of the boundary

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domainΩ⊂Rn\Omega \subset \mathbf {R}^n,n≥2n\geq 2, with boundary that is decomposed as∂Ω=D∪N\partial \Omega =D\cup N, withDDandNNdisjoint. We letΛ\Lambdadenote the boundary ofDD(relative to∂Ω\partial \Omega) and impose conditions on the dimension and shape ofΛ\Lambdaand the setsNNandDD. Under these geometric criteria, we show that there existsp0&gt;1p_0&gt;1depending on the domainΩ\Omegasuch that forppin the interval(1,p0)(1,p_0), the mixed problem with Neumann data in the spaceLp(N)L^p(N)and Dirichlet data in the Sobolev spaceW1,p(D)W^{1, p}(D)has a unique solution with the non-tangential maximal function of the gradient of the solution inLp(∂Ω)L^p(\partial \Omega ). We also obtain results forp=1p=1when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

The Mixed Problem for the Laplacian in Lipschitz Domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.

Estimates of the nontangential maximal function for solutions of a second-order elliptic equation

Estimates of the nontangential maximal function for solutions of a second-order elliptic equation

Hardy spaces associated with magnetic Schrödinger operators on strongly Lipschitz domains

Let p∈(0,1], Ω be a strongly Lipschitz domain in Rn and A:=−(∇−ia→)⋅(∇−ia→)+V a magnetic Schrödinger operator on L2(Ω) satisfying the Dirichlet boundary condition, where a→:=(a1,…,an)∈Lloc2(Ω,Rn) and 0≤V∈Lloc1(Ω). In this paper, the authors introduce the Hardy space HAp(Ω) by the Lusin area function associated with A and establish its equivalent characterization via the non-tangential maximal function associated with {e−tA}t>0. As applications, the authors obtain the boundedness of the Riesz transforms LkA−12, k∈{1,…,n}, from HAp(Ω) to Lp(Ω) for p∈(0,1] and the fractional integral A−γ from HAp(Ω) to HAq(Ω) for 0<p<q≤1 and γ:=n2(1p−1q), where Lk is the closure of ∂∂xk−iak in L2(Ω).

New real-variable characterizations of Musielak–Orlicz Hardy spaces

Let φ:Rn×[0,∞)→[0,∞) be such that φ(x,⋅) is an Orlicz function and φ(⋅,t) is a Muckenhoupt A∞(Rn) weight. The Musielak–Orlicz Hardy space Hφ(Rn) is defined to be the space of all f∈S′(Rn) such that the grand maximal function f∗ belongs to the Musielak–Orlicz space Lφ(Rn). Luong Dang Ky established its atomic characterization. In this paper, the authors establish some new real-variable characterizations of Hφ(Rn) in terms of the vertical or the non-tangential maximal functions, or the Littlewood–Paley g-function or gλ∗-function, via first establishing a Musielak–Orlicz Fefferman–Stein vector-valued inequality. Moreover, the range of λ in the gλ∗-function characterization of Hφ(Rn) coincides with the known best results, when Hφ(Rn) is the classical Hardy space Hp(Rn), with p∈(0,1], or its weighted variant.

On the Carleson duality

As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L_2-Whitney averages belongs to L_2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X, and characterize the pointwise multipliers from X to L_2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L_p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

Characterizations of Hardy spaces associated to higher order elliptic operators

In this paper, the authors first show that the classical Hardy space H1(Rn) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e−tP and e−tP, respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m (m⩾1). Moreover, the authors also prove that H1(Rn) can be characterized by the Riesz transforms ∇mP−1/2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space HL1(Rn) and give its characterizations by area integrals related to the semigroups e−tL and e−tL, respectively. Finally, authors give the (HL1,L1) boundedness of Riesz transforms, square functions and maximal functions associated with L.

Local Hardy spaces of Musielak-Orlicz type and their applications

Let φ: ℝ n × [0,∞) → [0,∞) be a function such that φ(x, ·) is an Orlicz function and $$\phi ( \cdot ,t) \in \mathbb{A}_\infty ^{loc} \left( {\mathbb{R}^n } \right)$$ (the class of local weights introduced by Rychkov). In this paper, the authors introduce a local Musielak-Orlicz Hardy space h φ(ℝ n ) by the local grand maximal function, and a local BMO-type space bmo φ (ℝ n ) which is further proved to be the dual space of h φ(ℝ n ). As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (ℝ n ), characterized by Nakai and Yabuta, is just the dual of $$L^1 \left( {\mathbb{R}^n } \right) + h_{\Phi _0 } \left( {\mathbb{R}^n } \right)$$ , where ϕ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ0 a Musielak-Orlicz function induced by ϕ. Characterizations of h φ (ℝ n ), including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h φ (ℝ n ), from which, the authors further deduce some criterions for the boundedness on h φ (ℝ n ) of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h φ (ℝ n ).

Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of $\mathbb{R}^n$

Let $\Omega$ be either $\mathbb{R}^n$ or an unbounded strongly Lipschitz domain of $\mathbb{R}^n$, and $\Phi$ be a continuous, strictly increasing, subadditive and positive function on $(0,\infty)$ of upper type 1 and of strictly critical lower type $p_{\Phi}\in(n/(n+1),1]$. Let $L$ be a divergence form elliptic operator on $L^2 (\Omega)$ with the Neumann boundary condition and the heat semigroup generated by $L$ have the Gaussian property $(G_{\infty})$. In this paper, the authors introduce the Orlicz-Hardy space $H_{\Phi,\,L}(\Omega)$ via the nontangential maximal function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$, and establish its equivalent characterization in terms of the Lusin area function associated with $\{e^{-t\sqrt{L}}\}_{t\ge0}$. The authors also introduce the "geometrical" Orlicz-Hardy space $H_{\Phi,\,z}(\Omega)$ via the classical Orlicz-Hardy space $H_{\Phi}(\mathbb{R}^n)$, and prove that the spaces $H_{\Phi,\,L}(\Omega)$ and $H_{\Phi,\,z}(\Omega)$ coincide with equivalent norms, from which, characterizations of $H_{\Phi,\,L}(\Omega)$, including the vertical and the nontangential maximal function characterizations associated with $\{e^{-tL}\}_{t\ge0}$, and the Lusin area function characterization associated with $\{e^{-tL}\}_{t\ge0}$, are deduced. All the above results generalize the well-known results of P. Auscher and E. Russ by taking $\Phi(t)\equiv t$ for all $t\in(0,\infty)$.

Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators

Let be a magnetic Schrödinger operator on , , where and . In this paper, the authors establish the equivalent characterizations of the Hardy space for , defined by the Lusin area function associated with , in terms of the radial maximal functions and the non-tangential maximal functions associated with and , respectively. This gives an affirmative answer to an open problem of Xuan Thinh Duong et al. [Ark. Mat. 44 (2006), 261–275]. The boundedness of the Riesz transforms , , from to is also presented, where is the closure of in and .

Conical square functions and non-tangential maximal funtions with respect to the gaussian measure

We study, in L 1 (R n ;) with respect to the gaussian measure, nontangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in L 1 -norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

REAL-VARIABLE CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BESSEL OPERATORS

Let λ &gt; 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and [Formula: see text] be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces Hp((0,∞),dmλ) associated with △λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dmλ(x) ≡ x2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

ORLICZ–HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES

Let [Formula: see text] be a metric space with doubling measure, L a nonnegative self-adjoint operator in [Formula: see text] satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type pω∈(0, 1] and ρ(t) = t-1/ω-1(t-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space [Formula: see text] via the Lusin area function associated to the heat semigroup, and the BMO-type space [Formula: see text]. The authors then establish the duality between [Formula: see text] and [Formula: see text]; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space [Formula: see text]. Characterizations of [Formula: see text], including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let [Formula: see text] and L = -Δ+V be a Schrödinger operator, where [Formula: see text] is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L-1/2is bounded from Hω, L(ℝn) to L(ω). Moreover, if there exist q1, q2∈(0, ∞) such that q1&lt;1&lt;q2and [ω(tq2)]q1is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space Hω, L(ℝn), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = tpfor all t ∈ (0, ∞) and p ∈ (0, 1).

The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions

In this paper we study uniformly elliptic equations withnon-compatible conditions, where $\Omega$ is a bounded Lipchitzdomain, and the right-hand side term and the boundary value of theelliptic equations belong to $L^p (p \geq 2)$ space. Then the optimalweighted $W^{2, p}$ estimates will be given by Whitney decompositionand $L^p$ estimates of non-tangential maximal function associatedto solutions of the elliptic equations.

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L 2 boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A 0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A 0‖ C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A 0‖ C . Our methods yield full characterization of weak solutions, whose gradients have L 2 estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in L 2 by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of ‖A−A 0‖ C and well-posedness for A 0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A 0 by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients A is an operational calculus to prove weighted maximal regularity estimates.