Non-tangential Maximal Function Research Articles

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a ball quasi‐Banach function space on , which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space , associated with , via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of , which proves to be a ball Campanato‐type function space associated with . All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of , the authors further prove that can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

The regularity problem in domains with lower dimensional boundaries

In the present paper we establish the solvability of the Regularity boundary value problem in domains with lower dimensional boundaries (flat and Lipschitz) for operators whose coefficients exhibit small oscillations analogous to the Dahlberg-Kenig-Pipher condition.The proof follows the classical strategy of showing bounds on the square function and the non-tangential maximal function. The key novelty and difficulty of this setting is the presence of multiple non-tangential derivatives. To solve it, we consider a cylindrical system of derivatives and establish new estimates on the “angular derivatives”.

Heat kernels and theory of Hardy spaces associated to Schrödinger operators on stratified groups

Let G be a stratified group and let ΔG be a sub-Laplacian on G. In this paper, we consider the Schrödinger operator L=ΔG+V, where the potential V is a nonnegative polynomial. We first prove the upper bound of the higher order derivatives of the heat kernel of L. We then establish a theory of Hardy spaces HLp(G) associated to L for the full range p∈(0,1]. Particularly, we prove that for any p∈(0,1], the Hardy space HLp(G) introduced in terms of nontangential or radial maximal functions associated to the semigroup e−tL admits a new atomic decomposition. Moreover, we provide the description of the dual spaces of these new Hardy spaces in terms of certain local Campanato spaces related to the potential V. As a byproduct, we obtain the maximal function characterizations for the local Hardy spaces associated to an arbitrary critical function.

Hardy spaces and quasiconformal maps in the Heisenberg group

We define Hardy spaces Hp, 0<p<∞, for quasiconformal mappings on the Korányi unit ball B in the first Heisenberg group H1. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of p0(K)>0 such that every K-quasiconformal map f:B→f(B)⊂H1 belongs to Hp for all 0<p<p0(K). Second, we give two equivalent conditions for the Hp membership of a quasiconformal map f, one in terms of the radial limits of f, and one using a nontangential maximal function of f. As an application, we characterize Carleson measures on B via integral inequalities for quasiconformal mappings on B and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from Rn to H1. A crucial difference between the proofs in Rn and H1 is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.

The Dirichlet-conormal problem for the heat equation with inhomogeneous boundary conditions

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base Ω⊂Rd and a time-dependent separation Λ. Under certain mild regularity assumptions on Λ, we show that for any q>1 sufficiently close to 1, the mixed problem in Lq is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space Lq1 and the Neumann data in Lq, there is a unique solution and the non-tangential maximal function of its gradient is in Lq on the lateral boundary of the domain. When q=1, a similar result is shown when the data is in the Hardy space. Under the additional condition that the boundary of the domain Ω is Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of m variables, where m=0,…,d−2, with respect to the Hausdorff distance, we also prove the unique solvability result for any q∈(1,(m+2)/(m+1)). In particular, when m=0, i.e., Λ is Reifenberg-flat of co-dimension 2, we derive the Lq solvability in the optimal range q∈(1,2). For the Laplace equation, such results were established in [25,24,5] and [13].

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u ) and L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-% \operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion [M. Akman, S. Hofmann, J. M. Martell and T. Toro, Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition, preprint 2021, https://arxiv.org/abs/1901.08261v3] is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaining a square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above.

The regularity problem for degenerate elliptic operators in weighted spaces

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let L_w be a degenerate elliptic operator with degeneracy given by a fixed weight w\in A_2(dx) in \mathbb{R}^n , and consider the associated block second order degenerate elliptic problem in the upper-half space \mathbb{R}_+^{n+1} . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces L^p(vdw) , where v is a Muckenhoupt weight with respect to the underlying natural weighted space (\mathbb{R}^n, wdx) . We recover earlier results in the non-degenerate case (when w\equiv 1 , and with or without weight v ). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L^2(dx) -solvability of the regularity problem for the block operator \mathbb{L}_\alpha u(x,t) = -|x|^{\alpha} \mathrm{\operatorname{div}}_x \big(|x|^{-\alpha } \,A(x) \nabla_x u(x,t)\big)-\partial_{t}^2 u(x,t) for any complex-valued uniformly elliptic matrix A and for all -\varepsilon&lt;\alpha&lt;{2n}/(n+2) , where \varepsilon depends just on the dimension and the ellipticity constants of A .

Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) −Extension

Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to L q ( ℝ + n + 1 , μ ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp –capacity, including its dual form, the Lp –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

The Fourier transform of anisotropic Hardy spaces with variable exponents and their applications

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors obtain that the Fourier transform of $f\in \mathcal{H}^{p(\cdot)}_A({\mathbb {R}}^n)$ coincides with a continuous function $F$ on $\mathbb{R}^n$ in the sense of tempered distributions. As applications, the authors further conclude a higher order convergence of the continuous function $F$ at the origin and then give a variant of the Hardy-Littlewood inequality in the setting of anisotropic Hardy spaces with variable exponents.

Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

Characterizations of Product Hardy Spaces in Bessel Setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

Molecular Decomposition of Anisotropic Hardy Spaces With Variable Exponents

Let A be an expansive dilation on ℝn, and p(·): ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Holder continuous condition. Let \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the authors establish its molecular decomposition, which is still new even in the classical isotropic setting (in the case A:= 2In×n). As applications, the authors obtain the boundedness of anisotropic Calderon-Zygmund operators from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to Lp(·)(ℝn) or from \(H_A^{p\left(\cdot \right)}\left({{\mathbb{R}^n}} \right)\) to itself.

The dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

We consider the mixed Dirichlet-conormal problem on irregular domains in Two types of regularity results will be discussed: the regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension 2 or is locally sufficiently close to a Lipschitz function of m variables, where For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.

Molecular characterizations of variable anisotropic Hardy spaces with applications to boundedness of Calderón–Zygmund operators

Let $$p(\cdot ):\ \mathbb {R}^n\rightarrow (0,\infty ]$$ be a variable exponent function satisfying the globally log-Holder continuous condition and A a general expansive matrix on $$\mathbb {R}^n$$ . Let $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ be the variable anisotropic Hardy space associated with A defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , the author establishes its molecular characterization with the known best possible decay of molecules. As an application, the author obtains a criterion on the boundedness of linear operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ , which is used to prove the boundedness of anisotropic Calderon–Zygmund operators on $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ . In addition, the boundedness of anisotropic Calderon–Zygmund operators from $$H_A^{p(\cdot )}(\mathbb {R}^n)$$ to the variable Lebesgue space $$L^{p(\cdot )}(\mathbb {R}^n)$$ is also presented. All these results are new even in the classical isotropic setting.

The Dirichlet problem in domains with lower dimensional boundaries

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb{R}^n \setminus \mathbb{R}^d$, with $d < n-1$. Following results of David, Feneuil and Mayboroda, we introduce an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all $q > 1$, provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d = n-1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n-1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.

Hardy spaces for the Dunkl harmonic oscillator

AbstractLet Δ and be the Dunkl Laplacian and the Dunkl harmonic oscillator respectively. We define the Hardy space associated with the Dunkl harmonic oscillator by means of the nontangential maximal function with respect to the semigroup . We prove that the space admits characterizations by relevant Riesz transforms and atomic decompositions. The atoms which occur in the atomic decompositions are of local type.

Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel–Lizorkin spaces

Let S(Rn) be the Schwartz class on Rn and S∞(Rn)≔ϕ∈S(Rn):∫Rnxαϕ(x)dx=0for any multi-indexα∈({0,1,…})n,and S′(Rn) and S∞′(Rn) be their dual spaces, respectively. Let a→≔(a1,…,an)∈[1,∞)n, p→≔(p1,…,pn)∈(0,1]n, and Ha→p→(Rn)⊂S′(Rn) be the anisotropic mixed-norm Hardy space, associated with an anisotropic quasi-homogeneous norm |⋅|a→, defined via the non-tangential grand maximal function. In this article, via the known atomic characterizations and Lusin area function characterizations of Ha→p→(Rn), the authors first establish a new atomic characterization of Ha→p→(Rn) in terms of S∞′(Rn). Applying this atomic characterization, the authors then obtain the Littlewood–Paley g-function characterization of Ha→p→(Rn) in terms of S∞′(Rn), which further induces the identification of Ha→p→(Rn) and certain anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces. As applications, the authors obtain the dual theorem on some anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces and the boundedness of Fourier multipliers as well as anisotropic pseudo-differential operators on Ha→p→(Rn). All these results are new even for isotropic mixed-norm Hardy spaces on Rn.

Notes on exterior Lp boundary value problems for nonhomogeneous elliptic operators

In terms of non-smooth domains and non-homogeneous elliptic operators, this paper is devoted to study non-tangential maximal function estimates for exterior Lp regular problems. We do not repeat the layer potential arguments (which is a possible way). Instead appealing to a radial maximal function we can employ the results on homogeneous operators to make the proof much easier. Nevertheless, this method relies on an exterior W1,p estimate, which is established in the course of the proof.

L$^2$ well-posedness of boundary value problems for parabolic systems with measurable coefficients

We prove the first positive results concerning boundary value problems in the upper half-space for second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green’s representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in L ^2 -Sobolev spaces together with invertibility of layer potentials, and perturbation results. On the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversal independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation.

Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

AbstractLet (𝒳,d,μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimensionω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spacesH*p,q(𝒳)H_*^{p,q}\left( \mathcal{X} \right)with the optimal rangep∈(ωω+η,∞)p \in \left( {{\omega \over {\omega + \eta }},\infty } \right)andq ∈(0, ∞]. When andp∈(ωω+η,1]p \in ({\omega \over {\omega + \eta }},1]q ∈(0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption ofμby fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the caseq ∈(0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.