Lusin Area Research Articles

Characterization of $$H^1$$ H 1 Sobolev Spaces by Square Functions of Marcinkiewicz Type

We establish characterization of $$H^1$$ Sobolev spaces by certain square functions of Marcinkiewicz type. The square functions are related to the Lusin area integrals. Also, in the one dimensional case, the non-periodic version of the function of Marcinkiewicz is applied to characterize weighted $$H^1$$ Sobolev spaces.

Littlewood–Paley characterizations of fractional Sobolev spaces via averages on balls

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

A Criterion for the Existence of Lp Boundary Values of Solutions to an Elliptic Equation

The paper is devoted to the study of the boundary behavior of solutions to a second-order elliptic equation. A criterion is established for the existence in Lp, p > 1, of a boundary value of a solution to a homogeneous equation in the self-adjoint form without lower order terms. Under the conditions of this criterion, the solution belongs to the space of (n − 1)- dimensionally continuous functions; thus, the boundary value is taken in a much stronger sense. Moreover, for such a solution to the Dirichlet problem, estimates for the nontangential maximal function and for an analog of the Lusin area integral hold.

Variable Anisotropic Hardy Spaces and Their Applications

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

AbstractIn this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .

Anisotropic variable Hardy–Lorentz spaces and their real interpolation

Let p(⋅):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition, q∈(0,∞] and A be a general expansive matrix on Rn. In this article, the authors first introduce the anisotropic variable Hardy–Lorentz space HAp(⋅),q(Rn) associated with A, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of HAp(⋅),q(Rn), respectively, in terms of the atom and the Lusin area function. As an application, the authors prove that the anisotropic variable Hardy–Lorentz space HAp(⋅),q(Rn) serves as the intermediate space between the anisotropic variable Hardy space HAp(⋅)(Rn) and the space L∞(Rn) via the real interpolation. This, together with a special case of the real interpolation theorem of H. Kempka and J. Vybíral on the variable Lorentz space, further implies the coincidence between HAp(⋅),q(Rn) and the variable Lorentz space Lp(⋅),q(Rn) when essinfx∈Rnp(x)∈(1,∞).

Mixed boundary value problems on cylindrical domains

We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients $A$ are close to coefficients $A_0$ that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if $A_0$ has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an $L^2$-bounded non-tangential maximal function or satisfies a Lusin area bound. To this end, we combine the first-order approach to elliptic systems with the Kato square root estimate for operators with mixed boundary conditions.

LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

Let$(X,d,\unicode[STIX]{x1D707})$be a metric measure space endowed with a distance$d$and a nonnegative, Borel, doubling measure$\unicode[STIX]{x1D707}$. Let$L$be a nonnegative self-adjoint operator on$L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup$e^{-tL}$satisfies a Gaussian upper bound. In this paper, we prove that for any$p\in (0,\infty )$and$w\in A_{\infty }$, the weighted Hardy space$H_{L,S,w}^{p}(X)$associated with$L$in terms of the Lusin (area) function and the weighted Hardy space$H_{L,G,w}^{p}(X)$associated with$L$in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duonget al.[‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’,J. Geom. Anal.26(2016), 1617–1646], who proved that$H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$for$p\in (0,1]$and$w\in A_{\infty }$by imposing an extra assumption of a Moser-type boundedness condition on$L$. Our result is new even in the unweighted setting, that is, when$w\equiv 1$.

-estimates for the nontangential maximal function of the solution to a second-order elliptic equation

The paper is concerned with the properties of the solution to a Dirichlet problem for a homogeneous second-order elliptic equation with -boundary function, . The same conditions are imposed on the coefficients of the equation and the boundary of the bounded domain as were used to establish the solvability of this problem. The -norm of the nontangential maximal function is estimated in terms of the -norm of the boundary value. This result depends on a new estimate, proved below, for the nontangential maximal function in terms of an analogue of the Lusin area integral. Bibliography: 31 titles.

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calder\'on-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

Variable weak Hardy spaces and their applications

Let p(⋅):Rn→(0,∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on Rn, WHp(⋅)(Rn), via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WHp(⋅)(Rn), respectively, by means of atoms, molecules, the Lusin area function, the Littlewood–Paley g-function or gλ⁎-function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ-order Calderón–Zygmund operators from Hp(⋅)(Rn) to WHp(⋅)(Rn) including the critical case when p−=n/(n+δ) or when p−=n/(n+γ), where p−:=essinfx∈Rnp(x).

Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates

Let p(⋅):Rn→(0,1] be a variable exponent function satisfying the globally log-Hölder continuous condition and L a non-negative self-adjoint operator on L2(Rn) whose heat kernels satisfying the Gaussian upper bound estimates. Let HLp(⋅)(Rn) be the variable exponent Hardy space defined via the Lusin area function associated with the heat kernels {e−t2L}t∈(0,∞). In this article, the authors first establish the atomic characterization of HLp(⋅)(Rn); using this, the authors then obtain its non-tangential maximal function characterization which, when p(⋅) is a constant in (0,1], coincides with a recent result by L. Song and L. Yan (2016) and further induces the radial maximal function characterization of HLp(⋅)(Rn) under an additional assumption that the heat kernels of L have the Hölder regularity.

Littlewood–Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls

Abstract In this paper, the authors characterize second-order Sobolev spaces W2,p(ℝn), with p ∊ [2,∞) and n ∊ N or p ∊ (1, 2) and n ∊ {1, 2, 3}, via the Lusin area function and the Littlewood–Paley g*λ -function in terms of ball means.

Parameterized Littlewood--Paley operators and their commutators on Herz spaces with variable exponents

The aim of this paper is to deal with Littlewood--Paley operators with real parameters, including the parameterized Lusin area integrals and the parameterized Littlewood--Paley $g_{\lambda}^{\ast}$-functions, and their commutators on Herz spaces with two variable exponents $p(\cdot),~q(\cdot)$. By using the properties of the Lebesgue spaces with variable exponents, the boundedness of the parameterized Littlewood--Paley operators and their commutators generated respectively by BMO function and Lipschitz function is obtained on those Herz spaces.

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.

Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a measurable function satisfying some decay condition and some locally log-H\"older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space $H^{p(\cdot)}(\mathbb R^n)$ in terms of the Littlewood-Paley $g$-function, the Lusin area function and the $g_\lambda^\ast$-function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_\lambda^\ast$-function. The $p(\cdot)$-Carleson measure characterization for the dual space of $H^{p(\cdot)}(\mathbb R^n)$, the variable exponent Campanato space $\mathcal{L}_{1,p(\cdot),s}(\mathbb R^n)$, in terms of the intrinsic function is also presented.

The Commutators of Intrinsic Square Functions on Weighted Herz Spaces

In this paper, we consider the boundedness properties of commutator operators generated by $$BMO(\mathbb {R}^n)$$ function and intrinsic square function including the Lusin area integral, Littlewood-Paley g-function and $$g^*_{\uplambda }$$ -function on weighted Herz spaces $$\dot{K}_q^{\alpha ,p}(\omega _1,\omega _2)(\mathbb {R}^n)$$ with general Muckenhoupt weights.

Existence and boundedness of multilinear Littlewood–Paley operators on Campanato spaces

In this paper, we established the boundedness of multilinear Littlewood–Paley operators on Campanato spaces, including the multilinear g-function, multilinear Lusin's area integral and multilinear Littlewood–Paley gλ⁎-function. We proved that if the above operators are finite for one point, then they are finite almost everywhere. Moreover, norm inequalities do hold for the above operators on the corresponding Campanato spaces.

Real-variable characterizations of Musielak-Orlicz-Hardy spaces associated with Schrödinger operators on domains

Let n >= 3, Omega be a strongly Lipschitz domain of R-n and L-Omega :=-Delta + V a Schrodinger operator on L-2(Omega) with the Dirichlet boundary condition, where Delta is the Laplace operator and the nonnegative potential V belongs to the reverse Holder class RHq0(R-n) for some q(0) > n/2. Assume that the growth function phi : R-n x [0, infinity) -> [0, infinity) satisfies that phi(x, .) is an Orlicz function, phi(., t) is an element of A(infinity)(R-n) (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index i(phi) is an element of (n/n+delta, 1], where delta := min {mu(0), 2- n/q(0)} and mu(0) is an element of (0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Delta on Omega. In this article, the authors first show that the heat kernels of L-Omega satisfy the Gaussian upper bound estimates and the Holder continuity. The authors then introduce the 'geometrical' Musielak-Orlicz-Hardy space H-phi,H-LRn,H-r(Omega) via H-phi,H-LRn (R-n), the Hardy space associated with L-Rn :=-Delta + Von R-n, and establish its several equivalent characterizations, respectively, in terms of the non-tangential or the vertical maximal functions or the Lusin area functions associated with L-Omega. All the results essentially improve the known results even on Hardy spaces H-LRn,r(p)(Omega) with p is an element of (n / (n + delta),1] (in this case, phi(x, t) := t(p) for all x is an element of Omega and t is an element of [0, infinity)). Copyright (C) 2016 John Wiley & Sons, Ltd.

POINTWISE ESTIMATES AND BOUNDEDNESS OF GENERALIZED LITTLEWOOD-PALEY OPERATORS IN BMO(ℝn)

In this paper, we study the generalized Littlewood-Paley operators. It is shown that the generalized g-function, Lusin area function and -function on any BMO function are either infinite everywhere, or finite almost everywhere, respectively; and in the latter case, such operators are bounded from BMO() to BLO(), which improve and generalize some previous results.