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Search: MSC category 42B25 ( Maximal functions, Littlewood-Paley theory )

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1. CMB Online first

Wang, Li-an Daniel
A Multiplier Theorem on Anisotropic Hardy Spaces
We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss.

Keywords:anisotropic Hardy space, multiplier, Fourier transform
Categories:42B30, 42B25, 42B35

2. CMB Online first

Saito, Hiroki; Tanaka, Hitoshi
The Fefferman-Stein type inequalities for strong and directional maximal operators in the plane
The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with an additional composition of the Hardy-Littlewood maximal operator in the plane.

Keywords:directional maximal operator, Fefferman-Stein type inequality, Hardy-Littlewood maximal operator, strong maximal operator
Categories:42B25, 42B35

3. CMB Online first

Sickel, Winfried; Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls
Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest.

Keywords:Besov space, Triebel-Lizorkin space, ball average, Calderón reproducing formula
Categories:42B25, 46E35, 42B35

4. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces
In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey space
Categories:42B20, 42B25

5. CMB Online first

Liu, Feng; Wu, Huoxiong
Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation
Categories:42B25, 46E35

6. CMB 2015 (vol 59 pp. 104)

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

7. CMB 2015 (vol 58 pp. 757)

Han, Yanchang
Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces
In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure.

Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

8. CMB 2015 (vol 58 pp. 808)

Liu, Feng; Wu, Huoxiong
On the Regularity of the Multisublinear Maximal Functions
This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity
Categories:42B25, 46E35

9. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse Hölder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained.

Keywords:Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

10. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces
Under the assumption that $\mu$ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$
Categories:42B25, 47B47, 42B20, 47A30

11. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$.

Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates
Categories:42B20, 42B25, 35S05, 47G30

12. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
Improved Range in the Return Times Theorem
We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$.

Keywords:Return Times Theorem, maximal multiplier, maximal inequality
Categories:42B25, 37A45

13. CMB 2011 (vol 55 pp. 303)

Han, Yongsheng; Lee, Ming-Yi; Lin, Chin-Cheng
Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces
In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$, where the decomposition converges in $L^2_w$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L^2_w$ and $0
Keywords:$A_p$ weights, atomic decomposition, Calderón reproducing formula, weighted Hardy spaces
Categories:42B25, 42B30

14. CMB 2010 (vol 54 pp. 100)

Fan, Dashan; Wu, Huoxiong
On the Generalized Marcinkiewicz Integral Operators with Rough Kernels
A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on $L^p$ and Triebel--Lizorkin spaces is established.

Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domain
Categories:42B20, , , , , 42B25, 42B30, 42B99

15. CMB 2010 (vol 53 pp. 491)

Jizheng, Huang; Liu, Heping
The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy--Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.

Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel
Categories:42B25, 43A62

16. CMB 2009 (vol 53 pp. 263)

Feuto, Justin; Fofana, Ibrahim; Koua, Konin
Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of Hardy–Littlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.

Keywords:fractional maximal operator, fractional integral, space of homogeneous type
Categories:42B35, 42B20, 42B25

17. CMB 2009 (vol 52 pp. 521)

Chen, Yanping; Ding, Yong
The Parabolic Littlewood--Paley Operator with Hardy Space Kernels
In this paper, we give the $L^p$ boundedness for a class of parabolic Littlewood--Paley $g$-function with its kernel function $\Omega$ is in the Hardy space $H^1(S^{n-1})$.

Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernel
Categories:42B20, 42B25

18. CMB 2006 (vol 49 pp. 3)

Al-Salman, Ahmad
On a Class of Singular Integral Operators With Rough Kernels
In this paper, we study the $L^p$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on $L^p$ provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spaces
Categories:42B20, 42B15, 42B25

19. CMB 2003 (vol 46 pp. 191)

Kim, Yong-Cheol
Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces
Let $\{A_t\}_{t>0}$ be the dilation group in $\mathbb{R}^n$ generated by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let $\varrho\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$ be a $A_t$-homogeneous distance function defined on $\mathbb{R}^n$. For $f\in \mathfrak{S}(\mathbb{R}^n)$, we define the maximal quasiradial Bochner-Riesz operator $\mathfrak{M}^{\delta}_{\varrho}$ of index $\delta>0$ by $$ \mathfrak{M}^{\delta}_{\varrho} f(x)=\sup_{t>0}|\mathcal{F}^{-1} [(1-\varrho/t)_+^{\delta}\hat f ](x)|. $$ If $A_t=t I$ and $\{\xi\in \mathbb{R}^n\mid \varrho(\xi)=1\}$ is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that $\mathfrak{M}^{\delta}_{\varrho}$ is well defined on $H^p(\mathbb{R}^n)$ when $\delta=n(1/p-1/2)-1/2$ and $0n(1/p-1/2)-1/2$ and $0
Categories:42B15, 42B25

20. CMB 2002 (vol 45 pp. 25)

Bloom, Steven; Kerman, Ron
Extrapolation of $L^p$ Data from a Modular Inequality
If an operator $T$ satisfies a modular inequality on a rearrangement invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between the indices of the space, then the Lebesgue inequality $\int |Tf|^p \leq C \int |f|^p$ holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form $\int \Phi (|Tf|) \leq \int \Phi (C |f|)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$ and $I(\Phi)$ as well.


21. CMB 2000 (vol 43 pp. 330)

Hare, Kathryn E.
Maximal Operators and Cantor Sets
We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on $L^2$ if the Cantor set has positive Hausdorff dimension.

Keywords:maximal functions, Cantor set, lacunary set
Categories:42B25, 43A46

22. CMB 1998 (vol 41 pp. 306)

Kolasa, Lawrence A.
Oscillatory integrals with nonhomogeneous phase functions related to Schrödinger equations
In this paper we consider solutions to the free Schr\" odinger equation in $n+1$ dimensions. When we restrict the last variable to be a smooth function of the first $n$ variables we find that the solution, so restricted, is locally in $L^2$, when the initial data is in an appropriate Sobolev space.

Categories:42A25, 42B25

23. CMB 1997 (vol 40 pp. 296)

Hare, Kathryn E.
A general approach to Littlewood-Paley theorems for orthogonal families
A general lacunary Littlewood-Paley type theorem is proved, which applies in a variety of settings including Jacobi polynomials in $[0, 1]$, $\su$, and the usual classical trigonometric series in $[0, 2 \pi)$. The theorem is used to derive new results for $\LP$ multipliers on $\su$ and Jacobi $\LP$ multipliers.

Categories:42B25, 42C10, 43A80

24. CMB 1997 (vol 40 pp. 169)

Cruz-Uribe, David
The class $A^{+}_{\infty}(\lowercase{g})$ and the one-sided reverse Hölder inequality
We give a direct proof that $w$ is an $A^{+}_{\infty}(g)$ weight if and only if $w$ satisfies a one-sided, weighted reverse H\"older inequality.

Keywords:one-sided maximal operator, one-sided $(A_\infty)$, one-sided, reverse Hölder inequality

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