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

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1. CMB Online first

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger 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 HÃ¶lder 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 SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

2. 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 HÃ¶rmander-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

3. 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 CalderÃ³n-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 estimatesCategories:42B20, 42B25, 35S05, 47G30

4. 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 inequalityCategories:42B25, 37A45

5. 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$weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 6. CMB 2010 (vol 54 pp. 100) Fan, Dashan; Wu, Huoxiong  On the Generalized Marcinkiewicz Integral Operators with Rough Kernels A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on$L^p$and Triebel--Lizorkin spaces is established. Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domainCategories:42B20, , , , , 42B25, 42B30, 42B99 7. CMB 2010 (vol 53 pp. 491) Jizheng, Huang; Liu, Heping  The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup In this paper, we discuss various maximal functions on the Laguerre hypergroup$\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$follow from the interpolation. Some applications are included. Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernelCategories:42B25, 43A62 8. CMB 2009 (vol 53 pp. 263) Feuto, Justin; Fofana, Ibrahim; Koua, Konin  Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams We give weighted norm inequalities for the maximal fractional operator$ \mathcal M_{q,\beta }$of HardyÂLittlewood and the fractional integral$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)$. The conditions on the weights are stated in terms of Orlicz norm. Keywords:fractional maximal operator, fractional integral, space of homogeneous typeCategories:42B35, 42B20, 42B25 9. CMB 2009 (vol 52 pp. 521) Chen, Yanping; Ding, Yong  The Parabolic Littlewood--Paley Operator with Hardy Space Kernels In this paper, we give the$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})$. Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernelCategories:42B20, 42B25 10. CMB 2006 (vol 49 pp. 3) Al-Salman, Ahmad  On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the$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$provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25 11. CMB 2003 (vol 46 pp. 191) Kim, Yong-Cheol  Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces Let$\{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

12. 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. Category:42B25

13. 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 setCategories:42B25, 43A46

14. CMB 1998 (vol 41 pp. 306)

Kolasa, Lawrence A.
 Oscillatory integrals with nonhomogeneous phase functions related to SchrÃ¶dinger 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

15. 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

16. CMB 1997 (vol 40 pp. 169)

Cruz-Uribe, David
 The class $A^{+}_{\infty}(\lowercase{g})$ and the one-sided reverse HÃ¶lder 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 HÃ¶lder inequalityCategory:42B25