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Search: MSC category 42B ( Harmonic analysis in several variables {For automorphic theory, see mainly 11F30} )

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1. CMB 2011 (vol 56 pp. 326)

Erdoğan, M. Burak; Oberlin, Daniel M.
 Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures. Keywords:Fourier transforms of fractal measures, Fourier restrictionCategories:42B10, 28A12

2. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
 Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof. Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limitsCategories:58G25, 81Q50, 35P20, 42B05

3. 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

4. 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

5. 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

6. 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$weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 7. CMB 2010 (vol 54 pp. 113) Hytönen, Tuomas P.  On the Norm of the Beurling-Ahlfors Operator in Several Dimensions The generalized Beurling-Ahlfors operator$S$on$L^p(\mathbb{R}^n;\Lambda)$, where$\Lambda:=\Lambda(\mathbb{R}^n)$is the exterior algebra with its natural Hilbert space norm, satisfies the estimate $$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad p^*:=\max\{p,p'\}$$ This improves on earlier results in all dimensions$n\geq 3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms. Categories:42B20, 60G46 8. 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 9. CMB 2010 (vol 54 pp. 172) Shayya, Bassam  Measures with Fourier Transforms in$L^2$of a Half-space We prove that if the Fourier transform of a compactly supported measure is in$L^2$of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support. Categories:42B10, 28A75 10. 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 11. 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)$. The conditions on the weights are stated in terms of Orlicz norm. Keywords:fractional maximal operator, fractional integral, space of homogeneous typeCategories:42B35, 42B20, 42B25 12. CMB 2009 (vol 52 pp. 521) Chen, Yanping; Ding, Yong  The Parabolic Littlewood--Paley Operator with Hardy Space Kernels In this paper, we give the$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})$. Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernelCategories:42B20, 42B25 13. CMB 2006 (vol 49 pp. 414) Jiang, Liya; Jia, Houyu; Xu, Han  Commutators Estimates on Triebel--Lizorkin Spaces In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces$\dot{F}^{s, q} _p$. Keywords:commutators, Triebel--Lizorkin spaces, paraproductCategories:42B, 46F 14. CMB 2006 (vol 49 pp. 3) Al-Salman, Ahmad  On a Class of Singular Integral Operators With Rough Kernels In this paper, we study the$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$provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions. Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spacesCategories:42B20, 42B15, 42B25 15. CMB 2005 (vol 48 pp. 260) Oberlin, Daniel M.  A Restriction Theorem for a \\$k$-Surface in$\mathbb R ^n$We establish a sharp Fourier restriction estimate for a measure on a$k$-surface in$\mathbb R ^n$, where$n=k(k+3)/2$. Keywords:Fourier restrictionCategory:42B10 16. CMB 2004 (vol 47 pp. 3) Al-Salman, Ahmad; Pan, Yibiao  Singular Integrals With Rough Kernels In this paper we establish the$L^p$boundedness of a class of singular integrals with rough kernels associated to polynomial mappings. Category:42B20 17. CMB 2003 (vol 46 pp. 191) Kim, Yong-Cheol  Weak Type Estimates of the Maximal Quasiradial Bochner-Riesz Operator On Certain Hardy Spaces Let$\{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

18. CMB 2002 (vol 45 pp. 46)

Dafni, Galia
 Local $\VMO$ and Weak Convergence in $\hone$ A local version of $\VMO$ is defined, and the local Hardy space $\hone$ is shown to be its dual. An application to weak-$*$ convergence in $\hone$ is proved. Categories:42B30, 46E99

19. 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

20. CMB 2001 (vol 44 pp. 121)

Wojciechowski, Michał
 A Necessary Condition for Multipliers of Weak Type $(1,1)$ Simple necessary conditions for weak type $(1,1)$ of invariant operators on $L(\rr^d)$ and their applications to rational Fourier multiplier are given. Categories:42B15, 42B20

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 setCategories:42B25, 43A46

22. CMB 2000 (vol 43 pp. 17)

Bak, Jong-Guk
 Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves Suppose that $\g \in C^2\bigl([0,\infty)\bigr)$ is a real-valued function such that $\g(0)=\g'(0)=0$, and $\g''(t)\approx t^{m-2}$, for some integer $m\geq 2$. Let $\Gamma (t)=\bigl(t,\g(t)\bigr)$, $t>0$, be a curve in the plane, and let $d \lambda =dt$ be a measure on this curve. For a function $f$ on $\bR^2$, let $$Tf(x)=(\lambda *f)(x)=\int_0^{\infty} f\bigl(x-\Gamma(t)\bigr)\,dt, \quad x\in\bR^2 .$$ An elementary proof is given for the optimal $L^p$-$L^q$ mapping properties of $T$. Categories:42A85, 42B15

23. CMB 2000 (vol 43 pp. 63)

Iosevich, Alex; Lu, Guozhen
 Sharpness Results and Knapp's Homogeneity Argument We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages imply certain geometric restrictions on the underlying hypersurface. We deduce that these bounds imply that a certain number of principal curvatures do not vanish. Category:42B99

24. CMB 1999 (vol 42 pp. 463)

Hofmann, Steve; Li, Xinwei; Yang, Dachun
 A Generalized Characterization of Commutators of Parabolic Singular Integrals Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1, \dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots \le\az_n$. Denote $|\az|=\az_1+\cdots+\az_n$. We characterize those functions $A(x)$ for which the parabolic Calder\'on commutator $$T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n} K(x-y)[A(x)-A(y)]f(y)\,dy$$ is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{-|\az|-1}K(x)$, $K$ is smooth away from the origin and satisfies a certain cancellation property. Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1-theorem, symbolCategory:42B20

25. CMB 1998 (vol 41 pp. 404)

Al-Hasan, Abdelnaser J.; Fan, Dashan
 $L^p$-boundedness of a singular integral operator Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be an $H^1$ function on the unit sphere satisfying the mean zero property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree $m$ satisfying $Q_m(0)=0$. We prove that the singular integral operator $$T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(|y|) \Omega(\,y) |y|^{-n} f \left( x-Q_m (|y|) y' \right) \,dy$$ is bounded in $L^p (\bR^n)$ for \$1 Keywords:singular integral, rough kernel, Hardy spaceCategory:42B20
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