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

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26. 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

27. 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

28. 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, paraproduct
Categories:42B, 46F

29. 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

30. 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 restriction

31. 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.


32. 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

33. 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

34. 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.


35. 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

36. 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

37. 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

38. 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.


39. 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, symbol

40. 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 space

41. CMB 1998 (vol 41 pp. 478)

Oberlin, Daniel M.
Convolution with measures on curves in $\bbd R^3$
We study convolution properties of measures on the curves $(t^{a_1}, t^{a_2}, t^{a_3})$ in $\hbox{\Bbbvii R}^3$.

Categories:42B15, 42B20

42. 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

43. 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

44. 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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