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Results 1 - 25 of 29 |
1. CMB 2011 (vol 56 pp. 326)
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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 restriction Categories:42B10, 28A12 |
2. CMB 2011 (vol 56 pp. 3)
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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 limits Categories:58G25, 81Q50, 35P20, 42B05 |
3. CMB 2011 (vol 55 pp. 646)
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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 |
4. CMB 2011 (vol 55 pp. 555)
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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 |
5. CMB 2011 (vol 55 pp. 708)
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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 |
6. CMB 2011 (vol 55 pp. 303)
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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 |
7. CMB 2010 (vol 54 pp. 113)
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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)
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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 |
9. CMB 2010 (vol 54 pp. 172)
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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)
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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 |
11. CMB 2009 (vol 53 pp. 263)
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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 HardyÂLittlewood 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 |
12. CMB 2009 (vol 52 pp. 521)
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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 |
13. CMB 2006 (vol 49 pp. 414)
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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 |
14. CMB 2006 (vol 49 pp. 3)
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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 |
15. CMB 2005 (vol 48 pp. 260)
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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 Category:42B10 |
16. CMB 2004 (vol 47 pp. 3)
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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)
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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
$0 Categories:42B15, 42B25 |
18. CMB 2002 (vol 45 pp. 46)
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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)
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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)
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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)
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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 2000 (vol 43 pp. 17)
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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)
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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)
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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 Category:42B20 |
25. CMB 1998 (vol 41 pp. 404)
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$L^p$-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 Category:42B20 |
