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Search: MSC category 42B20 ( Singular and oscillatory integrals (Calderon-Zygmund, etc.) )

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1. CJM Online first

 Sharp Localized Inequalities for Fourier Multipliers In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of LÃ©vy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator . Keywords:Fourier multiplier, martingale, laminateCategories:42B15, 60G44, 42B20

2. CJM Online first

Wu, Xinfeng
 Weighted Carleson Measure Spaces Associated with Different Homogeneities In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two CalderÃ³n-Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong-Stein. Keywords:composition of operators, weighted Carleson measure spaces, dualityCategories:42B20, 42B35

3. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
 Beltrami Equation with Coefficient in Sobolev and Besov Spaces Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$. Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, CalderÃ³n-Zygmund operatorsCategories:30C62, 35J99, 42B20

4. CJM 2012 (vol 65 pp. 299)

Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito
 On Multilinear Fourier Multipliers of Limited Smoothness In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of CalderÃ³n and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed. Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spacesCategories:42B15, 42B20

5. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
 Boundedness of CalderÃ³n-Zygmund Operators on Non-homogeneous Metric Measure Spaces Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a CalderÃ³n-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a CalderÃ³n-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition. Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, CalderÃ³n-Zygmund operator, maximal operatorCategories:42B20, 42B25, 30L99

6. CJM 2011 (vol 64 pp. 257)

Chen, Yanping; Ding, Yong; Wang, Xinxia
 Compactness of Commutators for Singular Integrals on Morrey Spaces In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right. Keywords:singular integral, commutators, compactness, VMO, Morrey spaceCategories:42B20, 42B99

7. CJM 2009 (vol 61 pp. 807)

Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo
 Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients We prove the $L^p(\mathbb{R}^d)$ ($1 Categories:42B20, 42B25 8. CJM 2006 (vol 58 pp. 154) Prestini, Elena  Singular Integrals on Product Spaces Related to the Carleson Operator We prove$L^p(\mathbb T^2)$boundedness,$1 Categories:42B20, 42B08

9. CJM 2004 (vol 56 pp. 655)

Tao, Xiangxing; Wang, Henggeng
 On the Neumann Problem for the SchrÃ¶dinger Equations with Singular Potentials in Lipschitz Domains We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon Keywords:Neumann problem, SchrÃ¶dinger equation, Lipschitz, domain, reverse HÃ¶lder class,$H^p$spaceCategories:42B20, 35J10 10. CJM 2003 (vol 55 pp. 504) Chen, Jiecheng; Fan, Dashan; Ying, Yiming  Certain Operators with Rough Singular Kernels We study the singular integral operator $$T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(|y|) \Omega(y') |y|^{-n-\alpha} f(x-y)\,dy,$$ defined on all test functions$f$,where$b$is a bounded function,$\alpha\geq 0$,$\Omega(y')$is an integrable function on the unit sphere$S^{n-1}$satisfying certain cancellation conditions. We prove that, for$1 Categories:42B20, 42B25, 42B15

11. CJM 2001 (vol 53 pp. 1031)

Sampson, G.; Szeptycki, P.
 The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l} {a_l(1-\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated. Categories:42B20, 46B70, 47G10

12. CJM 2000 (vol 52 pp. 381)

Miyachi, Akihiko
 Hardy Space Estimate for the Product of Singular Integrals $H^p$ estimate for the multilinear operators which are finite sums of pointwise products of singular integrals and fractional integrals is given. An application to Sobolev space and some examples are also given. Keywords:$H^p$ space, multilinear operator, singular integral, fractional integration, Sobolev spaceCategory:42B20

13. CJM 1998 (vol 50 pp. 29)

Ding, Yong; Lu, Shanzhen
 Weighted norm inequalities for fractional integral operators with rough kernel Given function $\Omega$ on ${\Bbb R^n}$, we define the fractional maximal operator and the fractional integral operator by  M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n-\alpha}} \int_{|\,y|1)\$, homogeneous of degree zero. Categories:42B20, 42B25