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

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1. CMB Online first

Li, Junfeng; Yu, Haixia
 Oscillatory Hyper-Hilbert Transform Associated with Plane Curves In this paper, the bounded properties of oscillatory hyper-Hilbert transform along certain plane curves $\gamma(t)$ $$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$ were studied. For a general curves, these operators are bounded in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their boundedness in $L^p(\mathbb{R}^{2})$ were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$. Keywords:oscillatory hyper-Hilbert transform, oscillatory integralCategories:42B20, 42B35

2. CMB Online first

Rocha, Pablo Alejandro
 A remark on certain integral operators of fractional type For $m, n \in \mathbb{N}$, $1\lt m \leq n$, we write $n = n_1 + \dots + n_m$ where $\{ n_1, \dots, n_m \} \subset \mathbb{N}$. Let $A_1, \dots, A_m$ be $n \times n$ singular real matrices such that $\bigoplus_{i=1}^{m} \bigcap_{1\leq j \neq i \leq m} \mathcal{N}_j = \mathbb{R}^{n},$ where $\mathcal{N}_j = \{ x : A_j x = 0 \}$, $dim(\mathcal{N}_j)=n-n_j$ and $A_1+ \dots+ A_m$ is invertible. In this paper we study integral operators of the form $T_{r}f(x)= \int_{\mathbb{R}^{n}} \, |x-A_1 y|^{-n_1 + \alpha_1} \cdots |x-A_m y|^{-n_m + \alpha_m} f(y) \, dy,$ $n_1 + \dots + n_m = n$, $\frac{\alpha_1}{n_1} = \dots = \frac{\alpha_m}{n_m}=r$, $0 \lt r \lt 1$, and the matrices $A_i$'s are as above. We obtain the $H^{p}(\mathbb{R}^{n})-L^{q}(\mathbb{R}^{n})$ boundedness of $T_r$ for $0\lt p\lt \frac{1}{r}$ and $\frac{1}{q}=\frac{1}{p} - r$. Keywords:integral operator, Hardy spaceCategories:42B20, 42B30

3. CMB Online first

Ding, Yong; Lai, Xudong
 On a singular integral of Christ-JournÃ© type with homogeneous kernel In this paper, we prove that the following singular integral defined by $$T_{\Omega,a}f(x)=\operatorname{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^d}\cdot m_{x,y}a\cdot f(y)dy$$ is bounded on $L^p(\mathbb{R}^d)$ for $1\lt p\lt \infty$ and is of weak type (1,1), where $\Omega\in L\log^+L(\mathbb{S}^{d-1})$ and $m_{x,y}a=:\int_0^1a(sx+(1-s)y)ds$ with $a\in L^\infty(\mathbb{R}^d)$ satisfying some restricted conditions. Keywords:CalderÃ³n commutator, rough kernel, weak type (1,1)Category:42B20

4. CMB 2017 (vol 60 pp. 571)

Li, Ji; Wick, Brett D.
 Weak Factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of Multilinear Riesz Transforms This paper provides a constructive proof of the weak factorization of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms. Keywords:Hardy space, BMO space, multilinear Riesz transform, weak factorizationCategories:42B35, 42B20

5. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
 Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively. Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey spaceCategories:42B20, 42B25

6. CMB 2016 (vol 59 pp. 834)

Liao, Fanghui; Liu, Zongguang
 Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations In this paper, using CalderÃ³n's reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel-Lizorkin and Besov spaces associated with Zygmund dilations. Keywords:Triebel-Lizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theoremCategories:42B20, 42B35

7. CMB 2015 (vol 59 pp. 104)

He, Ziyi; Yang, Dachun; Yuan, Wen
 Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means. Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-functionCategories:46E35, 42B25, 42B20, 42B35

8. CMB 2014 (vol 58 pp. 19)

Chen, Jiecheng; Hu, Guoen
 Compact Commutators of Rough Singular Integral Operators Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier transform estimates and approximation to the operator $T_{\Omega}$ by integral operators with smooth kernels, it is proved that if $b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain minimal size condition, then the commutator generated by $b$ and $T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for appropriate index $p$. The associated maximal operator is also considered. Keywords:commutator,singular integral operator, compact operator, maximal operatorCategory:42B20

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. CMB 2006 (vol 49 pp. 3)

 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

16. CMB 2004 (vol 47 pp. 3)

 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 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

18. 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

19. 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

20. 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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