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Boundedness of multiple Marcinkiewicz integral operators with rough kernels
Huoxiong Wu 대한수학회 2006 대한수학회지 Vol.43 No.3
This paper is concerned with giving some rather weak sizeconditions implying the L^p boundedness of the multipleMarcin-kiewicz integrals for some fixed 1<p<fz, which
Huoxiong Wu 대한수학회 2009 대한수학회지 Vol.46 No.3
In this paper, the author studies the k-th commutators of oscillatory singular integral operators with a BMO function and phases more general than polynomials. For 1<p<∞, the L^p-boundedness of such operators are obtained provided their kernels belong to the spaces L((log^+)L)^{k+1}(S^{n-1}). The results of the corresponding maximal operators are also established. In this paper, the author studies the k-th commutators of oscillatory singular integral operators with a BMO function and phases more general than polynomials. For 1<p<∞, the L^p-boundedness of such operators are obtained provided their kernels belong to the spaces L((log^+)L)^{k+1}(S^{n-1}). The results of the corresponding maximal operators are also established.
Wu, Huoxiong Korean Mathematical Society 2009 대한수학회지 Vol.46 No.3
In this paper, the author studies the k-th commutators of oscillatory singular integral operators with a BMO function and phases more general than polynomials. For 1 < p < $\infty$, the $L^p$-boundedness of such operators are obtained provided their kernels belong to the spaces $L(log+L)^{k+1}(S^{n-1})$. The results of the corresponding maximal operators are also established.
BOUNDEDNESS OF MULTIPLE MARCINKIEWICZ INTEGRAL OPERATORS WITH ROUGH KERNELS
Wu Huoxiong Korean Mathematical Society 2006 대한수학회지 Vol.43 No.3
This paper is concerned with giving some rather weak size conditions implying the $L^P$ boundedness of the multiple Marcin-kiewicz integrals for some fixed $1\;<\;p\;<\;{\infty}$, which essentially improve and extend some known results.
POINTWISE ESTIMATES AND BOUNDEDNESS OF GENERALIZED LITTLEWOOD-PALEY OPERATORS IN BMO(ℝ<sup>n</sup>)
Wu, Yurong,Wu, Huoxiong Korean Mathematical Society 2015 대한수학회보 Vol.52 No.3
In this paper, we study the generalized Littlewood-Paley operators. It is shown that the generalized g-function, Lusin area function and $g^*_{\lambda}$-function on any BMO function are either infinite everywhere, or finite almost everywhere, respectively; and in the latter case, such operators are bounded from BMO($\mathbb{R}^n$) to BLO($\mathbb{R}^n$), which improve and generalize some previous results.
POINTWISE ESTIMATES AND BOUNDEDNESS OF GENERALIZED LITTLEWOOD-PALEY OPERATORS IN BMO(Rn)
Yurong Wu,Huoxiong Wu 대한수학회 2015 대한수학회보 Vol.52 No.3
In this paper, we study the generalized Littlewood-Paley operators. It is shown that the generalized g-function, Lusin area function and g∗λ-function on any BMO function are either infinite everywhere, or finite almost everywhere, respectively; and in the latter case, such operators are bounded from BMO(Rn) to BLO(Rn), which improve and generalize some previous results.
VECTOR-VALUED INEQUALITIES FOR THE COMMUTATORS OF SINGULAR INTEGRALS WITH ROUGH KERNELS
Tang, Lin,Wu, Huoxiong Korean Mathematical Society 2011 대한수학회지 Vol.48 No.4
In this paper, we establish the vector-valued inequalities for the commutators of singular integrals with rough kernels. In particular, our results can essentially improve some well-known results.
CHARACTERIZATION OF FUNCTIONS VIA COMMUTATORS OF BILINEAR FRACTIONAL INTEGRALS ON MORREY SPACES
Mao, Suzhen,Wu, Huoxiong Korean Mathematical Society 2016 대한수학회보 Vol.53 No.4
For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.
Characterization of functions via commutators of bilinear fractional integrals on Morrey spaces
Suzhen Mao,Huoxiong Wu 대한수학회 2016 대한수학회보 Vol.53 No.4
For $b\in L^1_{\rm loc}(\mathbb{R}^n)$, let $\mathcal{I}_\alpha$ be the bilinear fractional integral operator, and $[b,\,\mathcal{I}_\alpha]_i$ be the commutator of $\mathcal{I}_\alpha$ with pointwise multiplication $b$ ($i=1,\,2$). This paper shows that if the commutator $[b, \mathcal{I}_\alpha]_i$ for $i=1$ or $2$ is bounded from the product Morrey spaces $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to the Morrey space $L^{q,\lambda}(\mathbb{R}^n)$ for some suitable indexes $\lambda,\,\lambda_1$, $\lambda_2$ and $p_1,\,p_2,\,q$, then $b\in BMO(\mathbb{R}^n)$, as well as that the compactness of $[b,\,\mathcal{I}_\alpha]_i$ for $i=1$ or $2$ from $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$ implies that $b\in CMO(\mathbb{R}^n)$ (the closure in $BMO(\mathbb{R}^n)$ of the space of $C^\infty(\mathbb{R}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO(\mathbb{R}^n)$ functions or $CMO(\mathbb{R}^n)$ functions in essential ways.
A NOTE ON MULTILINEAR PSEUDO-DIFFERENTIAL OPERATORS AND ITERATED COMMUTATORS
Wen, Yongming,Wu, Huoxiong,Xue, Qingying Korean Mathematical Society 2020 대한수학회보 Vol.57 No.4
This paper gives a sparse domination for the iterated commutators of multilinear pseudo-differential operators with the symbol σ belonging to the Hörmander class, and establishes the quantitative bounds of the Bloom type estimates for such commutators. Moreover, the C<sub>p</sub> estimates for the corresponding multilinear pseudo-differential operators are also obtained.