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Search: All articles in the CMB digital archive with keyword integral operator

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1. 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 space
Categories:42B20, 42B30

2. 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 space
Categories:42B20, 42B25

3. 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 operator

4. CMB 2011 (vol 56 pp. 593)

Liu, Congwen; Zhou, Lifang
On the $p$-norm of an Integral Operator in the Half Plane
We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

Keywords:Bergman projection, integral operator, $L^p$-norm, the upper half plane
Categories:47B38, 47G10, 32A36

5. CMB 2008 (vol 51 pp. 618)

Valmorin, V.
Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra ${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of $\overline{\C}$-valued $\C$-linear maps on the space ${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions
Categories:32A60, 45P05, 46F30

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