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Search: MSC category 42B30 ( $H^p$-spaces )

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1. CMB Online first

Liu, Yu; Qi, Shuai
 Endpoint estimates of Riesz transforms associated with generalized SchrÃ¶dinger operators In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized SchrÃ¶dinger operator. Keywords:SchrÃ¶dinger operator, fundamental solution, Riesz transformCategories:35J10, 42B20, 42B30

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

Wang, Li-an Daniel
 A Multiplier Theorem on Anisotropic Hardy Spaces We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss. Keywords:anisotropic Hardy space, multiplier, Fourier transformCategories:42B30, 42B25, 42B35

4. CMB 2015 (vol 58 pp. 507)

Hsu, Ming-Hsiu; Lee, Ming-Yi
 VMO Space Associated with Parabolic Sections and its Application In this paper we define $VMO_\mathcal{P}$ space associated with a family $\mathcal{P}$ of parabolic sections and show that the dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence. Keywords:Monge-Ampere equation, parabolic section, Hardy space, BMO, VMOCategory:42B30

5. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse HÃ¶lder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained. Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

6. CMB 2011 (vol 55 pp. 303)

Han, Yongsheng; Lee, Ming-Yi; Lin, Chin-Cheng
 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$weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 7. CMB 2010 (vol 54 pp. 100) Fan, Dashan; Wu, Huoxiong  On the Generalized Marcinkiewicz Integral Operators with Rough Kernels A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on$L^p$and Triebel--Lizorkin spaces is established. Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domainCategories:42B20, , , , , 42B25, 42B30, 42B99 8. CMB 2002 (vol 45 pp. 46) Dafni, Galia  Local$\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
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