Discrete Paraproduct Operators on Variable Hardy Spaces

Let$p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators$\unicode[STIX]{x1D70B}_{b}$on variable Hardy spaces$H^{p(\cdot )}(\mathbb{R}^{n})$, where$b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators$T$are bounded on$H^{p(\cdot )}(\mathbb{R}^{n})$if and only if$T^{\ast }1=0$, where$\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$and$\unicode[STIX]{x1D716}$is the regular exponent of kernel of$T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

Canonical Hankel wavelet transformation and Calderón’s reproducing formula

In this work, we have discussed some basic properties of canonical Hankel wavelet transformation. Further the Calder?n?s reproducing formula for linear canonical Hankel wavelet transformation is obtained.

Some Properties of Triebel–Lizorkin and Besov Spaces Associated with Zygmund Dilations

In this paper, using Calderón’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.

Smooth Decompositions of Triebel-Lizorkin and Besov Spaces on Product Spaces of Homogeneous Type

We introduce Triebel-Lizorkin and Besov spaces by Calderón’s reproducing formula on product spaces of homogeneous type. We also obtain smooth atomic and molecular decompositions for these spaces.

Calderón's reproducing formula for Hankel convolution

Calderón-type reproducing formula for Hankel convolution is established using the theory of Hankel transform.