Chin-Cheng Lin

Hardy spaces associated with different homogeneities and boundedness of composition operators

It is well known that standard Calderon-Zygmund singular integral operators with the isotropic and non-isotropic homogeneities are bounded on the classical H(R) and non-isotropic H h(R ), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderon-Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. It is interesting that such a Hardy space has surprisingly a multiparameter structure associated with the underlying mixed homogeneities arising from two singular integral operators under consideration. The Calderon-Zygmund decomposition and an interpolation theorem hold on such new Hardy spaces.

Hardy spaces and theTb theorem

AbstractIt is well-known that Calderón-Zygmund operators T are bounded on Hp for

A wavelet characterization for the dual of weighted Hardy spaces

\frac{n}{{n + 1}}< p \leqslant 1

Rough Bilinear Fractional Integrals

provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space Hp to a new Hardy space Hbp. To develop an Hbp theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the LP, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.

TWO-WEIGHT NORM INEQUALITIES FOR THE ROUGH FRACTIONAL INTEGRALS

In this article, we establish a new atomic decomposition for f ∈ Lw ∩ H p w , where the decomposition converges in Lw-norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on Lw and 0 < p ≤ 1, we obtain (i) if T is uniformly bounded in L p w-norm for all w-p-atoms, then T can be extended to be bounded from H p w to L p w ; (ii) if T is uniformly bounded in H p w-norm for all w-p-atoms, then T can be extended to be bounded on H p w; (iii) if T is bounded on H p w , then T can be extended to be bounded from H p w to L p w .

CARLESON MEASURE SPACES ASSOCIATED TO PARA-ACCRETIVE FUNCTIONS

We define the weighted Carleson measure space CMO p /ω using wavelets, where the weight function w belongs to the Muckenhoupt class. Then we show that CMO p / ω is the dual space of the weighted Hardy space H p / ω by using sequence spaces. As an application, we give a wavelet characterization of BMOω.

Summability of Hermite Series

The notion of local maximal operators and objects associated to them is introduced and systematically studied in the general setting of measure metric spaces. The locality means here some restrictions on the radii of involved balls. The notion encompasses different definitions dispersed throughout the literature. One of the aims of the paper is to compare properties of the ‘local’ objects with the ‘global’ ones (i.e. these with no restrictions on the radii of balls). An emphasis is put on the case of locality function of Whitney type. Some aspects of this specific case were investigated earlier by two out of three authors of the present paper.

Tb theorem on product spaces

The authors give the boundedness of the bilinear fractional integrals with rough kernels, which is an extension of a new result obtained by Kenig and Stein.