Kwok-Pun Ho

Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the φ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

Hardy's inequality and Hausdorff operators on rearrangement-invariant Morrey spaces

We generalize the Minkowski inequality, the Hardy–Littlewood–Pólya inequalities and the Hardy inequalities to rearrangement-invariant Morrey spaces. We obtain these results by extending the notion of Boyd’s indices to the rearrangementinvariant Morrey spaces. Our method also applied to establish the boundedness of Hausdorff operators on rearrangement-invariant Morrey spaces.

Generalized Boyd's indices and applications

Abstract A generalization of the Boyd indices to Banach function space is introduced. Some applications of these generalized Boyd´s indices to function spaces such as Littlewood–Paley characterization, Fefferman–Stein vector-valued inequalities and wavelet characterization of Banach function space are shown.

The Ahlfors–Beurling transform on Morrey spaces with variable exponents

ABSTRACT We establish the vector-valued inequalities of the Ahlfors–Beurling operator on Morrey spaces with variable exponents. As consequences of these inequalities, we have the boundedness of the Ahlfors–Beurling transform on Lebesgue spaces with variable exponents and Morrey spaces. The results obtained in this paper are new in the case of Morrey spaces.

Modular estimates of fractional integral operators and k-plane transforms

ABSTRACT The modular estimates for the fractional integral operators and the k-plane transforms are obtained in this paper. These estimates are obtained by using the modular estimates of Hardy operators and the modular interpolation theorem.

Hardy’s inequality on Hardy–Morrey spaces

Abstract We generalize the Hardy inequality to Hardy–Morrey spaces.

Exponential integrability of martingales

Abstract We obtain the exponential integrability of the maximal function, the quadratic variation and the conditional quadratic variation of bounded martingales and exponential integrable martingales.