Keng Hao Ooi

We study the thinness of sets with respect to weighted local Riesz capacities, where the associated weights are of local Muckenhoupt class. The thinness is characterized in terms of Wolff potentials. We also obtain the Kellogg property and Choquet theorem as applications.

We provide several characterizations of Sobolev multiplier spaces of Lorentz type and their preduals. Block decomposition and Köthe dual of such preduals are discussed. As an application, the boundedness of local Hardy-Littlewood maximal function on these spaces will be justified.

We verify a conjecture of D. R. Adams on a capacitary strong type inequality that generalizes the classical capacitary strong type inequality of V. G. Maz'ya. As a result, we characterize related function spaces as Köthe duals to a class of Sobolev multiplier type spaces. Moreover, using tools from nonlinear potential theory, weighted norm inequalities, and Banach function space theory, we show that these spaces are also isomorphic to more concrete spaces that are easy to use and fit in well with the modern theory of function spaces of harmonic analysis.

We characterize Sobolev multiplier spaces and their preduals in terms of Littlewood–Paley and Lusin area functions. The vector-valued and weighted norm estimates of Hardy–Littlewood maximal operator will be used as the main tools in such characterizations.

We address some vector valued maximal inequalities in a type of Sobolev multiplier spaces associated with Bessel capacities. We show that the vector valued maximal inequalities are of local type.