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In this paper we study the ($L$^{$p$}($u$^{$p$}),$L$^{$q$}($v$^{$q$})) boundedness of the higher order commutators$T$_{$Ω,α,b$}^{$m$}and$M$_{$Ω,α,b$}^{$m$}formed by the fractional integral operator$T$_{$Ω,α$}, the fractional maximal operator$M$_{$Ω,α$}, and a function$b(x)$in BMO($v$), respectively.

This paper examines the boundary behaviour of superharmonic functions on a half-space in terms of their behaviour along lines normal to the boundary. It is shown that, if the set of lines along which such functions grow quickly is (in a certain sense) metrically dense, then the set of lines along which they are bounded below is topologically small.

We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space$X$is reflexive if and only if it admits and equivalent norm such that there exists$x$_{0}≠0 in$X$and a weak-^{*}-open subset$A$of the dual space, satisfying that$x$^{*}⊗$x$_{0}attains its numerical radius. for each$x$^{*}in$A$.