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The concept of boundary layers, introduced by A. Volberg in [7], is generalized from subsets of the unit disk to subsets of general non-tangentially accessible (NTA) domains. Capacitary conditions of Wiener type series of both necessary and sufficient type for boundary layers are presented and the connection between boundary layers and minimally thin sets is studied.

We introduce a random recursive method for constructing random Salem sets in$R$^{d}. The method is inspired by Salem's construction [13] of certain signular monotonic functions.

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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$A$.