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.

Abstract. In this paper, we give some characterizations of matrices which have defect index one. Recall that an n -by-n matrix A is said to be of class S n (resp., S −1 n ) if its eigenvalues are all in the open unit disc (resp., in the complement of closed unit disc) and rank (I n − A * A) = 1 . We show that an n -by-n matrix A is of defect index one if and only if A is unitarily equivalent to U ⊕ C , where U is a k -by-k unitary matrix, 0 k < n , and C is either of class S n−k or of class S −1 n−k . We also give a complete characterization of polar decompositions, norms and defect indices of powers of S −1 n -matrices. Finally, we consider the numerical ranges of S −1 n -matrices and S n -matrices, and give a generalization of a result of Chien and Nakazato on tridiagonal matrices (cf. [3, Theorem 7]). Mathematics subject classification (2010): 47A12, 15A60.

As passivation layer and anti-reflection layer, silicon nitride (SiNx) thin film has been widely used in photovoltaic devices such as solar cells. The structure of SiNx film with thickness gradient can make full use of different wavelengths of sunlight. In this paper, we have studied this structure for the first time. While introducing a quartz layer by plasma-enhanced chemical vapor deposition (PECVD), we obtained a thin SiNx film in the center and gradually thicker toward the edge. The effects of PECVD process parameters, including deposition time, RF power, dielectric layer thickness, etc. on the thickness gradient of SiNx thin film are systematically studied. The film composition changing in the radial direction is also analyzed by ellipsometry. This study provides an instructive method for controlling the thickness gradient of SiNx films and plays an important role in using this structure to the solar cell application.