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In this paper, we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the Hörmander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators to provide sharp subelliptic Sobolev and Besov estimates for operators in the $ (\rho,\delta) $ (ρ , δ) -Hörmander classes. In contrast with the available results in the literature in the setting of compact Lie groups, we allow Fefferman-type estimates in the critical case $ \rho =\delta. $ ρ = δ. Interpolation properties between Besov spaces and Triebel–Lizorkin spaces are also investigated. [ABSTRACT FROM AUTHOR]

We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature. [ABSTRACT FROM AUTHOR]