In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, and , where is the weighted Lorentz space and is a rearrangement invariant space in . The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of weights.
We study the high-dimensional Hausdorff operators on the Morrey space and on the Campanato space. We establish their sharp boundedness on these spaces. Particularly, our results solve an open question posted by E. Liflyand (2013).
The main purpose of this paper is to consider a new definition of Hom-left-symmetric bialgebra. The coboundary Hom-left-symmetric bialgebra is also studied. In particular, we give a necessary and sufficient condition that -matrix is a solution of the Hom--equation by a cocycle condition.
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