| The bounded variation spaces are widely used in analysis,statistics,and image processing.The compactness criterion for subsets of bounded variation spaces is not simply formulated for a long time.Recently,using the concept of equivariation,Bugajewski and Gulgowski obtained a compactness criterion for subsets of the bounded variation space in the sense of Jordan.Inspired by that,the dissertation mainly studies the precompactness of Banach space-valued bounded variation spaces.The dissertation introduces several classes of Banach space-valued bounded variation spaces,such as Jordan variation spaces,Wiener variation spaces,Wiener-Young variation spaces,Waterman variation spaces,Schramm variation spaces,Riesz variation spaces,Korenblum variation spaces and Schramm-Korenblum variation spaces.By the notion of equivariation,we obtain sufficient conditions for precompact sets of above bounded variation spaces,parts of which are sufficient and necessary.The results are studied in univariate and bivariate cases,respectively.Finally,we obtain some sufficient conditions for precompact sets of bounded variation spaces of variable exponent,including univariate and bivariate cases. |
