After the economic downturn, many small business owners are finding it hard to keep their heads above water.

keep 在这里是动词，意思是"保持、维持"。 their 是形容词性物主代词，指代前文提到的"many small business owners"（许多小企业主）的。 heads 是 head 的复数形式，本义是"头部"，这里用作比喻。 above 是介词，表示"在...上方"。 water 是名词，意思是"水"，这里用作比喻。 keep one's head above water 是一个英语习语，字面意思是"保持头部在水面上"，比喻"勉强维持生计/经营"，特别指在经济困难时期。