You can't claim a home office deduction for the space in your home if you are able to accomplish the same tasks in the barn. The home office has to be the principal place of business, or a place where you regularly meet with customers.
You could use the square footage of the barn in relation to the square footage of the entire building structure of the property to determine how to allocate common expenses such as mortgage interest and property taxes.
I don't see a problem with depreciating the cost of the tennis court over it's estimated useful life and deducting that as a cost of the tennis business.
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