Can my neighbor put a sign that says "dog prohibited to poop" if we share the yard in a duplex?

Hi. I'm an owner of a half duplex and I have been having some problems with the other owner. The duplex was built side ways so the yard is on the side of his half, making my part of the yard that was decided all the way down to the side crossing his half duplex and his half yard. The parking lot is to the side of my half duplex. Here is where the most recent problem comes. The parking lot also has a green area. So I have been taking my dog out to pee or poop to that area. Now this neighbor has been putting some "dog prohibited to poop" flag like he was the owner of everything. What can I do? I feel harassed by him because he also told me that he may have to report me. If I stand out of my door, I'm stalking his wife. If I take my dog out, I'm intimidating them. The thing is that he has a dog long before I got mine, and I bought my half before he got his. Please I need help.