Hmmm, I didn't think of myself as having lower back issues until pretty far out, like a year and a half out. My observant PT, however, pointed out that my SI joint was weak and noisy on one side about six months out and she worked to fix that.
I did notice some strange lower ab weakness a year out when I discovered I couldn't do some lower ab exercises like mountain climbers very well. (Two years out I can do them.) Maybe that's part of what you feel? Some subtle ab weaknesses? Hip flexors and psoas are traumatized by the surgery and by the time you spent compensating with a bum hip, too.
The real lower back stiffness and weakness I didn't feel until I was doing a lot of walking and standing (of all things). When H and I were exploring new cities, walking everywhere, going to museums, standing and staring at exhibits, at the end of the day my back would be very stiff. This was as recently as a year and a half post op. That has gone away, too.
I'll agree not to push too hard this early. Let yourself heal. Yes, do keep a list of weaknesses and watch how they improve or don't improve. Some stuff goes away on its own, some new stuff pops up when you think you are fine. You can ask Dr Gross about them at your check ups. At three-four weeks, I wouldn't worry about lower back stiffness.
(Something to look forward to: I was so excited to bend over at six weeks. I envisioned myself sitting on the floor and picking things up off the floor with ease. Ha! I was still stiff and weak. Yeah, I could do it, but it wasn't pretty. Lee had to coach me on getting down on the floor.)