I would give up on lawn and adjust to having a shade garden there.

There are plenty of plants that can handle shaded and tree roots - pretty ones, drought tolerant ones, scented ones. Ferns, hostas, columbines, ... that's what I remember from my stay on the East Coast.

Thanks, lazygardens--I have a number of beds filled with hostas, many kinds of ferns, grasses that tolerate deep shade, liriope, hellebores, etc. and some larger pachysandra groundcover areas--all the usual shade garden suspects. I'm trying to preserve a smallish area of grass so that the back yard doesn't become a sort of undifferentiated bed. This spring I also had installed a sort of stone barrier to help with the runoff problem between the upper and lower sections, and tried re-seeding the upper section, with the result I outlined above. Still hanging onto my hope to continue to have a small grass area nearer the house...but at some point, if I can't figure this out, or if no one has any suggestions, I may have to let it all go to pachysandra, and install paths to get around the property. I'll upload pix when I can get a less contrasty shot.

A more shallow watering, particularly during summer, over tree roots isn't necessarily the worst idea. Those tree roots suck up water like nobody's business, so more frequent watering will be required in this case.

We're talking twice a week, maybe three times if the weather is blisteringly hot.

It's very hard to keep grass in a shady area! If you use synthetic fertilizer, feed normally--it boosts the grass too much and it's easy to put down excessive nitrogen on lawn that doesn't see enough sun to photosynthesize. Most of the benefit from fertilization goes to the trees, which isn't a mistake.

Organically, pour it on. This is less to feed the grass than it is to transform the soil to hold more water. And by all means, use the shampoo. Worst case, it can't hurt. Best case, it'll lighten and loosen the soils and hold more water!

Reseeding every fall is probably in your future, grasses really don't like heavy shade. Avoid bluegrass as it loathes even light shade, and favor the fescues as they do better and have deeper roots.

It's worth looking into grass varieties that tolerate a little more shade (no grass species tolerates full shade). You'll still need to reseed--just less often!

Red creeping fescue might be a good choice for you (it's green, not red!) It's pretty shade tolerant, but...well, you'll need to reseed.

Perennial ryegrass is a fast sprouter that'll look good for about a year, but will still need reseeding in fall. Fortunately, reseeding is not difficult and the grass develops so quickly that this might be the simplest answer for you. Plus rye is a nice grass, tolerant of drier conditions, and feels nice underfoot.

White Dutch clover is shade tolerant and adding some to your mix will give you a greenery cover at grass height that looks good even when the grass doesn't. You may not want clover, however--I sure understand that!

Can you limb up the tree to let more sunlight in?

Thanks for the followups, folks. I've limbed up the maple a few times in the past already, and I could probably get rid of one or two more very large limbs yet without jeopardizing the tree. I'm considering limbing up some other trees in the vicinity as well. I guess my dilemma right now is figuring out what to do about all those pebbles! Like--am I nuts to continue to try sweeping them up before I add a top dressing of compost and re-seed (with a mixture of all those great shade grasses--thanks morpheuspa!)? I've already filled one-and-a-half five gallon buckets, and I'm less than half done...I'm sure the neighbors think I've lost my marbles. I've come to the conclusion that aeration isn't going to be my answer--and the shampoo sounds like it should be a big help! But all those pebbles continue to worry me...there are SO MANY of them!! If they rise to the surface of the compost will I be back at square one? (I think the guy must have put down fill dirt or something--not topsoil...?) Thanks in advance for an opinion...

While they do have large rakes that can remove stones from soil, they're not very effective on pebbles (and you'd need to hire a landscaper).

Filling one 5 1/2 gallon bucket in half the property seems a bit excessive, yes, so I wouldn't say you're crazy in getting those stones out. I wouldn't go nuts over it, but if you can get the majority out, it'll certainly help.

Below-surface pebbles are too small to block roots, so you don't need to worry much about those. Surface ones are annoying, however, and can get sucked into your mower blades. That increases wear and causes nicks. Not to mention the problem of stepping on one barefoot.

I removed them from my soil, but I'd say I had a 2 gallon bucket or slightly less across the whole property.

There are some grass-like ground covers you might look at. Dwarf mondo grass comes to mind.

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That's all the taller it gets and it's always that dark shade of green.

Thank you all for your thoughtful and helpful input. I believe you're right in suggesting I try to open up the shade even more, and thank you for all the lawn grass suggestions. I'm also intrigued by the mondo grass idea--I've seen pictures of it in magazine articles, but have never noticed it for sale around here. It would make a striking patch at the edge of some ill-defined borders between the grass (non-grass, at the moment) area and perennial beds. I need to do some internet research. Thanks too for the advice about watering--it correlates with what I've been doing, though it's tedious, as I don't have an automatic system, and the grass areas are rather irregular and sprinklers have to be supplemented by hand watering. Hopefully, the shampoo method will allow enough permeability for water that I can back off a little, anyway, after reseeding and germination this fall. And finally, those pesky pebbles are going to get swept up, painstakingly, and will form the base over which I'm going to lay a new stone path. Lucky me! Thanks again, everyone!