What I have done and it may not work for you is select plants with long taproots rather than shallow rooted plants. I cut a hole in the root mass of the maple tree to provide access to the actual soil and then plant in that. In a year or so the root mass can grow back but with a long taproot nutrients can be accessed by the new plant. If the plant dies I then try something different. Bulbs such as small daffodils and hyachiths also do well under my maple trees.

I have successfully grown many types of plants under Maples for 40+ years. Some of my largest Hostas are there, as well as Dogwoods, Camellias, Hydrangeas, Epimediums, Ferns and Mosses, Japanese Maples, Helleborus, Native and Evergreen Azaleas, Iris, Columbine, Asarum, Terrestrial Orchids, Holly (Ilex), Hepatica, Chamaecyparis pisifera & obtusa, Arisaema, Gardenia and many other Native plants and shrubs.

The rules I follow are:
Add 4-6 inches of leafmould, compost, conifer bark, etc., to establish a raised bed above the shallow Maple roots, leaving a 6 inch air gap around the Maple trunk. When planting, cut whatever small roots are necessary and remove them. No other amendments are added to the planting hole.

Never added fertilizer to plants. An annual application of compost is provided to the entire area under the trees.

When irrigation is necessary, the entire area under and around the tree, well beyond the drip line, is watered.

I am growing Silver, Red, Sugar and 3 species of JMs (A. palmatum, A. japonicum, & A. shirasawanum), plus 4 Southern Magnolia, M. grandiflora, whose roots are much more of a problem than the Maples, but they are also underplanted with those plants that tolerate full evergreen shade.

Hope this can be of some assistance to you.
Good luck!

With proper care and attention - especially watering - it is possible, as ladywindsurfer has indicated, to garden beneath maples; I've been gardening among sugar maple roots for 30+ years. Maples are, however, among the most aggressive trees there are in terms of root growth. Their dense canopy of leaves also casts a very deep shade and acts as an umbrella preventing rain from reaching the ground. Oaks, on the other hand, may be the best of all trees for shade gardening - they are deep rooted, unlike maples, and their leaves will not smother everything beneath if not removed each fall.

So, if it were me, I'd remove the maples now while they are still small. Left alone they will become much harder to remove in a surprisingly short time. There are many plants with outstanding fall color that would flourish among the oaks. Japanese maples, for example, can provide spectacular color and have benign root systems. Sugar and other North American maples are undeniably beautiful in October, but are not well placed in relatively small residential settings.

Mainegrower, I would LOVE to put some Japanese maples under the oaks, but I'm zone 3/4 & not much grows here along those lines.

Not knowing about Maple roots I started a raised, rock edged flower garden on the south side of two large Maples four years ago.
The problem I had was their roots coming up into the raised garden. Over time the roots got thicker. Now it is almost impossible to plant an average potted plant without damaging roots.
Now I am into seed starting and there is no problem planting seedlings.
I also water everything more often and fertilize regularly.
We are in a severe drought this year and my flower garden looks as good as it did the other years - even better as the perennials are filling out.

According to Tallamy, oak supports 534 species of lepidoptera while maples support only 288. I love the look of maples and have many, but they are really water hogs. And, if you follow Tallamy's thinking about the food chain, oaks are better for the environment. I encourage all thoughtful gardeners to read his book.

In defense of maples, they are a major component of our eastern woodlands and under natural conditions the maple-beech-birch forests are diverse and beautiful ecosystems. In areas that get plentiful rainfall maples tend to succeed most oak species because they are more shade tolerant. Oaks hold there own on dry sites and in areas that occasionally burn (or if the maples are manually removed). Many species that naturally grow under maples are difficult to obtain because they are slow-growing. For example Trillium can take 7 years or more to bloom from seed.