I think the idea with lasagna gardening is to plant directly in the organic matter, so it doesn't matter much if the soil below stays hard.

I doubt very highly if you have any Bermuda in Montreal. If you do, it's the grass that greens up in about June and turns brown in September or so. Bermuda is a warm season grass that needs lots of sunlight and warmth to thrive.

I suspect the solution to helping soften the soil is to wet it thoroughly before laying down the cardboard. I think that may encourage the earthworms who will help to aerate the soil and improve it with their castings.

Having said that I know that trees will suck the moisture out of soil so not sure how long, or if, that works under a water hungry tree.

I don't do lasagna under trees but add compost and mulch and only expect really hardy perennials to grow there. The soil is quite soft but dries out quickly and needs frequent watering in the heat we've had recently. I've thot of only mulching and not having plants in these areas.

Don't know about Bermuda grass, we have either crab/couch/twitch grass, not sure which. I pull it constantly but I rarely get all the long roots so it just keeps on. Never till if you have those invasive grasses. Every tiny piece of root will grow so don't put it in your compost either. I have a friend who dug her entire garden and removed every last piece of root. Huge job and not one I'm going to do. I just try to pull when the grass is small. If you don't have perennials in this bed you can use a garden fork to loosen and pull the grass with roots attached.

It can take far more time than many people are wiling to wait for the Soil Food Web to correct the soil problems they have. If your soil is hard and compacted and lacks adequate soil organic matter piling a lot on top of that soil will not immediately correct that and it will take quite a bot of time for the Soil Food Web to 1. develop in that soil, and then 2. work that organic matter into the soil, maybe years rather than months.
Build up, low, Lasagna beds under that maple and plant in those Lasagna Beds while waiting for the soil underneath to get corrected.
You might have Bermuda grass growing there because even in Michigan some greatly misguided people have tried to grow it. Up here however it will seldom green up much before July and it goes dormant again (browns) in early September.

the idea of lasagne/raised beds is to plant into the newly formed medium, not the original soil the existed. once you get the beds working the worms will come and they will also improve that original soil to the degree that once the raised bed is established and you dig down you may not be able to tell where raised beds starts and original dirt ends.

anyhow pic's and ieas on our site.

len

Here is a link that might be useful: lens garden page

ismviv the grass and soil stays very hard underneath the mulch so it's difficult to plant. Any suggestions?

Use a pick to dig holes just big enough for the things you are planting.

To convert any patch of lawn to garden, I've had to strip and compost the turf and dig down about 18 inches (45 cm) to break up the compacted soil, remove rocks, and work in amendments. There aren't many short-cuts  laying a board or newspaper over the grass may kill it in a year but it will probably be a second year before the roots rot in place.

Easier techniques may work better on some soils (and grasses) than others. Mine is rocky and clay-ey, and has a good proportion of tenacious grasses that spread by runners.

The soil under your maple may be held together tightly by the tree's roots, which could also limit the depth at which you can plant there.

Thanks for all your ideas. I believe I probably didn't add a thick enough layer of soil/compost under the tree and possibly I've been miserly with it elsewhere as well. However, I'll try some of your ideas and see what happens.