The chemical in question, juglone, is mostly produced in the roots (which spread a distance equal to the height of most trees) and works as a germination inhibitor. Combined with shade, the juglone can make vegetable gardening quite difficult without imported soil in raised beds and without full sun. There are many native plants that will grow under black walnut and you can create what I call an Aviary with many fruit producing plants such as red elderberry, raspberries, bittersweet, virginia creeper, pawpaw, nanyberry and woodland plants such as baneberry, jack in the pulpit and false Solomon's seal.