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On some pasts questions on this site, I discovered that walnuts release Juglone, which is nasty for the growth of many plants.
I had/have a part of the vegetable garden partly under a Juglans regia, for more shadow tolerant vegetables. Note: still few meters away of trunk, and sun reach for good part of the day the vegetable garden. So I wondering if I can still plants some vegetables, and which one. On past, I had usually peas/green beans, potatoes and celery/parsley under the plant.
Potatoes aren't great near a walnut tree, but the others you mention should be fine, if they get sufficient sun. Vegetable plants sensitive to juglone include the nightshade family, so potatoes, tomatoes, aubergines and peppers are better away from the roots. You can grow root vegetables such as carrots, also squash/melon, if there's sufficient sun in the area, see here http://homeguides.sfgate.com/vegetables-can-grow-near-walnut-trees-54124.html
