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We develop characterizations for Sobolev spaces of potential type on gradedLie groups, by means of Littlewood-Paley square functions, and Strichartzfunctionals involving second-order differences. A key role is played by somemean value inequalities that may be of independent interest. Indeed, thepoint-wise first-order inequality is generalized in two directions: on$L^p$-spaces and by considering second-order differences.

Mean Value Inequalities and Characterizations of Sobolev spaces on graded groups