We prove the following theorem: Suppose the function f(x) belongs to L_q(ω, ℝⁿ), ω ⊂ ℝ^m, q∈(1, ∞), and satisfies the inequality {Mathematical expression} for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operator LL^* is elliptic. Then there exists a function p(x)∈W_q¹(ω) such that {Mathematical expression} Bibliography: 6 titles.