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We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms.

Let E 1 , E 2 be symmetric quasi Banach spaces on [0, α) (0 < α ≤ ∞). We collected and proved some properties of the space E 1 ⊙ E 2 , where ⊙ means the pointwise product of symmetric quasi Banach spaces. Under some natural assumptions, weAs application, we extend this results to the noncommutative symmetric quasi spaces and the noncommutative symmetric quasi Hardy spaces case. We also obtained the real case of Peter Jones' theorem for noncommutative symmetric quasi Hardy spaces. Keywords:Symmetric quasi Banach space, pointwise product of symmetric quasi Banach spaces, Noncommutative symmetric quasi space, Noncommutative symmetric quasi Hardy space, complex and real interpolation. 46L52, 47L51

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