On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

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On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems (2012, methodology): “¤ By Lemma 3.5 and the Leibniz formula, we have ∣∣∣∣Dαξ ( iξj λe−²|ξ| 2 P (ξ) λ + µ|ξ|2 )∣∣∣∣ ≤ Cα,σ,µ|ξ|1−|α|, ∣∣∣∣Dαξ { τ ∂ ∂τ ( iξj λe−²|ξ| 2 P (ξ) λ + µ|ξ|2 )}∣∣∣∣ ≤ Cα,σ,µ|ξ|1−|α| for j = 1, . . . , n and (λ, ξ) ∈ Σσ,0 × (Rn \ {0}), which combined with Lemma 3.6 below due to Shibata and Shimizu”