We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spacesFB˙p,q1-2β+3/p′. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical caseβ=1/2. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.

Let (X , d, µ) be a geometrically doubling metric space and assume that the measure µ satisfies the upper doubling condition. In this paper, the authors, by invoking a Cotlar type inequality, show that the maximal bilinear Calderón-Zygmund operators of type ω(t) is bounded fromMoreover, if w = (w 1 , w 2 ) belongs to the weight class A ρ p (µ), using the John-strömberg maximal operator and the John-strömberg sharp maximal operator, the authors obtain a weighted weak type estimate L p1 (w 1 ) × L p2 (w 2 ) → L p,∞ (v w ) for the maximal bilinear Calderón-Zygmund operators of type ω(t). By weakening the assumption of ω ∈ Dini(1/2) into ω ∈ Dini(1), the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces R n .

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