The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-LittlewoodSobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMOx(Rn) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V(t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in Lr(Rn) is proved.
Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation(BMO)functions can be obtained from their weighted variational estimates,we establish the similar variational estimates for the commutators of the BMO functions with rough singular integrals,which do not admit any weighted variational estimates.The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.
We consider the boundedness of the rough singular integral operator T_（？,ψ,h） along a surface of revolution on the Triebel-Lizorkin space F^α_（ p,q）（R^n） for Ω ∈ H^1（（S^n-1）） and Ω ∈ Llog^＋L（S^n-1） ∪_1
Let K be the Calderón-Zygmund convolution kernel on R^d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R^d)by T_af(x)=p.v.∫_(R^d)K(x-y)m_x,y^a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.
Let [b, T] be the commutator of parabolic singular integral T. In this paper, the authors prove that the boundedness of [b, T] on the generalized Morrey spaces implies b ∈ BMO（Rn, ρ）. The results in this paper improve and extend the Komori and Mizuhara＇s results.
