The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm...The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6-9].展开更多
The generalized maximal operator .44 in martingale spaces is considered. For 1 〈 p ≤ q 〈 ∞, the authors give a necessary and sufficient condition on the pair （μ, v） for M to be a bounded operator from martingal...The generalized maximal operator .44 in martingale spaces is considered. For 1 〈 p ≤ q 〈 ∞, the authors give a necessary and sufficient condition on the pair （μ, v） for M to be a bounded operator from martingale space L^P（μ） into L^q（μ） or weak-L^q（μ）, where μ is a measure on Ω × N and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.展开更多
基金supported by the National Natural Science Foundation of China (Nos.10871069,10871070)the Shanghai Leading Academic Discipline Project (No.B407)
文摘The authors consider a family of smooth immersions F（., t） ：M^n→R^n＋1 of closed hypersurfaces in R^n＋1 moving by the mean curvature flow F（p,t）/ t=-H（p,t）.v（p,t） for t E [0, T）. They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded, then the flow can extend past time T. The result is similar to that in [6-9].
基金supported by the National Natural Science Foundation of China (Nos. 10671147,11071190)
文摘The generalized maximal operator .44 in martingale spaces is considered. For 1 〈 p ≤ q 〈 ∞, the authors give a necessary and sufficient condition on the pair （μ, v） for M to be a bounded operator from martingale space L^P（μ） into L^q（μ） or weak-L^q（μ）, where μ is a measure on Ω × N and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.
