In this note we consider Wente＇s type inequality on the Lorentz-Sobolev space. If f∈L^p1,q1（Rn）,G∈L^p2,q2 （Rn） and div G ≡ 0 in the sense of distribution where 1/p1＋1/p2=1/q ＋1/q2=1,1〈p1,p2〈∞, it is k...In this note we consider Wente＇s type inequality on the Lorentz-Sobolev space. If f∈L^p1,q1（Rn）,G∈L^p2,q2 （Rn） and div G ≡ 0 in the sense of distribution where 1/p1＋1/p2=1/q ＋1/q2=1,1〈p1,p2〈∞, it is known that G. f belongs to the Hardy space H1 and furthermore ‖G· f‖N1≤C‖ f‖Lp1,q1（R2）‖G‖Lp2,q2（R2）Reader can see [9] Section 4 Here we give a new proof of this result. Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest. Finally, we use this inequality to get a generalisation of Bethuel＇s inequality.展开更多
In this note,the authors resolve an evolutionary Wente's problem associated to heat equation,where the special integrability of det▽u for u∈H^1(R^2,R^2)is used.
基金Supported by the National Natural Science Foundation of China(11271330,11371136,11471288)the Zhejiang Natural Science Foundation of China(LY14A010015)China Scholarship Council
文摘In this note we consider Wente＇s type inequality on the Lorentz-Sobolev space. If f∈L^p1,q1（Rn）,G∈L^p2,q2 （Rn） and div G ≡ 0 in the sense of distribution where 1/p1＋1/p2=1/q ＋1/q2=1,1〈p1,p2〈∞, it is known that G. f belongs to the Hardy space H1 and furthermore ‖G· f‖N1≤C‖ f‖Lp1,q1（R2）‖G‖Lp2,q2（R2）Reader can see [9] Section 4 Here we give a new proof of this result. Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest. Finally, we use this inequality to get a generalisation of Bethuel＇s inequality.
文摘In this note,the authors resolve an evolutionary Wente's problem associated to heat equation,where the special integrability of det▽u for u∈H^1(R^2,R^2)is used.
