Some Real-variable Characterizations Of Variable Anisotropic Hardy Spaces And Carleson Measure

The paper consists of two sections.In section 1,the paper first proves some approximation of identity results adapting to ellipsoid cover ?,the proof of this theorem is basically covers the gap about atomic decomposition characterization of variable anisotropic Hardy spaces.Then the finite atomic decomposition characterization of variable anisotropic Hardy spaces Hp((?))is obtained.Finally,as an application of the characterization of finite atoms,we establish some criterions of bounded sublinear operator from variable anisotropic Hardy Spaces to quasi-Banach Spaces.In section 2,the paper first show that the Carleson measure p is sufficient for which the integral defines a bounded operator from Hp((?))to Lp(Rn+1,?),0<p?1.Then it gives several equivalent Carleson measures adapted to multi-level ellipsoid covers.Finally,based on the above results,we obtain a specific Carleson measure induced by the highly anisotropic BMO functions.