The boundedness and compactness of the weighted differentiation composition operators from mixed-norm spaces to Bloch-type spaces are discussed in this paper.
Let F be a C^∞ curve in IR^n and μ be the measure induced by Lebesgue measure on F, multiplied by a smooth cut-off function. In this paper, we will prove some mixednorm estimates based on the average decay estimates...Let F be a C^∞ curve in IR^n and μ be the measure induced by Lebesgue measure on F, multiplied by a smooth cut-off function. In this paper, we will prove some mixednorm estimates based on the average decay estimates of the Fourier transform of μ.展开更多
In this paper, we characterize the boundedness and compactness of the weighted composi- tion operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the...In this paper, we characterize the boundedness and compactness of the weighted composi- tion operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.展开更多
Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(R^(n))be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(R^(n)).In this arti...Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(R^(n))be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(R^(n)).In this article,we show that the Fourier transform fcoincides with a continuous function g onℝn in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(R^(n))and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(R^(n))with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.展开更多
We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas ...We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.展开更多
We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial da...We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation.The phenomenon is a priori nontrivial due to the nonlocal structure of the equation.Our approach is based on Kato’s method using Picard’s interation,which can be apdated to the multi-dimensional case and other nonlinear non-local equations.We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.展开更多
基金item： Supported by the National Natural Science Foundation of China（60573040）
文摘The boundedness and compactness of the weighted differentiation composition operators from mixed-norm spaces to Bloch-type spaces are discussed in this paper.
基金Supported by Natural Science Fundation of Anhui Province (07021019)Education Committee of AnhuiProvince (KJ2007A009 KJ2008B244)
文摘Let F be a C^∞ curve in IR^n and μ be the measure induced by Lebesgue measure on F, multiplied by a smooth cut-off function. In this paper, we will prove some mixednorm estimates based on the average decay estimates of the Fourier transform of μ.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971153, 10671141)
文摘In this paper, we characterize the boundedness and compactness of the weighted composi- tion operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.
基金This work was partially supported by the National Natural Science Foundation of China(Grant Nos.11971058,12071197)the National Key Research and Development Program of China(Grant No.2020YFA0712900)Der-Chen Chang was partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.
文摘Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(R^(n))be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(R^(n)).In this article,we show that the Fourier transform fcoincides with a continuous function g onℝn in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(R^(n))and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(R^(n))with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.
基金supported by Simons Foundation(Grant No.580911(Stinga))Ministerio de Economía y Competitividad/Fondo Europeo de Desarrollo Regional from Government of Spain(Grant No.MTM2015-66157-C2-1-P(Torrea))。
文摘We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.
基金supported by the Simons Foundation,grant#354889。
文摘We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation.The phenomenon is a priori nontrivial due to the nonlocal structure of the equation.Our approach is based on Kato’s method using Picard’s interation,which can be apdated to the multi-dimensional case and other nonlinear non-local equations.We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.
