The authors investigate the conditions for the boundedness of Bergman type operators Ps,t in mixed norm space L_(p),q(φ)on the unit ball of C^(n)(n≥1),and obtain a sufficient condition and a necessary condition for ...The authors investigate the conditions for the boundedness of Bergman type operators Ps,t in mixed norm space L_(p),q(φ)on the unit ball of C^(n)(n≥1),and obtain a sufficient condition and a necessary condition for general normal functionφ,and a sufficient and necessary condition forφ(r)=(1-r 2)αlogβ(2(1-r)-1)(α>0,β≥0).This generalizes the result of Forelli Rudin on Bergman operator in Bergman space.As applications,a more natural method is given to compute the duality of the mixed norm space,solve the Gleason's problem for mixed norm space and obtain the characterization of mixed norm space in terms of partial derivatives.Moreover,it is proved thatf∈L(0)∞,q(φ)iff all the functions(1-|z|2)|α||α|fzα(z)∈L(0)∞,q(φ)for holomorphic functionf,1≤q≤∞.展开更多
Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some i...Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+|x|2f1(x)+…+|x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.展开更多
The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results,such as the Bochner-Martinelli formula,the existence theorem of the solutions to the non-ho...The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results,such as the Bochner-Martinelli formula,the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations,and the Hartogs theorem,are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular,the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable,several complex variables,and several quaternionic variables with suitable modifications.展开更多
In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤...In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤u≤∞, 1≤u, q≤∞. (iii) 1≤v≤2≤q≤∞, and 0<p≤u≤∞or 1≤p, u≤∞. The first case extends the result of Blasco, Jevtic, and Pavlovic in one variable. The third case generalizes partly the results of Jevtic, Jovanovic, and Wojtaszczyk to higher dimensions.展开更多
文摘The authors investigate the conditions for the boundedness of Bergman type operators Ps,t in mixed norm space L_(p),q(φ)on the unit ball of C^(n)(n≥1),and obtain a sufficient condition and a necessary condition for general normal functionφ,and a sufficient and necessary condition forφ(r)=(1-r 2)αlogβ(2(1-r)-1)(α>0,β≥0).This generalizes the result of Forelli Rudin on Bergman operator in Bergman space.As applications,a more natural method is given to compute the duality of the mixed norm space,solve the Gleason's problem for mixed norm space and obtain the characterization of mixed norm space in terms of partial derivatives.Moreover,it is proved thatf∈L(0)∞,q(φ)iff all the functions(1-|z|2)|α||α|fzα(z)∈L(0)∞,q(φ)for holomorphic functionf,1≤q≤∞.
基金supported by the National Natural Science Foundation of China(Grant No.10471134).
文摘Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+|x|2f1(x)+…+|x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.
基金supported by National Natural Science Foundation of China(Grant No.11371337)Research Fund for the Doctoral Program of Higher Education of China(Grant No.20123402110068)
文摘The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results,such as the Bochner-Martinelli formula,the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations,and the Hartogs theorem,are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular,the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable,several complex variables,and several quaternionic variables with suitable modifications.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471134)grants from Specialized Research Fund for the doctoral program of Higher Education(SRFDP20050358052)Program for New Century Excellent Talents in University(NCET-05-0539).
文摘In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤u≤∞, 1≤u, q≤∞. (iii) 1≤v≤2≤q≤∞, and 0<p≤u≤∞or 1≤p, u≤∞. The first case extends the result of Blasco, Jevtic, and Pavlovic in one variable. The third case generalizes partly the results of Jevtic, Jovanovic, and Wojtaszczyk to higher dimensions.
