正规权混合范数空间的有界投影及对偶空间

Bounded Projections and Duality on Regular Mixed Norm Spaces

令ω为正规权,对于单位圆盘D上由正规权诱导的加权混合范数空间L_(ω)^(p,q)(0<p,q≤∞),记A_(ω)^(p,q)=L_(ω)^(p,q)⋂H(D),其中H(D)为D上解析函数所成的空间。首先介绍了正规权的基本性质并对再生核函数做出了某些积分估计。其次利用前面得到的积分估计结果,得到了单位圆盘上混合范数正规权Bergman空间上投影算子Pω的有界性:当1≤p,q<∞时,Pω是从L_(ω)^(p,q)到Ap,qωq2的有界算子;当1≤p<∞时,Pω是从Lp,∞ω12到Ap,∞ω12的有界算子;当1<p,q<∞时,Pω是从Lp,qω到Ap,qω上的有界算子,其中Pω(f)(z)=∫Df(ξ)-Bω--z(ξ)ω(ξ)dA(ξ)。最后利用前面关于投影算子有界性的结果和推广的表示定理刻画了混合范数正规权Bergman空间的对偶空间。

Letωbe a regular weight,for the weight mixed norm space induced by the regular weight on the unit disk D,which is denoted by Lp,qω(0<p,q≤∞),write Ap,qω=Lp,qω⋂H(D),where H(D)denotes the space of all analytic functions in the unit disk.Firstly,the basic properties of normal weights are introduced and some integral estimates of reproducing kernel functions are given;secondly,by using the integral estimation results obtained,the boundedness of the projection operator on regular mixed norm spaces of the unit disk is obtained:when 1≤p,q<∞,Pωis bounded from Lp,qωq 2 onto Ap,qωq 2;when 1≤p,q<∞,Pωis bounded from Lp,∞ω12 onto Ap,∞ω12;when 1<p,q<∞,Pωis bounded from Lp,qωonto Ap,qω,where Pω(f)(z)=∫Df(ξ)Bωz(ξ)ω(ξ)dA(ξ).Finally,the dual spaces of Bergman space with mixed norm induced by regular weight is described by using the previous result on boundedness of projection operators and the generalized representation theorem.