势型算子的双加权不等式

Two Weighted Inequality for Potential Type Operators

研究了势型算子TΦf(x) = ∫RnΦ(x - y)f(y)dy 在Lpv(Rn)到Lqw (Rn)上有界的充分条件,其中Lpv(Rn),Lqw (Rn)分别是带权v 和w 的Lp ,Lq 空间。通过改进“二进制”分解定理证明了:当1≤p≤q< ∞,1< r< psp+ s- 1,s> 1,Φ(x)是非负函数,且Φ∈L1loc(Rn), Φ(t)= ∫｜z｜≤tΦr′(z)dz 1/r′。若对任何方体Q有Φ(l(Q))｜Q｜1q- 1p + 1r 1｜Q｜∫Qwqsdx1qs 1｜Q｜∫Qv- p′sdx1p′s ≤C成立,则算子TΦ是Lpv (Rn)到Lqw (Rn)上的有界算子,并且还证明了将Rn 换成齐性空间(X,d,μ)时,类似结论也成立。

In this paper,the author investigated the boundness from L p v(R n ) to L q w( R n ) of potential type operators defined as:T Φf(x)=∫ R n Φ(x-y)f(y) d y Having improved the descomposition theorem,the author proved if 1≤p≤q<∞,1<r<psp+s-1,s>1,Φ(x) is a nonnegative function and Φ∈L 1 loc (R n),(t)=∫ |z|≤t Φ r′ (z) d z 1/r′ ,if there is a constant C that for every cube Q the inequality (l(Q))|Q| 1q-1p+1r ×1|Q|∫ Qw -p′s d x 1p′s ≤C is satisfied,then T Φ is a bounded operator from L p v( R n ) to L q w( R n ),moreover,the relevant results were also given when replace R n with homogeneous space (X,d,μ) .