查看更多>>摘要：Let(→a):=(a1,...,an)∈[1,∞)n,p ∈(0,1),and α:=1/p-1.For any x ∈ Rn and t ∈[0,∞),letΦp(x,t):={t/1+(t[x]v(→a))1-p if vα ∉ N,t/1+(t[x]v(→a))1-p[log(e+|x|(→a))]p1f vα∈N,where[·](→a):=1+|·|(→a),|·|(→a)denotes the anisotropic quasi-homogeneous norm with respect to(→a),and v:=a1+…+an.Let Hp(→a)(Rn),L(→a)α(Rn),and HΦ(→a)p(Rn)be,respectively,the anisotropic Hardy space,the anisotropic Campanato space,and the anisotropic Musielak-Orlicz Hardy space associated with Φp on Rn.In this article,via first establishing the wavelet characterization of anisotropic Campanato spaces,we prove that for any f ∈ Hp(→a)(Rn)and g ∈ L(→a)α(Rn),the product of f and g can be decomposed into S(f,g)+T(f,g)in the sense of tempered distributions,where S is a bilinear operator bounded from Hp(→a)(Rn)×L(→a)α(Rn)to L1(Rn)and T is a bilinear operator bounded from Hp(→a)(Rn)×L(→a)α(Rn)to HΦ(→a)p(Rn).Moreover,this bilinear decomposition is sharp in the dual sense that any y ⊂ HΦ(→a)p(Rn)that fits into the above bilinear decomposition should satisfy(L1(Rn)+y)*=(L1(Rn)+HΦ(→a)p(Rn))*.As applications,for any non-constant b ∈ L(→a)α(Rn)and any sublinear operator T satisfying some mild bounded assumptions,we find the largest subspace of Hp(→a)(Rn),denoted by Hp(→a),b(Rn),such that the commutator[b,T]is bounded from Hp(→a),b(Rn)to L1(Rn).In addition,when T is an anisotropic Calderón-Zygmund operator,the boundedness of[b,T]from Hp(→a),b(Rn)to L1(Rn)(or to H1(→a)(Rn))is also presented.The key of their proofs is the wavelet characterization of function spaces under consideration.
万方数据