首页|Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators
Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators
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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.
School of Mathematics,China University of Mining and Technology,Xuzhou 221116,China
Laboratory of Mathematics and Complex Systems(Ministry of Education of China),School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China
National Natural Science Foundation of ChinaNational Natural Science Foundation of ChinaNational Natural Science Foundation of ChinaNatural Science Foundation of Jiangsu ProvincePostdoctoral Science Foundation of China
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