Orlicz-Hardy spaces and their duals

• Published in: Science China. Mathematics Vol. 57; no. 5; pp. 903 - 962
• Main Authors: Nakai, Eiichi, Sawano, Yoshihiro
• Format: Journal Article
• Language: English
• Published: Heidelberg Science China Press 01.05.2014
• Subjects: Applications of Mathematics; China; Functions (mathematics); Hardy空间; Infinity; Integrals; Mathematical analysis; Mathematics; Mathematics and Statistics; N-函数; Operators; Texts; Transforms; 允许函数; 原子分解; 对偶; 有界性; 生成子; 空间理论; atomic decomposition; 42B35; 42B30; 46E30; bounded mean oscillation; Campanato space; Orlicz space; Hardy space
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-014-4798-y

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H（Rn）in that the class of admissible functions is largely widened.We can deal with,for example,Ф（r）≡（rp1（log（e＋1/r））q1,0〈r≤1,r^p2 （log（e＋r））q2,r〉1,with p1,p2∈（0,∞）and q1,q2∈（.∞,∞）,where we shall establish the boundedness of the Riesz transforms on H（Rn）.In particular,is neither convex nor concave when 0〈p1〈1〈p2〈∞,0〈p21〈p1〈∞or p1=p2=1 and q1,q20.If（r）≡r（log（e＋r））q,then H（Rn）=H（logH）q（Rn）.We shall also establish the boundedness of the fractional integral operators I of order∈（0,∞）.For example,I is shown to be bounded from H（logH）1^1-α/n（Rn）to Ln/（n-α）（log L）（Rn）for 0〈α〈n.