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 rel...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.展开更多
基金supported by Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 24540159)Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 24540085)
文摘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.