R上分段单调和分段加倍的加倍权.pdf

2023,43A(5):15851594http:/RN?1?B2“_2Sr(1?X 033001;2?O?430062):T?R Aa?.k?R N?;gx?R N?;?R?.c:?;?;?.MR(2010)Ka:28A80a:O174.12zI:A?:1003-3998(2023)05-1585-101?R(m I0)?Borel?K,e3 c 1,?R(m I0)?fm I k(2I)c(I),2I I%I?m,K R(m I0)?c-?,?1.K?,?2.R?(x)?,XJ(x)U d=(x)dx?R?3.Beurling Ahlfors4d?Y,?Y.R?Y?u Lebesgue?Radon-Nikodym?3,5.A?A?2,6.?R(m I0)?,a 0,e?8 A R(m I0)k(A+a)(A),K 3 R(m I0)N4O;e?8 A R(m I0)k(A+a)(A),K 3 R(m I0)N4,A+a=x+a:x A v A+a R(m I0).N4ON4?N.XJ?|8km?,3km?zmN,K N?.Cruzuribe7y?N?Y?,Radon-Nikodym?A?,?R+N?Ap?9?.?R N?.vF:2022-04-18;?F:2023-03-23E-mail:78:Ig,7(11301162)!?e8(D20211005)!p?EM#8(2019L0963)p?g?8(2022RC10)Supported by the NSFC(11301162),the Hubei Provincial Department of Education(D20211005),the STIP(2019L0963)and the HLT(2022RC10)1586n?Vol.43 A?1 2?R O()?;1 3?R N?a?d;1 4?R?.?I,?P:?(x)?,P(I)=RI(x)dx 9W(t)=Zt0(x)dx,t 0;Z0t(x)dx,t 0.(1.1)|I|LmI?Lebesgue.e A R,P A=x:x A,8A?A?IAIA(t)=1,t A;0,t/A.2N?!?R N?,dIen.n2.1?d,n2.2?3|8?!5.n 2.18?R?Borel?K,K?=?3 C 1,?m I,J,k1C(J)(I)C(J).n 2.29?R?c-?.?m I,J,XJ dist(I,J)|I|,K(I)(J),p c,k,dist(I,J)=inf|x y|:x I,y J.en?R 4O?.n 2.1?3 R 4O?,K R?=?0 inftR(t)suptR(t)0,3 M 0,?t M k(t)?.l?(2M,M)=ZM2M(x)dx (0)(M).No.5?B?:RN?1587l?(2M,M)(0,M)?(0).d?59?5,n2.2?(g!?y.aqun2.1?y,N R 4?.n 2.2?3 R 4?,K R?=?0 inftR(t)suptR(t)y x 9 z y=y x=r.P I=x,y,J=y,z.y R?,=y3c 1(6u|I|,|J|),k1c(I)(J)c(I),(3.2)y(3.2),Ie2.1x 0 z 0.”5?x 0.uz 0?/aqy.d 3 0,+)4O,w,(I)(J);y(3.2),I23fy(J)c(I).1.1?x=0.d(3.1),Z2r0(x)dx Zr0(x)dx,l?Z2rr(x)dx (+1)Zr0(x)dx,1588n?Vol.43 A=?(J)+1(I).1.2?0 x r.P I0=0,y,J0=y,z+x,Kd 1.1,(J0)(1+1/)(I0).(3.3)q J J0,K(J)(J0).d x r,?(I0)=(0,x)+(x,y)2(I).2(3.3)?(J)2+1(I).1.3?0 r x.?n Z+,?x0:=y nr r.P z0=y+nr,I0=x0,y,J0=y,z0.Kd 1.2,(J0)2(1+1/)(I0).(3.4)3 0,+)4O,?d|J0|=n|J|,?(J)1n(J0).q|I0|=n|I|,K(I0)n(I).2(3.4)?(J)2+1(I).2x 0.”5?y 0.d x y.I23fy(3.2).2.1?y=0.d(3.1)?m W(r)W(r),Z0r(x)dx Zr0(x)dx,=(I)(J).aqk(J)(I).l?(I)(J)1(I).2.2?0 x y.d/2.1?(x,0)1(0,x).2d 3 0,+)4O,(x,y)(y,2y+x),(0,x)12(2y+x,z).l?(I)=(x,0)+(0,x)+(x,y)(1+1)(0,x)+(x,y)(1+1)12(2y+x,z)+(y,2y+x)12(1+1)(J).,d 1.2?(x+y,z)2(1+1)(x,x+y).(3 0,+)?4O59(3.1)?W(2t)W(t),(0,y)1 (y,2y)1 (x,x+y).l?(J)=(y,x+y)+(x+y,z)(x,x+y)+2(1+1)(x,x+y)(1 )(3+2)(0,y)(1 1)(3+2)(I).No.5?B?:RN?1589 2.3?0 y x.d 2.1?(0,y)(y,0)1(0,y),(0,x)(x,0)1(0,x).(3.5),d 3 0,+)4O?0,y y,2y.(3.5)?(I)=(x,0)+(0,y)1(0,x)+(0,y)(1+1)(y,2y)+1(y,x)(2+1)(J).,?0 x2 y,dd 1.2?(2y,z)2(1+1)(2y+x,2y)2(1+1)(0,2y).(3.6)d(3.1),(0,2y)12(y,0),(y,2y)1(1 1)(y,0).2(3.6)k(J)=(y,2y)+(2y,z)1(1 1)(y,0)+2(1+1)12(y,0)42(1+1)(I).?0 0,d?12t(2t)W(2t)=Zt0(x)dx+Z2tt(x)dx t(t)+(2t),l?(t)(2 1)(2t).uW(t)=Zt0(x)dx (2 1)Zt0(2x)dx=(112)Z2t0(x)dx=(112)W(2t).?=112,w,(0,12,?t R k(3.1).?dn 3.1 y R?.aqun 3.1 y?en 3.2.n 3.2?3 R?,e 3(,0 4O,3 0,+)4.K R?=?3 (1,2,?t R kW(t)W(2t);W(t)W(t),(3.8)W(t)?(1.1).aq 3.1,dn 3.2?e 3.2.3.2?3 R?,e 3(,0 4O,3 0,+)4.K R?=?3 1?t R k W(t)|t|(t),W(t)W(t).4?!?R?.3 R?,XJU d=(x)dx?|8k?m?,3km?zm?.ef R?3 R?,?!R?3 R E?.4.1?In=(8n,8n+1,n Z+0,-+(t)=1,0 t 1;+Xn=014nIIn(t),t 1,9(t)=1,t 0.+,O 0,+),(,0?.-(t)=+(t),t 0;(t),t 0.No.5?B?:RN?1591du(0,8n)(8n,0)=1+7n1Pk=02k8n=7 2n 68n 0,n .?dn 2.1 R?.en 4.1?R a?3 R E?7.n 4.1?+,O 0,+),(,0?,-(t)=+(t),t 0;(t),t 0,?0 t T0k1W(t)W(t);(iii)3 1,?T1,?2T1 R 2T1.Kdist(I,J)0 k(I)(J).aqk(J)(I).?=max,1,K1W(R)W(T1)W(R)W(T1).p 6u?.?y.5 4.1efn 4.1(iii)?R 2T1U R T1.4.2?(t)=1,t 0,-+(t)=11 t,0 t 1;1,t=1 2,+);1t 1,1 t 0;(t),t 0.1592n?Vol.43 A(t)R?.?0,?R=T1+?,T1=1,K(T1,R)(R,T1)=R?01/xdx?=2?+,?0.l?n 4.1(iii)?R 2T1U R T1.en4.1?R a?3 R E?.n 4.1?+,O 0,+),(,0?,(t)=+(t),t 0;(t),t 0.K R?=?e?(i),(ii),(iii):(i)+,O 0,+),(,0?;(ii)3?T0,?0 t T0k1W(t)W(t);(iii)3?T1(T0 T1 0,?T1 R|J|2?/aqy.c1,c2OL,+?.e3y(4.1).1|J+|T0.,PI+=2d,d I,K I+J?,d(ii)?(J)=(J)+(J+)(J)+(J+)=(+1)(J)+(|I|+d,d)(+1)c1(I+)+(|I|+d,d)2(+1)c1(I).,d 3(,0?,dn 2.2 36u c1?c3 1,k(J+2d)c3(J+d)c23(J+).(4.2)2(ii)?(J)=(J)+(J+)1c1(I+)+1(J+)1c1(I+)+1c23(J+2d)min1c1,1c23(I).?c=max2(+1)c1,1minc11,c23,K(4.1).No.5?B?:RN?1593 2T0|J+|T1.d 3(,0?,36u c1,T0,T1?c4 1 96u c2,T0,T1?c5 1,k(T1,0)c4(T0,0)(0,T1)c5(0,T0).(4.3),d(4.3)?(J)=(J)+(J+)c1(I+)+(0,T1)c1(I+)+c5(0,T0)c1(I+)+c1c5(T0,0).(4.4)?d T0|J+|,(4.2)(4.4)?(J)c1(I+)+c1c5c3(T0,0 d)2c1c5c3(I).?T0 d,d(4.4)?(J)c1(I+)+c1c5(J)(c1+c21c5)(I+)2c21c5(I).,d(ii),(4.3)(4.2)?(J)(J+)1(T0,0)1c4(T1,0)1c4c3(T1 d,d).(4.5)?T1|J|,“(4.5)?(J)1c4c3(I).?T1|J|,K|J|2|J+|T1 T1.-R=|J+|.d(ii)(4.3)?1c4(T1,0)(0,T1)c5(T1,0).(4.6),d(ii),(iii)(4.6)?(J)=(J)+(J+)(+1)(J+)=(+1)(0,T1)+(T1,R)(+1)(c5(T1,0)+(R,T1)(+1)c5(+1)(T1,d)+(+1)(R,T1)max(+1)2c5,(+1)(I).,d(ii),(iii)(4.2),(4.6)?(J)=(J)+(0,T1)+(T1,R)1c1(J d)+1c4(T1,0)+1(R,T1)1c1(J d)+1c4c3(T1 d,d)+1(R,T1)min1c1,1c4c3,1(I).1594n?Vol.43 A?c=max(+1)2c5,(+1),1minc11,1c13c14,1,K(4.1).?y.dn 4.1?e?(.n 4.2?R=I0I1In,m I0,In?m:O x0,xn,IiIi+1=xi,i=0,1,n1.e 3 R?,K 3 R?=?e?(i),(ii),(iii):(i)?i 0,1,n,3m Ii?;(ii)?i 0,1,n,3 i 1,Ti 0,?0 t Tik1i(xi,xi+t)(xi t,xi)i;(iii)3 1,T +,?T R k1(T,R)(R,T).z1 Heinonen J.Lectures on Analysis on Metric Spaces.New York:Springer Verlag,20012 Wik I.On Muckenhoupts classes of weight functions.Studia Math,1989,94(3):2452553 Semmes S.Bilipschitz mappings and strong Aweights.Ann Acad Sci Fenn Math,1993,18(2):2112484 Beurling A,Ahlfors L.The boundary correspondence under quasiconformal mapping.Acta Math,1956,96:1251425 Mattila P.Geometry of Sets and Measures in Euchidean Spaces.Cambridge:Cambridge University Press,19956 Fefferman C,Muckenhoupt B.Two nonequivalent conditions for weight functions.Proc Amer Math Soc,1974,45(1):991047 Cruzuribe D.Piecewise monotonic doubling measures.Rocky Mountain Journal of Mathematics,1996,26(2):5455848 Wu J M.Null sets for doubling and dyadic doubling measures.Ann Acad Sci Fenn Ser Math,1993,18:77919 Ruzhansky M.On uniform properties of doubling measures.Proc Amer Math Soc,2001,129:34133416Doubling Weights Which are Piecewise Monotonic or PiecewiseDoubling on R1Dang Yungui2Dai Yuxia2Yu Ximei(1Department of Mathematics,Lyuliang University,Shanxi Lvliang 033001;2Faculty of Mathematics and Statistics,Hubei University,Wuhan 430062)Abstract:The paper will study several kinds of doubling weights on R.We firstly give sufficientand necessary conditions on which monotonic weights are doubling.Secondly we describe doublingpiecewise monotonic weights.At last,we discuss doubling weights which are piecewise doubling on R.Key words:Weight function;Doubling weight;Doubling measure.MR(2010)Subject Classification:28A80