5 ?s￥'?é 1 !  [,  V  [, ?l O  [, ?" μK fx() ]ab gx() ]ab ]ab ??-??μ ￡ ü  [, 9 Vi Oμ fx gx() ()= gx() ]ab f x dx g x dx a b a b () () ∫∫ = b ￡  !? ),,2,1( picx i null== ) )()( xgxf ≠ b uW T?s  ? | ],[ ba bxxxxa nn =<<<<= ?110 null ],[ 1 iii xx ? ∈ξ 5 11 () () nn ii ii x ' (() ()) ii gfξξ ii gx fξξ == ?? ? ∑∑ i x= ?? ∑  ? V U?cμ { ' ∑ } i c ??￥l uW à ? p?b 2p : 12 sup ( ) , sup ( ) axb axb M gx M fx ≤≤ ≤≤ ==, ,0>?ε | 12 2( )pM M ε δ = + 5? δλ <?= ≤≤ }{max 1 i ni x H , ' (() () iii gfxξ ξε??< ∑ , ?[? V , V? 9 V , O? ? bfx() gx() f x dx g x dx a b a b () () ∫∫ = 2 ! ?  [, ? V h  è a üB?μ fx() gx() ]ab ? ? ? ? ? ? ?? ? ? ? ? ? ≠ ∫∫∫ b a b a b a dxxgdxxfdxxgxf )()()()( b 3  è ? ]2,0[,1)()( ∈== xxgxf 5 22 00 () 2, () 2f x dx g x dx= = ∫∫  2 0 ()() 2fxgxdx= ∫ ?[ b? ? ? ? ? ? ?? ? ? ? ? ? ≠ ∫∫∫ b a b a b a dxxgdxxfdxxgxf )()()()( 3 ￡ ü  ?i L ? o 1  ? ?i ü? ? cba ,, fxdx a b () ∫ fxdx a c () ∫ fxdx c b () ∫ f x dx f x dx f x dx a b a c c b () () () ∫∫∫ =+b 209 ￡  ? ! cba << 5 ? ^ ∫∫∫ += c b b a c a dxxfdxxfdxxf )()()( ∫ b a dxxf )( = b ∫ c a dxxf )( ∫∫∫ +=? b c c a c b dxxfdxxfdxxf )()()(  e f? V ?wb 4 ?/ s￥vl ò ?  xdx 0 1 ∫ xdx 2 0 1 ∫ ó ?  xdx 1 2 ∫ xdx 2 1 2 ∫ ? () 1 2 2 1 x dx ? ? ∫ ?  2 0 1 x dx ∫ ? sin xdx 0 2 π ∫ ? xdx 0 2 π ∫ b 3  1? H  ?[  b)1,0(∈x 2 xx > xdx 0 1 ∫ xdx 2 0 1 ∫  ? H  ?[ )2,1(∈x 2 xx < xdx 0 1 ∫ < xdx 2 0 1 ∫ b  ? )1,2( ??∈x H 2) 2 1 ( > x 7? )1,0(∈x H 22 < x  ?s?B?′? ? V¤ () 1 2 2 1 x dx ? ? ∫ > 2 0 1 x dx ∫ b  ? H  ?[ 0>x xx <sin sin xdx 0 2 π ∫ < xdx 0 2 π ∫ b 5 !   ?? ???1 0￡ üfx() [, ]ab fx()≥ 0 fxdx a b () ∫ > 0b ￡ ￡ üB ?^ ! ),(,0)( 00 baxxf ∈> b? 0)()(lim 0 0 >= → xfxf xx i D 0c > { } 00 0( min , )axbxδδ>< ?? P¤? ),( 00 δδ +?∈ xxx H ? ? b ? ^ cxf >)( () b a f xdx= ∫ 0 () x a f xdx δ? + ∫ 0 0 () x x f xdx δ δ + ? + ∫ 0 () b x f xdx δ+ ≥ ∫ 0 0 () x x f xdx δ δ + ? ∫ 2c 0δ≥>b 0 ()0fx > 0 x a=  0 x b= ￥ f ? V ? ?￡ üb ￡ ü=¨Q￡Eb ? 50)( = ∫ b a dxxf [,], () 0 t a tab fxdx? ∈= ∫ b??  [,  V? O ∫ = t a dxxftF )()( ]ab ],[),()( battftF ∈=′  ?[μ  D5 ! ±V7A?? ? b () 0ft≡ 0)( > ∫ b a dxxf 6 !  [,  ?? O ￡ ü  [, ?1 0b fx() ]ab fxdx a b 2 0() ∫ = fx() ]ab 210 ￡ ?   ?? V?fx() [, ]ab 2 ()f x  [,  ?? O b? 5' V¤?2 ?b ]ab 2 () 0fx≥ 7 !f ?   ?? = V? O ?@)(xf ],[ ba ),( ba )()( 2 2 bfdxxf ab ba a = ? ∫ + b ￡ üi ),( ba∈ξ  P¤ 0)( =′ ξf b ￡ ?s?B?′? ? [, ], 2 ab aη + ?∈  P¤ ()f η = )()( 2 2 bfdxxf ab ba a = ? ∫ +    [,)(xf ]bη ?¨ 3PMMF? ? (,) (,)babξ η? ∈? P¤ 0)( =′ ξf b 8 ! )(t?   ?? ],0[ a )(xf ),( +∞?∞ =¨ V? O b￡ ü 0)( ≥′′ xf ≤ ? ? ? ? ? ? ∫ a dtt a f 0 )( 1 ? ∫ a dttf a 0 ))(( 1 ? b ￡ | uW T?s],0[ a atttt nn =<<<<= ?110 0 null : ],[},{max, 1 1 1 iiii ni iii tttttt ? ≤≤ ? ∈?=?=? ξλ b? ? /j ? +FOTFO?? T  ? ?5  ¤? f  ∑∑ == ? ≤? ? ? ? ? ? ? n i i i n i i i a t f a t f 11 ))(()( ξ?ξ?  7 0→λ   ??? Tü?1  ≤ ? ? ? ? ? ? ∫ a dtt a f 0 )( 1 ? ∫ a dttf a 0 ))(( 1 ? b 9 !   ?? O??h ￡ ü ?ifx() ]1,0[ ]1,0[∈α ? ? ∫∫ ≥ 1 00 )()( dxxfdxxf α α b ￡ ￡ üBù5?N?￡ ü ?i ]1,0[∈α ? ? 211 ∫∫ ≥? 1 0 )()()1( α α αα dxxfdxxf b ?? T  ?¨s?B?′? ?5i 1 [0, ]x α∈ # 2 [,1]x α∈  P¤ 0 (1 ) ( )f xdx α α? ∫  1 (1 ) ( )f xα α? # 1 2 () (1 ) ( )f xdx f x α ααα=? ∫ b ??A ?μ  ?[¤? b)()( 21 xfxf ≥ ∫∫ ≥? 1 0 )()()1( α α αα dxxfdxxf ￡ ü= ! 5 b?  s?B?′? ? 1 00 () () ()F f xdx f xdx α αα=? ∫∫ 1 0 '( ) ( ) ( )Ff fxαα=? ∫ dx [0,1],ξ?∈ P¤ 1 0 () ()f fxdxξ = ∫ ' '( ) ( ) ( )Fffα αξ=?b ?? ??h  ?[?f 0 α ξ< < H '( ) 0F α ≥ ' ()F α ??9F ? 1ξ α<< H '( ) 0F α ≤ ' ()F α ??h b? (0) (1) 0FF= = ' V¤ ? ],1,0[∈?α ? ? () 0F α ≥ b ￡ ü ?? 0=α H ?? TA ?? ?b ? ]1,0(∈α H 7 tx α=  ? ¨ ??h ü¤? bf ∫∫∫ ≥= 1 0 1 00 )()()( dttfdttfdxxf ααα α 10 Young?? T ! yfx= () ^ [, )0 ∞ ?ì??9F￥ ??f ?  O : ?￥Qf ?10)0( =f xfy= ?1 ()b￡ ü + ∫ a dxxf 0 )( 1 0 () b f ydy ab ? ≥ ∫  b ab>>0, 0 ￡ 5￡? H?|? ?b )(afb = | uW T?s],0[ a axxxx nn =<<<<= ?110 0 null : ),,2,1,0()( nixfy ii null== 5 byyyy nn =<<<<= ?110 0 null : 11 , ?? ?=??=? iiiiii yyyxxx ? ^ ∑∑∑∑ = ? = ?? = ? = ? ?+?=?+? n i iii n i iii n i ii n i ii yyxxxyyyfxxf 1 1 1 11 1 1 1 1 )()()()( abyxyx nn =?= 00  : }{max 1 i ni x?= ≤≤ λ ? 0→λ H ￥K1 ∑∑ = ? = ? ?+? n i ii n i ii yyfxxf 1 1 1 1 )()( 212 + ∫ a dxxf 0 )( ∫ ? b dyyf 0 1 )( ?ü￡ ü ? H )(afb = + ∫ a dxxf 0 )( ∫ ? b dyyf 0 1 )( ab= b B? f ?/ ! 1 00 () () () ab F a f x dx f y dy ab ? = + ∫∫ ?5 '( ) ( )Fa fa b=?b:  V??()fT b= 0 aT< < H ??h  ? H ??9F ?[  ()Fa aT> ()Fa ()Fa aT= ) |?Kl′b?  ?￥) ? V?Kl′ V7 ?ü ^ ?1￡ ü￥b () 0FT = () 0Fa≥ ? ? H )(afb = + ∫ a dxxf 0 )( ∫ ? b dyyf 0 1 )( ab= ￥2 ?9 V°¤V+?m?  Ab 11 ￡ ü?s￥ ??? !f ? ?fx() fx() h = fx h()+  [,  V 5μ ]ab lim | () ()| h h a b fx fxdx → ?= ∫ 0 0b ￡ ??  [,  V V?ifx h ()= fx h()+ ]ab 0>δ  P¤  fx() ],[ δδ +? ba  Vb ! ()f xM≤ )],[( δδ +?∈ bax b ??  [,  V  fx() ]ab 0ε? > i uW [, ?s￥?s  P¤? ]abn P 8 ba nM ε? < H? ?  1 6 n ii i x ε ω = ?< ∑  ? ),,2,1( ni n ab x i null= ? =? b 6?? ba n δ ? < H: 10 , +n ωω sY ^  uW)(xf [, ba aa n ? ? ]? [, ] ba bb n ? + ￥??5 0 2Mω ≤  1 2 n Mω + ≤ b y1 1 11 |()()| |()()| ( )() i i nn bx hhi ax ii i f xfxdx fxfxdx f hf xξξ ? == ?= ?=+? ∑∑ ∫∫ ? 213 O? min , 8 ba h nM ε δ ? ? << ? ?? ? ? ] i H? 1 [, ii x xξ ? ∈  V? 21 1 1 [,][,][, iiiiii hxx xx xx]ξ ?? ? + +∈ ∪∪ ? 11 , n ba ba xa x b nn ?+ ? ? =? =+ V7 μ 11 )()( +? ++≤?+ iiiii fhf ωωωξξ ? ^ ∫ ? b a h dxxfxf |)()(| ε εε ωω ε ωωω =+< ? ++<?++≤ + = +?∑ 22 )( 2 )( 10 1 11 n ab x n n i iiii  ?[ lim | () ()| h h a b fx fxdx → ?= ∫ 0 0b 12 ! ?  [, ? V￡ ü?? T fx() gx() ]ab (1) Schwarz?? T    ∫∫∫ ?≤ ? ? ? ? ? ? b a b a b a dxxgdxxfdxxgxf )()()()( 22 2 (2)  Minkowski?? T  {}{ } { } 2 1 22 1 22 1 2 )()()]()([ ∫∫∫ +≤+ b a b a b a dxxgdxxfdxxgxf b ￡  ?? ?i￥ st 2 [() ()] 0 b a tf x g x dx+ ≥ ∫ ' 22 () b a tfxdx+ ∫ 2()() b a tfxgxdx+ ∫ 2 () 0 b a gxdx≥ ∫  ?[  ?Y T?1d?￥9ü ^? ?   ∫∫∫ ?≤ ? ? ? ? ? ? b a b a b a dxxgdxxfdxxgxf )()()()( 22 2  2 ? ()() b a f xgxdx≤ ∫ {} 1 2 2 () b a f xdx ∫ {} 1 2 2 () b a gxdx ∫ ¤? 2 () b a f xdx+ ∫ 2()() b a f xgxdx+ ∫ 2 () b a gxdx ∫  2 () b a f xdx≤+ ∫ {} 1 2 2 2() b a f xdx ∫ {} 1 2 2 () b a gxdx ∫ 2 () b a gxdx+ ∫  ' 214 2 [() ()] b a f xgxdx+ ∫ {}{} 2 11 22 22 () () bb aa f x dx g x dx ?? ??≤+ ?? ∫∫   H 7 üZ'¤? {}{ } { } 2 1 22 1 22 1 2 )()()]()([ ∫∫∫ +≤+ b a b a b a dxxgdxxfdxxgxf b 13 ! ?  [,  ?? O  ￡ üfx() gx() ]ab 0)( ≥xf 0)( >xg {})(max)()]([lim 1 xfdxxgxf bxa n b a n n ≤≤∞→ = ∫ b ￡ y1 [,   ?[μ]ab 0)( >xg 0()mgx M< ≤≤<+∞b: () max () axb A ffxξ ≤≤ == ?^ !  y10A> 0A= H? TA ?? ? b ? )()(lim ξ ξ fxf x = → , V? 0 Aε?<<  ],[],[ ba?? βα  P¤ ],[ βαξ ∈  O? ],[ βα∈x H? ? 0()A fx Aε<?< ≤? ^ {} 1 11 ( )[ ( )] ( ) ( ) [ ( )] b n n nn a A mfxgdxAMεβα??≤ ≤? ∫ bab ??? Hn →∞ 1 [( )] n m β α? 1→  1 [( )] n M ba? 1→  ?[ 0N? > ? H ? ? Nn >. 1 [( )] 1 n m A ε βα?>?D 1 2 [( )] 1 n Mb a A ε ?<+ V7? H? ? Nn >. {} 1 2()() b n n a AfxgxdA2ε ε?< <+ ∫ ' {} 1 ()() 2 b n n a fxgxdx A ε? < ∫  ?[ {} 1 lim [ ( )] ( ) max ( ) b n n an axb f xgxdx A fx →∞ ≤≤ == ∫ b 215