امتحانات في القياس و المكاملة

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1−x²) (x,√

1−x²)

−1 1

1

−1

u₁

u₁ u2

u2

x y

1

1 2

2015−2014

Ty`›A˜ Tns˜

xrhf˜

4 . . . . 2010/02/13 At› 1 5 . . . . 2010/06/09 At› 2 6 . . . . 2010/09/20 At› 3 7 . . . . 2011/02/06 At› 4 8 . . . . 2011/06/21 At› 5 9 . . . . 2011/06/29 At› 6 9 . . . . 2011/09/10 At› 7 10 . . . . 2012/03/15 At› 8 11 . . . . 2012/06/24 At› 9 12 . . . . 2012/09/06 At› 10 12 . . . . 2013/02/06 At› 11 13 . . . . 2013/06/01 At› 12 14 . . . . 2013/09/10 At› 13 15 . . . . 2014/02/06 At› 14 16 . . . . 2014/06/11 At› 15 16 . . . . 2014/06/23 At› 16 17 . . . . 2014/09/10 At› 17 20 . . . . 2010/02/13 At› ™ 18 23 . . . . 2010/09/20 At› ™ 19 24 . . . . 2011/06/21 At› ™ 20 27 . . . . 2011/09/10 At› ™ 21 30 . . . .2012/03/15 At› ™ 22 31 . . . . 2012/09/06 At› ™ 23 34 . . . . 2013/02/06 At› ™ 24 37 . . . . 2013/06/01 At› ™ 25 42 . . . .2013/09/10 At› ™ 26 45 . . . . 2014/02/06 At› ™ 27 49 . . . .2014/09/10 At› ™ 28

‰yR wm˜

A\ž) AyRA§C T` r˜ Tns˜ TblW˜ ›d’ ¨t˜ AžAt›¯ Ÿ› dˆ œ§rk˜ ¹CAql˜ dqž

.2014rbmtbF¤2010©rfy Ÿy A› ­rtf˜ ¨ Tbq˜A ­dAF°˜ Ayl`˜ TFCdm˜A (5 +A§Cw˜Ak .TlOfm˜ Ah˜wl CAbt¯ £@¡ Ÿ› {`b˜ AnqC dq˜¤

xAyq˜ T§r\n˜ ™S œh Ÿ› ¢nkm TbFAn› ­ A› xC dl˜ rwyF @¡   ¨ ryb• Anl›¤

. At›®˜ TbFAn› Tyfyk , Ab˜AV  A•   , rySt˜ Ÿ›¤ Tl›Akm˜ ¤

2010/02/13

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¹

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à

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éJ ®JºK. ÉÓA¾JË@ éK.AJºK.

Z +∞

0

sinx x

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x −e^−x

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ë

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IJk

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2010/06/09

 At› .

²

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š¤± Ÿ§rmt˜ .¹.²

.

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,

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ð

^{f(x) =e−|x|}

à

AK. á ¯QªÖÏ@ á®J®mÌ'@ áªK.AJË@

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ð

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áºJË .

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à

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é KPA ®Öß. á

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é JKAJ. JÓð ( HAJ.K@ àðYK. AêÊJ.®K ú æË@) éj.J JË@ è Yë ­J £ñK ½ JºÖß Éë ÐA ¢J KAK. @QÒ J‚Ó

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2010/09/20

 At› .

³

  © ,AyˆA`J rZAnt›  A• Ð .yw˜ xAyq ¤z› R<sup>2</sup> Ylˆ ¯wm• Ayqyq A`A ϕ Ÿky˜ þþ ry•@

An§d˜ ¢žr=p

x²₁+x²₂yϕ(x₁, x₂) =ϕ(r) Z

R²

ϕ(x₁, x₂)dx₁dx₂= 2π Z ∞

0

ϕ(r)r dr.

©¤rk˜ ­dw˜ ‘®‹ TAs› ¨¡σN) An§dlR^N ¨ ¯wm•¤ AyˆA`J rZAnt›ϕ‰A ™ Ÿ› T›Aˆ TfO¤

: (©dyl’±

Z

R^N

ϕ(x)dx= Z

R^N

ϕ(|x|)dx = σN

Z ∞

0

ϕ(r)r^N−1dr,

x= (x1, . . . , xN), r=|x|= q

x²₁+· · ·+x²_N.

 RYlˆ Ÿyr`m˜ v ¤u  Ayqyq˜  A`At˜ Ÿky˜ š¤± Ÿ§rmt˜ .¹.³

u(x) =−χ_[−1,0](x) +χ_]0,1](x) wa v(x) = (1 +x²)⁻¹.

. lim

x→+∞(u ? v)(x)Ÿyˆ œ u ? vl˜ º d s Y˜ ¨mtn§ψ(x) =e^−x2  RYlˆ ‘r`m˜ ψ¨qyq˜ ‰At˜   Ÿy ( ¨žA˜ Ÿ§rmt˜ .².³

?[1,+∞]“lŒm˜ šAm˜ Ÿ›p  A• Amh›L^p(R) Y˜ψ ¨mtn§ ™¡ .L¹(R)

£®ˆ ry•@t˜ ¤ ¨nyw Tn¡rbm Any`ts› s .I1 = Z

R

e^−x21dx1 = Z

R

e^−x22dx2 ‰Sn˜ ( .I1 Tmy’ Ahn› tntF ¤I₁² Tmy’

.(2 Ÿ› rb• ¨`ybV dˆ N y) I_N = Z

R^N

e^{−x21−x22−···−x2N}dx₁dx₂· · ·dx_N  µ ‰Sn˜ (þ .N  A• Amh›I_N T›Aˆ TfO¤ œI₃ s T§dyl’± ­rk˜ BR=B(0, R) Ÿkt˜¤ Ayqyq dˆ 0 < R <1 Ÿky˜ ( ˜A˜ Ÿ§rmt˜ .³.³  B_R¨ ‘r`m˜ θ‰At˜ Ÿky˜¤ .R rWq˜ Ož¤0 z•rm˜ ÐR^N ¨ Twtfm˜

θ(0) = 0 ¤ θ(x) =|x|^−N|ln|x||^−α, 0<|x|= q

x²₁+· · ·+x²_N < R

.L¹(B_R)Y˜ ¨mtn§θ  y .YW`› A ¨qyq dˆ1< αy .0< ε A• Amh›L^1+ε(BR)Y˜ ¨mtn§ ¯θ  b (

L^p(X,A, µ)Ÿ› ‰ w Ty˜Att›{f_n}^∞_n=1 Ÿkt˜¤ .Asyq› ºAS(X,A, µ)Ÿky˜ ‰ r˜ Ÿ§rmt˜ .⁴.³

{f_n}^∞_n=1 Ty˜Attm˜   |rfn˜ .^L∞(X,A, µ) Ÿ› ‰ w Ty˜Att›^{gn}^∞_n=1 Ÿkt˜¤ .1≤p <+∞ y

¨ –J þþµTCAqt›{g_n}^∞_n=1 Ty˜Attm˜  ¤ .ºASf˜ @¡ Ÿ›f ‰A wžL^p(X,A, µ) ¨ TCAqt›

‰›g‰A wžX kg_nk_L∞(X,A,µ)≤M, ∀n∈N^?,

¨ f gwž TCAqt› {f_ng_n}^∞_n=1 Ty˜Attm˜   Ylˆ A¡dnˆ Ÿ¡r .A›Am w› ¨qyq dˆ M y .L^p(X,A, µ) ].Tnmyh˜A CAqt˜ Tn¡rb› dts¤_{fngn−f g= (fn−f)gn+f(gn−g)} tk   –nkm§ : AJC[

2011/02/06

 At› .

⁴

.yw˜ xAyq¤ Tyl§Cwb˜ ¢ryK` (¢n› ºz © ¤ _R² ¤)_R ¤zž ¨l§ A› ™• ¨

.T = Z e

1

Z lnx 0

dy x(1 +y)

dx™›Akt˜ Ÿky˜ š¤± Ÿ§rmt˜ .¹.⁴ .Tl›Akm˜ yr Hk` (þ) œ ­rJAb› T s () .Tl›Akm˜   dy› œFC ()

¨qyq˜ ‰At˜ Ÿky˜¤R² ⊃C= [1,+∞[×[1,+∞[¢tnm˜ ry‹ ‰rm˜ Ÿky˜ ¨žA˜ Ÿ§rmt˜ .².⁴  C Ylˆ ‘r`m˜ f f(x, y) = x−y

(x+y)³·

.–tA Cr ?xwy’f ™¡

C dqm˜ As˜ ¢n› dftF ¤ ^∂

∂x

−x (x+y)²

¹z˜ “tKm˜ s ()

K = Z ∞

1

Z ∞

1

f(x, y)dx dy.

?K =L™¡ .L=R∞ 1

hR∞

1 f(x, y)dy i

dxC dqm˜ –˜@• s

xAy’ w¡λ₂ ,^L1(C,B_C, λ₂) ºASf˜ Y˜ ¨mtn§ ¯f ‰At˜   Yl`f˜ As˜ ¤ CAŒ}³A b () .yw˜

y ϕ(x) = (sinx)χ_I(x)   R Ylˆ ‘r`m˜ ϕ ¨qyq˜ ‰At˜ Ÿky˜ ˜A˜ Ÿ§rmt˜ .<sup>3</sup>.<sup>4</sup>   Tr`m˜ {ϕ_n}^∞_n=1 Tyqyq˜ ‰ wt˜ Ty˜Att› Ÿkt˜¤ .I =]0, π[ šAml˜ ­zymm˜ T˜ d˜ ¨¡ χI

.ϕn dnF ,suppϕnŸyˆ .R3x ,ϕn(x) =ϕ(x−nπ)

.3< n,ϕ_n T›Aˆ TfO¤ .ϕ₃ ,ϕ₂ ,ϕ₁ ,ϕ‰ wt˜ AžAy œFC ,œl`m˜ Hfž Ylˆ () .¢nyy` lW§ϕ∞‰A wžRYlˆ TVAsb CAqt^{ϕn}^∞_n=1Ty˜Attm˜   Ÿy ()

?^L1(R)¨ϕ∞wž Ty˜Attm˜ £@¡ CAq Ÿˆ ÐA›

.Ÿyqyq Ÿ§ dˆ1> b >0¤ 1> a >0Ÿky˜ ‰ r˜ Ÿ§rmt˜ .⁴.⁴

?Ayhtn›^R₀^ax^−αdx‰Fwm˜  Am§C ™›Ak  wk§ ¨• α¨qyq˜ d`˜ CAyt § y• ()

?Ayhtn›^R_b¹(1−x)^−βdx‰Fwm˜  Am§C ™›Ak  wk§ ¨•β¨qyq˜ d`˜ CAyt § y•¤ () Y˜ Aymtn› x^−α(1−x)^−β ‰At˜  wk§ ¨• β ¤α Ylˆ ŸyVrK˜ “bFAm› tntF (þ)

?^L1(0,1)

2011/06/21

 At› .

⁵

.¢FAy’¤ yw˜ ­ryK` ¢n› ºz © ¤ (R^N ¤)R¨l§A› ™• ¨ ¤z§

 Ayqyq˜  A`At˜ ¤ J = [−2,2] ¤ I = [−1,1]  A} rtm˜  ¯Am˜ Ÿky˜ š¤± Ÿ§rmt˜ .<sup>1</sup>.<sup>5</sup> .£dnF¤f ? g l˜ º d œg¤ f ©dnF Ÿyˆ .g(x) =χ<sub>J</sub>(x)¤ f(x) = 2xχ<sub>I</sub>(x) R3x A• Amh› 0< h(x)  A• Ð .L<sup>1</sup>(R) Y˜ ¨mtn§ Ayqyq A`AhŸky˜ ¨žA˜ Ÿ§rmt˜ .².⁵ .^L1(R)Y˜ ¨mtn§ ¯ (h‰At˜ wlq›)h⁻¹   b

]._h^−1/2_h^1/2º d˜ Ylˆ Tm¶®› Tn§Abt› “ybW –nkm§ : AJC[

  © ,AyˆA`J rZAnt›  A• Ð .<sub>R</sub><sup>N</sup> Ylˆ Abw›¤ AFwy’ Ayqyq A`AϕŸky˜ þþ ry•@

ϕ(x1, . . . , xN) =ϕ(r), r= q

x²₁+· · ·+x²_N

: (©dyl’± ©¤rk˜ ­dw˜ ‘®‹ TAs› ¨¡σN) An§d˜ ¢ž

Z

R^N

ϕ(x)dx = Z

R^N

ϕ(|x|)dx=σN

Z ∞

0

ϕ(r)r^N−1dr, x= (x₁, . . . , x_N), r=|x|=

q

x²₁+· · ·+x²_N.

Ÿ§ dˆ p ≥ 1 ¤ α 6= 0 Ÿky˜¤ .R^N ¨ T§dyl’¯ ­dw˜ ­r• ^U Ÿkt˜ ˜A˜ Ÿ§rmt˜ .³.⁵ :¥Akt˜ b .Ÿyyqyq

αp+N >0 ⇐⇒ ϕ(x) =|x|^α∈L^p(U).

(1)

αp+N <0 ⇐⇒ ψ(x) =|x|^α ∈L^p(^cU), ^cU =R^N \ U. (2)

‰Sn˜ .Ayqyq dˆ 1 < p < ∞ ¤ Ω =]0,1[ wtfm˜ šAm˜ Ÿky˜ ‰ r˜ Ÿ§rmt˜ .⁴.⁵ .Ω3x ,u_n(x) =n^1/pe^−nx

:^{un}^∞_n=1 Ty˜Attm˜   y .¤d`m˜ ‰At˜ wž Ω¨ TVAsb CAqt Ahnk˜ ­ ¤d› ry‹ (1 .<sub>N</sub><sup>?</sup>3n A• Amh› ,ku<sub>n</sub>k<sub>L</sub>p(Ω)≤M y 0< M A dw§ ¢ž Yn`m ,L^p(Ω)¨ ­ ¤d› (2 .¤d`m˜ ‰At˜ wž L<sup>p</sup>(Ω)¨ T§CAqt› ry‹ (3 .<sub>p</sub><sup>1</sup>+<sub>p</sub><sup>1</sup>0 = 1   ‘r`m˜ d`˜ w¡p⁰ y ,L^p0(Ω)3v A• Amh› ,lim

n→∞

Z

Ω

unv dx= 0 Ÿk˜ (4 ]._L^p

0(Ω)ºASf˜ ¨_Ω¨ T} rt› dnF Ф ­rmtsm˜ ‰ wt˜ TA•  dtF –nkm§ : AJC[

2011/06/29

 At› .

⁶

.¢FAy’¤ yw˜ ­ryK` ¢n› ºz © ¤ (_R^N ¤)_R¨l§A› ™• ¨ ¤z§

 Ayqyq˜  A`At˜ ¤ J = [−4,4] ¤ I = [−2,2]  A} rtm˜  ¯Am˜ Ÿky˜ š¤± Ÿ§rmt˜ .¹.⁶  Ay –˜@• œFC .£dnF¤f ? g l˜ º d œg¤f ©dnF Ÿyˆ .g(x) =χ_J(x)¤f(x) = 3x²χ_I(x)

.f ? g‰At˜

R3x A• Amh› 0< h(x)  A• Ð .L²(R) Y˜ ¨mtn§ Ayqyq A`AhŸky˜ ¨žA˜ Ÿ§rmt˜ .².⁶ .^L2(R) Y˜ ¨mtn§ ¯ (h‰At˜ wlq› ‰r›)h⁻²   b

α6= 0 Ÿky˜¤ .^U Tq}®›^U Ÿkt˜¤R^N ¨ T§dyl’¯ ­dw˜ ­r• ^U Ÿkt˜ ˜A˜ Ÿ§rmt˜ .³.⁶ :‰At˜ ¨mtn§ ¨•α d`˜ CAyt § y• .Ÿyyqyq Ÿ§ dˆ p≥1¤

?L^p(U)Y˜ ϕ(x) = (1− |x|)^α (1 .^cU =R^N \ U An¡ ?L^p(^cU)Y˜ ψ(x) = (|x| −1)^α (2

‰Sn˜ .Ayqyq dˆ1< p <∞¤ Ω =]−1,1[wtfm˜ šAm˜ Ÿky˜ ‰ r˜ Ÿ§rmt˜ .⁴.⁶

u_n(x) = (n|x|)^1/pe^−nx2, x∈Ω.

:{u_n}^∞_n=1Ty˜Attm˜ ™¡

?A› ‰A wžΩ¨ TVAsb TCAqt› ?­ ¤d› (1

?L^p(Ω)¨ ­ ¤d› (2

?L^p(Ω)¨ T§CAqt› (3 2011/09/10

 At› .

⁷

.¢n› ºz Ylˆ ¤ ¢l• ºASf˜ Ylˆ yw˜ xAy’ ¯ An¡ dtsž ¯ Ahž b .u_n(x) = x²

n⁴x⁴+ 1   R Ylˆ Tr`m˜ {u_n}^∞_n=1 Ty˜Attm˜ Ÿkt˜ š¤± Ÿ§rmt˜ .¹.⁷

!©C¤rR r§rbt˜ .¢nyy` ¨Œbn§u∞‰A wž ,[1,∞]3p A• Amh› ,^Lp(R)¨¤ TVAsb TCAqt›

  Tr`m˜ R² ­r•B Ÿkt˜ ¨žA˜ Ÿ§rmt˜ .².⁷

B = n

(x, y)∈R² |0≤x²+y² < 1 2

o

 B Ylˆ ‘r`m˜ ¤ AyˆA`J rZAntm˜ ¨qyq˜ ‰At˜ ϕrbt`ž , w› ¨qyq dˆ α™ Ÿ›¤

ϕ(x, y) = lnp

x²+y²

−α(x²+y²)^−1/2, (x, y)∈B\ {(0,0)}, ϕ(0,0) = 0.

?L²(B)Y˜ϕ‰At˜ ¨mtn§ ¨•α d`˜ CAyt § y•

   Ar`m˜  Ayqyq˜  A`At˜ ¤ I = [−1,1] ¨qyq˜ šAm˜ Ÿky˜ ˜A˜ Ÿ§rmt˜ .³.⁷ .f ? gl˜ º d Ÿyˆ .Cw•@m˜ šAml˜ ­zymm˜ T˜ d˜ ¨¡χI ,g(x) = _x2¹+1 ¤ f(x) = 2xχI(x)

  J Ylˆ Tr`m˜ {v<sub>n</sub>}<sub>n≥1</sub> Ty`At˜ Ty˜Attm˜ ¤ J = [0,1] šAm˜ Ÿky˜ ‰ r˜ Ÿ§rmt˜ .⁴.⁷ .v_n(x) =n/(n²x²+ 1)

.Ty˜Attm˜ £@h˜v∞TWysb˜ T§Ahn˜ Ÿyˆ (1

L¹([a,1]) ¨ C ¤ CAqt˜ ,Ÿk˜ .L¹(J) ¨ v∞ wž CAqt ¯ {v_n}_n≥1 Ty˜Attm˜   b (2 .0< a <1 A• Amh›

.^L1(J) ¨v∞ wž CAqt^{{x v}n}_n≥1 Ty˜Attm˜   b (3 .^J šAm˜ Ylˆ rmts›¤ w› ¨qyq ‰A Y˜^hz›r§ Ÿyy˜At˜ Ÿy˜ ¥s˜ ¨

‰At˜ wž CAqt ¯ ^{{h v}n}_n≥1 Ty˜Attm˜   b ,min

J h =g_m >0 ,h {yS  A• Ð (4 .L¹(J)ºASf˜ ¨ ¤d`m˜

.L¹(J) ºASf˜ ¨ ¤d`m˜ ‰At˜ wž CAqt{h v_n}n≥1   y ,h(0) = 0 A• Ð ,Ÿk˜ (5 2012/03/15

 At› .

⁸

.¢n› ºz Ylˆ ¤ ¢l• (R² ¤)RºASf˜ Ylˆ yw˜ xAy’ ¯ An¡ dtsž ¯

š¤± Ÿ§rmt˜ .¹.⁸ .D=

(x, y)∈R²|x²+y²≤9 ∧ y≥√

x²+ 1   ‘r`m˜ _R² ©wtsm˜ zy œFC () .R² Ylˆ šwm•f(x, y) =xyχ_D(x, y) ‰At˜   Ÿy .Dzyl˜ ­zymm˜ T˜ d˜ χ_D Ÿkt˜ ()

. yrt˜ @¡ Hk`¤ xœy Y˜ Tbsž Tl›Akm˜A @¡¤R<sup>2</sup> Ylˆf ‰At˜ ™›Ak s T˜ d˜ ¨¡χ<sub>I</sub> yϕ(x) = (1−x)χ<sub>I</sub>(x)  YW`m˜ ¨qyq˜ ‰At˜ ϕŸky˜ ¨žA˜ Ÿ§rmt˜ .².⁸ .ϕ_n(x) =nϕ(n(x+n))y^{ϕn}^∞_n=1Ty`At˜ Ty˜Attm˜ Ÿkt˜¤ .I = [0,1]Q rtm˜ šAml˜ ­zymm˜

.ϕn ‰At˜ dnF ,suppϕn Ÿyˆ¤ϕn T›Aˆ TfO¤ϕ2 ,ϕ1 ,ϕ‰ wt˜ AžAy œFC () .¢nyy` lW§ϕ∞‰A wž TVAsb TCAqt›^{ϕn}^∞_n=1 Ty˜Attm˜   Ÿy ()

?L¹(R)¨ϕ∞ wž{ϕ_n}^∞_n=1Ty˜Attm˜ CAqt ™¡ (þ)

˜A˜ Ÿ§rmt˜ .³.⁸

¤ 0 < x < 1 ™ Ÿ› g(x) = x−1

lnx   ‘r`m˜ g ‰At˜  wk§ ¨• β ¤ α ŸytA˜ Ÿyˆ () .[0,1]Q rtm˜ šAm˜ Ylˆ rmts›g(1) =β ¤ g(0) =α

.Yþ¶An˜ ™›Akt˜  dtFA ¢tmy’ As w¡ And¡¤ ww›T = Z 1

0

x−1

lnx dx Am§C ™›Ak  Ð

‘r`m˜ ¨qyq˜ ‰At˜ uŸky˜¤ .Ω =]0,1[×]0,1[  ‘r`m˜ _R²¨ wtfm˜ ºz˜ ΩŸky˜ () .Ω63(x, y) A• Ðu(x, y) = 0¤ Ω3(x, y) A• Ðu(x, y) =x^y  R² Ylˆ

.R² Ylˆ šwm•u  Ÿy (þ)

.K = Z

Ω

x^ydxdy Yþ¶An˜ ™›Akt˜ Ÿytyfyk s ( ) .T  Am§C ™›Ak As˜ Ah›dtF ¤ K ¤T Xr ¨t˜ T’®`˜ d .ξ<sub>n</sub>(x) = 1/(1 +x<sup>2</sup>)<sup>n</sup> y<sup>{ξ</sup>n}<sup>∞</sup><sub>n=1</sub> Ty`At˜ Ty˜Attm˜ Ÿkt˜ ‰ r˜ Ÿ§rmt˜ .⁴.⁸

.¢nyy` lW§ξ∞ ‰A wž TVAsb TCAqt›{ξ_n}^∞_n=1 Ty˜Attm˜   Ÿy ()

?^L∞(R)¨ C ¤ w¡ ™¡ ?[1,∞[3p A• Amh›^Lp(R)¨ C ¤ξ∞wž^{ξn}^∞_n=1CAq ™¡ () 2012/06/24

 At› .

⁹

¤ f(x) =e^xχ]−∞,0[(x)  RYlˆ Ÿyr`m˜ g¤ f  Ayqyq˜  A`At˜ Ÿky˜ š¤± š ¥s˜ .¹.⁹ .™y˜d˜ ¨ ­Cw•@m˜ Tˆwmml˜ ­zymm˜ T˜ d˜ Y˜χ ryK§ .g(x) =e^−xχ_]0,∞[(x)

.RŸ› xTWqž ™• dnˆ ¢tmy’ s œ dy ‘r`›f ? g l˜ º d ÐAm˜ ™’

¨žA˜ š ¥s˜ .².⁹  N^? 3n¨`ybV dˆ ™• ™ Ÿ› ‘r`m˜ ¨qyq˜ ‰At˜  Ay œFC ? .¹.².⁹

Tn(t) = 1

2{|t+n| − |t−n|}, t∈R.

? ¤d› ‰At˜ @¡ ™¡ .Tn(f) =Tn◦f ‰At˜ rbt`n˜¤ .[1,∞]3p‰›L<sup>p</sup>(R)3f Ÿky˜ ? .<sup>2</sup>.<sup>2</sup>.<sup>9</sup> .<sup>Lp</sup>(R)¨f wž TCAqt›<sup>{T</sup>n(f)}<sup>∞</sup><sub>n=1</sub> Ty`At˜ Ty˜Attm˜   b .T_n(f)¤f Ÿy`At˜ Ÿy  CA’

N^?3n™•¤L¹_loc(R)3f ‰A ™• ™ Ÿ› ‘r`m˜ ‰At˜ Y˜fnþ rKn˜ ˜A˜ š ¥s˜ .³.⁹ .R3x ,fn(x) = ⁿ₂ Rx+1/n

x−1/n f(t)dt  .ψ(x) =x χ_[0,∞[(x)‰At˜ T˜A ¨R3x7−→ψ_n(x) = ⁿ₂ Rx+1/n

x−1/n ψ(t)dt∈R‰At˜ Ÿyˆ ? .¹.³.⁹

ψ‰At˜ wž A\tžA TCAqt›{ψ_n}^∞_n=1Ty˜Attm˜   b œ .œl`m˜ Hfž Ylˆψ Ay¤ ¢žAy œFC .RYlˆ

Ty˜Attm˜   Ylˆ Ÿ¡r œ T} rt› f_n ‰ wt˜ dnF   Ÿyb ^Cc(R) 3 f  A• Ð ? .².³.⁹  A• Amh›L^p(R) Yf wž Ty˜Attm˜ £@¡ CAq nttF .RYlˆ f wž A\tžA CAqt {f_n}^∞_n=1

.[1,∞]3p

Á¤^∞> p >1Ÿky˜¤ yw˜ H§A’¤ ­ryK` ¤z›J .

=]0,+∞[šAm˜ Ÿky˜ ‰ r˜ š ¥s˜ .⁴.⁹ .J 3x,F(x) = ¹_xRx

0 f(t)dt  ‘r`m˜ F ‰At˜ Ÿky˜¤L^p(J)3f

.L¹63F   Ÿyb^{L1 3}f ‰›0< f  A• Ð. ? 1.4

  T¶zt˜A ®›Ak› bC_c(J)3f ‰›0≤f |rf. ? 1.4 Z ∞

0

F^p(x)dx=−p Z ∞

0

F^p−1(x)xF⁰(x)dx.

Hardy© CA¡ Tn§Abt› nttF ^RF^p−1f Ylˆ Cd˜w¡ Tn§Abt› “bWt œ xF⁰=f−F   ^¯

kFk_L^p ≤ p

p−1kfk_L^p.

2012/09/06

 At› .

¹⁰

.J = [0,+∞[‰Sž¤ .¢n› ºz Ylˆ ¤ ¢l• ºASf˜ Ylˆ yw˜ xAy’ ¯ An¡ dtsž ¯ ryK§ y f(x) =e^−xχ_J(x)   R Ylˆ ‘r`m˜ f ¨qyq˜ ‰At˜ Ÿky˜ š¤± š ¥s˜ .¹.¹⁰

Ož r•Ð ,As  ¤ ¤ dy ‘r`›f ? f l˜ º d ÐAm˜ ™’ .J šAml˜ ­zymm˜ T˜ d˜ Y˜χ_J

TWqž ™• dnˆ(f ? f)(x) s .–tA Cr ,supp (f ? f) £dnF ©wt§ ©@˜ © d`˜ œyqtsm˜

.RŸ›x

dˆαŸky˜¤ .(1,1),(1,0),(0,0)Xqn˜ dnˆ ¢F¤¦C ©@˜ lm˜ T Ÿky˜ ¨žA˜ Ÿ§rmt˜ .².¹⁰ .L=

Z

T

dx dy

(x+y+ 1)^α+1 ™›Akt˜ s .1Ÿˆ¤0Ÿˆ Aft› Ayqyq

J šAm˜ Ylˆ C rmtF¯A ”AqtJ²˜ ®A’¤ Ar`› Ayqyq A`A P Ÿky˜ ˜A˜ Ÿ§rmt˜ .³.¹⁰ .ACAqt›

Z ∞ 0

P⁰(x)e^−xdx™›Akt˜  wk§ y¤ lim

x→+∞P(x)e^−x= 0 “q§¤

.Ÿyl›Akt˜ Xr ¨t˜ T’®`˜ r•Ð ‰› CAqt›

Z ∞ 0

P(x)e^−xdx™›Akt˜   b .¹.³.¹⁰ .

Z ∞ 0

te⁻

√tdt™›Akt˜ s :“ybW .².³.¹⁰

].AqAs˜ Ahylˆ ™Om˜ T’®`˜  dtF C rk ¨ rk¤_x=

√tryŒtm˜ ™§dbt Ÿy`ts   –nkm§ : AJC[

‰ r˜ Ÿ§rmt˜ .⁴.¹⁰ .

Z ∞ 0

u₀(x)dx >0¤

Z ∞

−∞

u₀(x)dx= 0 “q§¤^L1(R)Y˜ ¨mtn§u₀ ‰A d¤ .¹.⁴.¹⁰ .

Z ∞ 0

u(x)dx > 0 ¤

Z ∞

−∞

u(x)dx = 0 “q§¤ ^L1(R) Y˜ ¨mtn§ Ayfy• A`A u Ÿky˜ .<sup>2</sup>.<sup>4</sup>.<sup>10</sup>   Ylˆ Ÿ¡r .R3x,u<sub>n</sub>(x) =nu(nx)  Tr`m˜ ^{un}^∞_n=1Ty`At˜ Ty˜Attm˜ Ÿkt˜¤

n→∞lim Z 1

−1

un(x)ϕ(x)dx= 0, ∀ϕ∈E,

.^[−1,1]Q rtm˜ šAm˜ Ylˆ ­rmtsm˜ ¤RYlˆ Tr`m˜ Tyqyq˜ ‰ wt˜ ºAS Y˜ z›r§Ey

  Ÿyb   –nkm§ : AJC[

Z 1

−1

un(x)ϕ(x)dx= Z n

−n

u(t)h ϕt

n

−ϕ(0)i

dt+ϕ(0) Z n

−n

u(t)dt

].rfO˜ Y˜  ¯¤¥§ Ÿymy˜ Ylˆ Ÿyl›Akt˜   Ab³ Tm¶®› Tn¡rb› dts œ

2013/02/06

 At› .

¹¹

.yw˜ xAyq¤ Tyl§Cwb˜ ¢ryK` (¢n› ºz © ¤ R² ¤)R ¤zž ¨l§ A› ™• ¨

.

Z 1

−1

Z

√1−x²

−√ 1−x²

u(x, y)dy

dx ¹An˜ ™›Akt˜ ¨ Tl›Akm˜   dy› œFC () š¤± Ÿ§rmt˜ .¹.¹¹ .u(x, y) = 1

1 +x²+y² ‰At˜ T˜A ¨ ™›Akt˜ @¡ s ()