ON DIFFERENTIATION OF A LEBESGUE INTEGRAL WITH RESPECT TO A PARAMETER

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1 Mh. Appl. 1 (2012, ON DIFFERENTIATION OF A LEBESGUE INTEGRAL WITH RESPECT TO A PARAMETER JIŘÍ ŠREMR Abr. The im of hi pper i o diu he bolue oninuiy of erin ompoie funion nd differeniion of Lebegue inegrl wih repe o prmeer. The reul obined re ueful when nlyzing rong oluion of pril differenil equion wih Crhéodory righ-hnd ide. 1. Inroduion nd noion Differeniion under inegrl ign i one of he very old queion in lulu of rel funion. For exmple, ondiion uffiien o enure h Leibniz rule i pplible, i.e., h y b f(x, y dx = b f(x, y y dx, (1.1 hve been inveiged lredy by Jordn, Hrnk, de l Vllée-Pouin, Hrdy, Young, nd oher (ee, e. g., urvey given in [2]. Thi rule nd i generlizion ply n imporn role in vriou pr of mhemi. In priulr, we re inereed in Crhéodory oluion o he pril differenil inequliy 2 γ(, x p(, xγ(, x + q(, x (1.2 x wih non-negive oeffiien p nd q inegrble on he rengle [, b] [, d] (ee, [3, Proof of Corollry 3.2(b]. I i known h uh oluion i, e. g., he funion γ(, x = Z,x (, ηq(, η dηd for (, x [, b] [, d], where Z,x denoe he o-lled Riemnn funion of he orreponding hrerii iniil vlue problem. However, Riemnn funion n be expliily wrien only in everl imple e nd hu we need o find noher oluion o (1.2 whih would be expreed effeively. By uing erin wo-dimenionl nlogy of he well-known Cuhy formul for ODE we rrive he funion γ(, x = q(, η e η p(ξ1,ξ2dξ2dξ1 dηd for (, x [, b] [, d]. ( MSC: primry 26A24, 26B05. Keyword: Differeniion of n inegrl wih repe o prmeer, bolue oninuiy of ompoie funion. The reerh w uppored by RVO:

2 92 J. ŠREMR We need o how h hi funion i boluely oninuou in he ene of Crhéodory 1 nd h ifie inequliy (1.2 lmo everywhere in [, b] [, d]. Le u menion h if he oeffiien p nd q re oninuou, he problem indied i no diffiul. If p nd q re dioninuou, he iuion i muh more omplied nd we hve no found ny reul pplible o hi priulr problem in he exiing lierure. In hi pper, we dp nd exend known reul in order o olve our problem. More preiely, we eblih Theorem 2.7 gurneeing he bolue oninuiy of he funion λ ϕ(λ f(, λ d nd giving formul for i derivive. Then, in Theorem 2.9, we inveige he queion on he exiene of pril derivive of he funion (λ, µ h(, λ, µ d. (1.4 The reul obined re pplied o olve he bove-menioned problem (ee Corollry 2.13 onerning pril differenil inequliy (1.2, beue he funion γ defined by relion (1.3 i priulr e of mpping (1.4. The following noion will be ued hroughou he pper: N, Q, nd R denoe he e of ll nurl, rionl, nd rel number, repeively, R + = [0, + [, nd for ny x R we pu [x] + = ( x + x/2 nd [x] = ( x x/2. If Ω R n i meurble e hen me Ω denoe he Lebegue meure of Ω nd L(Ω; R nd for he pe of Lebegue inegrble funion p: Ω R endowed wih he norm p L = p(x dx. Moreover, he pril derivive of he funion Ω u: Ω R he poin x Ω re denoed by u [k] (x 1,..., x n = u(x 1,..., x n x k for k {1,..., n}, u [k,l] (x 1,..., x n = 2 u(x 1,..., x n for k, l {1,..., n}. x k x l A l, AC ([α, β]; R nd for he e of boluely oninuou funion on he inervl [α, β] R. 2. Min reul I i well known h ombinion of boluely oninuou funion migh no be boluely oninuou. Therefore, before formuling of he min reul (nmely, Theorem 2.7 nd 2.9 we preen he following rher imple emen whih we will need ferwrd. Propoiion 2.1. Le ϕ AC ([, b]; R nd f AC ([, d]; R, where [, d] = ϕ([, b]. Pu F ( := f ( ϕ( for [, b]. (2.1 Then he following erion re ified: 1 Thi noion i defined in [1] (ee lo Lemm 3.1 below.

3 ( The relion ON DIFFERENTIATION UNDER INTEGRAL SIGN 93 F ( = f ( ϕ( ϕ ( for ll ϕ 1 (E 1 E 2 hold, where E 1 = {x [, d] : here exi f (x} nd E 2 = { [, b] : here exi ϕ (}. (b If he funion ϕ i monoone (no rily, in generl hen he funion F i boluely oninuou. ( If he funion ϕ i rily monoone hen F ( = f ( ϕ( ϕ ( for.e. [, b]. 2 (2.2 Remrk 2.2. Le he funion ϕ in Propoiion 2.1 be rily monoone. Then he e ϕ 1 (E 1 in he pr ( i meurble (wihou ny ddiionl umpion nd me ϕ 1 (E 1 = b if nd only if he invere funion ϕ 1 i boluely oninuou (ee, e. g., [4, Chper IX, 3, Theorem 3 nd 4]. Therefore, even in hi e, pr ( doe no follow, in generl, from pr (, beue he funion ϕ 1 migh no be boluely oninuou (ee [5, Seion 2]. Corollry 2.3. Le ϕ AC ([, b]; R be rily monoone funion nd g L([, d]; R, where [, d] = ϕ([, b]. Pu F ( = ϕ( Then he funion F i boluely oninuou nd g( d for [, b]. (2.3 F ( = g ( ϕ( ϕ ( for.e. [, b]. 3 (2.4 Condiion gurneeing h Leibniz rule (1.1 for he Lebegue inegrl i pplible ome priulr poin re well known. We menion here, for exmple, he following emen. Propoiion 2.4 ([2, Chper V, Seion 247]. Le he funion f : [, d] [, b] R ify he relion nd Moreover, le [, b] be uh h he funion d f(, x L([, d]; R for ll x [, b], (2.5 f(, AC ([, b]; R for.e. [, d], (2.6 f [2] L([, d] [, b]; R.4 (2.7 f [2] (, d: [, b] R i oninuou he poin. (2.8 2 In order o enure h relion (2.2 i meningful we pu f (x := α(x hoe poin x [, d], where he derivive of he funion f doe no exi, α: [, d] R beeing n rbirry funion. Oberve h hoie of he funion α h no influene on he vlue of he righ-hnd ide of equliy (2.2 (ee Lemm 3.2 below. 3 In order o enure h relion (2.4 i meningful we pu g(x := ω(x hoe poin x [, d], where he funion g i no defined, ω : [, d] R being n rbirry funion. Oberve h hoie of he funion ω h no influene on he vlue of he righ-hnd ide of equliy (2.4 (ee Lemm 3.2 below. 4 See Remrk 2.5.

4 94 J. ŠREMR Pu F (λ := d Then he funion F i differenible he poin nd F ( = f(, λ d for λ [, b]. (2.9 d f [2] (, d. Remrk 2.5. I follow from umpion (2.6 h here exi f [2] (, x for ll (, x Ω := {(, η : E, η A(}, where E [, d] wih me E = d nd, for ny E, we hve A( [, b] wih me A( = b. Noe h, in generl, he e Ω migh no be meurble. Clerly, in umpion (2.7 we require h he funion f [2] i defined (i.e., he pril derivive exi lmo everywhere in he rengle [, d] [, b]. I i worh menioning here h hi umpion follow, e. g., from he exiene of funion g L([, d] [, b]; R uh h f [2] g on Ω (ee Lemm 3.5 below. If we re no inereed in differenibiliy of he funion F priulr poin, oninuiy umpion (2.8 in Propoiion 2.4 n be omied nd hu we obin he following reul. Propoiion 2.6. Le f : [, d] [, b] R be funion ifying relion (2.5 (2.7. Then he funion F defined by formul (2.9 i boluely oninuou on he inervl [, b] nd F (λ = d f [2] (, λ d for.e. λ [, b]. (2.10 If we dd vrible upper boundry of he inegrl in (2.9, we obin Theorem 2.7. Le he funion ϕ AC ([, b]; R nd f : [, d] [, b] R be uh h relion (2.5 (2.7 hold nd ϕ([, b] = [, d]. Pu F (λ := ϕ(λ f(, λ d for λ [, b]. (2.11 Then he following erion re ified: ( There exi e E 1 [, d] nd E 2 [, b] uh h me E 1 = d, me E 2 = b, nd F (λ = f ( ϕ(λ, λ ϕ (λ + ϕ(λ f [2] (, λ d for ll λ ϕ 1 (E 1 E 2. (b If he funion ϕ i monoone (no rily, in generl hen he funion F i boluely oninuou on he inervl [, b]. ( If he funion ϕ i rily monoone hen F (λ = f ( ϕ(λ, λ ϕ (λ + ϕ(λ f [2] (, λ d for.e. λ [, b].5 ( In order o enure h relion (2.12 i meningful we pu f(, x := ω(, x hoe poin (, x [, d] [, b], where he funion f i no defined, ω : [, d] [, b] R being n rbirry funion. Oberve h hoie of he funion ω h no influene on he vlue of he righ-hnd ide of equliy (2.12 (ee Lemm 3.2 below.

5 ON DIFFERENTIATION UNDER INTEGRAL SIGN 95 Remrk 2.8. Le he funion ϕ in Theorem 2.7 i rily monoone. Anlogouly o Remrk 2.2 we n menion h relion (2.12 follow from pr ( if he invere funion ϕ 1 i boluely oninuou. In priulr, we hve d d f(, d = f(, + f [2] (, d for.e. [, b] whenever he funion f ifie relion (2.5 (2.7 wih = nd b = d. A we hve menioned bove, we need o how h he funion γ defined by formul (1.3 i Crhéodory oluion o differenil inequliy (1.2. In priulr, we hve o how h he funion γ i boluely oninuou on [, b] [, d] in he ene of Crhéodory whih, in view of Lemm 3.1, require o derive formul for pril derivive of he funion (1.4 wih repe o eh vrible. For hi purpoe we eblih he following emen whih will be pplied o prove Corollry 2.12 below. Theorem 2.9. Le h: [, b] [, b] [, d] R be funion uh h he relion nd h(, x, z L([, b]; R for ll (x, z [, b] [, d], (2.13 h(,, z AC ([, b]; R for.e. [, b] nd ll z [, d], (2.14 re ified. Pu h [2] (,, z L([, b] [, b]; R for ll z [, d]6 (2.15 H(λ, µ := Then he following erion re ified: h(, λ, µ d for ll (λ, µ [, b] [, d]. (2.16 ( For ny µ [, d] fixed, we hve H(, µ AC ([, b]; R nd H [1] (λ, µ = h(λ, λ, µ + h [2](, λ, µ d for.e. λ [, b]. (2.17 (b Le, in ddiion o (2.13 (2.15, here exi number k {0, 1} uh h nd ( 1 k h(, x, : [, d] R i non-dereing for ll x [, b] nd.e. [, x], ( 1 k h [2](, x, : [, d] R i non-dereing for.e. (, x [, b] [, b], x, (2.18 (2.19 h(x, x, : [, d] R i oninuou for.e. x [, b], (2.20 h [2](, x, d: [, d] R i oninuou for.e. x [, b]. ( See Remrk 2.10.

6 96 J. ŠREMR Then here exi e E 1 [, b] uh h me E 1 = b nd H [1] (λ, µ = h(λ, λ, µ + h [2] (, λ, µ d for ll λ E 1, µ [, d]. (2.22 ( Le, in ddiion o (2.13 (2.15 nd (2.18 (2.21, for ny x E 1 he funion h ify H nd h(x, x, AC ([, d]; R, (2.23 h [2](, x, AC ([, d]; R for.e. [, x], (2.24 h [2,3] (, x, L([, x] [, d]; R.7 (2.25 Then, for ny λ E 1 fixed, we hve H [1] (λ, AC ([, d]; R nd [1,2] (λ, µ = h [3] (λ, λ, µ + h [2,3] (, λ, µ d for ll µ E 2(λ, (2.26 where E 2 (λ [, d] i uh h me E 2 (λ = d. (d If, in ddiion o (2.13 (2.15, (2.18 (2.21, nd (2.23 (2.25, here i funion g L([, b] [, d]; R uh h g(x, z = h [3] (x, x, z + h [2,3] (, x, z d for ll x E 1 nd z E 2 (x, hen here exi H [1,2] lmo everywhere on [, b] [, d] nd (2.27 H [1,2] (λ, µ = g(λ, µ for.e. (λ, µ [, b] [, d]. (2.28 Remrk I follow from umpion (2.14 h, for ny z [, d] fixed, here exi h [2] (, x, z for ll (, x Ω z := {(, η : E z, η B z (}, where E z [, b] wih me E z = b nd, for ny E z, we hve B z ( [, b] wih me B z ( = b. Noe h, in generl, he e Ω z migh no be meurble. Clerly, in umpion (2.15 we require h, for every z [, d], he funion h [2](,, z i defined (i.e., he pril derivive exi lmo everywhere in he qure [, b] [, b]. I i worh menioning here h hi umpion follow, e. g., from he exiene of funion g z L([, b] [, b]; R uh h h [2] (,, z g z on Ω z (ee Lemm 3.5 below wih =, b = d, nd f h(,, z. Remrk Inluion (2.25 i underood in he ene, whih i nlogou o h onerning inluion (2.15 explined in Remrk Now we pply Theorem 2.9 o he funion γ defined by relion (1.3. Corollry Le he funion γ : [, b] [, d] R be defined by formul (1.3, where p, q L([, b] [, d]; R +. Then he following erion re ified: 7 See Remrk 2.11.

7 ON DIFFERENTIATION UNDER INTEGRAL SIGN 97 (i γ(, x AC ([, b]; R for every x [, d] nd he relion γ [1] (, x = q(, η dη ( + q(, η p(, ξ 2 dξ 2 e η p(ξ1,ξ2 dξ2dξ1 dηd η hold for.e. [, b] nd ll x [, d]. (ii γ(, AC ([, d]; R for every [, b] nd he relion γ [2] (, x = q(, x d ( + q(, η p(ξ 1, x dξ 1 e η p(ξ1,ξ2 dξ2dξ1 ddη hold for ll [, b] nd.e. x [, d]. (iii γ [1] (, AC ([, d]; R for.e. [, b] nd he relion γ [1,2] (, x = q(, x + q(, ηf(, η,, x e hold for.e. (, x [, b] [, d], where ( ( f(, η,, x := p(, x + p(ξ 1, x dξ 1 (iv γ [2] (, x AC ([, b]; R for.e. x [, d] nd he relion γ [2,1] (, x = q(, x + q(, ηf(, η,, x e η (2.29 (2.30 η p(ξ1,ξ2 dξ2dξ1 dηd (2.31 η p(, ξ 2 dξ 2. (2.32 p(ξ1,ξ2 dξ1dξ2 ddη (2.33 hold for.e. (, x [, b] [, d], where he funion f i defined by formul (2.32. (v γ [1,2], γ [2,1] L([, b] [, d]; R nd γ [1,2] (, x = γ [2,1](, x for.e. (, x [, b] [, d]. (2.34 Corollry Le p, q L([, b] [, d]; R +. Then he funion γ defined by relion (1.3 i Crhéodory oluion o differenil inequliy ( Auxiliry emen Lemm 3.1 ([6, Theorem 3.1]. Le u: [, b] [, d] R be funion of wo vrible. Then he following erion re equivlen: (1 The funion u i boluely oninuou on he rengle [, b] [, d] in he ene of Crhéodory. 8 (2 The funion u ifie he relion: ( u(, x AC ([, b]; R for every x [, d] nd u(, AC ([, d]; R, (b u [1](, AC ([, d]; R for.e. [, b], ( u [1,2] L([, b] [, d]; R. (3 The funion u ifie he relion: 8 Thi noion i defined in [1] (ee lo [6] nd referene herein.

8 98 J. ŠREMR (A u(, AC ([, d]; R for every [, b] nd u(, AC ([, b]; R, (B u [2](, x AC ([, b]; R for.e. x [, d], (C u [2,1] L([, b] [, d]; R. Lemm 3.2 ([4, Chper IX, 5, Lemm 2]. Le ϕ AC ([, b]; R be n inreing funion nd E [ϕ(, ϕ(b] be uh h me E = 0. Then me { [, b] : ϕ( E nd he relion ϕ ( = 0 doe no hold } = 0. Lemm 3.3 ([4, Chper IX, 5, Theorem]. Le ϕ AC ([, b]; R be n inreing funion nd h L([ϕ(, ϕ(b]; R. Then ϕ(b ϕ( h(x dx = b h ( ϕ( ϕ ( d. 9 (3.1 Lemm 3.4 ([6, Propoiion 3.5]. Le g L([, d] [, b]; R nd G(, x := where E [, d] wih me E = d. Then G [2](, x = g(, x g(, η dη for E, x [, b], (3.2 for.e. (, x [, d] [, b]. Lemm 3.5. Le he funion f : [, d] [, b] R ify f(, AC ([, b]; R for ll E [, d], me E = d, (3.3 nd here exi funion g L([, d] [, b]; R uh h f [2] (, x = g(, x for ll E nd x A(, (3.4 where A( [, b] wih me A( = b. Then he pril derivive f [2] exi lmo everywhere in [, d] [, b] nd f [2] (, x = g(, x for.e. (, x [, d] [, b]. (3.5 Proof. Aumpion (3.3 nd (3.4 yield h f(, x = f(, + f [2] (, η dη = f(, + g(, η dη nd hu deired relion (3.5 follow from Lemm 3.4. for ll E, x [, b], The nex lemm i dire generliion of he reul obined by Tolov in [7, 7] (ee lo [6, Proof of Propoiion 3.5(i]. nd Lemm 3.6. Le g : [, d] [, b] R be uh h g(, L([, b]; R + for.e. [, d] (3.6 g(, η dη L([, d]; R + for ll x [, b]. (3.7 9 In order o enure h relion (3.1 i meningful we pu h(x := α(x hoe poin x [ϕ(, ϕ(b], where he funion h i no defined, α: [ϕ(, ϕ(b] R being n rbirry funion. Oberve h hoie of he funion α h no influene on he vlue of he righ-hnd ide of equliy (3.1 (ee Lemm 3.2.

9 Pu ON DIFFERENTIATION UNDER INTEGRAL SIGN 99 G(, x := ( g(, η dη d for (, x [, d] [, b]. (3.8 Then here exi e E [, d] uh h me E = d nd G [1] (, x = g(, η dη for ll E nd x [, b]. (3.9 The following lemm onern he o-lled Crhéodory funion nd give reul whih i well known (ee, e. g., [1, 576]. Lemm 3.7. Le f : [, b] [, d] R 2 R be uh h f(,, α, β: [, b] [, d] R i meurble for ll (α, β R 2, (3.10 f(x, z,, : R 2 R i oninuou for.e. (x, z [, b] [, d], (3.11 nd le u, v : [, b] [, d] R be meurble funion. Then he funion h defined by he relion h(x, z := f ( x, z, u(x, z, v(x, z (3.12 i meurble on he rengle [, b] [, d]. A l, we formule lemm whih n be found in Crhéodory monogrph [1] (ee lo [6, Lemm 3.1]. Lemm 3.8. Le g L([, b] [, d]; R. Then he funion G defined by formul (3.2 i meurble on he rengle [, b] [, d]. 4. Proof of min reul Proof of Propoiion 2.1. ( The erion follow immediely from he rule for differeniion of ompoie funion. (b I n be proved eily by uing he definiion of boluely oninuou funion. ( Aume h he funion ϕ i inreing (if i i dereing, he proof i nlogou. Then Lemm 3.3 yield h f ( ϕ( ϕ ( L([, b]; R nd F ( F ( = f ( ϕ( f ( ϕ( = ϕ( ϕ( for ll [, b], whih give deired relion (2.2. f (x dx = f ( ϕ( ϕ ( d Proof of Corollry 2.3. A fir we pu g(x := ω(x for hoe x [, d] in whih he funion g i no defined, where ω i he funion from foonoe in our orollry. Pu f(x := g( d for x [, d]. I i ler F ( = f ( ϕ( for ll [, b], he funion f i boluely oninuou, nd f (x = g(x for ll x A, where A [, d] wih me A = d. On he oher hnd, by uing Propoiion 2.1, we ge e E [, b] uh h me E = b nd F ( = f ( ϕ( ϕ ( for ll E, (4.1

10 100 J. ŠREMR where we pu f (x := g(x hoe poin x [, d] in whih he derivive of he funion f doe no exi. Conequenly, we hve F ( = g ( ϕ( ϕ ( for ll E ϕ 1 (A. (4.2 However, i follow from Lemm 3.2 h { } me E : ϕ( A nd he relion ϕ ( = 0 doe no hold = 0 nd hu equliie (4.1 nd (4.2 yield he vlidiy of deired relion (2.4. Proof of Propoiion 2.6. By uing umpion (2.5 (2.7 nd Fubini heorem, we ge ( λ d ( d λ f [2] (, x d dx = f [2] (, x dx d = d [ f(, λ f(, ] d = F (λ F ( for ll λ [, b]. Conequenly, he funion F i boluely oninuou nd deired relion (2.10 hold beue we hve d f [2] (, d L([, b]; R. Proof of Theorem 2.7. ( Le H(µ, λ := µ f(, λ d for (µ, λ [, d] [, b]. Then F (λ = H ( ϕ(λ, λ for ll λ [, b] nd, in view of umpion (2.5 (2.7, we ge ( µ λ µ H(µ, λ = f [2] (, x dx d + f(, d = ( µ f [2] (, x d dx + µ f(, d for ll (µ, λ [, d] [, b]. Therefore, Lemm 3.6 gurnee h here exi e E 1 [, d] wih me E 1 = d uh h H [1] (µ, λ = f [2] (µ, x dx + f(µ, = f(µ, λ for ll µ E 1, λ [, b], nd h here i e E 2 [, b] uh h me E 2 = b, here exi ϕ (λ for every λ E 2, nd µ H [2] (µ, λ = f [2] (, λ d for ll µ [, d], λ E 2. Conequenly, we obin F (λ = H [1]( ϕ(λ, λ ϕ (λ + H [2] ( ϕ(λ, λ = f ( ϕ(λ, λ ϕ (λ + ϕ(λ f [2] (, λ d for ll λ ϕ 1 (E 1 E 2.

11 ON DIFFERENTIATION UNDER INTEGRAL SIGN 101 (b Aume h he funion ϕ i non-dereing (if i i non-inreing, he proof i nlogou nd le ε > 0 be rbirry. Then, in view of umpion (2.5 nd (2.7, here exi ω > 0 uh h f [2] (, x ddx < ε for ll E [, d] [, b], me E < ω (4.3 3 E nd f(, d < ε for ll I [, d], me I < ω. (4.4 I 3 Moreover, here exi number 0 < δ ω/(d uh h, for n rbirry yem { ] k, b k [ } m of muully dijoin ubinervl of [, b] ifying relion k=1 m k=1 (b k k < δ, we hve m ϕ(b k ϕ( k ω < mx{1, b }. (4.5 k=1 Now le { ] k, b k [ } m be n rbirry yem of muully dijoin ubinervl k=1 of [, b] wih propery m k=1 (b k k < δ. Then inequliy (4.5 hold, { [, d] [ k, b k ] } m nd { [ϕ( k=1 k, ϕ(b k ] [, b] } m form yem of non-overlpping rengle onined in [, d] [, b], nd { [ϕ( k, ϕ(b k ] } m k=1 i yem of nonoverlpping ubinervl of [, d]. Aording o umpion (2.5 (2.7, i i ey k=1 o verify h, for ny k = 1,..., m, we hve F (b k F ( k = = = ϕ(bk ϕ(k ϕ(k + ϕ(bk ϕ( k f(, b k d ( bk k ( bk k ϕ(k f [2] (, x dx f [2] (, x dx f(, k d d + d ( bk f [2] (, x dx d + nd hu, in view of relion (4.5, we ge m F (b k F ( k (, x ddx+ k=1 where me A 1 = me A 2 = me A 3 = A 1 f [2] A 2 f [2] m (d (b k k < (d δ ω, k=1 k=1 ϕ(bk ϕ( k ϕ(bk ϕ( k f(, b k d f(, d (, x ddx+ f(, d, A 3 m ( ϕ(bk ϕ( k ω(b (b < mx{1, b } ω, k=1 m ( ϕ(bk ϕ( k ω < mx{1, b } ω.

12 102 J. ŠREMR Conequenly, relion (4.3 nd (4.4 yield h m hu he funion F i boluely oninuou. k=1 F (bk F ( k < ε nd ( Aume h he funion ϕ i inreing (if i i dereing, he proof i nlogou. I follow from he umpion impoed on ϕ nd f h here exi ϕ ( nd f [2] (, x for.e. [, b] nd.e. (, x [, d] [, b], repeively. In order o enure h ll relion below re meningful we pu ϕ ( := 0 nd f [2] (, x := 0 hoe poin in whih he derivive indied do no exi. In uh wy, he funion ϕ nd f [2] re defined everywhere on [, b] nd [, d] [, b], repeively. Le E 1 [, b], me E 1 = b, be he e uh h f [2] (, x L([, d]; R for every x E 1. Pu h(λ, x := Clerly, we hve ϕ(λ Then, by uing Fubini heorem, we ge f [2] (, x d for ll λ [, b] nd x E 1. (4.6 h(λ, L([, b]; R for ll λ [, b]. (4.7 F (λ = = = ϕ(λ ϕ(λ ϕ(λ f(, d + f(, d + f(, d + ϕ(λ Moreover, Corollry 2.3 yield h ( λ f [2] (, x dx d ( ϕ(λ f [2] (, x d dx h(λ, x dx for ll λ [, b]. (4.8 h(, x AC ([, b]; R for ll x E 1, (4.9 h [1] (λ, x = f [2]( ϕ(λ, x ϕ (λ for ll x E 1 nd λ A(x, (4.10 where A(x [, b] wih me A(x = b, nd d dλ ϕ(λ Now we pu f 1 : [ f [2] f(, d = f ( ϕ(λ, ϕ (λ for.e. λ [, b]. (4.11 ] +, f 2 : [ f [2] ], nd h k (, x := f k ( ϕ(, x ϕ ( for ll (, x [, b] [, b], k = 1, 2. (4.12 Relion (4.9 nd (4.10 yield h h k (, x L([, b]; R + for ll x E 1. Moreover, in view of umpion (2.7, we hve f 1, f 2 L([, d] [, b]; R. Therefore, by virue of Fubini heorem nd Lemm 3.2 one n how h f k ( ϕ(, ϕ ( = h k (, L([, b]; R + for lmo every [, b]. Unforunely, i my hppen h h k L([, b] [, b]; R +. However, we n how h on every rengle onined in [, b] [, b] here exi boh iered inegrl nd heir vlue re equl

13 ON DIFFERENTIATION UNDER INTEGRAL SIGN 103 o eh oher. Indeed, le k {1, 2} be fixed nd [ 1, b 1 ] [ 2, b 2 ] [, b] [, b] be n rbirry rengle. Moreover, le { } { b2 Ω := [, d] : f k (, L([, b]; R, w( := 2 f k (, x dx for Ω, 0 for [, d] \ Ω. Then we hve me Ω = d nd w L([, d]; R nd hu, Lemm 3.3 yield h ϕ(b 1 ϕ( w( d = b w ( ϕ( ϕ ( d. However, wih repe o Lemm 3.2, one n verify h w ( ϕ( ϕ ( = b2 whih rrive he equliy ( ϕ(b1 b2 f k (, x dx d = ϕ( f k ( ϕ(, x ϕ ( dx for.e. [, b], b1 1 ( b2 ( f k ϕ(, x ϕ ( dx d. On he oher hnd, by uing Lemm 3.3 we ge ( b2 ϕ(b1 ( b2 b1 ( f k (, x d dx = f k ϕ(, x ϕ ( d dx. 2 ϕ( 1 2 Now ompring he l wo relion we obin he equliy ( b1 b2 ( b2 b1 h k (, x dx d = h k (, x d dx. ( I men h he funion h 1 nd h 2 ify ll umpion of Lemm 3.6 wih = nd d = b nd hu here exi e E 2 [, b] uh h me E 2 = b nd y z y h k (, x dx d = h k (y, x dx for ll y E 2 nd z [, b], k = 1, 2, ( y y h k (, x d dx = h k (, z d for ll y [, b] nd z E 2, k = 1, 2. (4.14 (4.15 Moreover, by virue of umpion (2.7 nd Lemm 3.6, we n ume wihou lo of generliy h E 2 i uh h he relion y 2 2 f [2] (, x d dx = f [2] (, y d y z 1 z 1 (4.16 hold well. for ll y E 2 nd z 1, z 2 [, d]

14 104 J. ŠREMR Le now E 1 E 2 be rbirry. Then, by uing relion (4.7, (4.9, (4.10, nd (4.13, we ge = h(λ, x dx h(, x dx ( λ h [1](, x d ( λ0 λ = h [1](, x d dx + dx + h(λ, x dx ( λ h [1](, x d dx = ( λ0 h [1] (, x d dx ( λ0 h 1 (, x dx d ( λ0 h 1 (, x d dx ( f [2] ( λ0 h 2 (, x dx d ( ϕ(, x ϕ ( d ( λ0 h 2 (, x d dx dx (4.17 for ll λ [, b]. Oberve h, in view of umpion (2.7 nd Lemm 3.3, for ny λ, y [, b] wih he propery 0 < (λ 2 (y (λ he relion ( 1 λ λ f ( [2] ϕ(, x ϕ ( d dx λ ( = 1 λ ϕ(λ f [2] λ λ (, x d dx 0 ϕ( gn(y λ ( ϕ(y f [2] (, x d dx ϕ( hold nd hu, by uing equliy (4.16, we ge ( 1 λ λ ( lim up ϕ(, x ϕ ( d dx λ + λ for ll y [, b], y >, nd ( 1 λ lim up λ λ f [2] f [2] ( ϕ(, x ϕ ( d dx ϕ(y ϕ( ϕ(λ0 ϕ(y f [2] (, d f [2] (, d

15 ON DIFFERENTIATION UNDER INTEGRAL SIGN 105 for ll y [, b], y <. Conequenly, we hve ( 1 λ λ ( lim ϕ(, x ϕ ( d dx = 0. (4.18 λ λ f [2] Therefore, by virue of ondiion (4.12, (4.14, (4.15, nd Lemm 3.3, i follow from relion (4.17 h d λ [ h(λ, x dx 1 λ ] λ0 dλ = lim h(λ, x dx h(, x dx λ=λ0 λ λ = h 1 (, x dx h 2 (, x dx = = + h 1 (, d h 2 (, d f [2]( ϕ(λ0, x ϕ ( dx + f [2]( ϕ(, λ0 ϕ ( d f [2]( ϕ(λ0, x ϕ ( dx + Thee equliie nd relion (4.8, (4.11 yield h F (λ = f ( ϕ(λ, ϕ (λ + + ϕ(λ f [2]( ϕ(λ, x ϕ (λ dx ϕ(λ0 f [2] (, d. f [2] (, λ d for ll λ A, (4.19 where A [, b], me A = b. I remin o how h he relion f ( ϕ(λ, ϕ (λ + hold for.e. λ A. Indeed, le f [2]( ϕ(λ, x ϕ (λ dx = f ( ϕ(λ, λ ϕ (λ (4.20 E 3 = { [, d] : f(, AC ([, b]; R }. Then, in view of umpion (2.6, we hve me E 3 = d nd Pu nd f(, + y f [2] (, x dx = f(, y for ll y [, b] nd E 3. (4.21 B 1 := { λ A : ϕ(λ E 3 }, B 2 := { λ A : ϕ(λ E 3 nd he relion ϕ (λ = 0 hold }, B 3 := { λ A : ϕ(λ E 3 nd he relion ϕ (λ = 0 doe no hold }. Then B 1 B 2 B 3 = A nd, by uing Lemm 3.2, we ge me B 3 = 0. Moreover, i i ler h, in view of (4.21, he relion (4.20 i ified for every λ

16 106 J. ŠREMR B 1 B 2. Conequenly, relion (4.20 hold lmo everywhere on A nd hu (4.19 gurnee he vlidiy of deired relion (2.12. Proof of Theorem 2.9. We fir exend he funion h ouide of [, b] [, b] [, d] by eing h(, x, z := 0. ( For ny µ [, d] fixed, he umpion of Theorem 2.7 re ified wih =, b = d, f(, h(,, µ, nd ϕ = id [,b] nd hu he erion follow immediely from Theorem 2.7(b,(. (b We n ume wihou lo of generliy h k = 0. Aording o umpion (2.20 nd (2.21, we n find e Ω 1 ], b[ of he meure b uh h nd h(x, x, : [, d] R i oninuou for ll x Ω 1 (4.22 h [2] (, x, d: [, d] R i oninuou for ll x Ω 1. (4.23 I follow from he erion ( h, for ny µ [, d], here exi e A(µ [, b] uh h me A(µ = b nd H [1] (λ, µ = h(λ, λ, µ + h [2](, λ, µ d for ll µ [, d], λ A(µ. (4.24 Pu Ω 2 = µ B A(µ, where B = ( [, d] Q {, d}. Sine he e B i ounble, he e Ω 2 i meurble nd me Ω 2 = b. Clerly, ondiion (4.24 yield h H [1] (λ, µ = h(λ, λ, µ + h [2] (, λ, µ d for ll λ Ω 2, µ B. (4.25 Now le Ω 1 Ω 2 be rbirry poin nd {l n } + n=1 be n rbirry equene of non-zero rel number uh h Pu g n (µ := 1 l n [ +l n lim l n = 0. (4.26 n + ] λ0 h(, + l n, µ d h(,, µ d Aording o relion (4.25 (4.27, we obin lim g n(µ = h(,, µ + n + for ll µ [, d], n N. (4.27 h [2] (,, µ d for ll µ B. (4.28 Oberve h, in view of umpion (2.13 (2.15, for ny µ [, d] we hve [ g n (µ = 1 λ0+l n ] λ0+l n h(, + l n, µ d + h [2](, x, µ dxd if l n > 0 l n

17 nd g n (µ = 1 l n ON DIFFERENTIATION UNDER INTEGRAL SIGN 107 [ h(,, µ d + l n l n l n h [2] (, x, µ dxd ] if l n < 0. Therefore, umpion (2.18 nd (2.19 yield h he funion g n (n N re non-dereing on [, d]. We will how h relion (4.28 hold for every µ [, d]. Indeed, le µ 0 [, d] nd ε > 0 be rbirry. Then, in view of relion (4.22 nd (4.23, here exi µ 1, µ 2 B uh h µ 1 µ 0 µ 2 nd h(,, µ 0 + h [2] (,, µ 0 d h(,, µ m h [2] (,, µ m d < ε (4.29 for m = 1, 2. 2 Moreover, by virue of limi (4.28, here exi n 0 N uh h g n(µ m h(,, µ m h [2] (,, µ m d < ε for n n 0, m = 1, 2. 2 (4.30 Now, by uing relion (4.29, (4.30, nd he monooniiy of he funion g n, we obin nd g n (µ 0 h(,, µ 0 h(,, µ 0 + h [2] (,, µ 0 d g n (µ 2 h(,, µ 2 + h(,, µ 2 + h(,, µ 0 < ε 2 + ε 2 = ε for n n 0 h [2] (,, µ 0 d g n (µ 0 h(,, µ 1 + h(,, µ 1 + h(,, µ 0 + < ε 2 + ε 2 = ε for n n 0 h [2] (,, µ 2 d h [2] (,, µ 2 d h [2] (,, µ 0 d h [2] (,, µ 1 d g n (µ 1 h [2] (,, µ 1 d h [2] (,, µ 0 d

18 108 J. ŠREMR nd hu g n(µ 0 h(,, µ 0 h [2] (,, µ 0 d < ε for n n 0. Conequenly, in view of rbirrine of µ 0 nd ε, he relion lim g n(µ = h(,, µ + n + h [2] (,, µ d for ll µ [, d] hold. Sine nd {l n } + n=1 were lo rbirry nd me Ω 1 Ω 2 = b, he l relion gurnee he vlidiy of deired equliy (2.22 wih E 1 = Ω 1 Ω 2. ( For ny λ E 1 fixed, he umpion of Propoiion 2.6 re ified wih f(, h(, λ, on [, λ] [, d] nd hu he erion follow from Propoiion 2.6. (d I follow immediely from Lemm 3.5 wih f H [1] on [, b] [, d]. Now we eblih ehnil lemm in order o implify he proof of Corollry Lemm 4.1. Le p, q L([, b] [, d]; R + nd h(, x, z := q(, e p(ξ1,ξ2 dξ2dξ1 d for ll E nd (x, z [, b] [, d], (4.31 where E [, b] wih me E = b. Then he funion h ifie relion (2.14, (2.15, nd here exi e Ω [, b] [, b] uh h me Ω = (b 2 nd h [2] (, x, z = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d (4.32 for ll (, x Ω, x, nd ll z [, d]. Proof. Le E nd z [, d] be rbirry. We pu f,z (, x := q(, e p(ξ1,ξ2 dξ2dξ1 for.e. [, d] nd ll x [, b]. Clerly, he funion f,z ifie ondiion (2.5, (2.6, nd f,z [2] (, x = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 for.e. [, d] nd ll x A(, (4.33 where A( [, b] wih me A( = b. Wih he funion f,z we oie he funion f,z(, 0 x := q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1. Clerly, he funion f 0,z i defined lmo everywhere on he rengle [, d] [, b]. Aording o he umpion p, q L([, b] [, d]; R + nd Lemm 3.8, we ee

19 ON DIFFERENTIATION UNDER INTEGRAL SIGN 109 h he funion f,z 0 i meurble on he rengle [, d] [, b]. Moreover, we hve ( d f,z(, 0 x q(, p(x, ξ 2 dξ 2 e p L for.e. (, x [, d] [, b] nd hu f,z 0 L([, d] [, b]; R. Hene, in view of equliy (4.33, he funion f,z [2] ifie ondiion (2.7 (ee Lemm 3.5 wih g f,z. 0 Conequenly, Propoiion 2.6 yield he vlidiy of relion (2.14 nd h [2] (, x, z = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d for ll E, z [, d], x B(, z, (4.34 where B(, z [, b] wih me B(, z = b. Now we will how h he funion h ifie ondiion (2.15. Indeed, for ny z [, d] fixed we pu ϕ z (x,, := q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1. Clerly, he funion ϕ z i defined lmo everywhere on he e [, b] [, b] [, d]. By uing he umpion p, q L([, b] [, d]; R + nd Lemm 3.8, we eily ge he meurbiliy of he funion ϕ z on he e [, b] [, b] [, d]. Moreover, i i ler h ( d ϕ z (x,, q(, p(x, ξ 2 dξ 2 e p L for.e. (x,, [, b] [, b] [, d] nd hu ϕ z L([, b] [, b] [, d]; R. Hene, Fubini heorem yield h, for ny z [, d], he funion ϕ z(,, d i inegrble on [, b] [, b] whih, ogeher wih equliy (4.34, enure he vlidiy of ondiion (2.15 (ee Lemm 3.5 wih =, b = d, nd g(, ϕ z(,, d nd h [2] (, x, z = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d (4.35 for ll z [, d] nd (, x C(z, where C(z E [, b] wih me C(z = (b 2. Pu Ω = z D C(z, where D = ( [, d] Q {, d}. Sine he e D i ounble, he e Ω i meurble nd me Ω = (b 2. Clerly, ondiion (4.35 yield h h [2] (, x, z = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d for ll (, x Ω nd z D. (4.36

20 110 J. ŠREMR Now le ( 0, x 0 Ω, 0 x 0, be rbirry poin nd {l n } + n=1 be n rbirry equene of non-zero rel number uh h relion (4.26 hold. Pu g n (z := 1 l n 0 q( 0, e 0 [e p(ξ1,ξ2 dξ2dξ1 0 +ln x 0 ] p(ξ1,ξ2 dξ2dξ1 1 d for ll z [, d], n N. (4.37 Aording o relion (4.26, (4.36, nd (4.37, we obin lim g n(z = n + q( 0, 0 p(x 0, ξ 2 dξ 2 e 0 p(ξ1,ξ2 dξ2dξ1 d (4.38 for ll z D. Noe lo h he funion g n (n N re non-dereing on [, d], beue he funion p nd q re non-negive nd 0 x 0. We will how h relion (4.38 hold for every z [, d]. Indeed, le z 0 [, d] nd ε > 0 be rbirry. By uing he inequliy e y2 e y1 e y2 (y 2 y 1 for ll y 1, y 2 R, y 1 y 2, (4.39 i n be eily verified h q( 0, p(x 0, ξ 2 dξ 2 e 0 q( 0, p(x 0, ξ 2 dξ 2 e ( d p(x 0, ξ 2 dξ 2 e p L ( d + q( 0, d e p L ( d + 0 z 0 z 0 p(ξ 1,ξ 2 dξ 2dξ 1 d 0 0 p(ξ1,ξ2 dξ2dξ1 d q( 0, d p(x 0, ξ 2 dξ 2 ( d q( 0, d p(x 0, ξ 2 dξ 2 e p L b 0 z p(ξ 1, ξ 2 dξ 2 dξ 1 for ll z [, d] nd hu here exi z 1, z 2 D uh h z 1 z 0 z 2 nd p(ξ q( 0, p(x 0, ξ 2 dξ 2 e 0 1,ξ 2 dξ 2dξ 1 d m m 0 m q( 0, p(x 0, ξ 2 dξ 2 e 0 p(ξ 1,ξ 2 dξ 2dξ 1 d < ε ( for m = 1, 2. Moreover, by virue of limi (4.38, here exi n 0 N uh h m m g x0 m n(z m q( 0, p(x 0, ξ 2 dξ 2 e 0 p(ξ 1,ξ 2 dξ 2dξ 1 d < ε 2 for n n 0, m = 1, 2. (4.41

21 ON DIFFERENTIATION UNDER INTEGRAL SIGN 111 Now, by uing inequliie (4.40, (4.41, nd he monooniiy of g n, for every n n 0 we obin p(ξ g n (z 0 q( 0, p(x 0, ξ 2 dξ 2 e 0 1,ξ 2 dξ 2dξ 1 d nd g n (z 2 q( 0, p(x 0, ξ 2 dξ 2 e q( 0, p(x 0, ξ 2 dξ 2 e 0 0 q( 0, < ε 2 + ε 2 = ε 0 q( 0, p(x 0, ξ 2 dξ 2 e 0 1 q( 0, q( 0, q( 0, < ε 2 + ε 2 = ε, nd hu we hve 0 g n(z 0 q( 0, p(x 0, ξ 2 dξ 2 e p(ξ 1,ξ 2 dξ 2dξ 1 d 2 p(ξ 1,ξ 2 dξ 2dξ 1 d 0 p(ξ 1,ξ 2 dξ 2dξ 1 d 0 p(ξ 1,ξ 2 dξ 2dξ 1 d g n (z 0 p(x 0, ξ 2 dξ 2 e p(x 0, ξ 2 dξ 2 e 0 0 p(x 0, ξ 2 dξ 2 e p(x 0, ξ 2 dξ 2 e p(ξ 1,ξ 2 dξ 2dξ 1 d g n (z p(ξ 1,ξ 2 dξ 2dξ 1 d 1 p(ξ 1,ξ 2 dξ 2dξ 1 d p(ξ 1,ξ 2 dξ 2dξ 1 d < ε for n n 0. Conequenly, in view of rbirrine of z 0 nd ε, he relion (4.38 hold for ll z [, d]. Sine he equene {l n } + n=1 w lo rbirry, we hve proved h h [2] ( x0 0, x 0, z = q( 0, p(x 0, ξ 2 dξ 2 e 0 for ll z [, d]. proof. Proof of Corollry Clerly γ(λ, µ = p(ξ1,ξ2 dξ2dξ1 d Menion on rbirrine of he poin ( 0, x 0 omplee he h(, λ, µ d for ll (λ, µ [, b] [, d], where he funion h i defined by formul (4.31 wih E [, b], me E = b. (i We fir menion h ondiion (2.13 hold. I follow from Lemm 4.1 h he funion h lo ifie ondiion (2.14, (2.15, nd (4.32, where Ω [, b] [, b] i uh h me Ω = (b 2. Conequenly, he umpion of Theorem 2.9( re fulfilled nd hu γ(, µ AC ([, b]; R for every µ [, d].

22 112 J. ŠREMR Now oberve h ondiion (2.18, (2.19 wih k = 0 nd (2.20 re ified beue we ume p, q L([, b] [, d]; R +. Moreover, in view of ondiion (4.32, here exi e A [, b] uh h me A = b nd h [2] (, x, z = q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d for ll x A, B(x, nd z [, d], where B(x [, x] i uh h me B(x = x. Therefore, for ny x A fixed we hve h [2] (, x, z 2 h [2] (, x, z 1 = z 1 q(, 1 1 q(, q(, p(x, ξ 2 dξ 2 z 1 p(x, ξ 2 dξ 2 [e e p(x, ξ 2 dξ p(ξ 1,ξ 2 dξ 2dξ 1 d e 2 p(ξ 1,ξ 2 dξ 2dξ 1 d p(ξ 1,ξ 2 dξ 2dξ 1 e 1 ] p(ξ 1,ξ 2 dξ 2dξ 1 d for.e. [, x] nd ll z 1, z 2 [, d]. Therefore, by uing inequliy (4.39, for every z 1 z 2 d we ge h [2] (, x, z 2 d h [2] (, x, z 1 d ( ( d b z2 p(x, ξ 2 dξ 2 e p L q(, dd 2 + q L e p L p(x, ξ 2 dξ 2 z ( 1 ( d b + q L e p L p(x, ξ 2 dξ 2 z 1 2 z 1 p(, dd Conequenly, relion (2.21 hold nd hu, ording o Theorem 2.9(b, here exi e E 1 A uh h me E 1 = b nd µ γ [1] (λ, µ = q(λ, d + µ for ll λ E 1 nd µ [, d]. ( µ q(, p(λ, ξ 2 dξ 2 e µ p(ξ1,ξ2 dξ2dξ1 dd (ii Sine we n hnge he order of he inegrion in relion (1.3, he erion follow immediely from he bove-proved pr (i by hnging he role of he vrible nd x..

23 ON DIFFERENTIATION UNDER INTEGRAL SIGN 113 (iii Le E 1 be he e ppering in he proof of pr (i nd x E 1 be n rbirry poin. Then we hve nd h [2] (, x, z = h(x, x, z = q(x, d for ll z [, d] q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 d for ll B(x nd z [, d], where B(x [, x] wih me B(x = x. Clerly, ondiion (2.23 hold. Le B(x be rbirry. We pu f,x (, z := q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 for.e. [, d] nd ll z [, d]. Then he funion f,x ifie ondiion (2.5, (2.6 (in whih =, b = d, nd f,x [2] [p(x, (, z = q(, z ( ] + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p(ξ1,ξ2 dξ2dξ1 for.e. [, d] nd ll x C(, (4.42 where C( [, d] wih me C( = d. Wih he funion f,x we oie he funion [ f,x(, 0 z := q(, p(x, z ( ] + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p(ξ1,ξ2 dξ2dξ1. Clerly, he funion f 0,x i defined lmo everywhere on he qure [, d] [, d]. Aording o he umpion p, q L([, b] [, d]; R + nd Lemm 3.8, we ee h he funion f 0,x i meurble on he qure [, d] [, d]. Moreover, we hve [ ( ( d ] b f,x(, 0 z q(, p(x, z + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p L for.e. (, z [, d] [, d] nd hu f 0,x L([, d] [, d]; R. Hene, in view of equliy (4.42, he funion f,x ifie ondiion (2.7 (ee Lemm 3.5 wih =, b = d, nd g f 0,x. Conequenly, Theorem 2.7 (wih =, b = d, nd ϕ = id [,d] yield he vlidiy

24 114 J. ŠREMR of relion (2.24 nd [ [2,3] (, x, z = q(, p(x, z ( ] + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p(ξ1,ξ2 dξ2dξ1 d h for ll B(x nd z D(, x, (4.43 where D(, x [, d] wih me D(, x = d. Now we will how h he funion h ifie ondiion (2.25. Indeed, we pu [ g x (, z := q(, p(x, z ( ] + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p(ξ1,ξ2 dξ2dξ1 d. Clerly, he funion g x i defined lmo everywhere on he rengle [, x] [, d]. Oberve h g x (, z = p(x, z e + p(ξ1,ξ2 dξ2dξ1 ( p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 ( p(ξ 1, z dξ 1 q(, e p(ξ1,ξ2 dξ2dξ1 d e p(ξ1,ξ2 dξ2dξ1 ( q(, p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1 q(, e p(ξ1,ξ2 dξ2dξ1 d e p(ξ1,ξ2 dξ2dξ1 d for.e. (, z [, x] [, d] whene, by uing he umpion p, q L([, b] [, d]; R + nd Lemm 3.8, we ge he meurbiliy of he funion g x on he rengle [, x] [, d]. Moreover, i i ler h [ ( ( d ] b g x (, z p(x, z + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p L d q(, d for.e. (, z [, x] [, d] nd hu g x L([, x] [, d]; R. Hene, in view of equliy (4.43, we ee h ondiion (2.25 hold (ee Lemm 3.5 wih b = x nd f(, h [2](, x,. Conequenly, Theorem 2.9( yield h γ [1] (λ, AC ([, d]; R for every λ E 1

25 ON DIFFERENTIATION UNDER INTEGRAL SIGN 115 nd γ [1,2] (λ, µ = q(λ, µ + µ [ ( ( µ ] q(, p(λ, µ + p(ξ 1, µ dξ 1 p(λ, ξ 2 dξ 2 (4.44 e µ p(ξ1,ξ2 dξ2dξ1 dd for ll λ E 1, µ E 2 (λ, where E 2 (λ [, d] i uh h me E 2 (λ = d. Finlly we pu [ ϕ(,, x, z := q(, p(x, z ( ( ] + p(ξ 1, z dξ 1 p(x, ξ 2 dξ 2 e p(ξ1,ξ2 dξ2dξ1. Clerly, he funion ϕ i defined lmo everywhere on he e [, b] [, d] [, b] [, d]. By uing he umpion p, q L([, b] [, d]; R + nd Lemm 3.8, i i ey o verify h he funion ϕ i meurble on he e [, b] [, d] [, b] [, d]. Moreover, we hve [ ( ( d ] b ϕ(,, x, z q(, p(x, z + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 e p L for.e. (,, x, z [, b] [, d] [, b] [, d] nd hu ϕ L([, b] [, d] [, b] [, d]; R. Now we exend he funion ϕ ouide of he e [, b] [, d] [, b] [, d] by eing ϕ(,, x, z := 0 nd we pu f(x, z, α, β := α β ϕ(,, x, z dd for.e. (x, z [, b] [, d] nd ll (α, β R 2. Then ondiion (3.10 nd (3.11 re ified nd ( ( d ] b f(x, z, x, z e p L q L [p(x, z + p(x, ξ 2 dξ 2 p(ξ 1, z dξ 1 for.e. (x, z [, b] [, d]. Conequenly, Lemm 3.7 yield he inegrbiliy of he funion g(x, z := q(x, z + f(x, z, x, z for.e. (x, z [, b] [, d]. (4.45 If we e g(x, z := γ [1,2] (x, z hoe poin (x, z {(, : E 1, E 2 (} in whih g i no defined 10 hen, in view of equliy (4.44, he funion g ifie ondiion (2.27. Therefore, Theorem 2.9(d yield h he pril derivive γ [1,2] exi lmo everywhere in he rengle [, b] [, d] nd h deired relion (2.31 hold for.e. (, x [, b] [, d]. 10 The e of uh poin h he meure equl o zero nd hu he funion g remin inegrble on [, b] [, d].

26 116 J. ŠREMR (iv Sine we n hnge he order of he inegrion in relion (1.3, he erion follow immediely from he bove-proved pr (iii by hnging he role of he vrible nd x. (v I follow from he inegrbiliy of he funion g defined by formul (4.45 nd he bove-proved equliie (2.31 nd (2.33. Proof of Corollry Aording o Corollry 2.12, we ge from Lemm 3.1 bolue oninuiy of he funion γ in he ene of Crhéodory. Sine he funion p nd q re non-negive, i follow from equliy (2.32 h he funion γ i oluion o differenil inequliy (1.2. Referene [1] C. Crhéodory, Vorleungen über reelle Funkionen, Verlg und Druk Von B. G. Teubner, Leipzig und Berlin, [2] E. W. Hobon, The Theory of Funion of Rel Vrible nd he Theory of Fourier erie, Vol II., Cmbridge Univeriy Pre, Cmbridge, [3] A. Lomidze, S. Mukhigulhvili nd J. Šremr, Nonnegive oluion of he hrerii iniil vlue problem for liner pril funionl-differenil equion of hyperboli ype, Mh. Compu. Modelling 47 (2008, No , [4] I. P. Nnon, Theory of Funion of Rel Vrible (in Ruin, Izd. Nuk, Moow, [5] S. Spăru, An boluely oninuou funion whoe invere funion i no boluely oninuou, Noe M. 23 (2004, [6] J. Šremr, Aboluely oninuou funion of wo vrible in he ene of Crhéodory, Eleron. J. Differ. Equ (2010, [7] G. P. Tolov, On he mixed eond derivive (in Ruin, M. Sb. 24 (66 (1949, Jiří Šremr, Iniue of Mhemi, Ademy of Siene of he Czeh Republi, Žižkov 22, Brno, Czeh Republi, e-mil: remr@ipm.z