ABSOLUTELY CONTINUOUS FUNCTIONS OF TWO VARIABLES IN THE SENSE OF CARATHÉODORY

Transcription

1 Eetroni Journa of Differentia Equations, Vo. 2010(2010), No. 154, pp ISSN: URL: or ftp ejde.math.txstate.edu ABSOLUTELY CONTINUOUS FUNCTIONS OF TWO VARIABLES IN THE SENSE OF CARATHÉODORY JIŘÍ ŠREMR Abstrat. In this note, the notion of absoute ontinuity of funtions of two variabes is disussed. We rea that the set of funtions of two variabes absoutey ontinuous in the sense of Carathéodory oinides with the ass of funtions admitting a ertain integra representation. We show that absoutey ontinuous funtions in the sense of Carathéodory an be equivaenty haraterized in terms of their properties with respet to eah of variabes. These equivaent haraterizations pay an important roe in the investigation of boundary vaue probems for partia differentia equation of hyperboi type with disontinuous right-hand side. We present severa statements whih are rather important when anayzing strong soutions of suh probems by using the methods of rea anaysis but, unfortunatey, are not formuated and proven preisey in the existing iterature, whih mosty deas with weak soutions or the ase where the right-hand side of the equation is ontinuous. 1. Introdution The notion of absoute ontinuity of funtions of a singe variabe has been generaized to more variabes in many ways. Eah of these onstrutions observes ony some of properties known in one dimensiona ase ike ontinuity, differentiabiity amost everywhere or integration by parts, the others are naturay ost. Let us mention absoute ontinuity due to Shwartz (see [11]), absoutey ontinuous funtions in the sense of Banah or in the sense of Tonei (see, e. g., [9, 10]), and n-absoute ontinuity introdued by Maý (see [7]). We are interested in the notion of absoute ontinuity of funtions of two variabes in the sense of Carathéodory (see [2]) in order to define a soution of a partia differentia equation of hyperboi type with disontinuous right-hand side. More preisey, if we onsider the hyperboi equation u(t, x) t x = f(t, x, u, u t, u x ) (1.1) on the retange [a, b] [, d], where f : [a, b] [, d] R 3 R is a Carathéodory funtion, then by a soution of the equation (1.1) is usuay understood a funtion 2000 Mathematis Subjet Cassifiation. 26B30, 26B05. Key words and phrases. Absoutey ontinuous funtion; Carathéodory sense; integra representation; derivative of doube integra Texas State University - San Maros. Submitted Marh 19, Pubished Otober 28, Supported by grants 201/06/0254 from the Czeh Siene Foundation, and AV0Z from the Aademy of Sienes of the Czeh Repubi. 1

2 2 J. ŠREMR EJDE-2010/154 u absoutey ontinuous on [a, b] [, d] in the sense of Carathéodory, whih satisfies the equaity (1.1) amost everywhere on the retange [a, b] [, d] (see, e. g., [1, 3, 5, 14]). The aim of this note is to rea the definition of an absoutey ontinuous funtion of two variabes introdued by Carathéodory in 1918 and to show that the set of those funtions oinides with the ass of funtions admitting a ertain integra representation. Some basi properties of suh funtions ike the existene of partia derivatives and differentiabiity amost everywhere are studied, e. g., in [2, 12, 13, 15] (see aso survey given in [4]). However, for the investigation of the initia and boundary vaue probems for the equation (1.1), we need to show that absoutey ontinuous funtions in the sense of Carathéodory an be equivaenty haraterized in terms of their properties with respet to eah of variabes. 2. Notation and definitions Throughout this paper, D = [a, b] [, d] denotes the retange in R 2 and Q(t, x) = [a, t] [, x] (t, x) D. (2.1) As usua, L ( D; R ) stands for the set of Lebesgue integrabe funtions on D. Further, AC ( [α, β]; R ) and L ( [α, β]; R ) are the sets of absoutey ontinuous and Lebesgue integrabe, respetivey, funtions on [α, β] R. For any measurabe set E R n (n = 1, 2), meas E denotes the Lebesgue measure of E. We first introdue a definition of an absoutey ontinuous funtion of two variabes (Definition 2.4), whih is based on the definition given by Carathéodory in Athough one of the onditions appearing in Definition 2.4 is sighty different from those stated in Carathéodory s monograph [2], both definitions desribe one and the same set of funtions (see Remarks 2.5 and 2.7). Let S(D) denote the system of a retanges [t 1, t 2 ] [x 1, x 2 ] ontained in D. For any retange P S(D), P denotes the voume of P. We say that retanges P 1, P 2 S(D) do not overap if the have no interior points in ommon. Furthermore, retanges P 1, P 2 S(D) are referred to be adjoining if they do not overap and P 1 P 2 S(D). Definition 2.1 ([6, 5.3]). A finite funtion F : S(D) R is said to be additive funtion of retanges if, for any adjoining retanges P 1, P 2 S(D), the reation hods. F (P 1 P 2 ) = F (P 1 ) + F (P 2 ) Definition 2.2 ([6, 7.3]). An additive funtion of retanges F : S(D) R is absoutey ontinuous if, for every ε > 0, there exists δ > 0 suh that if P 1,..., P k S(D) are mutuay non-overapping retanges with the property impies k P j δ j=1 k F (P j ) ε. j=1

3 EJDE-2010/154 ABSOLUTELY CONTINUOUS FUNCTIONS 3 The foowing statement gives a suffiient and neessary ondition for the funtion of retanges to be absoutey ontinuous. Theorem 2.3 ([6, 7.3]). The funtion of retanges F : S(D) R is absoutey ontinuous if and ony if there exists a funtion h L ( D; R ) suh that F (P ) = h(t, x) dt dx for P S(D). (2.2) P Now we are in position to define an absoutey ontinuous funtion of two variabes in the sense of Carathéodory. Having the funtion u: D R, we put F u ( [t1, t 2 ] [x 1, x 2 ] ) = u(t 1, x 1 ) u(t 1, x 2 ) u(t 2, x 1 ) + u(t 2, x 2 ) (2.3) for [t 1, t 2 ] [x 1, x 2 ] S(D). The funtion of retanges F u : S(D) R defined by the formua (2.3) is said to be a funtion of retanges assoiated with u. Definition 2.4. We say that a funtion u: D R is absoutey ontinuous on D in the sense of Carathéodory if the foowing two onditions hod: (a) the funtion of retanges F u assoiated with u is absoutey ontinuous; (b) the funtions u(a, ): [a, b] R and u(, ): [, d] R are absoutey ontinuous. Remark 2.5. In Carathéodory s monograph [2], the ondition (a) of the previous definition is sighty different. The onstrution of Carathéodory reads, roughy speaking, as foows. Let T(D) denote the system of a sets σ of the form σ = δ 1 δ p, where δ 1,..., δ p is a finite system of mutuay disjoint open retanges ontained in D, and et C be the system of a measurabe subset of D. An interva funtion Φ: T(D) R is said to be additive and absoutey ontinuous if there exists a set funtion G: C R that is additive, absoutey ontinuous, and suh that Φ(σ) = G(σ) for σ T(D). Having the funtion u: D R, aording to Carathéodory, we put Φ ( ]t 1, t 2 [ ]x 1, x 2 [ ) = u(t 1, x 1 ) u(t 1, x 2 ) u(t 2, x 1 ) + u(t 2, x 2 ) (2.4) for any ]t 1, t 2 [ ]x 1, x 2 [ D, and Φ(σ) = Φ(δ 1 ) + + Φ(δ p ) for σ = δ 1 δ p T(D). (2.5) The interva funtion Φ: T(D) R defined by the formuae (2.4) and (2.5) is aed an interva funtion assoiated with u. Definition 2.6 (Carathéodory, [2, 567]). A funtion u: D R is said to be absoutey ontinuous if the foowing two onditions hod: (a) the interva funtion Φ assoiated with u is absoutey ontinuous; (b) the funtions u(a, ): [a, b] R and u(, ): [, d] R are absoutey ontinuous. Remark 2.7. Carathéodory aso proved (see [2, 453, Satz 1]) that for an interva funtion Φ: T(D) R to be additive and absoutey ontinuous it is suffiient and neessary the existene of a funtion p L ( D; R ) suh that Φ(σ) = p(t, x) dt dx for σ T(D). σ

4 4 J. ŠREMR EJDE-2010/154 Consequenty, by virtue of Theorem 2.3, the funtion u: D R is absoutey ontinuous in the sense of Definition 2.4 if and ony if it is absoutey ontinuous in the sene of Definition Main resut The main resut of this note is the next theorem whih is party ontained in [2, 13, 15]. Theorem 3.1. The foowing three statements are equivaent: (1) the funtion u: D R is absoutey ontinuous on D in the sense of Carathéodory, i. e., (α) the funtion of retanges F u assoiated with u is absoutey ontinuous; (β) u(a, ) AC ( [, d]; R ) and u(, ) AC ( [a, b]; R ) ; (2) the funtion u: D R admits the integra representation t u(t, x) = e + f(s) ds + g(η) dη + h(s, η) ds dη (3.1) a Q(t,x) for (t, x) D, where e R, f L ( [a, b]; R ), g L ( [, d]; R ), h L ( D; R ), and the mapping Q is defined by the formua (2.1); (3) the funtion u: D R satisfies the foowing onditions: (a) u(, x) AC ( [a, b]; R ) for every x [, d], u(a, ) AC ( [, d]; R ) ; (b) u t (t, ) AC ( [, d]; R ) for amost every t [a, b]; () u tx L ( D; R ) ; Remark 3.2. It is ear that, using the Fubini theorem (see, e. g., [8, XII, 4, Thm. 1]), we get t t h(s, η) ds dη = h(s, η) dη ds = h(s, η) ds dη (t, x) D, Q(t,x) a (3.2) and thus the onditions (3a) (3) in Theorem 3.1 an be repaed by the symmetri ones: (A) u(t, ) AC ( [, d]; R ) for every t [a, b], u(, ) AC ( [a, b]; R ) ; (B) u x (, x) AC ( [a, b]; R ) for amost every x [, d]; (C) u xt L ( D; R ). Remark 3.3. It foows from Proposition 3.5 beow that an arbitrary funtion u absoutey ontinuous on D in the sense of Carathéodory has both mixed seondorder derivatives amost everywhere on the retange D and these derivatives are equivaent, i. e., u tx (t, x) = u xt (t, x) for a. e. (t, x) D. Moreover, by virtue of Lemma 3.6 beow, one an see that u has aso integrabe on D both first-order partia derivatives. Remark 3.4. It immediatey foows from the definition of an absoutey ontinuous funtion of a singe variabe that if v AC ( [a, b]; R ) then aso v AC ( [a, b]; R ). Unfortunatey, funtions of two variabes do not have this property. Indeed, the funtion u defined by the formua u(t, x) = t x (t, x) [0, 1] [0, 1] a

5 EJDE-2010/154 ABSOLUTELY CONTINUOUS FUNCTIONS 5 is absoutey ontinuous on [0, 1] [0, 1] in the sense of Carathéodory but u is not, beause the ondition (3b) of Theorem 3.1 does not hod for u. To prove Theorem 3.1 we need the foowing proposition. Proposition 3.5. Let h L ( D; R ) and u(t, x) = h(s, η) ds dη for (t, x) D, (3.3) Q(t,x) where the mapping Q is defined by the formua (2.1). Then: (i) there exists a set E 1 [a, b] suh that meas E 1 = b a and u t (t, x) = for t E 1 and x [, d]; (3.4) (ii) there exists a set E 2 [, d] suh that meas E 2 = d and u x (t, x) = t a h(s, x) ds for t [a, b] and x E 2 ; (iii) there exists a set E 3 D suh that meas E 3 = (b a)(d ) and u tx (t, x) = h(t, x) for (t, x) E 3 ; (3.5) (iv) there exists a set E 4 D suh that meas E 4 = (b a)(d ) and u xt (t, x) = h(t, x) for (t, x) E 4. Parts (i), (ii) are proved in [13, 7], the proofs of parts (iii), (iv) are based on the proof of Lemma 1 stated in [15]. For the sake of ompeteness we prove Proposition 3.5 in detai. Lemma 3.6. Let h L ( D; R ) and ϕ(t, x) = for t E, x [, d], (3.6) where E [a, b] is suh that meas E = b a. Then the funtion ϕ is measurabe on the retange D. This emma is stated in [2, 569]; we give here a sighty simper proof. Proof. It is ear that E [, d] is measurabe subset of the retange D with the measure equa to (b a)(d ), and thus the funtion ϕ is defined amost everywhere on D. First suppose that the funtion h is bounded, i. e., h(t, x) M for a. e. (t, x) D. It is we-known (see, e. g., [8, XII, 1]) that there exists a sequene {f n } + n=1 of funtions ontinuous on D suh that im f n(t, x) = h(t, x) for a. e. (t, x) D. (3.7) Without oss of generaity we an assume that f n (t, x) M (t, x) D, n N. (3.8) Aording to (3.7) and the Fubini theorem (see, e. g., [8, XI, 5, Cor. 1]), there exists a set F [a, b] suh that meas F = b a and the ondition im f n(t 0, x) = h(t 0, x) for a. e.x [, d]. (3.9)

6 6 J. ŠREMR EJDE-2010/154 hods for every t 0 F. Now we put ψ n (t, x) = f n (t, η) dη (t, x) D, n N. (3.10) It is easy to verify that the funtions ψ n are ontinuous on D. Indeed, by virtue of (3.8), for any (t 1, x 1 ), (t 2, x 2 ) D and n N we get ψ n (t 2, x 2 ) ψ n (t 1, x 1 ) = 2 d f n (t 2, η) dη 1 f n (t 1, η) dη f n (t 2, η) f n (t 1, η) dη + M x 2 x 1, and thus ontinuity of ψ n on D foows from ontinuity of f n. Consequenty, the funtions ψ n are measurabe on D. On the other hand, in view of (3.8) (3.10), the Lebesgue onvergene theorem yieds ϕ(t, x) = for t E F, x [, d]; i. e., = ϕ(t, x) = im f n (t, η) dη = im ψ n(t, x) for a. e. (t, x) D. im ψ n(t, x) Therefore, the funtion ϕ is measurabe on D (see, e. g., [8, XII, 1]). Now suppose that the funtion h is unbounded. For any k N, we put k for (t, x) D, h(t, x) < k, h k (t, x) = h(t, x) for (t, x) D, h(t, x) k, k for (t, x) D, h(t, x) > k. It is ear that the funtions h k (k N) are measurabe and bounded on D and, for any t 0 E, we have and h k (t 0, x) h(t 0, x) for a. e.x [, d], k N (3.11) h(t 0, x) = im h k(t 0, x) for a. e.x [, d]. (3.12) k + Moreover, aording to the above-proved, the funtions ϕ k (k N) defined by ϕ k (t, x) = h k (t, η) dη for t E, x [, d], k N are measurabe on D. On the other hand, in view of (3.11) and (3.12), it foows from the Lebesgue onvergene theorem that i. e., im k + h k (t, η) dη = ϕ(t, x) = for t E, x [, d], im ϕ k(t, x) for a. e. (t, x) D. k + Consequenty, the funtion ϕ is measurabe on D (see, e. g., [8, XII, 1]).

7 EJDE-2010/154 ABSOLUTELY CONTINUOUS FUNCTIONS 7 Proof of Proposition 3.5. We first extent the funtion h outside of D by setting h(t, x) = 0 (t, x) R 2 \ D. (i) Without oss of generaity we an assume that h(t, x) 0 for a. e. (t, x) D (in ontrary ase we represent the funtion h as the differene of its positive and negative parts). It foows from (3.2) and (3.3) that, for any x [, d], there exists a set A(x) ]a, b[ suh that meas A(x) = b a and u t (t, x) = for x [, d] and t A(x). (3.13) Put E 1 = x B A(x), where B = ( [, d] Q ) {, d}. Sine the set B is ountabe, the set E 1 is measurabe and meas E 1 = b a. Moreover, the ondition (3.13) yieds u t (t, x) = for t E 1 and x B. (3.14) We wi show that the ast reation hods for every x [, d]. Let t E 1 be arbitrary but fixed. Let, moreover, { n } + n=1 be an arbitrary sequene of nonzero numbers suh that Put ϕ n (x) = 1 n t+n Aording to (3.14) (3.16), we obtain t im ϕ n(x) = im n = 0. (3.15) h(s, η) dη ds for x [, d], n N. (3.16) for x B. (3.17) Sine the funtion h is assumed to be nonnegative, it is ear that the funtions ϕ n (n N) are nondereasing on [, d]. We wi show that the reation (3.17) hods for every x [, d]. Let x 0 [, d] and ε > 0 be arbitrary but fixed. Then there exist x 1, x 2 B suh that x 1 x 0 x 2 and 0 < ε x 1 2, < ε x 0 2. (3.18) Moreover, by virtue of (3.17), there exists n 0 N suh that xk ϕ n (x k ) < ε for n n 0, k = 1, 2. (3.19) 2 2 Using (3.18), (3.19), and the monotoniity of ϕ n, we obtain ϕ n (x 0 ) and 0 and thus 0 ϕ n (x 2 ) ϕ n (x 0 ) 0 2 x ϕn (x 0 ) < ε for n n0. x 0 < ε for n n 0 ϕ n (x 1 ) < ε for n n 0,

8 8 J. ŠREMR EJDE-2010/154 Consequenty, in view of arbitrariness of x 0 and ε, the reation im ϕ n(x) = for x [, d] hods. Sine the sequene { n } + n=1 was aso arbitrary, we have proved that u t (t, x) = for x [, d]. Mention on arbitrariness of t E 1 ompetes the proof of the part (i). (ii) The proof is anaogous to the previous ase and thus we omit it. (iii) Aording to the above-proved part (i), there exists a set E 1 [a, b] suh that meas E 1 = b a and the ondition (3.4) hods. Let E 3 = { u t (t, x + ) u t (t, x) (t, x) E 1 ], d[ : im = h(t, x) }. (3.20) 0 Then it is ear that, for any (t, x) E 3, there exists u tx (t, x) and the ondition (3.5) is satisfied. We wi show that the set E 3 is measurabe with the measure equa to (b a)(d ). Put W = E 1 ], d[ and ϕ k (t, x) = ψ k (t, x) = inf 0< <1/k { 1 sup 0< <1/k { 1 + x + x } } for (t, x) W, k N, for (t, x) W, k N. For any (t, x) W fixed, the funtion of variabe in braes is ontinuous, and thus the funtions ϕ k and ψ k an be defined as foows ϕ k (t, x) = ψ k (t, x) = inf 0< <1/k, Q { 1 sup 0< <1/k, Q { 1 + x + x } } for (t, x) W, k N, (3.21) for (t, x) W, k N. (3.22) It is ear that the set W is measurabe with the measure equa to (b a)(d ) and, for any Q fixed, the funtion of (t, x) in braes is measurabe on D (see Lemma 3.6), and thus it is measurabe on W. Consequenty, the funtions ϕ k and ψ k, whih are defined as the point-wise infimum and supremum of ountabe system of measurabe funtions, are measurabe on W (see [8, XII, 1]). Moreover, in view of (3.21) and (3.22), the reations ϕ 1 (t, x) ϕ 2 (t, x), ψ 1 (t, x) ψ 2 (t, x) (t, x) W hods. Consequenty, there exist point-wise (finite or infinite) imits ϕ(t, x) = im k + ϕ k(t, x), ψ(t, x) = im k + ψ k(t, x) (t, x) W.

9 EJDE-2010/154 ABSOLUTELY CONTINUOUS FUNCTIONS 9 Now it foows from measurabiity of the funtions ϕ k and ψ k that the imit funtions ϕ and ψ are aso measurabe. By virtue of (3.4), for any (t, x) D we obtain ( { 1 x+ }) ϕ(t, x) = im inf k + 0< <1/k x ( { 1 x+ }) = im inf δ 0+ 0< <δ x u t (t, x + ) u t (t, x) = im inf. 0 One an show in a simiar manner that ψ(t, x) = im sup 0 u t (t, x + ) u t (t, x) (t, x) W. Now it is ear that the set E 3 defined by the formua (3.20) satisfies E 3 = {(t, x) W : ϕ(t, x) = ψ(t, x) = h(t, x)}. This set is however measurabe, beause the funtion ϕ, ψ, and h are measurabe on W. Moreover, the ondition (3.4) guarantees that, for any t E 1, the setion E 3 (t) of the set E 3 has the measure equa to b a. Therefore, using the Fubini theorem (see, e. g., [8, XI, 5, Thm. 1]) we get meas E 3 = meas E 3 (t) dt = (d ) meas E 1 = (b a)(d ). E 1 (iv) The proof is anaogous to the previous ase, but the part (ii) has to be used instead of (i). Now we are in position to prove Theorem 3.1. Proof of Theorem 3.1. To prove the theorem it is suffiient to show that that the impiations (1) (2), (2) (3), and (3) (1) are satisfied. (1) (2): Suppose that the funtion u: D R is absoutey ontinuous on D in the sense of Carathéodory, i. e., the onditions (1α) and (1β) are fufied. Aording to the ondition (1β) and Theorem 2.3, there exists a funtion h L ( D; R ) suh that F u (P ) = h(t, x) dt dx for P S(D). In partiuar, we have P F u ( Q(t, x) ) = Q(t,x) h(s, η) ds dη (t, x) D, (3.23) where the mapping Q is defined by the formua (2.1). Furthermore, the reation (2.3) yieds ( ) F u Q(t, x) = u(a, ) u(t, ) u(a, x) + u(t, x) for (t, x) D. (3.24) On the other hand, the ondition (1α) guarantees that there exist f L ( [a, b]; R ) and g L ( [, d]; R ) suh that u(t, ) = u(a, ) + u(a, x) = u(a, ) + t a f(s) ds g(η) dη for t [a, b], for x [, d].

10 10 J. ŠREMR EJDE-2010/154 Now, using the ast two reations, it foows from (3.23) and (3.24) that the funtion u admits the integra representation (3.1) with e = u(a, ). (2) (3): Suppose that the funtion u: D R admits the integra representation (3.1), where e R, f L ( [a, b]; R ), g L ( [, d]; R ), and h L ( D; R ). By virtue of the reations (3.2), it is ear that the ondition (3a) hods. Moreover, aording to Proposition 3.5(i), there exists a set E 1 [a, b] suh that meas E 1 = b a and u t (t, x) = f(t) + for t E 1 and x [, d]. Therefore, the ondition (3b) is fufied. Finay, Proposition 3.5(ii) guarantees the vaidity of the ondition (3.5), where E 3 D is a measurabe set with the measure equa to (b a)(d ). Consequenty, the ondition (3) aso hods. (3) (1): Suppose that the funtion u: D R satisfies the onditions (3a) (3). Obviousy, the ondition (1β) hods. Let P = [t 1, t 2 ] [x 1, x 2 ] S(D) be an arbitrary retange. By virtue of (2.3), we have F u (P ) = u(t 1, x 1 ) u(t 1, x 2 ) u(t 2, x 1 ) + u(t 2, x 2 ) and thus, using the onditions (3a) (3), we obtain F u (P ) = = = t2 t 1 t2 t 1 t2 t 1 u s (s, x 2 ) ds t2 t 1 u s (s, x 1 ) ds [ us (s, x 2 ) u s (s, x 1 ) ] ds ( 2 x 1 ) u sη (s, η) dη ds = u sη (s, η) ds dη. P Therefore, in view of the ondition (3) and Theorem 2.3, it is ear that the funtion of retanges F u assoiated with u is absoutey ontinuous, i. e., the ondition (1α) hods. Consequenty, the funtion u is absoutey ontinuous on D in the sense of Carathéodory. Referenes [1] D. Bieawski; On the set of soutions of boundary vaue probems for hyperboi differentia equations, J. Math. Ana. App. 253 (2001), No. 1, [2] C. Carathéodory; Voresungen über ree funktionen, Verag und Druk Von B. G. Teubner, Leipzig und Berin, 1918 (in German). [3] K. Deiming; Absoutey ontinuous soutions of Cauhy probem for u xy = f(x, y, u, u x, u y), Ann. Mat. Pura App. 89 (1971), [4] O. Dzagnidze; Some new resuts on the ontinuity and differentiabiity of funtions of severa rea variabes, Pro. A. Razm. Math. Inst. 134 (2004), pp [5] T. Kiguradze; Some boundary vaue probems for systems of inear partia differentia equations of hyperboi type, Mem. Differentia Equations Math. Phys. 1 (1994), [6] S. Lojasiewiz; An introdution to the theory of rea funtions, Wiey Intersiene Pubiation, Chihester, [7] J. Maý; Absoutey ontinuous funtions of severa variabes, J. Math. Ana. App. 231 (1999), [8] I. P. Natanson; Theory of funtions of a rea variabe, Nauka, Mosow, 1974 (in Russian). [9] T. Rado, P. V. Reihederfer; Continuous transformations in anaysis, Springer-Verag, New York, [10] S. Saks; Theory of the integra, Monografie Matematyzne, Warszawa, [11] L. Shwartz; Théorie des ditributions I, Hermann, Paris, 1950 (in Frenh).

11 EJDE-2010/154 ABSOLUTELY CONTINUOUS FUNCTIONS 11 [12] G. P. Tostov; On the urviinear and iterated integra, Trudy Mat. Inst. Stekov. 35, 1950 (in Russian). [13] G. P. Tostov; On the mixed seond derivative, Mat. Sb. 24 (66) (1949), (in Russian). [14] S. Wazak; Absoutey ontinuous funtions of severa variabes and their appiation to differentia equations, Bu. Poish Aad. Si. Math. 35 (1987), No , [15] S. Wazak; On the differentiabiity of absoutey ontinuous funtions of severa variabes, remarks on the Rademaher theorem, Bu. Poish Aad. Si. Math. 36 (1988), No. 9 10, Jiří Šremr Institute of Mathematis, Aademy of Sienes of the Czeh epubi, Žižkova 22, Brno, Czeh Repubi E-mai address: sremr@ipm.z