International Mathematical Forum,, 2006, no 27, 33-323 On The Sobolev-type Inequality for Lebesgue Spaces with a Variable Exponent BCEKIC a,, RMASHIYEV a and GTALISOY b a Dicle University, Dept of Mathematics, 2280, Diyarbakır, Turkey b Inonu University, Dept of Mathematical Sciences, 44280, Malatya, Turkey Abstract This paper gives Sobolev-type inequality for the generalized Lebesgue space L p(x ( and corresponding Sobolev space W,p(x (, consisting of all f L p(x ( with first-order distributional derivatives in L p ( x (, where is a bounded domain R N (N 2, and p(x and q(x measurable function with q q ( x q + <, p + <N and [ sup q(x x p(x ] (q+ q (N N N In other words, our aim in this study is obtained pre-limit condition for measurable variable exponent functions in Sobolev-type inequality Mathematics Subject Classification: 46E35, 26D0 Keywords: Variable exponent, Sobolev-type inequality Introduction Let R N be an open set and p : [, be measurable function We define the variable exponent Lebesgue spaces L p ( x ( consisting of all measurable functions f : R such that λf ( x p ( x dx < for some λ>0 We can introduce the norm on L p ( x ( by f { p ( x, = inf λ>0: f ( x p ( x } dx λ and the L p ( x ( becomes a Banach spaces [5] Correspondig Authour : bilalc@dicleedutr This research is supported by the DUAPK 04-FF-40
34 CEKIC, MASHIYEV and ALISOY The corresponding Sobolev space W,p(x ( is the class of all functions f L p(x ( such that all generalized D i f, i =, 2,,n, belong to L p(x ( Endowed with the norm f,p (x = f p (x + f p (x it forms a Banach space OKovác and J Rákosník [5] obtained Sobolev-type inequality u q (x, c u p (x, ; u C 0 ( ( for continuous p (x function, where a positive constant c is independent of u and q (x Np(x N p (x ε = p (x ε, 0 <ε< N, and they showed that, in general, the Sobolev space W,p(x ( is not embedding in L p (x ( Obviously, q (x = Np(x equality is not valid for above N p (x embedding when p (x is discontinuous Naturally, q (x must be sufficiently less than Np(x N p (x Later DEEdmunds and JRákosník [2,3] proved W,p( x ( L p ( x ( for bounded Lipschitz domain, where p (x C 0, (and p (x q<n XLFan, JShen and DZhao [4] investigated Sobolev-type inequality for domain with cone property Recently, LDiening [] obtained above embedding for unbounded domain assuming that p (x is constant at infinity and satisfies (w-lip p (x p (y C log x y ; x, y R N, x y 2 That is, under mentioned conditions, Sobolev-type embedding is proved in case of limit by LDiening [] SSamko [6] also found the following result for measurable variable exponent functions in generalized Lebesgue space: Theorem [6] Let k(x L Q (B(0, 2Dwhere Q and D = diam The convolution operator K f = k(x yf(ydy is bounded from L p(x ( into L r(x (, r(x, if Q p + p +, r(x Q + p +
Sobolev-type Inequality 35 In this article, our aim is to show Sobolev-type inequality ( for Lebesgue spaces with a variable exponent where is assumed to be a bounded domain in R N and p (x is a measurable real-valued function (may be discontinuous defined on That is, we determine Pre-limit condition for measurable variable exponent functions in Sobolev-type inequality Notation: In this study, is the Lebesgue measure of a set in R N and g + = sup g ( x, g = inf g ( x, δ = q+ q and = N x x N 2 A Sobolev-type Inequality Theorem 2: Let R N, (N 2 be a bounded domain Let q q (x q + < and p + <N be such that sup x [ q(x p (x ] δ N for x Then, there exist a constant C(N,p,q > 0 such that u q ( x, C( N,p,q, u p ( x, ; u C 0 (, (2 where { C( N,p,q, = C( N,p,q N p + q + if C( N, p, q N p + + q if > Proof: We suppose that u q ( x, = and = We have = u ( x q (x dx = u ( x q (x dx + u ( x E 2 E 0 /E 2 = I + I 2, q (x dx where E t = { x : u (x >t} Then it is easy to see that I 2 = u (x q (x dx 2 q+ u (x dx (3 E 0 /E 2 and since u (x q (x q ( x u (x 2
36 CEKIC, MASHIYEV and ALISOY for x E 2, we can write I = u (x q (x dx 2 E 2 E 2 ( u( x dx q (x t q (x dt Also by Fubini s Theorem, we obtain I 2 ( dt t q (x q (x dx (4 E t By using Taylor s expansions of e x, from (4 we get I 2 ( t a dt t q (x a q (x dx =2 =2 t a dt t a dt E t ( E t ( n =0 ( n =0 ln n t n! ( q (x a n q (x ln n t dx n! E t q (x (q (x a n dx, (5 where a R, a < q On the other hand, by means of Gagliardo s inequality [7], for f C 0 (,w L loc ( ( f w(x dx C(N f w(x dx (6 we obtain ( q(x(q ( x a n f dx ( K (n,a f ( q ( x a n q(x dx (7 where K ( n, a=c(n ( q + n a, n =0,, 2, q a
Sobolev-type Inequality 37 ( u ( x If we take Z ( x = min t 2 t 2 in (7 then, we find q(x(q ( x a n dx,, t > 0 in place of the test function f E t K ( n, a t G t u ( q ( x a n q(x dx (8 where G t = { x : t 2 } < u ( x t Hence, from ( 5 and ( 8 we have I 2 t a dt ( n =0 K ( n, aln n t n! t ( n u ( q ( x a q(x dx G t By using Minkowski s inequalities for sums, we obtain I 2 t a dt ( G t u q(x [ n =0 K ( n,aln n t n! ( q (x a n ] dx =2C (N ( t a dt u q(x t ( q ( x a β ( a dx, (9 G t where β ( a = K n ( n, a Again by using Minkowski s inequality for inte-
38 CEKIC, MASHIYEV and ALISOY grals we have I 2 C ( N u q(x E dx 2 u ( x u (x t ( q (x a β ( a +a dt ( 2 q + C ( N 2 ( q + a β ( a + a ( q a β ( a +a E u u From ( 3 and ( 0, we get ( q ( x a β ( a + a dx (0 = I + I 2 C( N,q +,q,a +2 q+ and for a, E u u ( q ( x a β ( a + a dx u dx ( (q ( x a β (a + a N q ( x + q+ q C ( N, q +,q,a C (N, q +,q where C(N, q +,q is finite If we apply Fato s Lemma to (, then we write C u u q +δ dx + 2 q+ u dx /, (2 E where C>0 only depends on N, q + and q If we apply Young inequality db εd p +c (ε b p/ ; d>0, b>0,ε>0,p> to first term of inequality ( 2 and inequality ( 6 to second term for w(x =,
Sobolev-type Inequality 39 we obtain C C( ε u p ( x dx + ε u ( q +δ p / (x +2 q+ u dx, (3 + p ( x p / ( x where p ( x : =, p (x < By assumption of theorem, we have [ ] q(x N +δ p / (x q(x and by using Young s inequality again to last integral in the inequality (3, then we write C C ( ε u p( x dx + ε + C ( ε u p ( x dx + ε 2 q+ Hence, for sufficiently small ε>0 we obtain C u p(x dx + u p(x dx (4 ( If C u p( x dx <, then we have from ( 4 2 C u p ( x dx (5 ( To contrary, if C u p ( x dx ( u p ( x dx, it follows that 2C u p (x dx 2 u p(x dx and ( Therefore we have 2C (2C from ( 4 we can obtain inequality ( 5 again Now we suppose that u q(x, is any number Then, inequality (5 for function z = u u q(x,
320 CEKIC, MASHIYEV and ALISOY follows that C ( u u q(x, p (x dx Hence, u q(x, C ( N, p, q u p(x, (6 We suppose that is any positive number Then we consider the mapping x := Ty = x 0 + ry, where r>0, x 0 and x 0 is fixed It is easy to see that range of is as x y We assume that ũ ( y = u(x 0 + ry, p ( y = p(x 0 + ry and q ( y = q(x 0 + ry We choose r such that = ie r = C N N By the definition of norm, we obtain ( eq (y eu(y dy u eq(y, =If, by using inverse mapping we obtain q (x u(x dx ũ eq (y, /q+ Therefore, u q (x, C N q + ũ eq (y, (7 Again, with respect to definition of the norm of gradient of function, we find ( p ( x u dx = Then, we have u p ( x, ( ep ( y ũ dy r u p(x, and ũ N p u p(x, ep ( y dy (8 where As a result of ( 8, we obtain ũ ep (y, N p u p(x, (9
Sobolev-type Inequality 32 From ( 6, ( 7 and ( 9 for u, we have where As similarly, u q ( x, C (N,p,q N p + q + u p ( x,, u q ( x, C (N,p,q N p + + q u p ( x, where > The theorem is proved Remark Let D and E be any regular bounded domains in R 2 and E D Consider functions { 4/3 if x E p(x = if x D/E and q(x = { 2 if x E if x D/E It is clear that conditions of Theorem do not hold in this case But Theorem 2 holds Theorem 22 Let R N be an open bounded set with Lipschitz boundary, p(x w Lip and < p p + < N Then, for all measurable q : [, with q(x p (x δ (p = Np(x and some δ>0, there N p(x holds W,p( ( L q( ( ie the embedding is compact Proof: There exist m such that q m p, where 0 <p(p m < m(p p Note that m is a bounded exponent Let u n,u W,p( ( with u n u (weak lim We have to show that u n u in L q( ( (strong lim Theorem 2 implies chains of embeddings W,p(x W,m(x W,m L (m L q L q(x, (m q and from the generalized Hölder s inequality [], we obtain u n u m 0 or u n u
322 CEKIC, MASHIYEV and ALISOY in L q( ( Now, we consider of the classes u = u m,p(x W x j ( L p(x ( + m u x m j L p(x (, ( j =,, n Theorem 23 Let k h and kh< dist( k h,, if u Wx k,p(x j ( where p p + < and p(x w Lip, then Δ k x j,h u(x L p(x ( k h and we have Δ k k u x j,hu C k p(x, (k h x k j p(x, ( Proof For k =, (x,, x n h and Δ x,h u(x = we can write Δx,h u p(x, ( h h 0 h 0 h 0 u x (x + t, x 2,, x n dt, u x (x + t, x 2,, x n p(x,( h dt u p(x, x ( dt = C u x p(x, ( Therefore, for any k we obtain Δ k x j,h u p(x, (k h = Δ xj,h (Δk x j,h u p(x, (k h Δ k C x j,h u x j p(x,((k h C k u (k x j p(x, ( Theorem 23 is proved ACKNOWLEDGEMENTSThe authors would like to thank SGSamko for a useful discussion
Sobolev-type Inequality 323 References [] LDiening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p (x ( and W k, p (x ( Math Nachr, October 8, 2003, 2-5 [2] DE Edmunds, J Rákosník, Sobolev embedding with variable exponent Studia Math, 43 no3 2000, 267-293 [3] DE Edmunds, J Rákosník, Sobolev embedding with variable exponent II MathNachr, 246-247, 2002, 53-67 [4] XL Fan, JS Shen, D Zhao, Sobolev embedding theorems for spaces W k, p (x ( JMath Anal Appl, 262, 200, 749-760 [5] O Kovác, J Rákosník,On spaces L p (x ( and W k, p (x ( Czechoslovak MathJ, 6 99, 592-68 [6] SG Samko, Convolution and potential type operators in L p (x (R n Integr Transform and Special Funct v7, 3-4 998, 26-284 [7] E Sawyer, RL Wheeden, Weighted inequalities for Fractional Integrals on Euclidian and Homogeneous SpacesAmerican Journal of Math, 4, 992, 83-874 Received: October 4, 2005