APPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( )

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 200, APPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( ) Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno 4-24 Furue-higashi-machi, Nishi-ku, Hiroshima , Japan; fymaeda@h6.dion.ne.jp Hiroshima University, Graduate School of Science, Department of Mathematics Higashi-Hiroshima , Japan; yomizuta@hiroshima-u.ac.jp Hiroshima National College of Maritime Technology, General Arts Higashino Oosakikamijima Toyotagun , Japan; ohno@hiroshima-cmt.ac.jp Abstract. Our aim in this paper is to deal with approximate identities in generalized Lebesgue spaces L p( ) (log L) q( ) (R n ). As a related topic, we also study Young type inequalities for convolution with respect to norms in such spaces. Ω. Introduction Following Cruz-Uribe and Fiorenza [2], we consider two variable exponents p( ): R n [, ) and q( ): R n R, which are continuous functions. Letting Φ p( ),q( ) (x, t) = t p(x) (log(c 0 + t)) q(x), we define the space L p( ) (log L) q( ) (Ω) of all measurable functions f on an open set Ω such that ( Φ p( ),q( ) y, f(y) ) dy < λ for some λ > 0; here we assume (Φ) Φ p( ),q( ) (x, ) is convex on [0, ) for every fixed x R n. Note that (Φ) holds for some c 0 e if and only if there is a positive constant K such that (.) K(p(x) ) + q(x) 0 for all x R n (see Appendix). Further, we see from (Φ) that t Φ p( ),q( ) (x, t) is nondecreasing in t. We define the norm { ( f Φp( ),q( ),Ω = inf λ > 0: Φ p( ),q( ) y, f(y) ) } dy λ Ω for f L p( ) (log L) q( ) (Ω). Note that L p( ) (log L) q( ) (Ω) is a Musielak Orlicz space [9]. Such spaces have been studied in [2, 8, 0]. In case q( ) = 0 on R n, L p( ) (log L) q( ) (Ω) is denoted by L p( ) (Ω) ([7]). We assume throughout the article that our variable exponents p( ) and q( ) are continuous functions on R n satisfying: (p) p := inf x R n p(x) sup x R n p(x) =: p + < ; doi:0.586/aasfm Mathematics Subject Classification: Primary 46E30. Key words: Lebesgue spaces of variable exponent, approximate identity, Young s inequality for convolution.

2 406 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno C (p2) p(x) p(y) whenever x R n and y R n ; log(e + / x y ) C (p3) p(x) p(y) whenever y x /2; log(e + x ) (q) < q := inf x R n q(x) sup x R n q(x) =: q + < ; C (q2) q(x) q(y) whenever x R n and y R n log(e + log(e + / x y )) for a positive constant C. We choose p 0 as follows: we take p 0 = p if t p Φ p( ),q( ) (x, t) is uniformly almost increasing in t; more precisely, if there exists C > 0 such that s p Φ p( ),q( ) (x, s) Ct p Φ p( ),q( ) (x, t) whenever 0 < s < t and x R n. Otherwise we choose p 0 < p. Then note that t p 0 Φ p( ),q( ) (x, t) is uniformly almost increasing in t in any case. Let φ be an integrable function on R n. For each t > 0, define the function φ t by φ t (x) = t n φ(x/t). Note that by a change of variables, φ t L,R n = φ L,Rn. We say that the family {φ t } is an approximate identity if R φ(x) dx =. Define the n radial majorant of φ to be the function φ(x) = sup φ(y). y x If φ is integrable, we say that the family {φ t } is of potential-type. Cruz-Uribe and Fiorenza [] proved the following result: Then and Theorem A. Let {φ t } be an approximate identity. Suppose that either () {φ t } is of potential-type, or (2) φ L (p ) (R n ) and has compact support. for all f L p( ) (R n ). sup φ t f L p( ),R C f n L p( ),R n 0<t lim φ t f f L t +0 p( ),R = 0 n Our aim in this note is to extend their result to the space L p( ) (log L) q( ) (Ω) of two variable exponents. Theorem.. Let {φ t } be a potential-type approximate identity. If f L p( ) (log L) q( ) (R n ), then {φ t f} converges to f in L p( ) (log L) q( ) (R n ): lim φ t f f t 0 Φp( ),q( ),R n = 0. Theorem.2. Let {φ t } be an approximate identity. Suppose that φ L (p 0) (R n ) and has compact support. If f L p( ) (log L) q( ) (R n ), then {φ t f} converges to f in L p( ) (log L) q( ) (R n ): lim t 0 φ t f f Φp( ),q( ),R n = 0. We show by an example that the conditions on φ are necessary; see Remarks 3.5 and 3.6 below. Finally, in Section 4, we give some Young type inequalities for convolution with respect to the norms in L p( ) (log L) q( ) (R n ).

3 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces The case of potential-type Throughout this paper, let C denote various positive constants independent of the variables in question. Let us begin with the following result due to Stein []. Lemma 2.. Let p < and {φ t } be a potential-type approximate identity. Then for every f L p (R n ), {φ t f} converges to f in L p (R n ). We denote by B(x, r) the open ball centered at x R n and with radius r > 0. For a measurable set E, we denote by E the Lebesgue measure of E. The following is due to Lemma 2.6 in [8]. Lemma 2.2. Let f be a nonnegative measurable function on R n with f Φp( ),q( ),R n such that f(x) or f(x) = 0 for each x R n. Set J = J(x, r, f) = f(y) dy B(x, r) and Then L = L(x, r, f) = where C > 0 does not depend on x, r, f. Further we need the following result. B(x,r) Φ p( ),q( ) (y, f(y)) dy. B(x, r) B(x,r) J CL /p(x) (log(c 0 + L)) q(x)/p(x), Lemma 2.3. Let f be a nonnegative measurable function on R n such that ( + y ) n f(y) or f(y) = 0 for each y R n. Set J = J(x, r, f) = f(y) dy B(x, r) and Then L = L(x, r, f) = where C > 0 does not depend on x, r, f. B(x,r) Φ p( ),q( ) (y, f(y)) dy. B(x, r) B(x,r) J C { L /p(x) + ( + x ) n }, Proof. We have by Jensen s inequality ( J f(y) p(x) dy B(x, r) B(x,r) ( B(x, r) B(x,r) B(0, x /2) ( + B(x, r) = J + J 2. B(x,r)\B(0, x /2) ) /p(x) ) /p(x) f(y) p(x) dy ) /p(x) f(y) p(x) dy

4 408 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno We see from (p3) that ( J C B(x, r) B(x,r) B(0, x /2) f(y) p(y) dy) /p(x). Similarly, setting E 2 = {y R n : f(y) ( + x ) n }, we see from (p3) that ( ) /p(x) J 2 C f(y) p(y) dy B(x, r) {B(x,r)\B(0, x /2)} E 2 ( ) /p(x) + ( + x ) p(x)(n+) dy B(x, r) {B(x,r)\B(0, x /2)}\E { 2 ( ) } /p(x) C f(y) p(y) dy + ( + x ) (n+). B(x, r) B(x,r) Since f(y), f(y) p(y) CΦ p( ),q( ) (y, f(y)). Hence, we have the required estimate. By using Lemmas 2.2 and 2.3, we show the following theorem. Theorem 2.4. If {φ t } is of potential-type, then φ t f Φp( ),q( ),R n C φ L,R n f Φ p( ),q( ),R n for all t > 0 and f L p( ) (log L) q( ) (R n ). Proof. Suppose φ L,R n = and take a nonnegative measurable function f on R n such that f Φp( ),q( ),Rn. Write f = fχ {y R n :f(y) } + fχ {y R n :(+ y ) n f(y)<} + fχ {y R n :f(y) (+ y ) n } = f + f 2 + f 3, where χ E denotes the characteristic function of a measurable set E R n. Since φ t is a radial function, we write φ t (r) for φ t (x) when x = r. First note that φ t f(x) φt ( x y )f (y) dy = R n 0 ( B(x, r) B(x,r) so that Jensen s inequality and Lemma 2.2 yield ) f (y) dy B(x, r) d( φ t (r)), Φ p( ),q( ) (x, φ t f (x) ) ) Φ p( ),q( ) (x, f (y) dy B(x, r) d( 0 B(x, r) φ t (r)) B(x,r) ( ) C Φ p( ),q( ) (y, f (y)) dy B(x, r) d( B(x, r) φ t (r)) 0 = C( φ t g)(x), B(x,r)

5 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 409 where g(y) = Φ p( ),q( ) (y, f(y)). The usual Young inequality for convolution gives Φ p( ),q( ) (x, φ t f (x) ) dx C ( φ t g)(x) dx R n R n C φ t L,R n g L,R n C. Similarly, noting that f B(x,r) B(x,r) 2(y) dy and applying Lemma 2.3, we derive the same result for f 2. Finally, noting that φ t f 3 C φ t L,Rn C, we obtain Φ p( ),q( ) (x, φ t f 3 (x) ) dx C φ t f 3 (x) dx R n R n as required. We are now ready to prove Theorem.. C φ t L,R n f 3 L,R n C, Proof of Theorem.. Given ε > 0, we find a bounded function g in L p( ) (log L) q( ) (R n ) with compact support such that f g Φp( ),q( ),Rn < ε. By Theorem 2.4 we have φ t f f Φp( ),q( ),R n φ t (f g) Φp( ),q( ),R n + φ t g g Φp( ),q( ),R n + f g Φ p( ),q( ),R n Cε + φ t g g Φp( ),q( ),R n. Since φ t g g L,R n, φ t g g Φp( ),q( ),R n C φ t g g L,R n 0 by Lemma 2.. (Here C depends on g L,Rn.) Hence lim sup φ t f f Φp( ),q( ),R n Cε, t 0 which completes the proof. As another application of Lemmas 2.2 and 2.3, we can prove the following result, which is an extension of [4, Theorem.5] and [8, Theorem 2.7] (see also [6]). Let Mf be the Hardy Littlewood maximal function of f. Proposition 2.5. Suppose p >. Then the operator M is bounded from L p( ) (log L) q( ) (R n ) to L p( ) (log L) q( ) (R n ). Proof. Let f be a nonnegative measurable function on R n such that f Φp( ),q( ),R n and write f = f + f 2 + f 3 as in the proof of Theorem 2.4. Take < p < p and apply Lemmas 2.2 and 2.3 with p( ) and q( ) replaced by p( )/p and q( )/p, respectively. Then Φ p( ),q( ) (x, Mf (x)) C[Mg (x)] p and Φ p( ),q( ) (x, Mf 2 (x)) C { [Mg (x)] p + ( + x ) n }, where g (y) = Φ p( )/p,q( )/p (y, f(y)). As to f 3, we have Φ p( ),q( ) (x, Mf 3 (x)) C[Mf 3 (x)] p. Then the boundedness of the maximal operator in L p (R n ) proves the proposition.

6 40 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno Remark 2.6. If p >, then the function Φ p( ),q( ) is a proper N-function and our Proposition 2.5 implies that this function is of class A in the sense of Diening [5] (see [5, Lemma 3.2]). It would be an interesting problem to see whether class A is also a sufficient condition or not for the boundedness of M on L p( ) (log L) q( ) (R n ). 3. The case of compact support We know the following result due to Zo [2]; see also [, Theorem 2.2]. Lemma 3.. Let p <, /p + /p = and {φ t } be an approximate identity. Suppose that φ L p (R n ) has compact support. Then for every f L p (R n ), {φ t f} converges to f pointwise almost everywhere. Set p(x) = p(x)/p 0 and q(x) = q(x)/p 0 ; recall that p 0 [, p ] is chosen such that t p 0 Φ p( ),q( ) (x, t) is uniformly almost increasing in t. For a proof of Theorem.2, the following is a key lemma. Lemma 3.2. Let f be a nonnegative measurable function on R n with f Φp( ),q( ),R n such that f(x) or f(x) = 0 for each x R n and let φ have compact support in B(0, R) with φ L (p 0 ),R n. Set F = F (x, t, f) = φ t f(x) and Then G = G(x, t, f) = φ t (x y) Φ p( ),q( ) (y, f(y)) dy. R n F CG /p(x) (log(c 0 + G)) q(x)/p(x) for all 0 < t, where C > 0 depends on R. Proof. Let f be a nonnegative measurable function on R n with f Φp( ),q( ),R n such that f(x) or f(x) = 0 for each x R n and let φ have compact support in B(0, R) with φ L (p 0 ),R n. By Hölder s inequality, we have G φ t L (p 0 ),R n (R n Φ p( ),q( ) (y, f(y)) dy) /p0 t n/p 0. First consider the case when G. Since G t n/p 0, for y B(x, tr) we have by (p2) and by (q2) G p(y) G p(x)+c/ log(e+(tr) ) CG p(x) (log(c 0 + G)) q(y) C(log(c 0 + G)) q(x).

7 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 4 Hence it follows from the choice of p 0 that F G / p(x) (log(c 0 + G)) q(x)/p(x) R n φ t (x y) dy { f(y) + C φ t (x y) f(y) R G n / p(x) (log(c 0 + G)) q(x)/p(x) { } q(y) log(c 0 + f(y)) dy log(c 0 + G / p(x) (log(c 0 + G)) q(x)/p(x) ) CG / p(x) (log(c 0 + G)) q(x)/p(x) } p(y) (cf. the proof of [8, Lemma 2.6]). In the case G <, noting from the choice of p 0 that f(y) CΦ p( ),q( ) (y, f(y)) for y R n, we find F CG CG / p(x) CG / p(x) (log(c 0 + G)) q(x)/p(x). Now the result follows. Lemma 3.3. Suppose that φ L,R n. Let f be a nonnegative measurable function on R n with f Φp( ),q( ),Rn. Set I = I(x, t, f) = φ t (x y) f(y) dy {y R n : y > x /2} and H = H(x, t, f) = φ t (x y) Φ p( ),q( ) (y, f(y)) dy. R n If A > 0 and H H 0, then I C(H /p(x) + x A/p(x) ) for x > and 0 < t, where C > 0 depends on A and H 0. Proof. Suppose that φ L,Rn. Let f be a nonnegative measurable function on R n with f Φp( ),q( ),R n. Let x >. In the case H 0 H x A with A > 0, we have by (p3) for y x /2. Hence we find by (Φ) { I C H /p(x) + H p(y) CH p(x) C/ log(e+ x ) CH p(x) { f(y) H /p(x) CH /p(x). Next note from (p3) that {y R n : y > x /2} φ t (x y) f(y) } p(y) { } q(y) log(c0 + f(y)) dy} log(c 0 + H /p(x) ) x p(y) x p(x)+c/ log(e+ x ) C x p(x)

8 42 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno for y x /2. Hence, when H x A, we obtain by (Φ) { I C x A/p(x) + φ t (x y) f(y) { f(y) x A/p(x) C x A/p(x), which completes the proof. {y R n : y > x /2} } p(y) { log(c0 + f(y)) } q(y) dy} log(c 0 + x A/p(x) ) Theorem 3.4. Suppose that φ L (p 0) (R n ) has compact support in B(0, R). Then φ t f Φp( ),q( ),R n C φ L (p 0 ),R n f Φp( ),q( ),R n for all 0 < t and f L p( ) (log L) q( ) (R n ), where C > 0 depends on R. Proof. Let f be a nonnegative measurable function on R n such that f Φp( ),q( ),R n and let φ have compact support in B(0, R) with φ L (p 0 ),R n. Write We have by Lemma 3.2, f = fχ {y R n :f(y) } + fχ {y R n :f(y)<} = f + f 2. φ t f (x) C( φ t g(x)) p 0/p(x) (log(c 0 + φ t g(x))) q(x)/p(x), where g(y) = Φ p( ), q( ) (y, f(y)) = Φ p( ),q( ) (y, f(y)) /p 0, so that (3.) Φ p( ),q( ) (x, φ t f (x) ) C( φ t g(x)) p 0. Hence, since g L p 0 (R n ), the usual Young inequality for convolution gives Φ p( ),q( ) (x, φ t f (x) ) dx C ( φ t g(x)) p 0 dx R n R n Next we are concerned with f 2. Write C ( φ t L,R n g L p 0,R n)p 0 C. f 2 = f 2 χ B(0,R) + f 2 χ R n \B(0,R) = f 2 + f 2. Since φ t f 2 (x) C on R n, we have Φ p( ),q( ) (x, φ t f 2 (x) ) dx C. B(0,2R) Further, noting that φ t f 2 = 0 outside B(0, 2R), we find Φ p( ),q( ) (x, φ t f 2(x) ) dx C. R n Therefore it suffices to prove Φ p( ),q( ) (x, φ t f 2 (x) ) dx C. R n \B(0,2R) Thus, in the rest of the proof, we may assume that 0 f < on R n and f = 0 on B(0, R). Note that φ t (x y)f(y) dy = 0 B(0, x /2)

9 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 43 for x > 2R. Hence, applying Lemma 3.3, we have φ t f(x) p(x) C( φ t h(x) + x A ) for x > 2R, where h(y) = Φ p( ),q( ) (y, f(y)). Thus the integration yields φ t f(x) p(x) dx C, which completes the proof. R n \B(0,2R) We are now ready to prove Theorem.2. Proof of Theorem.2. Given ε > 0, choose a bounded function g with compact support such that f g Φp( ),q( ),Rn < ε. As in the proof of Theorem., using Theorem 3.4 this time, we have φ t f f Φp( ),q( ),R n Cε + φ t g g Φp( ),q( ),R n. Obviously, g L p 0 (R n ). Hence by Lemma 3., φ t g g almost everywhere in R n. Since there is a compact set S containing all the supports of φ t g, φ t g g Φp( ),q( ),R n C φ t g g L p + +,R n with C depending on S, and the Lebesgue convergence theorem implies φ t g g L p + +,Rn 0 as t. Hence which completes the proof. lim sup φ t f f Φp( ),q( ),R n Cε, t 0 Remark 3.5. In Theorem.2 (and in Theorem A), the condition φ L (p ) (R n ) cannot be weakened to φ L q (R n ) for q < (p ). In fact, for given p > and q < (p ), we can find a smooth exponent p( ) on R n such that p = p, f L p( ) (R n ) and φ L q (R n ) having compact support for which φ f L p( ),R n =. For this, let a R n be a fixed point with a > and let p 2 satisfy (p ) + p 2 < q. Then choose a smooth exponent p( ) on R n such that p(x) = p for x B(0, /2), p(x) = p 2 for x B(a, /2), p = p and p(x) = const. outside B(0, a + ). Take Then φ j = j n/q χ B(a,j ) and f j = j n/p χ B(0,j ), j = 2, 3,.... φ j L q,r n = C < and f j L p( ),R n = f j L p,b(0,/2) = C <. Note that if x B(a, j ), then φ j f j (x) = j n/q+n/p B(a, j ) B(x, j ) Cj n/q+n/p j n,

10 44 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno so that {φ j f j (x)} p(x) dx R n B(a,j ) {φ j f j (x)} p(x) dx Cj p 2(n/q+n/p n) j n = Cj p 2n(/q /(p ) /p 2 ). Now consider φ = j 2 φ 2 j and f = j=2 j 2 f 2 j. Then φ L q (R n ) and f L p( ) (R n ). On the other hand, {φ f(x)} p(x) dx j 4 {φ 2 j f 2 j(x)} p(x) dx R n R n Cj 4 2 p 2nj(/q /(p ) /p 2 ) as j. Hence, φ f L p( ),R n =. Remark 3.6. Cruz-Uribe and Fiorenza [] gave an example showing that it can occur j=2 lim sup φ t f L p( ),R = t 0 for f L p( ) (R) when φ does not have compact support. By modifying their example, we can also find p( ) and φ L (p ) (R), whose support is not compact, such that φ f L p( ),R C f L p( ),R does not hold, namely there exists f N (N =, 2,...) such that f N L p( ),R and lim φ f N L N p( ),R =. For this purpose, choose p >, p 2 > p and a > such that p /p 2 ap + 2 > 0, and let p( ) be a smooth variable exponent on R such that p(x) = p for x 0, p(x) = p 2 for x and p p(x) p 2 for 0 < x <. Set φ = j= χ j, where χ j = χ [ j, j+j a ). Then j+j a φ(x) q dx = χ j (x) q dx = j a C(a) < R j= j j for any q > 0. Further set f N = N /p 2 χ [,N+]. Note that for j + j a < x < 0 and j N χ j f N (x) x+j x+j j a χ j (x y)f N (y) dy = N /p 2 j a,

11 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 45 so that {φ f N (x)} p(x) dx 0 R j= 0 N j=2 N p /p 2 { p χ j f N (x)} dx j j a {χ j f N (x)} p dx N j=2 j ap (j j a ) CN p /p 2 ap +2 (N ). 4. Young type inequalities Cruz-Uribe and Fiorenza [] conjectured that Theorem A remains true if φ satisfies the additional condition (4.) φ(x y) φ(x) y when x > 2 y. x n+ Noting that this condition implies sup φ(x) φ(z) C2 nj, x,z B(0,2 j+ )\B(0,2 j ) we see that lim x φ(x) = 0 since φ L (R n ) and (4.2) φ(x) C x n. if φ satisfies (4.). In this connection we show Theorem 4.. Let p >. Suppose that φ L (R n ) L (p 0) (B(0, R)) and φ satisfies (4.2) for x R. Then φ f Φp( ),q( ),R n C( φ L,R n + φ L (p 0 ),B(0,R) ) f Φ p( ),q( ),R n for all f L p( ) (log L) q( ) (R n ). Remark 4.2. Theorem 4. does not imply an inequality φ t f Φp( ),q( ),R n C f Φ p( ),q( ),R n with a constant C independent of t (0, ] even if φ satisfies (4.2) for all x, because { φ t L (p 0 ),B(0,R) } 0<t is not bounded. Proof of Theorem 4.. Let f be a nonnegative measurable function on R n such that f Φp( ),q( ),R n. Suppose that φ satisfies (4.2) for x R and φ L,R n + φ L (p 0 ),B(0,R). Decompose φ = φ + φ, where φ = φχ B(0,R). We first note by Theorem.2 that φ f Φp( ),q( ),R n C. Hence it suffices to show that For this purpose, write φ f Φp( ),q( ),R n C. f = fχ {y R n :f(y) } + fχ {y R n :f(y)<} = f + f 2,

12 46 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno as before. Then we have by (4.2) and (Φ) φ f (x) C R n \B(x,R) CR n R n f (y) dy x y n f (y) dy CR n R n Φ p( ),q( ) (y, f(y)) dy C. Noting that φ f 2, we obtain Φ p( ),q( ) (x, φ f(x)) dx C. B(0,R) Next, let h(y) = Φ p( ),q( ) (y, f(y)). Then φ h(x) CR n R n h(y) dy CR n. If x R n \ B(0, R), then we have by (4.2) and Lemma 3.3 φ f(x) φ (x y) f(y) dy + B(0, x /2) C { x n C R n \B(0, x /2) φ (x y) f(y) dy f(y) dy + ( φ h(x) ) /p(x) + x A/p(x) B(x,3 x /2) } { Mf(x) + ( φ h(x) ) /p(x) + x A/p(x) with A > n. Now it follows from Proposition 2.5 that { Φ p( ),q( ) (x, φ f(x) ) dx C R n \B(0,R) as required. Φ p( ),q( ) (x, Mf(x)) dx R n \B(0,R) + φ h(x) dx + R n C, Theorem 4.3. Let p /p + θ <, < p < p, s = θ p and r(x) = θ p(x). R n \B(0,R) } } x A dx Take ν = p / p, if t p / p Φ p( )/ p,q( ) (x, t) is uniformly almost increasing in t; otherwise choose ν < p / p. Suppose that φ L (R n ) L s (R n ) L sν (B(0, R)) and φ satisfies φ(x) C x n/s for x R. Then φ f Φr( ),q( ),R n C( φ L,R n + φ L s,r n + φ L sν,b(0,r) ) f Φ p( ),q( ),R n for all f L p( ) (log L) q( ) (R n ).

13 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 47 Proof. Suppose that φ L,R n + φ L s,r n + φ L sν,b(0,r) and φ satisfies φ(x) C x n/s for x R. Let f be a nonnegative measurable function on R n such that f Φp( ),q( ),R n, and decompose f = f + f 2, where f = fχ {x R n : f(x) }. Let r = θ and = + p s r. p + By our assumption, s. It follows from Young s inequality for convolution that φ f 2 L r,r n φ L s,r n f 2 L p,r n. Here note that s < s, so that φ L s,r n φ L,R n + φ L s,rn. Since 0 f 2 <, f 2 L p +,R n C f Φp( ),q( ),R n C. Thus, noting that φ f 2 and we see that r(x) r = θ p(x) θ p 0, (4.3) φ f 2 Φr( ),q( ),R n C φ f 2 L r,r n C. On the other hand, we have by Hölder s inequality ( ) ( θ)/ p ( φ f (x) φ(x y) s f (y) p dy φ(x y) s dy R n R ( ) n θ/ p (4.4) f (y) p dy R ( n ( θ)/ p C φ s f p (x)). ) / p Noting that φ s L (R n ) L ν (B(0, R)), φ s satisfies (4.2) for x R and f p Φp( )/ p,q( ),Rn C, we find by Theorem 4. Since (4.4) implies it follows that φ s f p Φp( )/ p,q( ),R n C. Φ r( ),q( ) (x, φ f (x)) CΦ p( )/ p,q( ) (x, φ s f p (x)), Thus, together with (4.3), we obtain as required. φ f Φr( ),q( ),R n C. φ f Φr( ),q( ),R n C, Remark 4.4. Cruz-Uribe and Fiorenza [] conjectured that Theorem A remains true if φ satisfies the additional condition (4.). If p >, this conjecture was shown to be true by Cruz-Uribe, Fiorenza, Martell and Pérez in [3], using an extrapolation theorem ([3, Theorem.3 or Corollary.]). Using our Proposition 2.5, we can prove the following extension of [3, Theorem.3]:

14 48 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno Proposition 4.5. Let F be a family of ordered pairs (f, g) of nonnegative measurable functions on R n. Suppose that for some 0 < p 0 < p, f(x) p 0 w(x) dx C 0 g(x) p 0 w(x) dx R n R n for all (f, g) F and for all A -weights w, where C 0 depends only on p 0 and the A -constant of w. Then f Φp( ),q( ),R n C g Φ p( ),q( ),R n for all (f, g) F such that g L p( ) (log L) q( ) (R n ). Then, as in [3, p. 249], we can prove: Theorem 4.6. Assume that p >. If φ is an integrable function on R n satisfying (4.), then φ t f Φp( ),q( ),R n C f Φ p( ),q( ),R n for all t > 0 and f L p( ) (log L) q( ) (R n ). If, in addition, φ(x) dx =, then lim t 0 φ t f f Φp( ),q( ),R n = Appendix For p, q R and c e, we consider the function Φ(t) = Φ(p, q, c; t) = t p( log(c + t) ) q, t [0, ). In this appendix, we give a proof of the following elementary result: Theorem 5.. Let X be a non-empty set and let p( ) and q( ) be real valued functions on X such that p(x) p 0 < for all x X. Then, the following () and (2) are equivalent to each other: () There exists c 0 e such that Φ(p(x), q(x), c 0 ; ) is convex on [0, ) for every x X; (2) There exists K > 0 such that K(p(x) ) + q(x) 0 for all x X. This theorem may be well known; however, the authors fail to find any literature containing this result. This theorem is a corollary to the following with Proposition 5.2. () If ( + log c)(p ) + q 0, then Φ is convex on [0, ). (2) Given p 0 > and c e, there exists K = K(p 0, c) > 0 such that Φ is not convex on [0, ) whenever p p 0 and q < K(p ). Proof. By elementary calculation we have Φ (t) = t p 2 (c + t) 2( log(c + t) ) q 2 G(t) G(t) = p(p )(c + t) 2( log(c + t) ) 2 + 2pqt(c + t) log(c + t) qt 2 log(c + t) + q(q )t 2 for t > 0. Φ(t) is convex on [0, ) if and only if G(t) 0 for all t (0, ).

15 Approximate identities and Young type inequalities in variable Lebesgue Orlicz spaces 49 () If q 0, then G(t) qt ( 2p(c + t) t ) log(c + t) qt 2 qt ( 2pc + 2(p )t ) 0 for all t (0, ), so that Φ is convex on [0, ). If ( + log c)(p ) q < 0, then { p q G(t) = p (c + t) log(c + t) + t p pq2 p t2 qt 2 log(c + t) + q(q )t 2 ( q)t 2 ( pq p ) + log c (q ) ) 0 = ( q)t 2 ( q p + log c + for all t (0, ), so that Φ is convex on [0, ). (2) If p = and q < 0, then G(t) = qt ( (t + 2c) log(c + t) + (q )t ) as t. Hence Φ is not convex on [0, ). Next, let < p p 0 and q = k(p ) with k > 0. Then G(t) p = p( (c + t) log(c + t) kt ) 2 + k ( log(c + t) k + ) t 2 p 0 ( (c + t) log(c + t) kt ) 2 + k ( log(c + t) k + ) t 2. Let λ = /(2p 0 ). Then 0 < λ <. If k > (log c)/λ, there is (unique) t k > 0 such that log(c + t k ) = λk. Note that t k /k as k. We have G(t k ) p p ( ) 2 0 (c + tk )λk kt k + k(λk k + )t 2 k { (p0 = kt 2 k ( λ) ) ( λ)k + 2p 0 cλ( λ) k + p 0 c 2 λ 2 k }. t k t 2 k Since p 0 ( λ) = /2, it follows that there is K = K(c, p 0 ) > (log c)/λ such that G(t k ) < 0 whenever k K. Hence Φ is not convex if < p p 0 and q K(p ). Acknowledgement. The authors are grateful to the referee for his/her valuable comments. References [] Cruz-Uribe, D. and A. Fiorenza: Approximate identities in variable L p spaces. - Math. Nachr. 280, 2007, [2] Cruz-Uribe, D. and A. Fiorenza: L log L results for the maximal operator in variable L p spaces. - Trans. Amer. Math. Soc. 36, 2009, [3] Cruz-Uribe, D., A. Fiorenza, J. M. Martell, and C. Pérez: The boundedness of classical operators on variable L p spaces. - Ann. Acad. Sci. Fenn. Math. 3, 2006, } 2

16 420 Fumi-Yuki Maeda, Yoshihiro Mizuta and Takao Ohno [4] Cruz-Uribe, D., A. Fiorenza, and C. J. Neugebauer: The maximal function on variable L p spaces. - Ann. Acad. Sci. Fenn. Math. 28, 2003, ; Ann. Acad. Sci. Fenn. Math. 29, 2004, [5] Diening, L.: Maximal function on Musielak Orlicz spaces. - Bull. Sci. Math. 29, [6] Diening, L., P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura: Maximal functions in variable exponent spaces: limiting cases of the exponent. - Ann. Acad. Sci. Fenn. Math. 34:2, 2009, [7] Kováčik, O., and J. Rákosník: On spaces L p(x) and W k,p(x). - Czechoslovak Math. J. 4, 99, [8] Mizuta, Y., T. Ohno, and T. Shimomura: Sobolev s inequalities and vanishing integrability for Riesz potentials of functions in L p( ) (log L) q( ). - J. Math. Anal. Appl. 345, 2008, [9] Musielak, J.: Orlicz spaces and modular spaces. - Lecture Notes in Math. 034, Springer- Verlag, Berlin, 983. [0] Orlicz, W.: Über konjugierte Exponentenfolgen. - Studia Math. 3, 93, [] Stein, E. M.: Singular integrals and differentiability properties of functions. - Princeton Univ. Press, Princeton, 970. [2] Zo, F.: A note on approximation of the identity. - Studia Math. 55, 976, 22. Received 25 June 2009