Maximal functions for Lebesgue spaces with variable exponent approaching 1

Transcription

1 Hiroshima Math. J. 36 (2006), Maximal functions for Leesgue spaces with variale exponent approaching 1 Dedicated to Professor Fumi-Yuki Maeda on the occasion of his seventieth irthday Toshihide Futamura and Yoshihiro Mizuta (Received Novemer 25, 2004) (Revised Octoer 3, 2005) Astract. Our aim in this paper is to deal with maximal functions for Leesgue spaces with variale exponent approaching Introduction Let R n denote the n-dimensional Euclidean space. We denote y x; rþ the open all centered at x of radius r. For a locally integrale function f on R n, we consider the maximal function 1 Mf xþ ¼sup j f yþjdy: r>0 jx; rþj x; rþ Following Orlicz [4] and Kováčik and Rákosník [2], we consider a positive continuous function pþ : R n!½1; yþ and a measurale function f satisfying j f yþj p yþ dy < y: In this paper we are concerned with pþ satisfying the following log-hölder condition prþ ¼1 a loglog1=rþþ log1=rþ log1=rþ for 0 < r a r 0 < 1=e, where a > 0 and is a real numer; set p0þ ¼1 and prþ ¼pr 0 Þ for r > r 0. For a ounded open set in R n, consider pxþ ¼pdxÞÞ; where dxþ denotes the distance of x from the oundary q of Mathematics Suject Classification. Primary 46E30, Key words and phrases. maximal functions, Leesgue spaces with variale exponent.

2 24 Toshihide Futamura and Yoshihiro Mizuta Cruz-Urie, Fiorenza and Neugeauer [1] proved the maximal operator M is not ounded on L pþ Þ if inf pxþ ¼1. Recently, Hästö [3] has proved that the maximal operator M is ounded from L pþ Þ to L 1 Þ when a > 1 and satisfies a certain regular condition. Our aim in this note is to show that the same conclusion is still valid for a ¼ Maximal functions Throughout this paper, let C denote various constants independent of the variales in question. Let e a ounded open set in R n, and consider a positive continuous function pþ on such that pxþ ¼1 loglog1=dxþþþ 1Þ log1=dxþþ log1=dxþþ when 0 < dxþ a r 0 < 1=e, where is a real numer; assume always that pxþ > 1 when dxþ > 0. For simplicity, we denote the Leesgue measure of E y jej. Let us egin with the following elementary lemmas. Lemma 2.1. Let e a ounded open set in R n. For 0 < k a n and r > 0, set r ¼fx A : dxþ < rg and assume that 2Þ j r j a Cr k ; or the Minkowski n kþ-content of q is finite. Then dxþ k log1 dxþ 1 ÞÞ a dx < y for every a > 1. Lemma 2.2. Set jrþ ¼ loglog1=rþþ log1=rþ log1=rþ for a real numer. Then there exists r 0 > 0 such that (i) j 0 rþ > 0 when 0 < r < 2r 0 ; (ii) j 00 rþ < 0 when 0 < r < 2r 0 ; (iii) js tþ a jsþjtþ when 0 < s; t < r 0. For a locally integrale function f on, we consider the maximal function Mf defined y 1 Mf xþ ¼sup j f yþjdy; jj V

3 Maximal functions for variale Leesgue spaces 25 where the supremum is taken over all alls ¼ x; rþ. Define the L pþ Þ norm y ( ) f yþ p yþ k f k pþ ¼kfk pþ; ¼ inf l > 0 : l dy a 1 and denote y L pþ Þ the space of all measurale functions f on with k f k pþ < y. Lemma 2.3. Suppose the Minkowski n 1Þ-content of q is finite. If f is a measurale function on with k f k pþ a 1, then j f xþj log1 jfxþjþdx a C: Proof. Consider the set 0 ¼fx A : j f xþj < dxþ 1 log1=dxþþþ a g for a > 2. If x A 0 and dxþ < r 0 < 1=eÞ, then j f xþj log1 jf xþjþ a CdxÞ 1 log1=dxþþþ a1 : Hence we have y Lemma 2.1 j f xþj log1 jfxþjþdx a C: r0 V 0 If x 0 and dxþ < r 0 < 1=eÞ, then dxþ Cj f xþjlogj f xþjþ a Þ 1 ; so that Lemma 2.2 yields log logcj f xþjlogj f xþjþ j f xþj pxþ a Þ j f xþj exp logcj f xþjlogj f xþjþ a Þ logcj f xþjlogj f xþjþ a logj f xþj Þ Cj f xþj exp loglogj f xþj logclogj f xþjþa ÞÞ logj f xþj logclogj f xþjþ a logj f xþj Þ loglogj f xþjþ Cj f xþj exp logj f xþj logj f xþj ¼ Cj f xþj logj f xþj: Here note that logt sþ log t t s t a C log t 2 a C t t when 1 < s a C log t:

4 26 Toshihide Futamura and Yoshihiro Mizuta Hence it follows that j f xþj log1 jfxþjþdx a C j f xþj pxþ dx a C: r0 n 0 Finally, since pxþ p 0 > 1 when dxþ r 0, we find j f xþj log1 jfxþjþdx a C j f xþj pxþ dx C a C: n r0 Consequently, the required assertion is proved. r Now we are ready to show our main result, which gives an improvement of Hästö [3]. Theorem 2.4. Suppose the Minkowski n 1Þ-content of q is finite. Then kmf k 1 a Ck f k pþ for all f A L pþ Þ: This is a consequence of Lemma 2.3 and the well-known fact of maximal functions (see Stein [5]). Remark 2.5. Theorem 2.4 is sharp in the following sense: for instance, if pxþ ¼1 loglog1=dxþþþ logloglog1=dxþþþþ log1=dxþþ log1=dxþþ when 0 < dxþ a r 0 < 1=e and inf fx:dxþ>r0 g pxþ > 1, then we can find L pþ Þ for which j f xþj log1 jfxþjþdx ¼ y; where ¼ 0; 1Þ and dxþ ¼1 jxj for x A. For this purpose, letting log 1Þ t ¼ log t and log m1þ t ¼ loglog mþ tþ for m ¼ 1; 2;..., we consider the function f xþ ¼dxÞ 1 log1=dxþþþ 2 log 2Þ 1=dxÞÞÞ 1 for x A with dxþ < r 0 ; set f xþ ¼0 when dxþ r 0. Then r0 f xþ log1 f xþþdx C t 1 log 1Þ 1=tÞÞ 1 log 2Þ 1=tÞÞ 1 dt ¼ y: Further, we have for t ¼ dxþ < r 0 0 f xþ pxþ 1 a explog1=tþþlog 2Þ 1=tÞÞ=log1=tÞ log 3Þ 1=tÞÞ=log1=tÞÞÞ ¼log 1Þ 1=tÞÞlog 2Þ 1=tÞÞ 1 ; f A

5 Maximal functions for variale Leesgue spaces 27 so that r0 f xþ pxþ dx a C t 1 log 1Þ 1=tÞÞ 1 log 2Þ 1=tÞÞ 2 dt < y: 0 3. Variale exponent approaching 1 at a point Suppose pþ satisfies inf fx:jxj>r0 g pxþ > 1 and pxþ ¼1 1 n loglog1=jxjþþ log1=jxjþ log1=jxjþ for 0 < jxj a r 0 < 1=e, where is a real numer. Of course, p0þ ¼1 as efore. Theorem 3.1. If k f k pþ a 1, then j f xþj log1 jfxþjþdx a C; and hence kmf k 1 a Ck f k pþ for all f A L pþ Þ: Proof. As in the proof of Lemma 2.3, we consider the set 0 ¼fx A : j f xþj < jxj n log1 jxj 1 ÞÞ a g for a > 2. Then we have j f xþj log1 jfxþjþdx a C jxj n log1 jxj 1 ÞÞ a1 dx 0 a C 1 0 t 1 log1 t 1 ÞÞ a1 dt < y with the aid of Lemma 2.1. If x A n 0, then we see that jxj Cj f xþj 1=n logj f xþjþ a=n Þ 1 ; which yields j f xþj pxþ Cj f xþj logj f xþj: Hence we otain j f xþj log1 jfxþjþdx a C j f xþj pxþ dx a C; n 0 n 0 as required. r Remark 3.2. As stated in Hästö [3] we have a general result: Consider a compact suset F of a ounded open set, and denote y

6 28 Toshihide Futamura and Yoshihiro Mizuta dxþ ¼distx; FÞ the distance of x from F. For 0 a m < n and 0 < r < r 0 with r 0 small enough let r ¼fxA : dxþ < rg and assume 3Þ j r j a Cr n m : Further pþ is a continuous function on such that 4Þ pxþ ¼1 1 n m loglog1=dxþþþ log1=dxþþ log1=dxþþ when 0 < dxþ a r 0 < 1=e for some real numer and inf fx:dxþ>r0 g pxþ > 1. Then we claim that if f is a locally integrale function on satisfying k f k pþ a 1, then j f xþj log1 jfxþjþdx a C; so that M : L pþ Þ!L 1 Þ is ounded. To prove this, as in proofs of Theorems 2.4 and 3.1, it su ces to consider the set 0 ¼fx A : j f xþj < dxþ m n log1=dxþþ a g for a > 2. References [ 1 ] D. Cruz-Urie, A. Fiorenza and C. J. Neugeauer, The maximal function on variale L p spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), , 29 (2004), [ 2 ] O. Kováčik and J. Rákosník, On spaces L pxþ and W k; pxþ, Czechoslovak Math. J. 41 (1991), [3] P. Hästö, The maximal operator in Leesgue spaces with variale exponent approaching 1, to appear in Math. Nachr. [ 4 ] W. Orlicz, Üer konjugierte Exponentenfolgen, Studia Math. 3 (1931), [ 5 ] E. M. Stein, Singular integrals and di erentiaility properties of functions, Princeton Univ. Press, Princeton, Toshihide Futamura Department of Mathematics Daido Institute of Technology Nagoya , Japan address: futamura@daido-it.ac.jp Yoshihiro Mizuta The Division of Mathematical and Information Sciences Faculty of Integrated Arts and Sciences Hiroshima University Higashi-Hiroshima , Japan address: mizuta@mis.hiroshima-u.ac.jp