THE MAXIMAL OPERATOR ON GENERALIZED ORLICZ SPACES

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1 THE MAXIMAL OPERATOR ON GENERALIZED ORLICZ SPACES PETER A. HÄSTÖ ASTRACT. In thi note I preent a ufficient condition for the boundedne of the maximal operator on generalized Orlicz pace. The reult include a pecial cae the optimal condition for Orlicz pace a well a the eentially optimal condition for variable exponent Lebegue pace and the double-phae functional.. INTRODUCTION Generalized Orlicz pace L ϕ have been tudied ince the 940. A major ynthei of thi reearch i given in the monograph of Muielak [3] from 983, hence the alternative name Muielak Orlicz pace. Thee pace are imilar to the better known Orlicz pace, but defined by a more general function ϕx, t which may vary with the location in pace: the norm i defined by mean of the integral R n ϕx, fx dx, wherea in an Orlicz pace ϕ would be independent of x, ϕ fx. The pecial cae of variable exponent Lebegue pace L p, i.e. ϕx, t := t px, wa introduced by Orlicz [7] already in 93, but lay dormant for many year. However, in the beginning of the new millennium, there wa an exploion in the number of L p - paper. It wa Diening [5] who opened the floodgate by proving the boundedne of the maximal operator under natural and eentially optimal condition on the exponent ee alo [4, 5, 8]. Thi reult allowed for the development of harmonic analyi and related differential equation in the L p etting. In thi note I preent the analogue of thi reult for L ϕ with a treamlined proof which i a implification even in the Orlicz cae cf. Section 3. Furthermore, thi general reult ha optimal or near optimal condition in three important pecial cae: Orlicz pace, where the optimal condition of Gallardo [9] i recovered; 2 Variable exponent pace, where the log-hölder condition i recovered cf. [8] regarding the optimality; 3 The double phae functional ϕx, t = t p + axt q of Mingione and collaborator [, 2, 3], where the harp condition for the regularity of minimizer i recovered, namely q p < + α n with a Cα Theorem 4.7. I hope that the reult and technique in thi note will allow mot of the reult that have been derived in L p over the pat 5 year to be etablihed in L ϕ a well. With thee technique, the Riez potential ha been conidered in [7] and the Dirichlet energy integral in [8]. Maeda, Mizuta, Ohno and Shimomura [0,, 6] have alo recently tudied the boundedne of the maximal operator in L ϕ. Their reult are pecial cae of our, Date: September 4, Mathematic Subject Claification. 46E30; Key word and phrae. Hardy Littlewood maximal opertor, maximal function, Orlicz pace, variable exponent Lebegue pace, generalized Orlicz pace, Muielak Orlicz pace.

2 2 PETER A. HÄSTÖ a they deal only with doubling ϕ and have other retricting aumption a well, ee Section 2. and 5. Related differential equation have been tudied recently by aroni, Colombo and Mingione [, 2, 3] and Giannetti and Paarelli di Napoli [4]. 2. ACKGROUND Definition 2.. A convex, left-continuou function ϕ: [0, [0, ] with ϕ0 = lim t 0 + ϕt = 0, and lim t ϕt = i called a Φ-function. The et of Φ-function i denoted by Φ. Definition 2.2. The et ΦR n conit of thoe ϕ: R n [0, [0, ] with ϕy, Φ for every y R n ; 2 ϕ, t L 0 R n, the et of meaurable function, for every t 0. Alo the function in ΦR n will be called Φ-function. In ub- and upercript the dependence on x will be emphaized by ϕ : L ϕ Orlicz v L ϕ Muielak Orlicz. Definition 2.3. Let ϕ ΦR n and define ϱ ϕ for f L 0 R n by ϱ ϕ f := ϕx, fx dx R n The generalized Orlicz pace, alo called Muielak Orlicz pace, i defined a the et equipped with the Luxemburg norm L ϕ R n = {f L 0 R n : lim λ 0 ϱ ϕ λf = 0} f ϕ := inf { x } λ > 0: ϱ ϕ. λ Two function ϕ and ψ are equivalent if there exit L uch that ψx, t L ϕx, t ψx, Lt for all x and t. Equivalent Φ-function give rie to the ame pace with comparable norm. For further propertie of uch pace ee [6, Chapter 2] and [3]. The notation f g mean that there exit a contant C > 0 uch that f Cg. The Hardy Littlewood maximal operator i defined for f L 0 R n by Mfx := up fy dy, r>0 x,r where x, r i the ball with center x and radiu r, and ffl denote the average integral. For a convex function ϕ Jenen inequality tate that ϕ fx dx ϕ fx dx. A A 2.. Example. Our theorem applie e.g. to the following Φ-function, with uitable p: ϕ x, t = t px log + t, ϕ 2 x, t = t p + axt q, ϕ 3 x, t = t p + axt p loge + t, ϕ 4 x, t = e pxt, ϕ 5 x, t = e t t px, ϕ 6 x, t = χ, t. PDE related to ϕ [4] and ϕ 2 and ϕ 3 [, 2, 3] have been tudied recently. The latter three function are not doubling and therefore not covered by [0,, 6], wherea the reult of thi paper work. Our definition of Φ-function preuppoe convexity, in contrat to that of [0,, 6]. However, their i a empty generalization, a we how in Section 5 that any Φ-function atifying their condition Φ Φ5 i equivalent to a convex Φ-function.

3 THE MAXIMAL OPERATOR ON GENERALIZED ORLICZ SPACES 3 3. ORLICZ SPACES Lemma 3.. Let ϕ: [0, [0, ] left-continuou function with ϕ0 = lim t 0 + ϕt = 0, and lim t ϕt =. If ϕ i increaing, then there exit ψ Φ equivalent to ϕ. Proof. Let ψ be the greatet convex minorant of ϕ. Since 0 ψ ϕ, it follow that ψ0 = lim t 0 + ψt = 0. Suppoe that ϕ > 0. Then ϕt t ϕ for t >. Thu the function t ϕ i a convex minorant of ϕ on [0, and ince ψ i the greatet convex minorant we conclude that ψt t ϕ. It follow that lim t ψt =. Furthermore, thi inequality implie that ψ2 ϕ. Since alo ψ ϕ, we ee that ϕ ψ. Finally, ince ψ i convex, it i continuou except at the poible left-mot point t with ψ = for > t. We force ψ to be left-continuou by redefining ψ = lim t ψt. The propertie above till hold; for ψ2 ϕ we need the leftcontinuity of ϕ. Lemma 3.2. Let ϕ Φ and β > be uch that β ϕ i increaing. Then there exit ψ Φ equivalent to ϕ uch that ψ /β i convex. Proof. The function ϕ /β atifie all the aumption of Lemma 3.. Hence there exit ξ Φ uch that ξ ϕ /β. Set ψ := ξ β. Since β >, ψ Φ and further ψ ϕ, a required. Corollary 3.3. Let ϕ Φ and β > be uch that β ϕ i increaing. Then i bounded. M : L ϕ R n L ϕ R n Proof. Let ψ Φ be a in Lemma 3.2. It uffice to how that M : L ψ R n L ψ R n. Since ψ /γ i convex, it follow from Jenen inequality that ψɛmf = ψ γ γ ɛmf M ψ γ γ. ɛf Let f L ψ R n and ɛ := f ψ o that ϱ ψɛf. Since M i bounded in L γ R n, R n M ψ γ ɛf γ dx Hence ϱ ψ ɛmf, which implie that ɛmf ψ Mf ψ = f ɛ ψ, which complete the proof. R n ψ γ ɛf γ dx = R n ψɛf dx. 4. GENERALIZED ORLICZ SPACES. Dividing by ɛ, we find that For R n define ϕ t := inf x ϕx, t and ϕ + t := up x ϕx, t. We will ue the following aumption for ome common contant σ > 0. The econd correpond in the L p cae to local log-hölder continuity. A0 There exit β > 0 uch that ϕx, β and ϕx, σ for every x R n. A There exit β 0, uch that ϕ + βt ϕ t for every t [ σ, ϕ ] and every ball with ϕ σ.

4 4 PETER A. HÄSTÖ A2 There exit β > 0 and h L weak Rn L R n uch that, for every t [0, σ], ϕx, βt ϕy, t + hx + hy. Remark 4.. Thee condition are invariant under equivalence of Φ-function, which can be een a follow. Suppoe that ϕ ψ with contant L and ϕ atifie A0 A2. From A0 we obtain ψx, β/l ϕx, β and ψx, Lσ ϕσ, o ψ atifie A0 with contant β/l and σ := Lσ in place of σ. Suppoe that t [ σ, ψ ]. Then ψ + β L 2 t ϕ + β L t ϕ t L ψ t ince t L [ σ, ϕ ] o that A of ϕ could be ued. Thu A hold for ψ, a well. For A2 we etimate, when t [0, σ ], ψx, βt/l 2 ϕx, βt/l ϕy, t/l + hx + hy ψy, t + hx + hy. Remark 4.2. Suppoe that ϕ t := lim x ϕx, t exit for every t [0, ] and that hx := up ϕx, t ϕ t L weakr n L R n. t [0,] Then by the triangle inequality we obtain that ϕx, t ϕx, t ϕ t + ϕ t ϕy, t + ϕy, t hx + hy + ϕy, t o A2 hold. Thi aumption correpond to Nekvinda [5] decay condition in L p. Lemma 4.3. Let ϕ ΦR n. Then ϕ atifie the Jenen-type inequality ϕ f dx ϕ f dx. 2 Proof. Let ψ be the greatet convex minorant of ϕ. Since t ϕ t i increaing, we t conclude a in Lemma 3. that ϕ ψ2. y Jenen inequality for ψ, ϕ f dx ψ f dx ψf dx ϕ f dx. 2 Lemma 4.4. Let ϕ ΦR n atify aumption A0 A2. If i a ball and f L ϕ R n with ϱ ϕ fχ { f >σ}, then ϕ x, β f dy + 4 ϕy, f dy + hx + hy dy. σ Proof. Fix a ball. Aume without lo of generality that f 0, and denote f := fχ {f>σ} and f 2 := f f. Since ϕ i convex and increaing, ϕ x, β f dy ϕ x, β f dy + ϕ x, β f 2 dy. 4 2 Conider firt the part f when ϕ σ and define { ϕx, σt when t σ, ϕx, t := ϕx, t when t > σ. Since ϕ i convex, ϕ ϕ, and ince f 0, σ, ϕy, f y = ϕy, f y. Therefore it uffice to prove the econd inequality in ϕ x, β f dy ϕ x, β f dy ϕy, f dy = ϕy, f dy. 2 2

5 THE MAXIMAL OPERATOR ON GENERALIZED ORLICZ SPACES 5 Note that ϕ atifie A on all of [ 0, ϕ ]. y Lemma 4.3, ϕ f dy ϕ 2 f dy ϕy, f dy. Therefore we can ue A and Lemma 4.3 to conclude that ϕ x, β β f dy ϕ + f dy ϕ f dy Suppoe then that ϕ σ. Now f dy ϕ σf dy ϕy, f dy. y convexity, A0 and convexity again, we conclude that ϕ x, β f dy ϕx, β f dy ϕy, σf dy σ For f 2 we ue the convexity of ϕx, and A2: ϕ x, β f 2 dy ϕx, βf 2 dy ϕy, f 2 dy + Adding the etimate for f and f 2, we conclude the proof. ϕy, f dy. ϕy, f dy. hx + hy dy. Taking the upremum over ball in the previou lemma, and noticing that hx Mhx, we obtain the following corollary: Corollary 4.5. Let ϕ ΦR n atify aumption A0 A2 and let f L ϕ R n with ϱ ϕ fχ { f >σ}. Then ϕx, β 4 Mf M ϕ, f + Mhx. Theorem 4.6. Let ϕ ΦR n atify aumption A0 A2. Suppoe that β > i uch that β ϕx, i increaing for every x R n. Then the maximal operator i bounded, M : L ϕ R n L ϕ R n. Proof. Let ψx, Φ be related to ϕx, a in Lemma 3.2, for every x R n. Since the Φ-function ϕ and ψ are equivalent, it uffice to how that M : L ψ R n L ψ R n. Note that ψ alo atifie aumption A0 A2, by Remark 4.. Let f L ψ R n and chooe ɛ > 0 uch that ϱ ψ ɛf. When t > σ, ψx, t by A0, o that ψx, t /γ ψx, t. Thu ϱ ψ ɛfχ /γ { ɛf >σ} and we can apply Corollary 4.5 to ɛf with the Φ-function ψ /γ : ψ x, β ɛmfx γ M ψ γ 4, ɛf x + Mhx. Raiing both ide to the power γ and integrating, we find that ψx, β ɛmfx dx M ψ γ 4, ɛf x γ dx + Mhx γ dx. R n R n R n Note that h L weak Rn L R n L γ R n. Since M i bounded on L γ R n, we obtain that ψx, β ɛmfx dx ψ γ γ 4 x, ɛf dx + hx γ dx = ϱ ψ ɛf + h γ γ. R n R n R n

6 6 PETER A. HÄSTÖ Hence ϱ ψ β ɛmf, and the proof i completed by a caling argument like Corollary A an example and application we conider the double-phae Φ-function tudied by aroni, Colombo and Mingione [, 2, 3]. Note that the bound q + α i the ame p n a that obtained by thee reearcher for ome of their reult, the trict inequality i required. Theorem 4.7. Let Ω R n be open and bounded and ϕx, t := t p + axt q, q > p >. If a C α Ω i non-negative, then the maximal operator i bounded on L ϕ Ω when q p + α n. Proof. We extend a by McShane extenion to a function in C α R n. Thi extenion can be multiplied by a mooth cut-off function H C 0 R n which equal in Ω. Since q > p > and a 0 in R n, it follow that t t p ϕx, t i increaing. We how that A0 A2 hold with σ =. For A0, we note that ϕx, + a. If K i the upport of H, then ϕ t p in R n \ K, o A2 hold hold with h := a χ K. Let u how that alo condition A hold. Note firt that ϕx, t max{t p, axt q }. Denote a + := up z az and a := inf z az. It uffice to how that max{t p, a + tq } max{t p, a tq } when ϕ t <. We prove the inequality in the even greater range tp <. The inequality t p max{t p, a tq } i trivial, o we only have to how that a + max{t p q, a }. Uing the upper bound on t, we ee that it i ufficient to prove that a + q p max{ p q p, a } diam p n + a. In view of the definition of α, thi follow from the aumption a C α. Note that the revere implication doe not hold, i.e. A doe not imply that a C α Ω. Indeed, if we chooe a = χ Ω + χ E for ome meaurable E Ω, then a i dicontinuou but ϕ + 2ϕ. On the other hand, the aumption i harp in the ene that if q > + α, then a p n Cα doe not imply A, a hown by the example ax = x α. Remark 4.8. aroni, Colombo and Mingione [2] conidered the border-line p = q double-phae functional ϕx, t := t p + axt p loge + t. Alo in thi cae their regularity aumption a C log Ω implie the condition A, o our theorem yield the boundedne of the maximal operator. Remark 4.9. It wa pointed out in the introduction that in many ituation the condition on ϕ are optimal or near optimal. However, in term of interaction of the variability in x and t the preent aumption are far from optimal. Thi i een by conidering ϕx, t = t p wx, the weighted Lebegue pace. In thi cae the optimal aumption for the boundedne i that w A p [2]. The difference i that in thi well-tructured ituation it i poible to trade integrability of the function and dependence of ϕ on x. Thi i not poible in the general cae, indeed not even in the L p -cae. It i of coure poible to write ϕx, t = ψx, twx and have a more general condition for the weight cf. [6, Section 5.8], but thi i not a very elegant olution. Whether a better olution exit i a quetion for future reearch.

7 THE MAXIMAL OPERATOR ON GENERALIZED ORLICZ SPACES 7 5. REMARKS ON ALTERNATIVE CONDITIONS In the paper [0,, 6], Maeda, Mizuta, Ohno and Shimomura conidered Muielak Orlicz pace with ix condition on the Φ-function. The firt four condition are, for ome contant D > : Φ ϕ: [0, [0, i continuou, ϕ0 = lim ϕt = 0, and lim ϕt =. t 0 + t Φ2 ϕx, D. D Φ3 ϕ i almot increaing, i.e. ϕ ϕt for every > t. D t Φ4 ϕ i doubling, i.e. ϕ2t Dϕt for every t > 0. We notice that aumption Φ i otenibly weaker than the aumption in thi note, ince convexity i not aumed a priori. However, we how below that any function atifying thee condition i equivalent to a convex function. Aumption Φ4 doe not correpond to any aumption in thi note. 5.. Almot increaing and non-convex. Aumption Φ3 eem to be le tringent than the one ued in thi paper, ince it follow from convexity that ϕ i increaing, not merely almot increaing. ϕt Let ϕ atify aumption Φ3 and define ψ := up t. Clearly ψ i increaing and ϕ ψ. y condition Φ3, up t D ϕ. Therefore t t ϕt ϕ ψ Dϕ ϕd 2, o the function are equivalent. Thu, there i no added generality in conidering almot increaing function intead of increaing function. Furthermore, ince ψ i increaing there exit a convex ξ Φ i equivalent to ψ by Lemma Decay condition. Condition Φ5 in [0,, 6] i eentially the ame a A in thi paper. However, their decay condition Φ6 eem more general, until we combine it with Φ2, which i a tronger verion of A0. The former condition i a follow: Φ6 there exit a function g L R n and a contant uch that 0 gx < for all x R n and ϕx, t ϕx, t ϕx, t whenever x x and gx t. Let u how how thi relate to condition A2. Firt we define ϕ t := lim up ϕx, t. x If t [gx, ], then ϕx, t ϕ t. If t [0, gx], then ϕx, t Dϕx, t DAgx by condition Φ2 and Φ3. Hence ϕx, t ϕ t + DAgx for all t [0, ]. Similarly we may etablih that and thu we conclude that a required by aumption A2. ϕ t ϕy, t + DAgy, ϕx, t ϕy, t + gx + gy,

8 8 PETER A. HÄSTÖ ACKNOWLEDGEMENT I thank Tetu Shimomura for comment and for pointing out ome hort-coming of the original proof and Fumi-Yuki Maeda for a counter-example. The original verion incorrectly omitted the aumption A0, although it wa implicitly ued. I alo thank the referee for pointing out an additional application of the main theorem. REFERENCES [] P. aroni, M. Colombo and G. Mingione: Harnack inequalitie for double phae functional. Nonlinear Anal , [2] P. aroni, M. Colombo and G. Mingione: Non-autonomou functional, borderline cae and related function clae. St Peterburg Mathematical Journal, to appear. [3] M. Colombo and G. Mingione: Regularity for Double Phae Variational Problem. Arch. Ration. Mech. Anal , no. 2, [4] D. Cruz-Uribe, A. Fiorenza and C. Neugebauer: The maximal function on variable L p pace. Ann. Acad. Sci. Fenn. Math., , ; , [5] L. Diening: Maximal function on generalized Lebegue pace L p. Math. Inequal. Appl , [6] L. Diening, P. Harjulehto, P. Hätö and M. Růžička: Lebegue and Sobolev pace with variable exponent. Lecture Note in Mathematic, 207. Springer, Heidelberg, 20. [7] P. Harjulehto and P. Hätö: Riez potential in generalized Orlicz Space, Preprint 205. [8] P. Harjulehto, P. Hätö and R. Klén: Generalized Orlicz pace and PDE, Preprint 205. [9] D. Gallardo: Orlicz pace for which the Hardy Littlewood maximal operator i bounded. Publ. Mat , no. 2, [0] F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura: oundedne of maximal operator and Sobolev inequality on Muielak-Orlicz-Morrey pace. ull. Sci. Math , [] F.-Y. Maeda, Y. Mizuta, T. Ohno and T. Shimomura: Approximate identitie and Young type inequalitie in Muielak-Orlicz pace. Czecholovak Math. J , no. 4, [2]. Muckenhoupt: Weighted norm inequalitie for the Hardy maximal function. Tran. Amer. Math. Soc , [3] J. Muielak: Orlicz pace and modular pace. Lecture Note in Mathematic, 034. Springer, erlin, 983. [4] F. Giannetti and A. Paarelli di Napoli: Regularity reult for a new cla of functional with nontandard growth condition. J. Differential Equation , [5] A. Nekvinda: Hardy-Littlewood maximal operator on L px R n. Math. Inequal. Appl , [6] T. Ohno and T. Shimomura: Trudinger inequality for Riez potential of function in Muielak- Orlicz pace. ull. Sci. Math , no. 2, [7] W. Orlicz: Über konjugierte Exponentenfolgen. Studia Math. 3 93, [8] L. Pick and M. Růžička: An example of a pace L px on which the Hardy-Littlewood maximal operator i not bounded. Expo. Math , DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF OULU, FINLAND AND DEPART- MENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF TURKU, FINLAND addre: peter.hato@oulu.fi URL: phato/