ON A MAXIMAL OPERATOR IN REARRANGEMENT INVARIANT BANACH FUNCTION SPACES ON METRIC SPACES

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1 Vasile Alecsandri University of Bacău Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics Vol ), No., ON A MAXIMAL OPRATOR IN RARRANGMNT INVARIANT BANACH FUNCTION SPACS ON MTRIC SPACS MARCLINA MOCANU Abstract. We introduce a maximal operator for functions defined on a doubling metric measure space, belonging to a rearrangement Banach function space. We provide some estimates for the distribution function of this operator, generalizing results proved by Bastero, Milman and Ruiz 999) in the uclidean case and by Costea and Miranda 202) for Newtonian Lorentz spaces on metric spaces.. Introduction The maximal operators, in the first place the Hardy-Littlewood maximal operator, are an essential tool in real analysis and harmonic analysis and have applications to PD s and geometric analysis and ergodic theory, also playing an important role in understanding differentiability properties of functions and of singular integrals. In analysis on metric measure spaces, maximal operators have been used, in the case of a doubling measure, to study pointwise behaviour of Sobolev-type functions, to prove density of Lipschitz functions in Sobolev-type spaces, etc. Keywords and phrases: metric measure space, rearrangement invariant Banach function space, Hardy-Littlewood) maximal operator 200) Mathematics Subject Classification: 4630,

2 50 MARCLINA MOCANU In the following, X, d, µ) is a metric measure space, i.e. a metric space X, d) endowed with a Borel regular measure µ, that is finite and positive on balls. The classical Hardy-Littlewood maximal function operator is defined by Mfx) = sup f dµ, where f L µbx,r)) loc X). r>0 Bx,r) The corresponding non-centered maximal function operator is given by M fx) = sup f dµ for f L loc X), where the µb) B x B supremum is taken over all balls B X which contain x. Note that M fx) = sup fχ B B x χ B L X) L X) and that Mf M f for every f L loc X). Moreover, M f and Mf are comparable if the measure µ is doubling. The maximal function theorem extends to doubling metric measure spaces [7, Theorem 2.2], showing that the Hardy- Littlewood maximal function operator maps L X) to weak- L X) and L X) and L p X) to L p X) for p >, i.e. the following inequalities hold:.) µ {x X : Mf x) > }) C and X f dµ for all > 0, for f L X) Mf p dµ C p f p dµ for f L p X), p >. X X Here the constant C and C p depend only on the doubling constant of µ and, respectively, on p >. For the case of Lorentz spaces L p,q X) with p, q < Stein [3] defined the non-centered) maximal operator M p,q fx) = sup fχ µb) /p B L p,q X), see also [6]. B x Bastero, Milman and Ruiz [2] defined for a Banach function space on R n the maximal operator M fx) = sup fχ Q x χ Q Q for f, where the supremum is taken over all cubes Q R n which contain x and have sides parallel to the coordinate axes. Under the assumption that is a rearrangement invariant space satisfying a lower Φ estimate, where Φ is the fundamental function of, Bastero, Milman and Ruiz [2]

3 ON A MAXIMAL OPRATOR IN R. I. BANACH... 5 generalized.) proving that µ {x X : M f x) > }) C f for all > 0. In this paper we introduce a maximal operator for functions defined on a doubling metric measure space, belonging to a rearrangement Banach function space. This operator is analogue to the above operator M introduced by Bastero, Milman and Ruiz [2]. We provide some estimates for the distribution function of this operator, generalizing the above result proved by Bastero, Milman and Ruiz [2] in the uclidean case and a similar result proved by Costea and Miranda [6] for Newtonian Lorentz spaces on metric spaces. 2. Preliminaries Denote the open balls, respectively the closed balls in the metric space X, d) by Bx, r) = {y X : dy, x) < r} and Bx, r) = {y X : dy, x) r}. We use the abbreviation B x, r) = B x, r), whenever x X and r, > 0. The measure µ on the metric space X, d) is said to be doubling if there exists a constant C µ such that µ 2B) C µ µ B) for every ball B X. A doubling measure can have as atoms only singletons consisting of isolated points [9]. Under the assumption that the measure µ is doubling, the space X, d, µ) is proper i.e. every closed ball of the space is compact) if and only if it is complete. We use the definition given by Bennet and Sharpley [3] for Banach function spaces over the σ finite measure space X, µ). Let X, µ) be a σ finite measure space and M + X) be the set of µ measurable non-negative functions on X. Definition. [3] A function ρ : M + X) [0, ] is called a Banach function norm if, for all functions f, g, f n n ) in M + X), for all constants a 0 and for all measurable sets A X, the following properties hold: if f = 0 µ a.e.; ρaf) = aρf); ρf + g) ρf) + ρg). P2) If 0 g f µ a.e., then ρg) ρf). P3) If 0 f n f µ a.e., then ρf n ) ρf). P)ρf) = 0 if and only P4) If µa) <, then ρχ A ) <. P5) If µa) <, then f dµ C A ρf), for some constant C A 0, + ) depending only on A and ρ.

4 52 MARCLINA MOCANU The collection of all µ measurable functions f : X [, + ] for which ρ f ) < is called a Banach function space on X. For f define f B = ρ f ). We identify two functions that coincide µ a.e. If f, g : X R such that ρ f ) < and f = g µ a.e, then g is µ measurable and ρ g ) = ρ f ) <. Moreover, by Definition P5) and the σ finiteness of µ, it follows that f and g are finite µ a.e., hence f g = 0 µ a.e. and therefore f g = 0. Orlicz spaces and Lorentz spaces are important examples of Banach function spaces, both generalizing Lebesgue spaces. Definition 2. [3, Definition I.3.] A function f is said to have absolutely continuous norm in if and only if fχ k 0 for every sequence k ) k of measurable sets satisfying ) µ lim sup k = 0. k The space is said to have absolutely continuous norm if every f has absolutely continuous norm. We recall that an Orlicz space L Ψ X) has absolutely continuous norm if the Young function Ψ is doubling and that the p, q) norm of a Lorentz space L p,q X) with < p < and q < is absolutely continuous [6], while in L X) the only function having an absolutely continuous norm is the null function. 3. Rearrangement invariant Banach function spaces and lower Φ estimates The notion of rearrangement invariant Banach function space, set up by Hardy and Littlewood, has many applications in real and harmonic analysis, playing a central role in interpolation theory and being involved in the study of singular integrals, isoperimetric inequalities, in the theory of elliptic second order PD s. Let f : X R be a µ measurable function. The distribution function of f is defined by d f t) = µ {x X : f x) > t}), where t 0. The nonincreasing rearrangement of f is f t) = inf {s 0 : d f s) t}, t 0. A Banach function space, ) is said to be rearrangement invariant if f = g implies f = g whenever f, g.

5 ON A MAXIMAL OPRATOR IN R. I. BANACH Let be a rearrangement invariant space over X, µ). The fundamental function of is defined by Φ t) = χ A, t 0, where A X is any µ measurable set with µ A) = t. The definition is unambiguous, since the characteristic functions of two sets with equal measures have the same distribution function. Lemma. [3, Corollary II. 5.3]Let be a rearrangement invariant space over a resonant measure space X, µ). Then the fundamental function Φ satisfies: Φ is increasing, vanishes only at the origin, is continuous except perhaps at the origin) and t Φ t) is decreasing. t Recall that X, µ) is resonant if µ is σ finite and nonatomic. The most well-known examples of rearrangement invariant Banach function spaces are the Orlicz spaces L Ψ X) and the Lorentz spaces L p,q X) with q p < [3, Theorem IV.4.3]. Let A X be a measurable set of finite positive measure. If =L Ψ X) is an Orlicz space with the Luxemburg norm, then χ A = /Ψ /µ A)), see [, page 7, xample 9]. If =L p,q X) is a Lorentz space with the p, q norm, where q p <, then χ A = c p, q) µ A) /p, where c p, q) = p/q) /q. If =L X), then χ A = sgn µ A), this time even for a null set. Definition 3. [2] Let Φ : [0, ) [0, ) be an increasing bijection. The rearrangement invariant Banach function space is said to satisfy a lower Φ estimate if there exists a positive constant M < such that for every finite family {f i : i =,..., n} of functions with disjoint supports, n n ) 3.) f i MΦ Φ f i ). i= It is easy to see that satisfies a lower Φ estimate if and only if there exists a positive constant M < such that for every pairwise disjoint finite family {A i : i =,..., n} of measurable sets and every function f we have n ) fχ A MΦ Φ fχ Ai ), where A = n A i. i= i= For Φ t) = t /p a lower Φ estimate is known as a lower p estimate [8]. In the case where =L p,q X) is a Lorentz space, i=

6 54 MARCLINA MOCANU then satisfies a lower Φ estimate with M =, for Φ t) = t /p if q p as proved by Chung, Hunt and Kurtz [4], respectively for Φ t) = t /q if < p < q < as proved by Costea and Maz ya [5]. Now we consider the case of an Orlicz space =L Ψ X) with the Luxemburg norm L Ψ X), where the Young function Ψ : [0, ) [0, ) is bijective. The fundamental function of the given Orlicz space is Φ t) = and has the inverse Ψ t ) Φ s) = Ψ. Denote F t) = for t 0, ). Then the s) Ψ t ) lower Φ estimate 3.) for Φ = Φ has the form 3.2) F n n ) f i F f i M L i= L Ψ X). Ψ X) i= Recall that an N function is a continuous Young function Ψ : [0, ) [0, ) satisfying Ψ t) = 0 only if t = 0, Ψ t) lim t t Ψ t) = and lim t 0 t = 0. Lemma 2. If Ψ : [0, ) [0, ) is an N-function and F t) = Ψ t ) for t 0, ), then inequality 3.2) holds with M = 4 for every finite family {f i : i =,..., n} L Ψ X) of functions with disjoint supports. Proof. Since Ψ : [0, ) [0, ) is convex, the function t Ψt) t is increasing not necessarily strictly increasing, in other terms is nondecreasing) on [0, ). Then the function t F t) is increasing, therefore F is superadditive. It follows that t n ) n ) 3.3) F f i L Ψ X) F f i L Ψ X) i= Let {f i : i =,..., n} L Ψ X) be a family of functions with disjoint supports. By Lemma 3.2 in [], n n 4 f i f i L L Ψ X). Ψ X) i= Since F is increasing, by the above inequality and by 3.3) we get n ) n F f i L Ψ X) F 4 f i, i= i= L Ψ X) i= i=

7 q.e.d. ON A MAXIMAL OPRATOR IN R. I. BANACH Corollary. If Ψ : [0, ) [0, ) is an N-function, then the Orlicz space =L Ψ X) satisfies a lower Φ estimate. 4. stimates for the distribution function of the maximal operator Recall that Bastero, Milman and Ruiz [2] defined for a Banach function space on R n the maximal operator M fx) = sup fχ Q x χ Q Q for f, where the supremum is taken over all cubes Q R n which contain x and have sides parallel to the coordinate axes. They have shown that M f Mf for every f, since the conditional expectation operator is a norm one projection in any rearrangement invariant space. We will introduce an analogue of the maximal operator defined by Bastero, Milman and Ruiz [2], corresponding to a Banach function space. Definition 4. The non-centered) maximal operator associated to a Banach function space on a metric measure space X, d, µ) is given by M fx) = sup fχ B B x χ B for f L loc X), where the supremum is taken over all balls B X which contain x. In order to compare the above maximal operator and the noncentered Hardy-Littlewood operator, we need the following wellknown result, which is Theorem II.4.8 from [3]. Theorem. [3] Let X, µ) be a resonant measure space and let A i ) i be a countable an at most countable) collection of pairwise disjoint subsets of X, each with finite positive measure. Let A = X \ A i. For each extended-real-valued function g i= which is measurable and finite µ a.e., let T g = gχ A + gdµ χ Ai. µ A i ) i= Let be a rearrangement invariant Banach function space over X, µ). Then T g g A i

8 56 MARCLINA MOCANU for every g. Proposition. If X, d, µ) is a metric measure space such that X, µ) is resonant and if is a rearrangement invariant Banach function space on X, then M f M f for every f. Proof. Fix f. Let B X be a ball. Take A = B and let A = X \ ) B. Let g = f χ B. Then g and T g = f dµ χ B. According to Theorem, µb) B µ B) B f dµ χ B fχ B. Let x X. Taking in the above inequality the supremum over all the balls B x, we get M f x) M f x), as desired. The following theorem extends Theorem from [2] to the setting of metric measure spaces. Theorem 2. Let be a rearrangement invariant Banach function space on a doubling metric measure space X, d, µ) without isolated points. Let Φ and M the corresponding fundamental function and maximal operator, respectively. If satisfies a lower Φ estimate, then there exists a positive constant C < such that for every f we have 4.) Φ µ {x X : M f x) > })) C f. Moreover, if has absolutely continuous norm, then 4.2) lim Φ µ {x X : M f x) > })) = 0. Proof. Since µ is doubling and X, d) has no isolated point, the measure µ is nonatomic [9], but µ is also σ finite, therefore X, µ) is resonant. For each R > 0 consider the restricted maximal function operator defined by { } M R fx) = sup fχ B χ B : x B, B X is a ball, diamb R. Fix > 0. Denote G = {x X : M fx) > } and, given R > 0, G R = { x X : M R fx) > }. Note that 0 < R

9 ON A MAXIMAL OPRATOR IN R. I. BANACH R 2 implies G R G R 2 and that G = R>0 G R, hence, by the continuity from below of the measure µ, it follows that 4.3) χ G = lim χ G R R and µ G ) = lim µ ) G R. R Let R > 0. For each x G R, by the definitions of GR and MR f, we can find a ball B x = B y x, r x ), containing x and with diameter at most R, such that fχ Bx > χ Bx. Note that B x G R. By the definition of the fundamental function of, the above inequality writes as fχ Bx > Φ µ B x )), hence 4.4) µ B x ) < Φ fχ B x ). The family { } B x : x G R is a cover of G R. According to the basic 5r ) covering theorem [7, Theorem.2], this family has a countable subfamily {B xi : i } of pairwise disjoint balls such that { } 5B xi : x G R is a cover of G R. Then µ ) G R µ 5B xi ) C µ ) 3 µ B xi ) i= i= i= ) C µ ) 3 Φ fχ Bxi. We used the doubling property of the measure µ and 4.4). Since Φ is increasing and t Φ t) is decreasing, the above t inequalities imply )) ) Φ µ G R ) Φ C µ ) 3 Φ fχbxi i= C µ ) 3 Φ Φ fχbxi ) ). i= Since satisfies a lower Φ estimate with a constant M) and the balls B xi : i are pairwise disjoint, we have ) ) Φ Φ fχbxi M fχ B xi fχ M G R. i= It follows that 4.5) Φ µ G R )) C i= fχ G R,

10 58 MARCLINA MOCANU where we denoted C = C µ ) 3 /M. Now we let R in 4.5). Using the second part of 4.3) and the continuity of Φ Lemma ) in the left-hand side, respectively the first part of 4.3) and the property P3) of a Banach function norm in the right-hand side, we obtain 4.6) Φ µ G )) C fχ G. Since fχ G fχ G, by property P2) of a Banach function norm, 4.) follows from 4.6). The estimate 4.) implies lim Φ µ G )) = 0, hence lim µ G ) = 0, by Lemma. Assuming that has absolutely continuous norm, it follows that lim fχ G = 0. Now, using 4.6), we get 4.2). Remark. For =L p,q X) with q p the above theorem becomes Lemma 6.3 from [6], which for p = q = yields Lemma 4.2 from [2], in view of Proposition. Also, Lemma 4.2 from [2] yields lim p µ {x X : M f x) > }) = 0 for every f L p X), a fact that was generalized in Lemma 6.8 from [4], which implies lim Ψ ) µ {x X : M f x) > }) = 0 for every f L Ψ X), where Ψ is a doubling Young function and µ is a doubling measure. Remark 2. By Corollary, every Orlicz space =L Ψ X) corresponding to an N function Ψ satisfies a lower Φ estimate. References [] N. Aïssaoui, Capacitary type estimates in strongly nonlinear potential theory and applications, Rev. Mat. Complut ), no. 2, [2] J. Bastero, M. Milman and F. J. Ruiz, Rearrangements of Hardy- Littlewood maximal functions in Lorentz spaces, Proc. Amer. Math. Soc ), [3] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, vol. 29, Academic Press, London, 988. [4] H.M. Chung, R.A. Hunt, D.S. Kurtz, The Hardy-Littlewood maximal function on Lp, q) spaces with weights, Indiana Univ. Math. J., 3 982), no., [5] Ş. Costea and V. Maz ya. Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities, Sobolev Spaces in Mathematics II. Applications in Analysis and Partial Differential quations, p. 03-2, Springer, 2009).

11 ON A MAXIMAL OPRATOR IN R. I. BANACH [6] Ş. Costea and M. Miranda, Newtonian Lorentz metric spaces, Illinois J. Math. Volume 56, Number 2 202), [7] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer- Verlag, New York, 200. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer Verlag, 979. [9] R. A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math ), [0] M. Mocanu, A generalization of Orlicz-Sobolev spaces on metric measure spaces via Banach function spaces, Complex Var. lliptic qu., 55-3)200), [] M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker, [2] N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana ), no.2, [3].M. Stein, ditor s Note: The differentiability of functions in R n, Ann. of Math. 3398), [4] H. Tuominen, Orlicz-Sobolev spaces on metric measure spaces, Ann. Acad. Sci. Fenn., Diss ) 86 pp. Vasile Alecsandri University of Bacău, Department of Mathematics and Informatics, Calea Mărăşeşti, Bacău 6005, Romania mmocanu@ub.ro