Morrey-Sobolev Spaces on Metric Measure Spaces
aa r X i v : . [ m a t h . C A ] D ec Morrey-Sobolev Spaces on Metric Measure Spaces
Yufeng Lu, Dachun Yang ∗ and Wen Yuan Abstract
In this article, the authors introduce the Newton-Morrey-Sobolev spaceon a metric measure space ( X , d, µ ). The embedding of the Newton-Morrey-Sobolevspace into the H¨older space is obtained if X supports a weak Poincar´e inequalityand the measure µ is doubling and satisfies a lower bounded condition. Moreover,in the Ahlfors Q -regular case, a Rellich-Kondrachov type embedding theorem is alsoobtained. Using the Haj lasz gradient, the authors also introduce the Haj lasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with theHaj lasz-Morrey-Sobolev space when µ is doubling and X supports a weak Poincar´einequality. In particular, on the Euclidean space R n , the authors obtain the coinci-dence among the Newton-Morrey-Sobolev space, the Haj lasz-Morrey-Sobolev spaceand the classical Morrey-Sobolev space. Finally, when ( X , d ) is geometrically dou-bling and µ a non-negative Radon measure, the boundedness of some modified (frac-tional) maximal operators on modified Morrey spaces is presented; as an application,when µ is doubling and satisfies some measure decay property, the authors furtherobtain the boundedness of some (fractional) maximal operators on Morrey spaces,Newton-Morrey-Sobolev spaces and Haj lasz-Morrey-Sobolev spaces. In 1996, via introducing the notion of Haj lasz gradients, Haj lasz [13] obtained an equiva-lent characterization of the classical Sobolev space on R n , which becomes an effective wayto define Sobolev spaces on metric spaces. From then on, several different approachesto introduce Sobolev spaces on metric measure spaces were developed; see, for example,[26, 11, 37, 16, 14, 22, 41, 27].Throughout the paper, ( X , d, µ ) denotes a metric measure space with a non-trivialBorel regular measure µ , which is finite on bounded sets and positive on open sets. Let f be a measurable function on X . Recall that a non-negative function g on X is called Mathematics Subject Classification . Primary 46E35; Secondary 42B25, 42B35, 30L99.
Key words and phrases . Sobolev space, Morrey space, upper gradient, Haj lasz gradient, metric measurespace, maximal operatorDachun Yang is supported by the National Natural Science Foundation of China (Grant No. 11171027).Wen Yuan is supported by the National Natural Science Foundation of China (Grant No. 11101038)and the Alexander von Humboldt Foundation. This project is also partially supported by the SpecializedResearch Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) andthe Fundamental Research Funds for Central Universities of China (Grant No. 2012LYB26). ∗ Corresponding author. Yufeng Lu, Dachun Yang and Wen Yuan a Haj lasz gradient of f if there exists a set E ⊂ X such that µ ( E ) = 0 and, for all x, y ∈ X \ E , | f ( x ) − f ( y ) | ≤ d ( x, y )[ g ( x ) + g ( y )] . The
Haj lasz-Sobolev space M ,p ( X ) with p ∈ [1 , ∞ ] is then defined to be the space of allmeasurable functions f ∈ L p ( X ) which have Haj lasz gradients g ∈ L p ( X ). The norm ofthis space is defined by k f k M ,p ( X ) := k f k L p ( X ) + inf k g k L p ( X ) , where the infimum is taken over all Haj lasz gradients g of f . It was proved in [13] that,when X = R n and p ∈ (1 , ∞ ] , M ,p ( R n ) coincides with the classical Sobolev space W ,p ( R n ).Over a decade ago, based on the notions of upper gradients and weak upper gradients,Shanmugalingam [37, 38] introduced another type of Sobolev spaces on metric measurespaces, which are called Newtonian spaces or Newton-Sobolev spaces. These spaces werealso proved to coincide with the Haj lasz-Sobolev spaces if X supports some Poincar´einequality and the measure is doubling. Now we recall their definitions.Recall that we call γ a curve if it is a continuous mapping from an interval into X . Acurve γ is said to be rectifiable if its length is finite. All rectifiable curve can be arc-lengthparameterized. Without loss of generality, we may assume that all curves appearing inthis article are always treated as arc-length parameterized .Let p ∈ [1 , ∞ ) and Γ be a family of non-constant rectifiable curves on X . Recall thatthe admissible class F (Γ) for Γ is defined by(1.1) F (Γ) := (cid:26) ρ ∈ [0 , ∞ ] : ρ is Borel measurable and Z γ ρ ( s ) ds ≥ γ ∈ Γ (cid:27) . If Γ contains a constant curve, then F (Γ) = ∅ . The p -modulus of Γ is then defined byMod p (Γ) := inf ρ ∈ F (Γ) k ρ k pL p ( X ) , where the infimum is taken over all admissible functions ρ in F (Γ). We let the infimum overthe empty set always be infinity. Let f be a measurable function on X . A non-negativefunction g is called an upper gradient of f if, for any curve γ ∈ Γ rect ,(1.2) | f ◦ γ (0) − f ◦ γ ( l ( γ )) | ≤ Z γ g ( s ) ds, where Γ rect is the class of all non-constant rectifiable curves in X . Moreover, if theinequality (1.2) holds for all the curves except for a family of curves of p -modulus zero,then we call g a p -weak upper gradient of f . The notion of p -weak upper gradient wasintroduced by Heinonen and Koskela in [15]; see also [19] and [37, 38].For all p ∈ [1 , ∞ ), denote by the symbol e N ,p ( X ) the space of all measurable functions f ∈ L p ( X ) which have p -weak upper gradients g ∈ L p ( X ) and, for all f ∈ e N ,p ( X ), let k f k e N ,p ( X ) := k f k L p ( X ) + inf k g k L p ( X ) , orrey-Sobolev Spaces on Metric Measure Spaces p -weak upper gradients g of f . The Newton-Sobolevspace N ,p ( X ) is then defined to be the quotient space N ,p ( X ) := e N ,p ( X ) / ∼ withthe norm k · k N ,p ( X ) := k · k e N ,p ( X ) , where ∼ is an equivalence relation defined by setting,for all f , f ∈ e N ,p ( X ), f ∼ f if k f − f k e N ,p ( X ) = 0. It was proved in [37, Theorem4.9] that the Newton-Sobolev space coincides with the Haj lasz-Sobolev space if ( X, µ )supports some Poincar´e inequality and the measure µ is doubling. We refer the reader to[37, 19, 7, 12, 6] for more properties about these spaces.Recently, there were some attempts to study Newtonian type spaces in more generalsettings. Durand-Cartagena in [10] introduced and studied the Newtonian space N , ∞ ( X )in the limit case p = ∞ . Tuominen [40] considered Newtonian type spaces associatedwith Orlicz spaces by replacing the Lebesgue norm in the definition of N ,p ( X ) withOrlicz norms. Using Lorentz spaces instead of Lebesgue spaces, Costea and Miranda [9]introduced Newtonian type spaces related to Lorentz spaces. Mal´y [29, 30] studied theNewtonian type spaces associated with a general quasi-Banach function lattice X , namely,a quasi-Banach function space X satisfying that, if f ∈ X and | g | ≤ | f | almost everywhere,then g ∈ X and k g k X ≤ k f k X .Let 0 < p ≤ q ≤ ∞ . Recall that the Morrey space M qp ( X ) (see [33]) is defined to bethe space of all measurable functions f on X such that(1.3) k f k M qp ( X ) := sup B ⊂ X [ µ ( B )] /q − /p (cid:20)Z B | f ( x ) | p dµ ( x ) (cid:21) /p < ∞ , where the supremum is taken over all balls in X . In recent years, Morrey spaces and theMorrey versions of many classical function spaces such as Hardy spaces and Besov spaces,namely, the spaces defined via replacing Lebesgue norms by Morrey norms in their norms,attract more and more attentions and have proved useful in the study of partial differentialequations and harmonic analysis; see, for example, [1, 2, 3, 4, 34, 32, 28, 31, 43] and theirreferences.The main purpose of this article is to develop a theory of Newtonian type spaces basedon Morrey spaces, namely, Newton-Morrey-Sobolev spaces, as well as the Haj lasz-Morrey-Sobolev spaces on metric measure spaces.We begin with the following generalized modulus based on Morrey spaces. Definition 1.1.
Let 1 ≤ p ≤ q < ∞ and Γ be a collection of rectifiable curves. The Morrey-modulus of Γ is defined byMod qp (Γ) := inf ρ ∈ F (Γ) k ρ k p M qp ( X ) , where F (Γ) is defined as in (1.1). Definition 1.2.
Let f be a measurable function and g a non-negative Borel measurablefunction. If the inequality (1.2) holds true for all non-constant rectifiable curves in X except a family of curves of Morrey-modulus zero, then g is called a Mod qp -weak uppergradient of f . Yufeng Lu, Dachun Yang and Wen Yuan
Via these Mod qp -weak upper gradients, the Newton-Morrey-Sobolev space is introducedas follows. Definition 1.3.
Let 1 ≤ p ≤ q < ∞ . The space ^ N M qp ( X ) is defined to be the set of all µ -measurable functions f such that k f k ^ NM qp ( X ) < ∞ , where k f k ^ NM qp ( X ) := k f k M qp ( X ) + inf k g k M qp ( X ) with the infimum being taken over all Mod qp -weak upper gradients g of f . The Newton-Morrey-Sobolev space
N M qp ( X ) is then defined as the quotient space ^ N M qp ( X ) (cid:14) (cid:26) f ∈ ^ N M qp ( X ) : k f k ^ NM qp ( X ) = 0 (cid:27) with k f k NM qp ( X ) := k f k ^ NM qp ( X ) . It is easy to see that k · k NM qp ( X ) is a norm. Moreover, when p = q , the space N M qp ( X )is just the Newton-Sobolev space N ,p ( X ) introduced by Shanmugalingam [37]. We alsoremark that, since Morrey spaces are Banach function lattices, these Newton-Morrey-Sobolev spaces are special cases of the Newtonian type spaces associated with quasi-Banachfunction lattices considered by Mal´y [29, 30].This article is organized as follows. In Section 2, we show that the Newton-Morrey-Sobolev space is non-trivial by proving that the set of Lipschitz functions with boundedsupport is contained in the Newton-Morrey-Sobolev space N M qp ( X ) (see Theorem 2.4below), but not dense in some examples (see Remark 2.5 below), which is different fromthe Newton-Sobolev space. Moreover, in Remark 4.8 below, we even show that the set ofLipschitz functions is not dense in N M qp ( R n ) when 1 < p < q < ∞ .In Section 3, the embedding of the Newton-Morrey-Sobolev space into the H¨older spaceis obtained when X supports a weak Poincar´e inequality, the measure µ is doubling andsatisfies a lower bounded condition (see Theorem 3.1 below). Moreover, if the space X isAhlfors Q -regular and supports a weak Poincar´e inequality, via proving the boundednessof some fractional integrals on Morrey spaces, we also obtain a Rellich-Kondrachov typeembedding theorem of the Newton-Morrey-Sobolev space (see Theorem 3.6 below). Bothembedding properties on Newton-Morrey-Sobolev spaces generalize the corresponding re-sults for Newton-Sobolev spaces obtained by Shanmugalingam in [37, Theorems 5.1 and5.2].In Section 4, using the Haj lasz gradient, we introduce the Haj lasz-Morrey-Sobolev spaceon metric measure spaces and show that, when X supports a weak Poincar´e inequality andthe measure µ is doubling, the Newton-Morrey-Sobolev space coincides with the Haj lasz-Morrey-Sobolev space (see Theorem 4.6 below). This generalizes the result on the relationbetween Newton-Sobolev spaces and Haj lasz-Sobolev spaces obtained by Shanmugalingamin [37, Theorem 4.9]. In particular, when X = R n and 1 < p ≤ q < ∞ , both the Newton-Morrey-Sobolev space N M qp ( R n ) and the Haj lasz-Morrey-Sobolev space HM qp ( R n ) areproved to coincide with the classical Morrey-Sobolev space on R n (see Theorem 4.7 below). orrey-Sobolev Spaces on Metric Measure Spaces X is a doubling metric measure space satisfying the relative 1-annular decay propertyand the measure lower bound condition (see Theorem 5.13 below). If X supports a weakPoincar´e-inequality, and the measure is doubling and satisfies the measure lower boundcondition, then the boundedness of discrete (fractional) maximal operators on Newton-Morrey-Sobolev spaces is also obtained (see Theorem 5.14 below). All these conclusionsgeneralize the corresponding known results on Newton-Sobolev spaces and Haj lasz-Sobolevspaces by Heikkinen et al. in [17, 18].At the end of this section, we make some conventions on notation. Throughout thepaper, we denote by C a positive constant which is independent of the main parameters,but it may vary from line to line. The symbols A . B and A & B means A ≤ CB and A ≥ CB , respectively, where C is a positive constant. If A . B and B . A , then we write A ≈ B . If E is a subset of X , we denote by χ E its characteristic function . In this section, we consider some basic properties of Newton-Morrey-Sobolev spacesincluding their completeness and non-triviality. Throughout this section, we only assumethat µ is a non-trivial Borel regular measure .Recall that the Newton-Morrey-Sobolev space is a special case of the Newtonian spacesbased on quasi-Banach function lattice X introduced in [29]. The following result is aspecial case of [29, Theorem 7.1]. Theorem 2.1.
For all ≤ p ≤ q < ∞ , the space N M qp ( X ) is a Banach space. The next lemma is usually called the truncation lemma , which shows how a Mod qp -weakupper gradient behaves when multiplying a characteristic function. Its proof is similar tothose of [9, Lemmas 4.6 and 4.7], the details being omitted. Lemma 2.2.
Let f ∈ N M qp ( X ) and g , g ∈ M qp ( X ) be two Mod qp -weak upper gradientsof f . (i) If f is a constant on a closed set E , then g := g χ X \ E is also a Mod qp -weak uppergradient of f . (ii) If E is closed in X , then h := g χ E + g χ X \ E is also a Mod qp -weak upper gradient of f . We also need the following conclusion.
Yufeng Lu, Dachun Yang and Wen Yuan
Proposition 2.3.
Let ≤ p ≤ q < ∞ . For any set E ⊂ X with finite measure, k χ E k M qp ( X ) is bounded by a positive constant multiple of [ µ ( E )] /q with the positive con-stant independent of E .Proof. Notice that k χ E k M qp ( X ) = sup B ⊂ X [ µ ( B )] /q (cid:20) µ ( B ∩ E ) µ ( B ) (cid:21) /p . If µ ( B ) ≥ µ ( E ) /
2, then by p ≤ q , we have[ µ ( B )] /q (cid:20) µ ( B ∩ E ) µ ( B ) (cid:21) /p . [ µ ( E )] /q − /p [ µ ( B ∩ E )] /p . [ µ ( E )] /q . If µ ( B ) ≤ µ ( E ) /
2, then [ µ ( B )] /q (cid:20) µ ( B ∩ E ) µ ( B ) (cid:21) /p . [ µ ( E )] /q . This finishes the proof of Proposition 2.3.Recall that
N M pp ( X ) = N ,p ( X ), which is a non-trivial space, namely, the space N ,p ( X ) contains more than just the zero function and might be a proper subspace of L p ( X ) if X has enough rectifiable paths (see [37]). The following conclusion shows that,even when q > p ≥ N M qp ( X ) is also a non-trivial space. In what follows, Lip b ( X )denotes the set of all Lipschitz functions on X with bounded support . Theorem 2.4.
Let ≤ p ≤ q < ∞ . Then, Lip b ( X ) ⊂ N M qp ( X ) ⊂ N ,p loc ( X ) , where N ,p loc ( X ) denotes the collection of functions which belong to N ,p ( B ) for any ball B ⊂ X .Proof. To show the first embedding, let B be a ball in X . By Proposition 2.3, we knowthat χ B ∈ M qp ( X ) and k χ B k M qp ( X ) . [ µ ( B )] /q < ∞ . Recall, by our conventions on notation at the end of Section 1, that the symbol . meansthat the implicit positive constant here is independent of B .Now let f ∈ Lip b ( X ) with supp ( f ) ⊂ B and L be the Lipschitz constant of f , whichmeans that, for all x, y ∈ X , | f ( x ) − f ( y ) | ≤ Ld ( x, y ) . Since f ∈ Lip b ( X ), we know that there exists a positive constant M such that | f | ≤ M χ B . Hence, by (1.3), we see that f ∈ M qp ( X ) and k f k M qp ( X ) . M [ µ ( B )] /q . orrey-Sobolev Spaces on Metric Measure Spaces γ , it holds true that | f ◦ γ ( ℓ (0)) − f ◦ γ ( ℓ ( γ )) | ≤ L d ( γ ( ℓ (0)) , γ ( ℓ ( γ ))) ≤ Z γ L ds.
Hence L is an upper gradient of f . Then, by Lemma 2.2, Lχ B is a Mod qp -weak uppergradient of f , which further implies that f ∈ N M qp ( X ) and k f k NM qp ( X ) . ( M + L )[ µ ( B )] /q . Thus, Lip b ( X ) ⊂ N M qp ( X ).The second embedding follows directly from definitions, together with [29, Corollary5.7]. Indeed, let f ∈ N M qp ( X ). Then, by [29, Definition 2.4 and Corollary 5.7], we knowthat k f k NM qp ( X ) = k f k M qp ( X ) + inf k h k M qp ( X ) < ∞ , where the infimum is taken over all the upper gradients of f . From this, we deduce that f has an upper gradient h ∈ M qp ( X ) and hence h ∈ L p ( E ) for any ball E ⊂ X . Sinceit is obvious that f ∈ L p ( E ), by [29, Definition 2.4 and Corollary 5.7] again, we obtainthat f ∈ N ,p ( E ), which, together with the arbitrariness of E ⊂ X and the definition of N ,p loc ( X ), implies that f ∈ N ,p loc ( X ) and hence completes the proof of Theorem 2.4. Remark 2.5.
We point out that Lip b ( X ) might not be dense in N M qp ( X ) when p < q .Indeed, even the set of Lipschitz functions, Lip( X ) , might not be dense in N M qp ( X ) when p < q . This behavior of N M qp ( X ) (non-density of Lipschitz functions) is different fromthe Newton-Sobolev space N ,p ( X ) = N M pp ( X ), since Lip( X ) is dense in N ,p ( X ) (see[37, Theorem 4.1]). A counterexample in the Euclidean setting is given in Remark 4.8below. Let α ∈ (0 ,
1] and C ,α ( X ) denote the α -H¨older space on X , namely, the space of allfunctions f satisfying that, for all x, y ∈ X , | f ( x ) − f ( y ) | ≤ C [ d ( x, y )] α , where C is a positive constant independent of x and y .It is well known that, when X = R n , the following Sobolev embeddings hold true: W ,p ( R n ) ֒ → L np/ ( n − p ) ( R n ) if p < n, (3.1)and W ,p ( R n ) ֒ → C , − n/p ( R n ) if p > n, (3.2)where the symbol ֒ → means continuous embedding. The generalizations of (3.1) and (3.2)to the Newton-Sobolev space and the Haj lasz-Sobolev space on metric measure spaces were Yufeng Lu, Dachun Yang and Wen Yuan obtained in [37] and [15, 16], respectively. This section is devoted to the correspondingSobolev embedding theorems for Newton-Morrey-Sobolev spaces.Recall that a space X is said to support a weak (1 , p ) -Poincar´e inequality if there existpositive constants C and τ ≥ B in X and all pairs offunctions f and ρ defined on τ B , whenever ρ is an upper gradient of f in τ B and f isintegrable on B , then(3.3) 1 µ ( B ) Z B | f ( x ) − f B | dµ ( x ) ≤ C diam( B ) (cid:26) µ ( τ B ) Z τB [ ρ ( x )] p dµ ( x ) (cid:27) /p , where above and in what follows, f B denotes the integral mean of f on B , namely,(3.4) f B = 1 µ ( B ) Z B f ( y ) dµ ( y ) , diam( B ) the diameter of B and τ B the ball with the same center as B but τ times theradius of B . In particular, if τ = 1, then we say that X supports a (1 , p ) -Poincar´einequality .It is well known that the Euclidean space supports a (1 , p )-Poincar´e inequality. Formore information on Poincar´e inequalities, we refer the reader to [20, 21, 16] and theirreferences.A measure µ on X is said to be doubling if there exists a positive constant C such that,for all balls B in X , it holds true that µ (2 B ) ≤ Cµ ( B ). As a generalization of (3.2) toNewton-Morrey-Sobolev spaces, we have the following conclusion. Theorem 3.1.
Let ≤ p ≤ q < ∞ and Q ∈ (0 , q ) . Assume that ( X , d, µ ) is a metricmeasure space, with doubling measure µ , and supports a weak (1 , p ) -Poincar´e inequality.If there exists a positive constant C such that µ ( B ( x, r )) ≥ Cr Q for all x ∈ X and < r < X ) , then N M qp ( X ) ֒ → C , − Q/q ( X ) . Proof.
By the same reason as that stated in the proof of [37, Theorem 5.1], we only needto show that, if f ∈ N M qp ( X ) and x, y are Lebesgue points of f , then | f ( x ) − f ( y ) | . [ d ( x, y )] − Q/q k f k NM qp ( X ) . To this end, let B := B ( x, d ( x, y )) , B − := B ( y, d ( x, y )) and, for all i > B i = 12 B i − and B − i = 12 B − i +1 . Let B := B ( x, d ( x, y )). Since x, y are Lebesgue points, it follows that | f ( x ) − f ( y ) | ≤ X i ∈ Z | f B i − f B i +1 | . Let ρ be an upper gradient of f such that k f k M qp ( X ) + k ρ k M qp ( X ) . k f k NM qp ( X ) . orrey-Sobolev Spaces on Metric Measure Spaces r i be the radius of the ball B i . Then, by this, (1.3), the doubling condition of µ andthe weak (1 , p )-Poincar´e inequality, together with µ ( τ B i ) & r Qi , we see that, when i ∈ N , | f B i − f B i +1 | . µ ( B i ) Z B i | f B i − f ( x ) | dµ ( x )(3.5) . diam( B i ) (cid:26) µ ( τ B i ) Z τB i [ ρ ( x )] p dµ ( x ) (cid:27) /p . r i [ µ ( τ B i )] − /q k ρ k M qp ( X ) . r − Q/qi k ρ k M qp ( X ) . − i (1 − Q/q ) [ d ( x, y )] − Q/q k f k NM qp ( X ) . Similarly, for all i ≤ −
2, we also have | f B i − f B i +1 | . i (1 − Q/q ) [ d ( x, y )] − Q/q k f k NM qp ( X ) . On the other hand, by the H¨older inequality and the doubling condition of µ , we seethat | f B − − f B | ≤ µ ( B − ) Z B − | f B − f ( z ) | dµ ( z ) . µ ( B ) Z B | f B − f ( x ) | dµ ( x )and then, similar to (3.5), we further conclude that | f B − − f B | . [ d ( x, y )] − Q/q k f k NM qp ( X ) . Meanwhile, by the same method as above, we also find that | f B − f B | . [ d ( x, y )] − Q/q k f k NM qp ( X ) . Thus, combining the above estimates, by Q ∈ (0 , q ), we see that | f ( x ) − f ( y ) | . [ d ( x, y )] − Q/q "X i ∈ Z −| i | (1 − Q/q ) k f k NM qp ( X ) . [ d ( x, y )] − Q/q k f k NM qp ( X ) , which completes the proof of Theorem 3.1. Remark 3.2.
Theorem 3.1 generalizes [37, Theorem 5.1] by taking p = q .Next we give a Rellich-Kondrachov type embedding theorem for N M qp ( X ) when p issmall, which can be seen as a generalization of (3.1). We begin with the following notionof the Ahlfors Q -regular measure spaces; see, for example, [19]. Definition 3.3.
Let Q ∈ (0 , ∞ ). A metric measure space X is said to be Ahlfors Q -regular (or Q -regular ), if there exists a constant C ≥ x ∈ X and any r ∈ (0 , X )) , C r Q ≤ µ ( B ( x, r )) ≤ Cr Q . Yufeng Lu, Dachun Yang and Wen Yuan
Let L ( X ) be the collection of all locally integrable functions on X . The Hardy-Littlewood maximal operator M is defined by setting, for all f ∈ L ( X ) and x ∈ X ,(3.6) M f ( x ) := sup B ∋ x µ ( B ) Z B | f ( y ) | dµ ( y ) , where the supremum is taken over all balls B in X containing x . The following statementshows that the operator M is bounded on Morrey spaces. For its proof, we refer the readerto [5] for example. Lemma 3.4.
Let ( X , d, µ ) be a metric space with doubling measure µ and < p ≤ q ≤ ∞ .Then there exists a positive constant C such that, for all f ∈ M qp ( X ) , k M f k M qp ( X ) ≤ C k f k M qp ( X ) . We also need the following boundedness of fractional integral operators on Morreyspaces.
Proposition 3.5.
Let X be Ahlfors Q -regular with Q ∈ (0 , ∞ ) , < p ≤ q < ∞ and α > such that q < Q/α . Then, the fractional integral I α is bounded from M qp ( X ) to M q ∗ p ∗ ( X ) , where p ∗ := QpQ − qα , q ∗ := QqQ − qα and I α is defined by setting, for all f ∈ M qp ( X ) and x ∈ X , I α ( f )( x ) := Z X f ( y )[ d ( x, y )] Q − α dµ ( y ) . Proof.
Without loss of generality, we may assume that f ∈ M qp ( X ) is non-negative. Forany x ∈ X , fix δ > I α ( f )( x ) = Z B ( x,δ ) f ( y )[ d ( x, y )] Q − α dµ ( y ) + Z X \ B ( x,δ ) f ( y )[ d ( x, y )] Q − α dµ ( y )=: b ( α ) δ ( x ) + g ( α ) δ ( x ) . By the H¨older inequality, (1.3) and the Ahlfors Q -regular property of X , together with q < Q/α , we see that g ( α ) δ ( x ) = ∞ X j =0 Z B ( x, j +1 δ ) \ B ( x, j δ ) f ( y )[ d ( x, y )] Q − α dµ ( y )(3.7) . ∞ X j =0 (cid:0) j δ (cid:1) α − Q (cid:2) µ (cid:0) B (cid:0) x, j +1 δ (cid:1)(cid:1)(cid:3) − /p (Z B ( x, j +1 δ ) [ f ( y )] p dµ ( y ) ) /p . ∞ X j =0 (cid:0) j δ (cid:1) α − Q (cid:2) µ (cid:0) B (cid:0) x, j +1 δ (cid:1)(cid:1)(cid:3) − /q k f k M qp ( X ) ≈ ∞ X j =0 (cid:0) j δ (cid:1) α − Q (cid:0) j δ (cid:1) (1 − /q ) Q k f k M qp ( X ) . δ α − Q/q k f k M qp ( X ) . orrey-Sobolev Spaces on Metric Measure Spaces b δ , let A j := B j \ B j +1 := B ( x, − j δ ) \ B ( x, − j − δ ) for all j ∈ N ∪ { } =: Z + .Then, by the Ahlfors Q -regular property of X , together with α >
0, we see that, for all x ∈ X , b ( α ) δ ( x ) = X j ∈ Z + Z A j f ( y )[ d ( x, y )] Q − α dµ ( y ) ≈ X j ∈ Z + (cid:0) − j δ (cid:1) α − Q Z B j f ( y ) dµ ( y )(3.8) . δ α X j ∈ Z + − jα µ ( B j ) Z B j f ( y ) dµ ( y ) . δ α M ( f )( x ) . Combining (3.7) and (3.8), we have I α ( f )( x ) . δ α M ( f )( x ) + δ α − Q/q k f k M qp ( X ) . Now let δ := k f k q/Q M qp ( X ) [ M ( f )( x )] − q/Q . Then for any x ∈ X , I α ( f )( x ) . k f k αq/Q M qp ( X ) [ M ( f )( x )] − αq/Q , which, together with Lemma 3.4, further implies that k I α ( f ) k M q ∗ p ∗ ( X ) . k f k qα/Q M qp ( X ) (cid:13)(cid:13)(cid:13) [ M ( f )] − qα/Q (cid:13)(cid:13)(cid:13) M q ∗ p ∗ ( X ) ≈ k f k qα/Q M qp ( X ) k M ( f ) k − qα/Q M qp ( X ) . k f k M qp ( X ) . This finishes the proof of Proposition 3.5.Now we have the following Rellich-Kondrachov type embedding result, which generalizes[37, Theorem 5.2] by taking p = q , α = 1 and X being bounded. Theorem 3.6.
Let ≤ r < p ≤ q < ∞ . Let < p/r ≤ q/r < Q/α < ∞ , α ∈ (0 , r ) ∩ (0 , Q ) and X be an Ahlfors Q -regular metric measure space supporting a weak (1 , r ) -Poincar´e inequality. Then there exists a positive constant C such that, for all functions f ∈ N M qp ( X ) , upper gradients ρ of f and R ∈ (0 , ∞ ) , k f − f B ( · ,R ) k M Q ∗ qQ ∗ p ( X ) ≤ CR − α/r k ρ k M qp ( X ) , where Q ∗ := QrQr − qα .Proof. Let f ∈ N M qp ( X ) and ρ be an upper gradient of f . For any Lebesgue point x for f , we write B := B ( x, R ) and B i := B ( x, − i R ) for all i ∈ N . Since an Ahlfors Q -regularspace is doubling, by the weak (1 , r )-Poincar´e inequality (namely, the inequality (3.3) with p replaced by r ) and r > α , we see that | f ( x ) − f B ( x,R ) | ≤ ∞ X i =0 | f B i − f B i +1 | . ∞ X i =0 µ ( B i ) Z B i | f ( z ) − f B i | dµ ( z )2 Yufeng Lu, Dachun Yang and Wen Yuan . ∞ X i =0 diam( B i )[ µ ( τ B i )] /r (cid:26)Z τB i [ ρ ( z )] r dµ ( z ) (cid:27) /r ≈ ∞ X i =0 diam( B i )(2 − i τ R ) Q/r (cid:26)Z τB i [ ρ ( z )] r dµ ( z ) (cid:27) /r . ∞ X i =0 − i R (2 − i τ R ) α/r (cid:26)Z τB i [ ρ ( z )] r [ d ( x, z )] Q − α dµ ( z ) (cid:27) /r . R − α/r (cid:26)Z X [ ρ ( z )] r [ d ( x, z )] Q − α dµ ( z ) (cid:27) /r ≈ R − α/r [ I α ( ρ r )( x )] /r . Applying Proposition 3.5, together with 1 < p/r ≤ q/r < Q/α , we conclude that k f − f B ( · ,R ) k M Q ∗ qQ ∗ p ( X ) . R − α/r (cid:13)(cid:13)(cid:13) [ I α ( ρ r )] /r (cid:13)(cid:13)(cid:13) M Q ∗ qQ ∗ p ( X ) ≈ R − α/r k I α ( ρ r ) k /r M Q ∗ q/rQ ∗ p/r ( X ) . R − α/r k ρ r k /r M q/rp/r ( X ) ≈ R − α/r k ρ k M qp ( X ) , which completes the proof of Theorem 3.6. Remark 3.7. (i) Let 1 < r < p < ∞ , 1 < p/r < Q < ∞ , and X be an Ahlfors Q -regularmetric measure space supporting a weak (1 , r )-Poincar´e inequality. Then, by Theorem3.6 with α = 1, we see that there exists a positive constant C such that, for all functions f ∈ N ,p ( X ), upper gradients ρ of f and R ∈ (0 , ∞ ), k f − f B ( · ,R ) k L QprQr − p ( X ) ≤ CR − /r k ρ k L p ( X ) , which has its own interest. However, it is not clear whether the above conclusion stillholds true for the case r = 1 or not, since, we had to use Theorem 3.6 with α = 1 and, tothis end, we need r > α = 1.(ii) We also remark that Theorem 3.6 generalizes the classical result for Newton-Sobolevspaces in [37, Theorem 5.2]. Indeed, if we further assume that X is bounded, then weknow that f X = f B ( x, diam( X )) for almost all x ∈ X . Thus, it follows, from (i), that,under the same assumptions on Q, r, p as in (i), there exists a positive constant C suchthat, for all functions f ∈ N ,p ( X ) and upper gradients ρ of f , k f − f X k L QprQr − p ( X ) ≤ C [diam( X )] − /r k ρ k L p ( X ) , which is just [37, Theorem 5.2].(iii) The condition on the weak (1 , r )-Poincar´e inequality in Theorem 3.6 can be replacedby the weak (1 , In this section, we introduce Morrey-Sobolev spaces associated with Haj lasz gradientsand consider the relation between the Haj lasz-Morrey-Sobolev space and the Newton-Morrey-Sobolev space. orrey-Sobolev Spaces on Metric Measure Spaces Definition 4.1.
Let 0 < p ≤ q ≤ ∞ . The Haj lasz-Morrey-Sobolev space HM qp ( X )is defined to be the space of all measurable functions f that have a Haj lasz gradient h ∈ M qp ( X ). The norm of f ∈ HM qp ( X ) is defined as k f k HM qp ( X ) := k f k M qp ( X ) + inf k h k M qp ( X ) , where the infimum is taken over all Haj lasz gradients h of f .We remark that HM qp ( X ) when p = q is just the Haj lasz-Sobolev space M ,p ( X ) of[13]. Moreover, k f k HM qp ( X ) = 0 if and only if f = 0 almost everywhere.To consider the relation between the Haj lasz-Morrey-Sobolev space and the Newton-Morrey-Sobolev space, we need the following technical lemma, which is a special case of[29, Lemma 5.6]. Lemma 4.2.
Let ≤ p ≤ q < ∞ and g be a Mod qp -weak upper gradient of f . Then,for any ε ∈ (0 , ∞ ) , there exists a function g ε , which is an upper gradient of f , such that k g ε − g k M qp ( X ) ≤ ε and g ε ≥ g everywhere on X . Applying Lemma 4.2, we obtain the following conclusion.
Theorem 4.3.
Let < p ≤ q < ∞ . If X supports a weak (1 , p ) -Poincar´e inequality andthe measure µ is doubling, then N M qp ( X ) ֒ → HM qp ( X ) . Proof.
Let f ∈ N M qp ( X ). By [25, Theorem 1.0.1], we see that X supports a weak (1 , r )-Poincar´e inequality for some r ∈ (1 , p ). By Lemma 4.2, there exists an upper gradient g of u such that k f k M qp ( X ) + k g k M qp ( X ) . k f k NM qp ( X ) . Since X supports a weak (1 , r )-Poincar´e inequality for some r ∈ (1 , p ), by [16, Theorem3.2], we know that there exists a set E ⊂ X with µ ( E ) = 0 such that, for all x, y ∈ X \ E , | f ( x ) − f ( y ) | . d ( x, y ) n [ M ( g r )( x )] /r + [ M ( g r )( y )] /r o . Hence a positive constant multiple of h := [ M ( g r )] /r is a Haj lasz gradient of f . Then, byLemma 3.4, we know that k f k HM qp ( X ) . k f k M qp ( X ) + k h k M qp ( X ) . k f k M qp ( X ) + k g k M qp ( X ) . k f k NM qp ( X ) , which completes the proof of Theorem 4.3. Remark 4.4.
When p = q , under the same assumptions as in Theorem 4.3, it was provedby Shanmugalingam in [37, Theorem 4.9] that N M pp ( X ) = HM pp ( X ) with equivalentnorms.Next we turn to consider the inverse embedding of Theorem 4.3.4 Yufeng Lu, Dachun Yang and Wen Yuan
Theorem 4.5.
Let ≤ p ≤ q < ∞ . Then, HM qp ( X ) ֒ → N M qp ( X ) . Proof.
Let f ∈ HM qp ( X ). Then, there exists a Haj lasz gradient h ∈ M qp ( X ) of f suchthat | f ( x ) − f ( y ) | ≤ d ( x, y )[ h ( x ) + h ( y )] , x, y ∈ X \ E, for some E of measure 0, and k f k M qp ( X ) + k h k M qp ( X ) . k f k HM qp ( X ) . It was proved in [24,Theorem 1.1] that, if f, g ∈ L ( X ) and g is a Haj lasz gradient of f , then there exist e f and e g such that e f = f and e g = g almost everywhere, and 8 e g is an upper gradient of e f .Since e f = f in HM qp ( X ), we identify f and e f . In this sense, 8 e h is an upper gradient of f . Therefore, k f k NM qp ( X ) . k f k M pq ( X ) + (cid:13)(cid:13)(cid:13)e h (cid:13)(cid:13)(cid:13) M qp ( X ) ∼ k f k M qp ( X ) + k h k M qp ( X ) . k f k HM qp ( X ) , which completes the proof of Theorem 4.5.Combining Theorems 4.3 and 4.6, we have the following conclusion. Theorem 4.6.
Let < p ≤ q < ∞ . If X supports a weak (1 , p ) -Poincar´e inequality andthe measure µ is doubling, then N M qp ( X ) = HM qp ( X ) with equivalent norms. Next we consider the relations among the Haj lasz-Morrey-Sobolev space, the Newton-Morrey-Sobolev space and the classical Morrey-Sobolev space on R n . Let 1 ≤ p ≤ q < ∞ .Recall that the classical Morrey-Sobolev space
W M qp ( R n ) is defined by W M qp ( R n ) := (cid:8) f ∈ M qp ( R n ) : |∇ f | ∈ M qp ( R n ) (cid:9) , where ∇ f denotes the weak derivative of f . The norm of f ∈ W M qp ( R n ) is given by k f k W M qp ( R n ) := k f k M qp ( R n ) + k|∇ f |k M qp ( R n ) . Observe that
W M pp ( R n ) is just the Sobolev space W ,p ( R n ). Theorem 4.7.
Let < p ≤ q < ∞ . Then, W M qp ( R n ) = N M qp ( R n ) = HM qp ( R n ) with equivalent norms.Proof. Observe that the conclusion of Theorem 4.7 when 1 < p = q < ∞ is just [13,Theorem 1]. Thus, in what follows of this proof, we always assume that 1 < p < q < ∞ .By Theorem 4.6, it suffices to prove that W M qp ( R n ) ֒ → HM qp ( R n ) and N M qp ( R n ) ֒ → W M qp ( R n ) . orrey-Sobolev Spaces on Metric Measure Spaces W M qp ( R n ) ֒ → HM qp ( R n ). Let f ∈ W M qp ( R n ). By the definitionof W M qp ( R n ), we see that |∇ f | ∈ L p ( Q ) for all cubes Q in R n and then, following theargument as in [13, p. 404], we know that, for all Lebesgue points x, y ∈ R n of f , | f ( x ) − f ( y ) | . | x − y | [ M ( |∇ f | )( x ) + M ( |∇ f | )( y )] . Hence a positive constant multiple of M ( |∇ f | ) is a Haj lasz gradient of f . Moreover, byDefinition 4.1 and Lemma 3.4, we further see that k f k HM qp ( R n ) ≤ k f k M qp ( R n ) + k M ( |∇ f | ) k M qp ( R n ) . k f k M qp ( R n ) + k|∇ f |k M qp ( R n ) ≈ k f k W M qp ( R n ) . This shows that
W M qp ( R n ) ⊂ HM qp ( R n ).Next we prove N M qp ( R n ) ⊂ W M qp ( R n ). Let f ∈ N M qp ( R n ). Then, by Definition 1.3,there exists g ∈ M qp ( R n ) such that g is a weak upper gradient of f . Moreover, for anyball B ⊂ R n , we have g ∈ L p ( B ), which implies that f ∈ N ,p ( B ) . By [6, Theorem A.2],we know that, for i ∈ { , . . . , n } and almost every x ∈ B , ∂f∂x i ( x ) exists and | ∂f∂x i ( x ) | iscontrolled by g ( x ). Since B ⊂ R n is arbitrary, it follows that, for almost every x ∈ R n , (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂x i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ g ( x ) , which, together with g ∈ M qp ( R n ), implies that ∂f∂x i ∈ M qp ( R n ). Furthermore, we have |∇ f | ∈ M qp ( R n ), from which, together with f ∈ M qp ( R n ), we deduce that f ∈ W M qp ( R n ).This finishes the proof of Theorem 4.7.We remark that Theorem 4.7 when p = q goes back to the equivalence between Sobolevspaces and Haj lasz-Sobolev spaces on R n obtained in [13]. Remark 4.8. (i) We remark that, for all < p < q < ∞ , the set C ( R n ) of functionshaving continuous derivatives up to order is not dense in N M qp ( R n ). To see this, byTheorem 4.7, we only need to consider W M qp ( R n ). For simplicity, we only consider thecase n = 1. Let φ ∈ C ∞ c ( R ) such that 0 ≤ φ ≤ φ ≡ − , φ ≡ − , c .Write g ( x ) := | x | − /q φ ( x ) for all x ∈ R . Then, the function g ′ ( x ) := (1 − /q ) x − /q φ ( x ) + x − /q φ ′ ( x ) , x ∈ (0 , ∞ );0 , x = 0; − (1 − /q )( − x ) − /q φ ( x ) + ( − x ) − /q φ ′ ( x ) , x ∈ ( −∞ , g . Since it is known that | x | α χ ( − , ( x ) ∈ M qp ( R ) if and only if α ≥ − /q , we then see that g ∈ W M qp ( R ).Now we apply an approach from [44, pp. 587-588] to show that g can not be approx-imated by C ( R ) functions in W M qp ( R ). Indeed, it suffices to prove that g ′ can not beapproximated by continuous functions in M qp ( R ). To see this, for any continuous function6 Yufeng Lu, Dachun Yang and Wen Yuan h , write N := sup x ∈ ( − , | h ( x ) | p < ∞ . Notice that φ ≡ − , R ∈ (0 , Z R − R (cid:12)(cid:12) g ′ ( x ) − h ( x ) (cid:12)(cid:12) p dx ≥ − p Z R − R (cid:12)(cid:12) g ′ ( x ) (cid:12)(cid:12) p dx − N R.
Notice that Z R − R (cid:12)(cid:12) g ′ ( x ) (cid:12)(cid:12) p dx ≥ (1 − /q ) p Z R x − p/q dx = (1 − /q ) p − p/q R − p/q . We know that Z R − R (cid:12)(cid:12) g ′ ( x ) − h ( x ) (cid:12)(cid:12) p dx ≥ − p (1 − /q ) p − p/q R − p/q − N R = R − p/q (cid:20) − p (1 − /q ) p − p/q − N R p/q (cid:21) . Hence, taking R small enough such that2 − p (1 − /q ) p − p/q − N R p/q ≥ − p − (1 − /q ) p − p/q , we then see that (cid:13)(cid:13) g ′ − h (cid:13)(cid:13) p M qp ( R ) & R p/q − Z R − R (cid:12)(cid:12) g ′ ( x ) − h ( x ) (cid:12)(cid:12) p dx & − p − (1 − /q ) p − p/q > . This implies the above claim.(ii) We point out that the key property we used in (i) is the locally boundedness ofcontinuous functions, which ensures that the number N is finite. If we replace continuousfunctions h by any locally bounded functions, then the subsequent argument remains true.From this observation, together with the well-known fact that any Lipschitz function f on R n is differentiable almost everywhere and the absolute value | ∂ i f | of its weak derivative ∂ i f is dominated by its Lipschitz constant L f almost everywhere, we deduce that g cannot be approximated by any Lipschitz function f in the norm of W M qp ( R n ). Therefore,the set of Lipschitz functions is not dense in N M qp ( R n ) when < p < q < ∞ . This section is devoted to the boundedness of (fractional) maximal operators on Morreytype spaces over metric measure spaces.In Subsection 5.1, for a geometrically doubling metric measure space ( X , d, µ ) in thesense of Hyt¨onen [23], we show, in Theorem 5.8 below, that the modified maximal operator orrey-Sobolev Spaces on Metric Measure Spaces M ( β )0 (see (5.1) below) is bounded on the modified Morrey space M q, ( k ) p ( X ), which, when( X , d, µ ) := ( R n , | · | , µ ) with µ being a Radon measure satisfying the polynomial growthcondition (also called the non-doubling measure), was introduced by Sawano and Tanaka[36]. As an application, the boundedness of the fractional maximal operator M ( β ) α on thisspace is also obtained in Proposition 5.10 below.In Subsection 5.2, if µ is a doubling measure, as applications of Theorem 5.8 andProposition 5.10, we show the boundedness of the fractional maximal operator M α onMorrey spaces (see Corollary 5.11 below), from which, we further deduce, in Corollary5.12 below, the boundedness of the fractional maximal operator f M α on Morrey spaceswhen µ further satisfies the measure lower bound condition (see (5.9) below). If µ isdoubling, satisfies (5.9) and has the relative 1-annular decay property (see (5.12) below),we then obtain the boundedness of f M α on HM qp ( X ) (see Theorem 5.13 below). Finally,we prove that, if µ is doubling and satisfies (5.9), and X supports a weak (1 , p )-Poincar´einequality, then the discrete fractional maximal function M ∗ α is bounded on N M qp ( X ) (seeTheorem 5.14 below). M q, ( k ) p ( X ) In 2010, Hyt¨onen [23] introduced the notion of geometrically doubling metric measurespaces which include both spaces of homogeneous type and the Euclidean spaces withnon-doubling measures satisfying the polynomial growth condition as special cases; seealso the monograph [42] for some recent developments of this subject.Now we recall the following notion of the geometrically doubling from [23], which isalso known as metrically doubling (see, for example, [19, p. 81]).
Definition 5.1.
A metric space ( X , d ) is said to be geometrically doubling , if there exists N ∈ N such that any given ball contains no more than N points at distance exceedinghalf its radius.From the geometrically doubling property, we deduce the following conclusion, whichis used later on. Proposition 5.2.
Let ( X , d ) be a geometrically doubling metric space. Then, for any ball B ( x, r ) ⊂ X , with x ∈ X and r ∈ (0 , ∞ ) , and any n ≥ n > , there exist r ∈ (0 , ∞ ) and e N balls { B ( x i , r ) } e Ni =1 such that n B ( x i , r ) ⊂ n B ( x, r ) for all i ∈ { , . . . , e N } and B ( x, r ) ⊂ e N [ i =1 B ( x i , r ) , where e N ∈ N depends only on n , n and the constant N in Definition 5.1.Proof. Let n and n be as in Proposition 5.2, and k := (cid:22) log n + 1 n − (cid:23) + 1 , Yufeng Lu, Dachun Yang and Wen Yuan where ⌊ t ⌋ denotes the maximal integer not more than t ∈ R . We claim that, for any y ∈ X and ball B ( x, r ) ⊂ X with x ∈ X and r ∈ (0 , ∞ ), if B ( y, r k ) ∩ B ( x, r ) = ∅ , then n B ( y, r k ) ⊂ n B ( x, r ). Indeed, by choosing z ∈ B ( y, r k ) ∩ B ( x, r ) and observing that k > log n +1 n − , we have d ( x, y ) ≤ d ( x, z ) + d ( z, y ) < (cid:18) k (cid:19) r < (cid:16) n − n k (cid:17) r. Thus, for all w ∈ n B ( y, r k ), d ( w, x ) ≤ d ( w, y ) + d ( y, x ) < n r k + (cid:16) n − n k (cid:17) r = n r, which shows the above claim. Then, by repeating the proof that (1) implies (2) in [23,Lemma 2.3], we obtain the desired conclusion, which completes the proof of Proposition5.2.Now we recall the definition of the modified Morrey space, which, when ( X , d, µ ) :=( R n , | · | , µ ) with µ being a Radon measure satisfying the polynomial growth condition,was originally introduced by Sawano and Tanaka [36]. Definition 5.3.
Let k ∈ (0 , ∞ ), 1 ≤ p ≤ q < ∞ and X be a metric measure space. The modified Morrey space M q, ( k ) p ( X ) is defined as M q, ( k ) p ( X ) := n f ∈ L p loc ( X ) : k f k M q, ( k ) p ( X ) < ∞ o , where k f k M q, ( k ) p ( X ) := sup B ( x,r ) ⊂ X [ µ ( B ( x, kr ))] /q − /p "Z B ( x,r ) | f ( y ) | p dµ ( y ) /p , where the supremum is taken over all balls B ( x, r ), with x ∈ X and r ∈ (0 , ∞ ), of X .Recall that a geometrically doubling metric measure space ( X , d, µ ) means that ( X , d )is geometrically doubling and µ is a non-negative Radon measure on ( X , d ). Proposition 5.4.
Let ( X , d, µ ) be a geometrically doubling metric measure space and ≤ p ≤ q < ∞ . Then, the space M q, ( k ) p ( X ) is independent of the choice of k ∈ (1 , ∞ ) .Proof. Let k , k ∈ (1 , ∞ ). We need to show that M q, ( k ) p ( X ) and M q, ( k ) p ( X ) coincidewith equivalent norms. To this end, without loss of generality, we may assume that k < k . By Definition 5.3, we easily find that M q, ( k ) p ( X ) ⊂ M q, ( k ) p ( X ) . Thus, we stillneed to show the inverse embedding. Let B be a ball in X . By Proposition 5.2, thereexist e N balls { B i } e Ni =1 with the same radius such that, for all i ∈ { , . . . , e N } , k B i ⊂ k B and B ⊂ ∪ e Ni =1 B i , where e N depends only on k , k and N in Definition 5.1. By these, wesee that[ µ ( k B )] /q − /p (cid:20)Z B | f ( x ) | p dµ ( x ) (cid:21) /p ≤ e N X i =1 [ µ ( k B )] /q − /p (cid:20)Z B i | f ( x ) | p dµ ( x ) (cid:21) /p orrey-Sobolev Spaces on Metric Measure Spaces ≤ e N X i =1 [ µ ( k B i )] /q − /p (cid:20)Z B i | f ( x ) | p dµ ( x ) (cid:21) /p ≤ e N k f k M q, ( k p ( X ) . By the arbitrariness of B and Definition 5.3, we conclude that k f k M q, ( k p ( X ) ≤ e N k f k M q, ( k p ( X ) , which further implies that M q, ( k ) p ( X ) ⊂ M q, ( k ) p ( X ) and hence completes the proof ofProposition 5.4.Recall that, for α ∈ [0 ,
1] and β ∈ [1 , ∞ ), the modified fractional maximal operator M ( β ) α is defined by setting, for all f ∈ L ( X ) and x ∈ X ,(5.1) M ( β ) α f ( x ) := sup r> [ µ ( B ( x, βr ))] α − Z B ( x,r ) | f ( y ) | dµ ( y ) . In particular, we write M α := M (1) α .The maximal operator M ( β )0 where β ∈ (1 , ∞ ) is bounded on the modified Morreyspaces. To prove this, we need the following technical lemma. Lemma 5.5.
Let β ∈ (1 , ∞ ) and ( X , d ) be a geometrically doubling metric space. Supposethat B := { B ( x λ , r λ ) } λ ∈ Λ such that sup λ ∈ Λ r λ < ∞ . Then, there exist J β ∈ N , dependingonly on β and N in Definition 5.1, and sub-families of balls of B , B i := { B ( x λ , r λ ) } λ ∈ Λ i with i ∈ { , . . . , J β } , such that (i) for each i ∈ { , . . . , J β } , B i consists of disjoint balls; (ii) for any λ ∈ Λ , there exists λ ′ ∈ ∪ J β i =1 Λ i such that B ( x λ , r λ ) ⊂ B ( x λ ′ , βr λ ′ ) . Proof.
Let R := sup λ ∈ Λ r λ and, for all j ∈ Z + ,(5.2) A j := n B ( x λ , r λ ) : λ ∈ Λ , R p β − j − < r λ ≤ R p β − j o . Here, we need β ∈ (1 , ∞ ) and, otherwise, A j = ∅ for all j ∈ Z + . Let D ⊂ A be amaximal subset in A such that, for any two distinct balls B ( x λ , r λ ) and B ( x λ ′ , r λ ′ ) in D , it holds true that d ( x λ , x λ ′ ) > R ( √ β − B ( x λ , r λ ) ∈ A , there exists B ( x λ ′ , r λ ′ ) ∈ D such that d ( x λ , x λ ′ ) ≤ R ( √ β − r λ ≤ R and βr λ ′ > R √ β , it follows that B ( x λ , r λ ) ⊂ B ( x λ ′ , βr λ ′ ) . Let E be the collection of balls B ( x λ , r λ ) which belong to B and satisfy that, for some B ( x λ ′ , r λ ′ ) ∈ D , B ( x λ , r λ ) ⊂ B ( x λ ′ , βr λ ′ ). Obviously, A ⊂ E . Yufeng Lu, Dachun Yang and Wen Yuan
Let m ≥
1. We now define D m and E m recursively. Suppose that D j and E j for j ∈ { , . . . , m − } has already been defined. Let D m ⊂ A m \ ∪ m − j =0 E j be a maximal subsetsatisfying that, for all distinct balls B ( x λ , r λ ) and B ( x λ ′ , r λ ′ ) in D m , it holds true that d ( x λ , x λ ′ ) > R p β − j ( p β − . Let E m be the collection of balls B ( x λ , r λ ) which belong to B and satisfy that, for some B ( x λ ′ , r λ ′ ) ∈ D m , B ( x λ , r λ ) ⊂ B ( x λ ′ , βr λ ′ ). Notice that, for any ball B ∈ A m , we haveeither B ∈ E j for some j ∈ { , . . . , m − } or B ∈ A m \ ∪ m − j =1 E j . In the first case, we canfind a ball B ( x λ ′ , r λ ′ ) ∈ D j such that B ⊂ B ( x λ ′ , βr λ ′ ). In the second case, we can find B ( x λ ′ , r λ ′ ) ∈ D m such that B ⊂ B ( x λ ′ , βr λ ′ ).Due to the geometrically doubling condition, we can partition each D j into disjointsub-families, D j, , . . . , D j,L β , where L β is a positive constant depending only on β and thegeometrically doubling constant N , since, for any j ∈ Z + and any B := B ( x λ , r λ ) ∈ D j ,there are at most L β balls in D j intersect B . Indeed, let F j be the collection of balls B ′ := B ( x λ ′ , r λ ′ ) which belong to D j and intersect B . Let y ∈ B ′ and z ∈ B ∩ B ′ . Then, d ( y, x λ ) ≤ d ( y, x λ ′ ) + d ( x λ ′ , z ) + d ( z, x λ ) < r λ ′ + r λ ≤ R p β − j . Thus, B ′ ⊂ B ( x λ , R p β − j ) . On the other hand, by the choice of D j , we know that d ( x λ , x λ ′ ) > R p β − j ( p β − B (cid:18) x λ , √ β − r λ (cid:19) \ B (cid:18) x λ ′ , √ β − r λ ′ (cid:19) = ∅ . By the geometrically doubling property of X and [23, Lemma 2.3], we see that thereexists a constant L β , depending on N and β , such that F j has no more than L β balls.Let N β ∈ N satisfy(5.3) 1 + 2 q β − N β < p β. We claim that, if j ≥ j + N β , then, for any pair of balls, ( B , B ) ∈ D j × D j , B and B do not intersect. To see this, assume that B ∩ B = ∅ and x ∈ B ∩ B . Then, by(5.3), for any y ∈ B , we have d ( y, x ) ≤ d ( y, x ) + d ( x, x ) < r ( B ) + r ( B ) ≤ R p β − j + R p β − j ≤ R p β − j (2 q β − N β + 1) ≤ R p β − j − β < βr ( B ) , where x i and r ( B i ) denote the center and the radius of B i , for i ∈ { , } , respectively.Thus, B ⊂ βB and hence belongs to E j , which contradicts to the definition of D j , since D j ∩ E j = ∅ . Thus, the above claim holds true.Therefore, if, for i ∈ { , . . . , N β } and n ∈ { , . . . , L β } , let B i,n := ∞ [ j =0 D N β j + i,n . orrey-Sobolev Spaces on Metric Measure Spaces { B i,n : i ∈ { , . . . , N β } , n ∈ { , . . . , L β }} are the desired families, which completesthe proof of Lemma 5.5.The boundedness of the modified maximal operator on L p ( X ) could be deduced fromthe above lemma by borrowing some ideas used in the proof of [39, Section 3.1, Theorem1]. We give some details as follows. Theorem 5.6.
Let β ∈ (1 , ∞ ) and ( X , d, µ ) be a geometrically doubling metric measurespace. (i) Then, there exists a positive constant C such that, for all λ ∈ (0 , ∞ ) and f ∈ L ( X ) , (5.4) µ (cid:16)n x ∈ X : M ( β )0 ( x ) > λ o(cid:17) ≤ Cλ k f k L ( X ) . (ii) Let p ∈ (1 , ∞ ] . Then, there exists a positive constant C such that, for all f ∈ L p ( X ) , (5.5) (cid:13)(cid:13)(cid:13) M ( β )0 f (cid:13)(cid:13)(cid:13) L p ( X ) ≤ C k f k L p ( X ) . Proof.
The boundedness of M ( β )0 on L ∞ ( X ) is obvious. Next we only prove (i), since (ii)can be deduced from (i) and the L ∞ ( X )-boundedness of M β via interpolation.Let E λ := n x ∈ X : M ( β )0 f ( x ) > λ o . Then, by the definition of E λ , for any x ∈ E , there exists a ball B x such that(5.6) 1 µ ( βB x ) Z B x | f ( y ) | dµ ( y ) > λ. For all k ∈ Z + , let B ( k ) be the collection of all balls B x for x ∈ E , whose radius r ( B x ) ∈ (0 , k ], and E ( k ) λ := { x ∈ E λ : B x ∈ B ( k ) } . Then, E λ = ∪ k ∈ Z + E ( k ) λ and E ( k ) λ ⊂ E ( k +1) λ forany k ∈ Z + .For each k ∈ Z + , by Lemma 5.5, we can find J β ∈ N , independent of k , and sub-families B ( k ) i ⊂ B ( k ) , i ∈ { , . . . , J β } such that [ B ∈B ( k ) B ⊂ J β [ i =1 [ B ∈B ( k ) i βB, where βB denotes the ball with the same center as B but β times the radius of B . Thus,by the fact that E ( k ) λ increasingly converges to E λ as k → ∞ , and the disjointness of ballsin B ( k ) i over i , we see that µ ( E λ ) = lim k →∞ µ (cid:16) E ( k ) λ (cid:17) ≤ lim k →∞ µ [ B ∈B ( k ) B ≤ lim k →∞ µ L β [ i =1 [ B ∈B ( k ) i βB Yufeng Lu, Dachun Yang and Wen Yuan ≤ lim k →∞ L β X i =1 X B ∈B ( k ) i µ ( βB ) ≤ lim k →∞ L β X i =1 X B ∈B ( k ) i λ Z B | f ( y ) | dµ ( y ) . L β λ k f k L ( X ) . This finishes the proof of Theorem 5.6.
Remark 5.7. (i) It is worth pointing out that Theorem 5.6 also holds true for the non-centered maximal operator, whose proof is similar, the details being omitted.(ii) We should point out that Lemma 5.5 and Theorem 5.6 are generously provided tous by Professor
Yoshihiro Sawano from Tokyo Metropolitan University of Japan.(iii) Lemma 5.5 and Theorem 5.6 in the case β = 1 are still unknown.Then we have the following conclusion, which generalizes [36, Theorem 2.3], whereinthe corresponding result on the non-doubling measure satisfying the polynomial growthcondition on R n was obtained. The proof of Theorem 5.8 is similar to that of [36, Theorem2.3], and one key tool used in the proof is the L p ( µ )-boundedness in Theorem 5.6. For thesake of convenience, we give the details. Theorem 5.8.
Let X be a geometrically doubling metric measure space, < p ≤ q < ∞ , β ∈ (1 , ∞ ) and k ∈ (1 , ∞ ) . Then, there exists a positive constant C such that, for all f ∈ M q, ( k ) p ( X ) , (5.7) (cid:13)(cid:13)(cid:13) M ( β )0 f (cid:13)(cid:13)(cid:13) M q, ( k ) p ( X ) ≤ C k f k M q, ( k ) p ( X ) . Proof.
By Proposition 5.4, it suffices to consider the case that k := ββ +1 >
1. Let f ∈M q, ( k ) p ( X ) and B ⊂ X be a ball. Define e β := β +7 β − > f := f χ e βB and f := f − f .Then, by Definition 5.3, together with 1 < p ≤ q < ∞ , we have1[ µ ( k e βB )] /p − /q (cid:26)Z B h M ( β )0 f ( y ) i p dµ ( y ) (cid:27) /p ≤ µ ( k e βB )] /p − /q (cid:26)Z X h M ( β )0 f ( y ) i p dµ ( y ) (cid:27) /p . µ ( k e βB )] /p − /q (cid:26)Z e βB | f ( y ) | p dµ ( y ) (cid:27) /p . k f k M q, ( k ) p ( X ) , where we used the fact that M ( β )0 is bounded on L p ( µ ), for p ∈ (1 , ∞ ) (see Theorem 5.6).To estimate f , observe that, if B ⊂ X is a ball satisfying that B ∩ B = ∅ and B ∩ ( X \ e βB ) = ∅ , then the radius r B > e β − r B = β − r B , where r B and r B denote,respectively, the radii of B and B , and hence B ⊂ β +12 B . Therefore, we see that, forany x ∈ B , M ( β )0 f ( x ) ≤ sup x ∈ B µ ( βB ) Z B | f ( y ) | dµ ( y ) ≤ sup B ⊂ β +12 B µ ( βB ) Z B | f ( y ) | dµ ( y ) orrey-Sobolev Spaces on Metric Measure Spaces ≤ sup B ⊂ B µ ( ββ +1 B ) Z B | f ( y ) | dµ ( y ) = sup B ⊂ B µ ( kB ) Z B | f ( y ) | dµ ( y ) , and hence, by this, the H¨older inequality and Definition 5.3, we further see that1[ µ ( k e βB )] /p − /q (cid:20)Z B h M ( β )0 f ( y ) i p dµ ( y ) (cid:21) /p ≤ [ µ ( B )] /p [ µ ( k e βB )] /p − /q sup B ⊂ B µ ( kB ) Z B | f ( y ) | dµ ( y ) ≤ [ µ ( B )] /p [ µ ( k e βB )] /p − /q sup B ⊂ B [ µ ( B )] − /p µ ( kB ) (cid:20)Z B | f ( y ) | p dµ ( y ) (cid:21) /p ≤ sup B ⊂ B [ µ ( B )] /p [ µ ( k e βB )] /p − /q [ µ ( B )] − /p µ ( kB ) [ µ ( kB )] /p − /q k f k M q, ( k ) p ( X ) ≤ sup B ⊂ B [ µ ( B )] − /p +1 /q [ µ ( kB )] − /p +1 /q k f k M q, ( k ) p ( X ) ≤ k f k M q, ( k ) p ( X ) , where the last inequality follows from the fact that B ⊂ B and k, e β >
1. This estimatefor f , together with the previous estimate for f and Proposition 5.4, further implies that (cid:13)(cid:13)(cid:13) M ( β )0 f (cid:13)(cid:13)(cid:13) M q, ( k ) p ( X ) ≈ (cid:13)(cid:13)(cid:13) M ( β )0 f (cid:13)(cid:13)(cid:13) M q, ( k e β ) p ( X ) . k f k M q, ( k ) p ( X ) , which completes the proof of Theorem 5.8. Remark 5.9.
It is still unknown whether the conclusions of Theorems 5.6 and 5.8 holdtrue or not when β = 1. Indeed, it is known that M (1)0 might not be bounded on L p ( X )with p ∈ (1 , ∞ ) when µ is not doubling; see, for example, [35].Using Theorem 5.8, we further have the following boundedness of the modified fractionalmaximal operator M ( β ) α on the modified Morrey space. Proposition 5.10.
Let X be a geometrically doubling metric measure space, < p ≤ q < ∞ , β ∈ (1 , ∞ ) , α ∈ (0 , /q ) and k ∈ (1 , ∞ ) . Then, there exists a positive constant C such that, for all f ∈ M q, ( k ) p ( X ) , (cid:13)(cid:13)(cid:13) M ( β ) α f (cid:13)(cid:13)(cid:13) M e q, ( k ) e p ( X ) ≤ C k f k M q, ( k ) p ( X ) , where e p := p − αq and e q := q − αq .Proof. By Proposition 5.4, it suffices to consider the case that k = β ∈ (1 , ∞ ).For any ball B ( x, r ) ⊂ X , with x ∈ X and r >
0, and f ∈ M q, ( β ) p ( X ), we know, bythe H¨older inequality, (1.3) and (3.6), that[ µ ( B ( x, βr ))] α − Z B ( x,r ) | f ( y ) | dµ ( y )4 Yufeng Lu, Dachun Yang and Wen Yuan = [ µ ( B ( x, βr ))] α − "Z B ( x,r ) | f ( y ) | dµ ( y ) αq "Z B ( x,r ) | f ( y ) | dµ ( y ) − αq ≤ [ µ ( B ( x, βr ))] α − − p ) αq "Z B ( x,r ) | f ( y ) | p dµ ( y ) αqp "Z B ( x,r ) | f ( y ) | dµ ( y ) − αq = ( µ ( B ( x, βr ))] − pq Z B ( x,r ) | f ( y ) | p dµ ( y ) ) αqp " µ ( B ( x, βr )) Z B ( x,r ) | f ( y ) | dµ ( y ) − αq ≤ k f k αq M q, ( β ) p ( X ) h M ( β )0 f ( x ) i − αq , which, together with (5.1), implies that, for all x ∈ X , M ( β ) α f ( x ) ≤ k f k αq M q, ( β ) p ( X ) h M ( β )0 f ( x ) i − αq . Then, by Theorem 5.8, we see that (cid:13)(cid:13)(cid:13) M ( β ) α f (cid:13)(cid:13)(cid:13) M e q, ( β ) e p ( X ) ≤ k f k αq M q, ( β ) p ( X ) (cid:13)(cid:13)(cid:13)(cid:13)h M ( β )0 f i − αq (cid:13)(cid:13)(cid:13)(cid:13) M e q, ( β ) e p ( X ) . k f k M q, ( β ) p ( X ) , which completes the proof of Proposition 5.10. HM qp ( X ) and N M qp ( X ) Recently, Heikkinen et al. [17, 18] studied the boundedness of some (fractional) maximaloperators on the Newton-Sobolev space and the Haj lasz-Sobolev space over metric measurespaces. In this section, we consider the corresponding problem for Newton-Morrey-Sobolevspaces and Haj lasz-Morrey-Sobolev spaces.Throughout this section, we always assume that the measure µ is doubling . We call ameasure is doubling if there exists a constant C ∈ [1 , ∞ ) such that, for all x ∈ X and r > µ ( B ( x, r )) ≤ C µ ( B ( x, r )) (doubling property) . Recall that it is well known that any space of homogeneous type is also geometricallydoubling (see [8, pp. 66-67]). Moreover, since µ is doubling, we see that, for any β ∈ (1 , ∞ )and α ∈ [0 , C , depending on β and α , such that, forall f ∈ L ( X ) and x ∈ X , M α f ( x ) ≤ CM ( β ) α f ( x ). From these facts, Theorem 5.8 andProposition 5.10, we immediately deduce the following conclusion. Corollary 5.11.
Let < p ≤ q < ∞ and α ∈ [0 , /q ) . Then, there exists a positiveconstant C , depending on α , p and q , such that, for all f ∈ M qp ( X ) , k M α f k M e q e p ( X ) ≤ C k f k M qp ( X ) , where e p := p − αq and e q := q − αq . orrey-Sobolev Spaces on Metric Measure Spaces f M α on modified Morrey spaces.Recall that a measure µ is said to satisfy the measure lower bound condition , if thereexists a positive constant C such that, for any x ∈ X and r ∈ (0 , ∞ ),(5.9) µ ( B ( x, r )) ≥ Cr Q for some Q ∈ (0 , ∞ ).Recently, if µ satisfies (5.9), Heikkinen et al. [18] established the boundedness from L p ( X ) to L s ( X ) for p ∈ (1 , Q ) and s := QpQ − αp of the following modified (fractional)maximal function f M α , defined by setting, for any α ∈ [0 , f ∈ L ( X ) and x ∈ X ,(5.10) f M α f ( x ) := sup r> r α µ ( B ( x, r )) Z B ( x,r ) | f ( y ) | dµ ( y ) . It is easy to see that, in the present setting, there exists a positive constant C , dependingon α and Q , such that, for all f ∈ L ( X ) and x ∈ X , f M α f ( x ) ≤ CM α f ( x ), which,together with Corollary 5.11, implies the following conclusion. Corollary 5.12.
Let < p ≤ q < ∞ and α ∈ [0 , /q ) . Assume that µ satisfies (5.9) .Then, there exists a positive constant C , depending on α , p and q , such that, for all f ∈ M qp ( X ) , (5.11) (cid:13)(cid:13)(cid:13) f M α f (cid:13)(cid:13)(cid:13) M e q e p ( X ) ≤ C k f k M qp ( X ) , where e p := p − αq and e q := q − αq . Recall that X is said to satisfy the relative -annular decay property , if there exists apositive constant C such that, for all x ∈ X , R ∈ (0 , ∞ ) and h ∈ (0 , R ),(5.12) µ ( B ∩ [ B ( x, R ) \ B ( x, R − h )]) ≤ C hr B µ ( B )for all balls B with radius r B < R ; see, for example, [18, (2.5)].Now we turn to the boundedness of the (fractional) maximal operator f M α on Haj lasz-Morrey-Sobolev spaces. Theorem 5.13.
Assume that µ satisfies (5.9) and X has the relative 1-annular decayproperty (5.12) . Let < p ≤ q < ∞ and α ∈ [0 , Q/q ) . Then, for any f ∈ HM qp ( X ) , f M α f ∈ HM q ∗ p ∗ ( X ) , where p ∗ := QpQ − αq and q ∗ := QqQ − αq . Moreover, there exists a positiveconstant C , depending only on the doubling constant, Q , p , q and α , such that, for all f ∈ HM qp ( X ) , (cid:13)(cid:13)(cid:13) f M α f (cid:13)(cid:13)(cid:13) HM q ∗ p ∗ ( X ) ≤ C k f k HM qp ( X ) . Yufeng Lu, Dachun Yang and Wen Yuan
Proof.
The proof is similar to that of [18, Theorem 4.5]. We present some details. Let f ∈ HM qp ( X ) and g ∈ M qp ( X ) be a Haj lasz gradient of f such that k g k M qp ( X ) . k f k HM qp ( X ) .It is easy to see that g is also a Haj lasz gradient of | f | . Let r ∈ (1 , p ) and define e g := h f M αr ( g r ) i /r . By an argument similar to that used in the proof of [18, Theorem 4.5], we know that e g is a Haj lasz gradient of f M α ( | f | ), as well as f M α f , since f M α ( | f | ) = f M α f . Moreover, by p/r >
1, (1.3) and (5.11), we see that(5.13) k e g k M q ∗ p ∗ ( X ) = (cid:13)(cid:13)(cid:13) f M αr ( g r ) (cid:13)(cid:13)(cid:13) /r M q ∗ /rp ∗ /r ( X ) . k g r k /r M q/rp/r ( X ) ≈ k g k M qp ( X ) . Combining (5.13) and Definition 4.1, we obtain the desired conclusion and then completethe proof of Theorem 5.13.We point out that Theorem 5.13 when p = q goes back to [18, Theorem 4.5].Now we recall the discrete (fractional) maximal operator M ∗ α introduced in [17, Section5]. Let { B ( x i , r ) } i ∈ N be a ball covering of X such that { B ( x i , r ) } i ∈ N are of finite overlap.Since X is doubling, the overlap number N depends only on the doubling constant andis independent of r . Let { ϕ i } i ∈ N be a partition of unity related to { B ( x i , r ) } i ∈ N such that0 ≤ ϕ i ≤ ϕ i = 0 on X \ B ( x i , r ), ϕ i ≥ v on B ( x i , r ) and ϕ i is Lipschitz function withLipschitz constant L/r , where L ∈ (0 , ∞ ) and v ∈ (0 ,
1] are constants depending only onthe doubling constant, and P i ∈ N ϕ i ≡
1. The discrete convolution of u ∈ L ( X ) at thescale 3 r is defined by setting, for all x ∈ X , u r ( x ) := X i ∈ N ϕ i ( x ) u B ( x i , r ) , where u B ( x i , r ) denotes the integral mean of u on B ( x i , r ) (see (3.4)). Now, let { r j } j ∈ N be a sequence of the positive rational numbers, and { B ( x i,j , r j ) } i ∈ N for each j is a ballcovering of X as above. Then, the discrete (fractional) maximal function M ∗ α u of u isdefined by setting, for all x ∈ X , M ∗ α u ( x ) := sup j ∈ N r αj | u | r j ( x ) . Similar to the proof of [17, Theorem 6.3], we obtain the following result on the boundednessof M ∗ α on Newton-Morrey-Sobolev spaces. Theorem 5.14.
Let µ satisfy (5.9) , < p ≤ q < ∞ and α ∈ [0 , Q/q ) . Assume that X is complete and supports a weak (1 , p ) -Poincar´e inequality. Then, for any f ∈ N M qp ( X ) ,it holds true that M ∗ α f ∈ N M q ∗ p ∗ ( X ) with p ∗ := Qp/ ( Q − αq ) and q ∗ := Qq/ ( Q − αq ) .Moreover, there exists a positive constant C , independent of f , such that k M ∗ α f k NM q ∗ p ∗ ( X ) ≤ C k f k NM qp ( X ) . orrey-Sobolev Spaces on Metric Measure Spaces Proof.
Let f ∈ N M qp ( X ) and g ∈ M qp ( X ) be a Mod qp -weak upper gradient of f such that(5.14) k g k M qp ( X ) ≤ k f k NM qp ( X ) . By [17, Lemma 5.1] and (5.11), we have(5.15) k M ∗ α f k M q ∗ p ∗ ( X ) . k f k M qp ( X ) . By the same reason as that used in the proof [17, Theorem 6.3], observing that thepointwise Lipschitz constant of a function is also an upper gradient of that function, wesee that a positive constant multiple of ( M ∗ αθ g θ ) /θ is a Mod qp -weak upper gradient of M ∗ α f ,where θ lies in (1 , p ) such that the weak (1 , θ )-Poincar´e inequality is supported by X . By g θ ∈ M q/θp/θ ( X ) and p/θ >
1, together with [17, Lemma 5.1] and (5.11), we know that(5.16) (cid:13)(cid:13)(cid:13)(cid:13)h M ∗ αθ ( g θ ) i /θ (cid:13)(cid:13)(cid:13)(cid:13) M q ∗ p ∗ ( X ) . k g k M qp ( X ) . Combining (5.14), (5.15) and (5.16), we obtain k M ∗ α f k NM q ∗ p ∗ ( X ) . k M ∗ α f k M q ∗ p ∗ ( X ) + (cid:13)(cid:13)(cid:13)(cid:13)h M ∗ αθ ( g θ ) i /θ (cid:13)(cid:13)(cid:13)(cid:13) M q ∗ p ∗ ( X ) . k f k M qp ( X ) + k g k M qp ( X ) . k f k NM qp ( X ) , which completes the proof of Theorem 5.14.We remark that Theorem 5.14 when p = q goes back to [17, Theorem 6.3]. Acknowledgements.
The authors would like to deeply thank Professor Pekka Koskela,Professor Nagewari Shanmugalingam and Dr. Renjin Jiang for some helpful discussionson the subject of this article, and Dr. Luk´aˇs Mal´y for delivering a copy of his Ph. D thesisto us. The authors would also like to express their deep thanks to Professor YoshihiroSawano, who generously provides us the proofs of Lemma 5.5 and Theorem 5.6 of thisarticle, and to the referee for his/her carefully reading and so many helpful and usefulcomments which essentially improves this article.
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