Fractional maximal functions in metric measure spaces

We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to H\"older continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.


Introduction
Fractional maximal operators are standard tools in partial differential equations, potential theory and harmonic analysis. In the Euclidean case they are studied for example in [1], [2], [3], [17], [18], [19], [22], and in the metric setting in [8], [9], [10], [15], [16], [24], [26], [27]. In the Euclidean case, the fractional maximal operator M α , defined by M α u(x) = sup r>0 r α |B(x, r)| B(x,r) |u(y)| dy, has similar smoothing properties as the Riesz potential, see Kinnunen and Saksman [17]. In this paper, we generalize the Euclidean results of [17] to the metric setting under an annular decay property, which places certain restrictions on the geometry of the space X. More precisely, in Theorem 4.2 we show that the fractional maximal function of an L p -function, p > 1, belongs to a (pointwise fractional) Sobolev space M s,p (X), where s depends on the annular decay. Another example of a smoothing property is given in Theorem 4.5, where we show that the fractional maximal operator maps the Sobolev space M 1,p (X) to a slightly better Sobolev space M 1,p * (X), where p * is the conjugate exponent of p. The proof of this result is based on an unpublished proof of MacManus [21] for the Sobolev boundedness of the usual Hardy-Littlewood maximal operator. We also study the action of the fractional maximal function in Campanato spaces. In this context our main result is Theorem 3.1, which states that the fractional maximal function maps functions in Campanato spaces to Hölder continuous functions, provided that the space satisfies the annular decay property. The result is new even in the Euclidean case. This is again analogous to the well known properties of the Riesz potential, studied in the metric setting for example in [6] and [7]. Note Date: January 31, 2013. 2010 Mathematics Subject Classification. 42B25, 46E35. The research is supported by the Academy of Finland, grants no. 135561 and 252108. 1 here that the Campanato estimates for the Riesz potential do not immediately imply the corresponding oscillation estimates for the fractional maximal function. In particular, Theorem 3.1 shows that the fractional maximal operator maps Hölder continuous functions to Hölder continuous functions with a better exponent, and functions of bounded mean oscillation to Hölder continuous functions.
A part of the motivation for our work comes from [15], where it was shown that similar mapping properties hold for the so called discrete fractional maximal operator even without the annular decay property. In contrast to those results, we conclude this paper with two examples which verify that there is a real obstruction in the study of fractional maximal functions in metric measure spaces. We modify the example given by Buckley in [4] for the standard Hardy-Littlewood maximal function and show that the fractional maximal function of a Lipschitz continuous function may fail to be continuous if the space does not satisfy annular decay. The same is actually true also for the noncentered fractional maximal function, for which the main results given in Sections 3 and 4 hold as well, as is briefly noted at the end of each theorem.

Notation and preliminaries
We assume that X = (X, d, µ) is a metric measure space equipped with a metric d and a Borel regular, doubling outer measure µ, for which the measure of every open set is positive and the measure of each bounded set is finite. The doubling property means that there exists a fixed constant c d > 0, called the doubling constant, such that for every ball B(x, r) = {y ∈ X : d(y, x) < r}. We say that the measure µ satisfies a lower bound condition if there exist constants Q ≥ 1 and c l > 0 such that for all x ∈ X and r > 0. We follow the standard procedure that the letter C denotes a positive constant whose value is not necessarily the same at each occurrence.
The fractional maximal function. Let α ≥ 0. The fractional maximal function of a locally integrable function u is Annular decay properties. Let 0 < δ ≤ 1. We say that the metric measure space X satisfies the δ-annular decay property, if there exists a constant C > 0 such that for all x ∈ X, R > 0, and 0 < h < R, we have Similarly, we say that X satisfies the relative δ-annular decay property, if there exist a constant C > 0 such that for all x ∈ X, R > 0, and 0 < h < R, we have for all balls B with radius r B ≤ 3R. Note that the relative condition (2.5) implies the δ-annular decay property (2.4). If X is a geodesic space (or more generally a length space), then X satisfies the relative annular decay for some δ > 0. See for instance [4], [14,Chapter 9] and [25] for this fact and for more information on these and related conditions. Buckley studied in [4] the action of the usual maximal operator M on C 0,β (X), the space of β-Hölder continuous functions equipped with the seminorm He showed that if a doubling space X satisfies the δ-annular decay property, then is bounded whenever 0 < β ≤ δ. He also gave an example of a doubling space where the maximal function of a Lipschitz function is not continuous; see Section 5.

Campanato spaces
In this section, we study the action of the fractional maximal operator M α on Campanato spaces L p,β (X). A locally integrable function u belongs to L p,β (X), if where the supremum is taken over all x ∈ X and r > 0. It is known that if β > 0, where C is independent of p, see for example [23]. Here BMO(X) = L 1,0 (X) is the space of functions of bounded mean oscillation.
Theorem 3.1. Assume that X satisfies the δ-annular decay property (2.4) and that 0 < α ≤ δ and 0 ≤ α + β ≤ δ. If u ∈ L p,β (X) and M α u ≡ ∞, then M α u ∈ C 0,α+β (X), and there is a constant C > 0, depending only on the doubling constant and constant of the annular decay, such that Note that Theorem 3.1 implies that are bounded operators, when restricted to functions satisfying M α u ≡ ∞. If the measure µ satisfies the lower bound condition (2.2) and α − Q/p ≥ 0, then also Notice that (3.4), (3.5) and (3.6), where α − Q/p > 0, are analogous to the well known properties of the Riesz potential, studied in the metric setting for example in [6] and [7]. For the proof Theorem 3.1, we need a lemma which is proved using a chaining argument.
If β = 0, then The constant C depends only on the doubling constant and C 0 .
Proof of Theorem 3.1. Since |u| ∈ L p,β (X) and |u| L p,β (X) ≤ C u L p,β (X) , we may assume that u ≥ 0. Let r > 0 and define v : . We begin by proving the claim for v. Let x, y ∈ X. Case 1. Assume first that r ≤ 2d(x, y). Let B = B(x, 2d(x, y)), B x = B(x, r) and B y = B(y, r). If β < 0, then, by (3.7), Similarly, if β = 0, estimate (3.8) implies that where the last inequality follows by the boundedness of the function f (t) = t −α log t for t ≥ 1.
If β > 0, then u is β-Hölder continuous and so For each i, by (3.7) and the Hölder inequality, we have (3.10) To estimate the measure of ∆ ′ , we first use the annular decay property and doubling to obtain and similarly, because µ(B x ) is comparable with µ(B y ) by the doubling property and the assumption r > 2d(x, y), Using (3.9), the facts that ∆ ⊂ ∪ i 2B i and B i ⊂ ∆ ′ for all i, (3.10), disjointedness of balls B i and (3.11), we obtain where the last inequality follows because r > d(x, y) and 0 < α < δ. Assume then that β ≥ 0 and s > 1. Let q = s/(s − 1). By (3.9), the Hölder inequality, the facts that ∆ ⊂ 2B y and µ(B x ) is comparable with µ(B y ), and (3.11), we have Since, by (3.1) and Since r > d(x, y) and 0 < α + β ≤ δ, the estimate above implies Finally, we prove the claim for M α u. We may assume that M α u(x) ≥ M α u(y). Let ε > 0, and let r > 0 be such that r α u B(x,r) > M α u(x) − ε. Then, by the first part of the proof, The claim follows by letting ε → 0.

Remark 3.3.
A modification of the proof above shows that the result holds also for the noncentered fractional maximal function Let x, y ∈ X. We may assume that u ≥ 0 and M α u(x) ≥ M α u(y). Let ε > 0. Then there exists a ball B(z, r) containing x such that Since y ∈ B(z, r + d(x, y)), we have that Arguments similar to those in the proof of Theorem 3.1 imply that The claim follows by letting ε → 0.

Sobolev spaces
In this section, we show that the fractional maximal operator M α maps L p -spaces, p > 1, to Sobolev spaces, and Sobolev spaces to slightly better Sobolev spaces. We prove the results for Sobolev spaces M s,p (X), defined by a pointwise equation. These spaces were introduced by Haj lasz in [12] for s = 1, and the fractional versions by Yang in [28].
Let s > 0. We say that a measurable function g ≥ 0 is a generalized s-gradient of a measurable function u, g ∈ D s (u), if there is a set E ⊂ X with µ(E) = 0 such that for all x, y ∈ X \ E. The Sobolev space M s,p (X), 1 ≤ p < ∞, consists of functions u ∈ L p (X) for which there exists a function g ∈ L p (X) ∩ D s (u). The space M s,p (X), equipped with the norm where the infimum is taken over all functions g ∈ L p (X) ∩ D s (u), is a Banach space [13,Theorem 8.3]. It follows from (4.1) that every u ∈ M s,p (X) and g ∈ D s (u) satisfy the Poincaré inequality where C depends only on s. Hence the assumption that X satisfies a Poincaré inequality is not needed.
We will use the following Sobolev type theorem for the fractional maximal operator. As in [11], it can be proven easily using the Hardy-Littlewood maximal function theorem; see also [5] or [8].
Theorem 4.1. Assume that the measure µ satisfies the lower bound condition (2.2). Let p > 1 and 0 < α < Q/p. There is a constant C > 0, depending only on the doubling constant, constant in the measure lower bound, p and α, such that The following theorem is a generalization of the main result of [17] to the metric setting. It shows that the fractional maximal operator is a smoothing operator. More precisely, the fractional maximal function of an L p -function u has a generalized gradient, and both M α u and the generalized gradient belong to a higher Lebesgue space than u.
Theorem 4.2. Assume that the measure µ satisfies the lower bound condition (2.2) and that X satisfies the δ-annular decay property (2.4). Assume that u ∈ L p (X) with 1 < p < Q. Let Then there is a constant C > 0 depending only on the doubling constant and the constant of the δ-annular decay property such that C M α−δ u is a generalized δgradient of M α u. Moreover, where C depends only on the doubling constant, the constant in the measure lower bound, p and α.
Proof. We may assume that u ≥ 0. Let x, y ∈ X. Assume that M α u(x) ≥ M α u(y). Let ε > 0 and let r > 0 such that If r ≤ d(x, y), then, as u ≥ 0, If r > d(x, y), we write a = d(x, y) and use the doubling property to obtain u dµ + ε.
The δ-annular decay property together with the assumption r > d(x, y) implies that By letting ε → 0 and changing the roles of x and y, we have that Hence M α−δ u is a generalized δ-gradient of M α u. The norm estimates for M α u and M α−δ u follow from Theorem 4.1.

Remark 4.3.
With the assumptions of Theorem 4.2, M α u ∈ M δ,q loc (X) and for all open sets A ⊂ X with µ(A) < ∞.
Remark 4.4. The above proof with the ball B(x, r) replaced by a ball B(z, r) for which Mu(x) < r α u B(z,r) + ε and B(y, r + a) replaced by the ball B(z, r + a) shows that the result holds also for the noncentered fractional maximal function.
In the next theorem, we show that if u is a Sobolev function, then its fractional maximal function belongs to a Sobolev space with the Sobolev conjugate exponent. The proof is a modification of the result that the usual Hardy-Littlewood maximal operator is bounded in Sobolev spaces if the underlying space satisfies the relative 1-annular decay property. Since the original proof of MacManus [21] (for α = 0) is unpublished, we give in the below proof all the details. Theorem 4.5. Assume that the measure µ satisfies the lower bound condition and X satisfies the relative 1-annular decay property (2.5). Let p > 1, u ∈ M 1,p (X) and 0 < α < Q/p. Then M α u ∈ M 1,p * (X) with p * = Qp/(Q − αp) and there is a constant C > 0, depending only on the doubling constant, the constant in the measure lower bound, p and α, such that Proof. Let u ∈ M 1,p (X) and let g ∈ L p (X) be a generalized gradient of u. We may assume that u ≥ 0 since |u| ∈ M 1,p (X) and g is a generalized gradient of |u|. Fix 1 < q < p and defineg = (M αq (g q )) 1/q .
Since p/q > 1, Theorem 4.1 implies that If we can show thatg is a generalized gradient of M α u, then this together with Theorem 4.1 for u implies norm estimate (4.4).
We are going to show that for almost all x, y ∈ X and all r > 0. This implies that for almost every x, y ∈ X, which proves our theorem. The proof of (4.5) is easy if r ≤ 3 d(x, y): Since g is a generalized gradient of u, for almost all z ∈ B(x, r) and w ∈ B(y, r). By integrating both sides with respect to z and w and using the Hölder inequality, we obtain . Now (4.5) follows by multiplying both sides by r α and using the definition ofg.
Suppose then that r > 3 d(x, y). Let ∆ y = B(y, r)\B(x, r), ∆ x = B(x, r)\B(y, r) and ∆ = ∆ y ∪ ∆ x . As in the proof of Theorem 3.1, we have and hence y)). Since the balls B(x, r) and B(y, r) have comparable measures and ∆ ⊂ A, We want to show that the right-hand side is bounded by C d(x, y) ( B(x,5r) g q dµ) 1/q , which implies (4.5). For that, we prove the following estimate for the integral over the annulus. Claim: To prove the claim, define r k = 3 k r and B k (z) = B(z, r k ), k ∈ Z.
We use a standard chaining argument, Poincaré inequality (4.3), and the fact that B(x, r) ⊂ B 1 (z) for each z ∈ A to see that for all Lebesgue points z ∈ A. (Since almost every point is a Lebesgue point of u, this holds for almost all z ∈ A.) Integration of both sides over A and a use of the Fubini theorem yield Since k ≤ 1 we have that K(w) = 0 when w / ∈ B(x, 5r). This implies that where, by that fact that χ B k (z) (w) = χ B k (w) (z), If z ∈ B k (w), then the balls B k (z) and B k (w) have comparable measures. Thus from which we obtain that It follows from the relative 1-annular decay that each term in the above sum is bounded by C min{r k , d(x, y)}. Moreover, for the indices k ≤ 1 for which B k (w) ∩ A = ∅, the terms are zero. Now The first term at most Cd(x, y). An upper bound for the second term is To see this, we consider two cases. Assume first that |r − d(w, x)| < 3d(x, y). Since r k > d(x, y), we have that which implies that the amount of such indices in the sum is less than The second case, |r − d(w, x)| ≥ 3 d(x, y) can be split to two parts. When w ∈ B(x, r − 3d(x, y)), the requirement that B k (w) ∩ A = ∅ implies that and hence we have at most such terms. (Note that the condition B k (w) ∩ A = ∅ gives the number of terms, not the condition r k > h.) Similarly, when w ∈ 5B(x, r) \ B(x, r + 3d(x, y)), we must have y)), and the upper bound for the number of terms follows just as in the first case. This implies that the second term is bounded above by (4.7). Since 0 ≤ d(w, x) < 5r, we have that log 5r |r−d(w,x)| ≥ log 5 4 , and hence the claim follows. Now, by the claim and the Hölder inequality, where q ′ is the conjugate exponent of q and To estimate the integral of L q ′ over B(x, 5r), we define for each i = 0, 1, . . .
The sets A i are disjoint, and on each A i we have Moreover, the set {w ∈ B(x, 5r) : |r − d(w, x)| = 0} has measure zero by the relative annular decay, and so It follows that , r)) .
For i ≥ 2, the set A i consists of two annuli of thickness 4 −i r centered at x. The inner and outer radii of these annuli are comparable to r. Thus the relative 1-annular decay implies that µ(A i ) ≤ C 4 −i µ(B(x, r)). The same estimate is trivial when i = 0 or 1. It follows that This sum converges, which can be seen for example by a ratio test, and we have that Estimate (4.5) and hence the theorem follows from this.
We close this section by considering the more general case where u belongs to a fractional Sobolev space and X satisfies the relative δ-annular decay property. Using similar arguments as in the proof above, we obtain the following results.
Theorem 4.6. Assume that X satisfies the relative δ-annular decay property (2.5). Let 1 < q < p, s > 0, u ∈ M s,p (X) and g ∈ D s (u). Then there is a constant C > 0 such that the following holds. a) If s < δ, theng Proof. It suffices to prove the claim for the functions x → r α u B(x,r) , r > 0. Fix x, y ∈ X and r > 0. Suppose first that r ≤ 3d(x, y). If s ≤ δ, the desired estimate follows as in the proof of Theorem 4.5. If s > δ, we need a simple chaining argument. Let k be the smallest integer such that 2 k r ≥ 4d(x, y) and let z ∈ {x, y}. Then the doubling property, the fact that B(x, 3d(x, y)) ⊂ B(z, 2 k r) and the Poincaré inequality (4.3) imply that Hence, using the selection of k, we have that ≤ Cd(x, y) δ M α+s−δ g(z), and so |r α u B(x,r) − r α u B(y,r) | ≤ r α |u B(x,r) − u B(x,3d(x,y)) | + r α |u B(y,r) − u B(x,3d(x,y)) | ≤ Cd(x, y) δ (M α+s−δ g(x) + M α+s−δ g(y)) .
Suppose then that r > 3d(x, y). By following the proof of Theorem 4.5, we obtain estimate The relative δ-annular decay implies that where K 1 = {k ≤ 1 : r k ≤ d(x, y)} and K 2 = {k ≤ 1 : r k > d(x, y), B k (w) ∩ A = ∅}. It follows that If s < δ or s > δ, the claim follows by combining the above estimate with (4.8) and multiplying the resulting inequality by r α . In the case s = δ, we argue as in the proof of Theorem 4.5.
Corollary 4.7. Assume that the measure µ satisfies the lower bound condition and X satisfies the relative δ-annular decay property (2.5). Let p > 1, s > 0, u ∈ M s,p (X) and 0 < α < Q/p. If s ≤ δ, then M α u ∈ M s,p * (X) with p * = Qp/(Q − αp) and there is a constant C > 0, independent of u, such that M α u M s,p * (X) ≤ C u M s,p (X) .
Proof. The claim follows from Theorems 4.6 and 4.1.
For related results concerning the discrete fractional maximal operator, see [16].
Remark 4.8. The usual modifications in the proofs show that Theorems 4.5 and 4.6 remain true for the noncentered fractional maximal function.

Examples
We modify the example given by Buckley [4] a little bit and show that the fractional maximal function of a Lipschitz function may fail to be continuous. 2 + π 4 = π 8 + π > M u(0).
The first part of the above argument does not work for M α , because for α large enough, r α B(0,r) |u| dµ = r α 2r + π 2 B(0,1) |u| dµ no longer maximizes as r → 1+. This difficulty can be overcome by modifying the measure. Let w : X → R be a weight such that w(x) = 1 + π 2 , when x > 1 and w(x) = 1 otherwise. Then the measure ν, defined by is doubling and satisfies ν(B(0, r)) = (2 + π 2 )r, for r > 1. If α > 1, it follows that M α u ≡ ∞. If α ≤ 1, then, for r > 1, The rest of the argument is the same as above.