Smoothing properties of the discrete fractional maximal operator on Besov and Triebel--Lizorkin spaces

Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in general setting, but they are new already in the Euclidean case.


Introduction
Maximal functions are standard tools in harmonic analysis. They are usually used to estimate absolute size, but recently there has been interest in studying their regularity properties, see [1], [2], [3], [11], [12], [13], [15], [17], [18], [19], [21], [22], [23], [25], [26], [28]. A starting point was [17], where Kinnunen observed that the Hardy-Littlewood maximal operator is bounded on W 1,p (R n ) for 1 < p ≤ ∞. In [22] and [23] Korry extended this result by showing that the maximal operator preserves also fractional Sobolev spaces as well as Besov and Triebel-Lizorkin spaces. Another kind of extension was given in [20], where Kinnunen and Saksman showed that 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, is bounded from W 1,p (R n ) to W 1,p * (R n ), where p * = np/(n − αp), and from L p (R n ) toẆ 1,q (R n ), where q = np/(n−(α −1)p) andẆ 1,q (R n ) is the homogenous Sobolev space. These results indicate that M α has similar smoothing properties as the Riesz potential. It is natural to ask whether these results can be seen as special cases of the behaviour of the fractional maximal operator on Besov and Triebel-Lizorkin spaces. In this paper we show that this is indeed the case, and that all these results can be obtained by the same rather simple method. Instead of the standard fractional maximal operator, we consider its variant, the so-called discrete fractional maximal operator M * α . This allows us to present our results in a setting of doubling metric measure spaces. In this generality, the standard fractional maximal operator behaves quite badly. Indeed, one can construct spaces, where the fractional maximal function of a Lipschitz function fails to be continuous, see [3] and [15]. Since M * α and M α are comparable, for practical purposes it does not matter which one we choose. The discrete fractional maximal operator was introduced in [18] and further studied in [21], [1] and [13].
Among the many possible definitions of Besov and Triebel-Lizorkin spaces, the most suitable for our purposes is the one based on Haj lasz type pointwise inequalities. This approach, introduced by Koskela, Yang and Zhou in [24], provides a new point of view to the classical Besov and Triebel-Lizorkin spaces. On the other hand, it allows these spaces to be defined in the setting of metric measure spaces.
By employing this definition, we can prove very general results using only simple "telescoping" arguments and Poincaré type inequalities. As special cases, we obtain versions of the results of Kinnunen and Saksman as well as those of Korry, see Remark 3.4 and Theorems 4.5 and 4.6. We prove our results in doubling metric measure spaces but they are new even in Euclidean spaces. Our main results (Theorems 4.3 and 4.4) imply that if α ≥ 0 and 0 < s+α < 1, then M * α is bounded fromḞ s p,q (R n ) toḞ s+α p,q (R n ) for n/(n + s) < p, q < ∞ and fromḂ s p,q (R n ) toḂ s+α p,q (R n ) for n/(n + s) < p < ∞, 0 < q < ∞, see Section 4 for the definition of Triebel-Lizorkin and Besov spaces.

Preliminaries and notation
We assume that X = (X, d, µ) is a metric measure space equipped with a metric d and a Borel regular outer measure µ, which satisfies 0 < µ(U) < ∞ whenever U is nonempty, open and bounded. We assume that the measure is doubling, that is, 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}.
The doubling condition implies that for every 0 < r ≤ R and y ∈ B(x, R) for some C and Q > 1 that only depend on c D . In fact, we may take Q = log 2 c d . For the boundedness of the fractional maximal operator in L p , we have to assume, in Theorems 2.1 and 3.3.(b), that the measure µ satisfies the lower bound condition (2.3) µ(B(x, r)) ≥ c l r Q with some constant c l > 0 for all x ∈ X and r > 0. Throughout the paper, C will denote 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 The following Sobolev type inequality for the fractional maximal operator follows easily from the boundedness of the Hardy-Littlewood maximal operator in L p , for the proof, see [4], [6] or [15].
Theorem 2.1. Assume that the measure lower bound condition holds. If p > 1 and 0 < α < Q/p, then there is a constant C > 0, depending only on the doubling constant, constant in the measure lower bound, p and α, such that for every u ∈ L p (X) with p * = Qp/(Q − αp).

Remark 2.2.
If u is only locally integrable, then M α u may well be identically infinite. However, if M α u(x 0 ) < ∞ for some x 0 ∈ X, then M α u(x) < ∞ for almost every x. This follows from the estimate combined with the doubling condition and the fact that The discrete fractional maximal function. We begin the construction of the discrete maximal function with a covering of the space. Let r > 0. Since the measure is doubling, there are balls B(x i , r), i = 1, 2, . . . , such that Then we construct a partition of unity subordinate to the covering B(x i , r), i = 1, 2, . . . , of X. Indeed, there is a family of functions , ϕ i is Lipschitz with constant L/r with ν and L depending only on the doubling constant, and The discrete convolution of a locally integrable function u at the scale 3r is for every x ∈ X, and we write u α r = r α u r . Let r j , j = 1, 2, . . . be an enumeration of the positive rationals and let balls B(x i , r j ), i = 1, 2, . . . be a covering of X as above. The discrete fractional maximal function of u in X is for every x ∈ X. For α = 0, we obtain the Hardy-Littlewood type discrete maximal function M * studied in [18], [21] and [1]. The discrete fractional maximal function is easily seen to be comparable to the standard fractional maximal function, see [13].

Fractional Haj lasz spaces
Let u be a measurable function and let s ≥ 0. A nonnegative measurable function g is an s-Haj lasz gradient of u if there exists E ⊂ X with µ(E) = 0 such that for all x, y ∈ X \ E, The collection of all s-Haj lasz gradients of u is denoted by D s (u). A homogeneous Haj lasz spaceṀ s,p (X) consists of measurable functions u such that The space M 1,p (X), a counterpart of a Sobolev space in metric measure space, was introduced in [9], see also [10]. The fractional spaces M s,p (X) were introduced in [30] and studied for example in [16] and [14]. Notice that M 0,p (X) = L p (X).
The pointwise definition of the Haj lasz spaces implies the validity of Sobolev-Poincaré type inequalities without the assumption that the space admits any weak Poincaré inequality.  7]). Let s ∈ [0, ∞) and let p ∈ (0, Q/s). There exists a constant C such that for all measurable functions u with g ∈ D s (u), all x ∈ X and r > 0, Moreover, if p ≥ Q/(Q + s) and g ∈ D s (u) ∩ L p (X), then (3.2) implies that u is locally integrable and that For the case s = 1, see [9] and [10].
In the next theorem, we use the following simple result. If u i , i ∈ N, are measurable functions with a common s-Haj lasz gradient g and u = sup i u i is finite almost everywhere, then g is an s-Haj lasz gradient of u.
We begin by proving the claims for u α r . Let r > 0, let g ∈ D s (u) and let x, y ∈ X.
Assume first that r ≥ d(x, y). Let I xy be a set of indices i for which x or y belongs to B(x i , 6r). Then, for each i ∈ I xy , B(x i , 3r) ⊂ B(x, 10r) ⊂ B(x i , 17r). This together with the doubling condition, the properties of the functions ϕ i , the fact that there are bounded number of indices in I xy and Poincaré inequality (3.3) implies that If 0 < s + α ≤ 1, then by (3.4) and the assumption r ≥ d(x, y), we have that If s + α > 1, then by (3.4), This shows that Haj lasz gradient inequality (3.1) with desired exponent holds when r ≥ d(x, y). Assume then that r < d(x, y). Let R = d(x, y). Then B(y, r) ⊂ B(x, 2R) and where I x is a set of indices i for which x belongs to B(x i , 6r) and I y the corresponding set for y. Let k ∈ N be the smallest integer such that 2 k r ≥ R.
Assume first that 0 < s + α ≤ 1. If i ∈ I x , then (3.6) By the doubling condition and Poincaré inequality (3.3), we have and, by the doubling condition, Poincaré inequality (3.3), the fact that r ≤ 2 i 9r for all i, and the selection of k, Similarly we obtain that If i ∈ I y , we use balls B(y, 2 i 9r) instead of balls B(x, 2 i 9r) in (3.6). Estimates corresponding (3.7) and (3.8) are as above (x replaced by y) and, corresponding to (3.9), Now, by (3.5)-(3.10) and the fact R = d(x, y), we have If s + α > 1, then similar estimates as above show that if i ∈ I x ∪ I y , then (3.11) These estimates together with (3.5) and the fact that there are bounded number of indices in I x and I y imply that Haj lasz gradient inequality (3.1) with desired exponent holds when r < d(x, y). The claim for u α r follows from the estimates above and for M * α u from the discussion before the theorem.
for all u ∈Ṁ s,p (X) with M * α u ≡ ∞. b) If 1 < s + α ≤ 1 + Q/p and the measure lower bound condition holds, there exists a constant C > 0 such that where q = Qp/(Q − (s + α − 1)p), for all u ∈Ṁ s,p (X) with M * α u ≡ ∞. Proof. a) Let Q/(Q + s) ≤ t < p. By Theorem 3.2, the function C(M g t ) 1/t is an (s + α)-gradient of M * α u. Since g ∈ L p (X), the claim follows from the boundedness of the Hardy-Littlewood maximal operator in L q (X) for q > 1. b) Let Q/(Q + s) ≤ t < p. By Theorem 3.2, the function (M t(s+α−1) g t ) 1/t is a 1-gradient of M * α u. Since g ∈ L p (X), the claim follows from Theorem 2.1.

Remark 3.4.
In the cases s = 0 and s = 1 of Theorem 3.3.b), we obtain counterparts of the results of Kinnunen and Saksman. Remark 3.5. As a special case of Theorems 3.2 and 3.3 we obtain boundedness results for the discrete maximal operator M * inṀ s,p (X). If 0 < s ≤ 1, theñ g = C(M g t ) 1/t is an s-Haj lasz gradient of M * u for all t ≥ Q/(Q + s) and for all u ∈Ṁ s,p (X), p > Q/(Q + s).

Haj lasz-Besov and Haj lasz-Triebel-Lizorkin spaces
Let u be a measurable function and let s ∈ (0, ∞). Following [24], we say that a sequence of nonnegative measurable functions (g k ) k∈Z is a fractional s-Haj lasz gradient of u if there exists E ⊂ X with µ(E) = 0 such that |u(x) − u(y)| ≤ d(x, y) s (g k (x) + g k (y)) for all k ∈ Z and all x, y ∈ X \ E satisfying 2 −k−1 ≤ d(x, y) < 2 −k . The collection of all fractional s-Haj lasz gradients of u is denoted by D s (u).
Proof. Let k ∈ Z and let x, y ∈ X such that 2 −k−1 ≤ d(x, y) < 2 −k . We will show that |u α r (x) − u α r (y)| ≤ Cd(x, y) s+α (g k (x) +g k (y)), where C is independent of r and k.
Assume first that d(x, y) > r. Then where I x is a set of indices i for which x belongs to B(x i , 6r) and I y the corresponding set for y. Let m ∈ Z be such that 2 and hence Poincaré inequality (4.2) implies that Suppose then that d(x, y) ≤ r. Let I xy be a set of indices i for which x or y belongs to B(x i , 6r). Let l be such that 2 −l−1 < 10r ≤ 2 −l . Using the doubling condition, the properties of the functions ϕ i , the fact that there are bounded number of indices in I xy and Poincaré inequality (4.2), we have that This together with (4.4) implies that By splitting the sum in two parts and using the estimates l ≤ j + 2 and l ≤ k, we obtain ∞ j=l−2 which implies the claim for u α r . The claim for M * α u follows similarly as in the proof of Theorem 3.2. Theorem 4.3. Let 0 < s + α < 1 and Q/(Q + s) < p, q < ∞. Then there exists a constant C > 0 such that for all u ∈Ṁ s p,q (X) with M * α u ≡ ∞.
Proof. Let δ = 1 2 (1 − (s + α)), ε = 1 2 max{s, s + Q−Qr r }, ε ′ = 1 2 (ε + s), where r = min{p, q}, and let t = Q/(Q + ε). Then 0 < ε < ε ′ < s and Q/(Q + s) < t < min{p, q}. By Theorem 4.2, (Cg k ) defined by (4.3) is a fractional (s + α)-Haj lasz gradient of M * α u. It suffices to show that (g k ) ∈ L p (X, l q ). We estimate the L p (X, l q ) norm of the other part can be estimated similarly. If q ≥ 1, we have, by the Hölder inequality, that If q < 1, we obtain the same estimate by using the elementary inequality ( j a j ) q ≤ j a q j for a j ≥ 0. By the Fefferman-Stein vector valued maximal function theorem from [5] (for a metric space version, see for example [27] or [8]), we obtain now the desired estimate 1/t L p/t (X, l q/t ) ≤ C (g t k ) k∈Z 1/t L p/t (X, l q/t ) = C (g k ) k∈Z L p (X, l q ) .
It suffices to show that (g k ) l q (L p (X)) ≤ C (g k ) l q (L p (X)) . By the Hardy-Littlewood maximal theorem, If q ≥ 1, we have by the Hölder inequality, If q < 1, we use the inequality ( j a j ) q ≤ j a q j instead of the Hölder inequality. The second part of (g k ) can be estimated similarly.  a) If Q/(Q + s) < p, q < ∞, then there exist a constant C > 0 such that M * u Ṁ s p,q (X) ≤ C u Ṁ s p,q (X) , whenever u ∈Ṁ s p,q (X) and M * u ≡ ∞. b) If 1 < p, q < ∞, then there exist a constant C > 0 such that M * u M s p,q (X) ≤ C u M s p,q (X) , for all u ∈ M s p,q (X). Theorem 4.6. Let 0 < s < 1. a) If Q/(Q + s) < p < ∞ and 0 < q < ∞, there exist a constant C > 0 such that M * u Ṅ s p,q (X) ≤ C u Ṅ s p,q (X) for all u ∈Ṅ s p,q (X) with M * u ≡ ∞. b) If 1 < p < ∞ and 0 < q < ∞, there exist a constant C > 0 such that M * u N s p,q (X) ≤ C u N s p,q (X) for all u ∈ N s p,q (X).