Potential Analysis https://doi.org/10.1007/s11118-022-10006-z Sobolev Embeddings for Fractional Hajłasz-Sobolev Spaces in the Setting of Rearrangement Invariant Spaces Joaquim Martı́n1 · Walter A. Ortiz1 Received: 28 April 2021 / Accepted: 10 April 2022 © The Author(s) 2022 Abstract We obtain symmetrization inequalities in the context of Fractional Hajłasz-Sobolev spaces in the setting of rearrangement invariant spaces and prove that for a large class of measures our symmetrization inequalities are equivalent to the lower bound of the measure. Keywords Sobolev inequality · Fractional Hajłasz-Sobolev spaces · Metric measure spaces Mathematics Subject Classification (2010) Primary 46E35 1 Introduction Let us consider a metric measure space (Ω, d, μ) where μ is a Borel measure on (Ω, d) such 0 < μ(B) < ∞, for every ball B in Ω. We will always assume μ(Ω) = ∞ and μ({x}) = 0 for all x ∈ Ω. Let X be a rearrangement invariant (r.i.) space on Ω (see Section 2.2.1 below). In this paper, we introduce the fractional Hajłasz-Sobolev spaces M s,X (Ω) for s > 0, and we will focus on understanding the relation between Sobolev embeddings theorems for spaces M s,X (Ω) and the growth properties of the measure μ. Let s > 0 and let X be a r.i. space on Ω. We say that f ∈ M s,X (Ω), if f ∈ X, and there exits a non-negative measurable function g ∈ X such that |f (x) − f (y)| ≤ d(x, y)s (g(x) + g(y)) μ − a.e. x, y ∈ Ω. (1) J. Martı́n is partially supported by Grants PID2020-113048GB-I00 and PID2020-114167GB-I00 both funded by MCIN/AEI/10.13039/501100011033 W.A. Ortiz is partially supported by Grant 2017SGR395 (AGAUR, Generalitat de Catalunya) Joaquim Martı́n jmartin@mat.uab.cat Walter A. Ortiz waortiz@mat.uab.cat 1 Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain J. Martı́n, W.A. Ortiz A function g satisfying (1) will be called a s−gradient of f . We denote by D s (f ) the collection of all s−gradients of f . The homogeneous Hajłasz-Sobolev space Ṁ s,X (Ω) consists of all functions f ∈ X for which f Ṁ s,X (Ω) = inf g X g∈D s (u) is finite. The Hajłasz-Sobolev space M s,X (Ω) is Ṁ s,X (S) ∩ X equipped with the norm f M s,X (Ω) = f X + f Ṁ s,X (Ω) . When X = Lp (Ω), 1 ≤ p ≤ ∞, we shall write M s,p (Ω) instead of M s,X (Ω). Remark 1 In the context of metric spaces, the spaces M 1,p (Ω) were first introduced by Hajłasz (see [11] and [12]). They play an important role in the area of analysis called analy- sis on metric spaces and a lot of papers have focused on this subject (see for example [3, 13– 16], and the references quoted therein). When the measure μ is doubling1 , spaces M 1,X (Ω) have been considered in some particular cases, for example, Hajłasz-Lorentz-Sobolev p,q Φ spaces M 1,L (Ω) (see [19]) and Musielak-Orlicz-Hajłasz-Sobolev spaces M 1,L (Ω), Φ where L is an Orlicz space (see [34]). Also in the doubling case, fractional spaces M s,p (Ω) were introduced and studied in [35] (see also [17] and [18]). For p > 1, M 1,p (Rn ) = W 1,p (Rn ) (see [11]), whereas for p = 1, M 1,1 (Rn ) coin- cides with the Hardy.Sobolev space H 1,1 (Rn ) (see [24, Thm 1]) and if 0 < s < 1, then M s,p (Rn ) = Bp,∞s (Rn ) (see [35]). Notice that in Rn , if s > 1, then M s,p (Rn ) is trivial (contains only constant functions). However, if Ω is a fractal, then M s,p (Ω) for s > 1 may be non-trivial (see [17]). It is well known that the lower bound for the growth of the measure μ(B(x, r)) ≥ br α , (2) implies Sobolev embedding theorems for Hajłasz-Sobolev spaces M 1,p (see [11] and [12]). The converse problem, i.e. when the embedding M 1,p (X) ⊂ Lq (X), q > p (3) implies a lower bound for the growth of the measure, has been considered by several authors (see [10, 18, 20–23] and the references quoted therein). In the recent paper [2], R. Alvarado, P. Górka and P. Hajłasz show that in fact if (3) holds with q = αp/(α − p), then lower bound for the growth (2) holds. The purpose of this paper is to obtain an analogous result for M s,X spaces. This will be done by obtaining pointwise estimates between the special difference f ∗∗ (t)−f ∗ (t) (called the oscillation2 of f ) and the function g (see Theorem 1 below), i.e. we will see that for a wide range of measures, condition (2) implies f ∗∗ (t) − f ∗ (t) ≤ Ct s/α g ∗∗ (t), (4) for every f ∈ M s,L1 +L∞ and g ∈ D s (f ). Moreover, if 0 < s ≤ 1, then (4) implies (2). 1μ is said to be doubling provided there exists a constant C > 0 such that μ(2B) ≤ Cμ(B) for all balls B ⊂ Ω. 2 Here f ∗ is the decreasing rearrangement of f , f ∗∗ (t) = 1 t ∗ t 0 f (s)ds, for all t > 0, (see Section 2.2). Sobolev Embeddings for Fractional... Symmetrization inequalities imply Sobolev inequalities in the setting of rearrangement invariant spaces. Indeed, from (4) we obtain: for any r.i. space X with upper Boyd3 index ᾱX < 1, we have t −s/α (f ∗∗ (t) − f ∗ (t)) X ≤ c g X , where c = c(s, α, X). Notice that we avoid one common drawback of the usual approaches to Sobolev inequal- ities which require the choice of specific norms before one starts the analysis. Instead, we work with pointwise symmetrization inequalities which are *universal* and it is the inequalities themselves that select the *correct* spaces. For example, in the particular case of X = Lp (see Corollary 1 below) we obtain that if 1 > s/α > p1 , then4 t −s/α (f ∗∗ (t) − f ∗ (t)) Lp t −s/α f ∗∗ (t) Lp = f Lp ∗ ,p where ps∗ = αp α−sp , i.e. ∞ 1 p dt 1/p 1/p − α1 tp f ∗∗ (t) ≤C gp . 0 t Ω On the other hand, since p < ps∗ we have that αp αp ,p L α−sp ⊂ L α−sp in particular, if s = 1, then we get 1/p1∗ ∞ p 1/p 1/p p1∗ 1 − α1 dt |f | dμ ≤ tp f ∗∗ (t) ≤C g p dμ . Ω 0 t Ω Remark 2 The technique to obtain Sobolev oscillation type inequalities has been developed by M. Milman and J. Martı́n (see [27, 28] and [29]) and provide a considerable simpli- fication in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces. The paper is organized as follows. In Section 2, we introduce the notation and the stan- dard assumptions used in the paper, in Section 3, we will obtain oscillation type inequalities for spaces M s,X , we will see that they are equivalent to the lower bound for the growth of the measure and will obtain Sobolev type embedding of M s,X into a rearrangement invari- ant spaces. Finally, in the Appendix A we will give some properties of the measures we will be working with. 2 Preliminaries In this section we establish some further notation and background information and we provide more details about metrics spaces and r.i. spaces that we will working with. 3 The restriction on the Boyd indices is only required to guarantee that the inequality g ∗∗ X ≤ cX g X , holds for all g ∈ X. 4 As usual, the symbol f g will indicate the existence of a universal constant c > 0 (independent of all parameters involved) so that (1/c)f ≤ g ≤ c f , while the symbol f g means that f ≤ c g, and f g means that f ≥ c g. J. Martı́n, W.A. Ortiz 2.1 Metric Spaces Let (Ω, d) be a metric space. As usual a ball B in Ω with a center x and radius r > 0 is a set B = B(x, r) := {y ∈ Ω; d(x, y) < r}. Throughout the paper by a metric measure space we mean a triple (Ω, d, μ), where μ is a Borel measure on (Ω, d) such 0 < μ(B) < ∞, for every ball B in Ω, we also assume that μ(Ω) = ∞ and μ({x}) = 0 for all x ∈ Ω. We will say that a measure μ is α−lower bounded if there are b, α > 0 such that μ(B(x, r)) ≥ br α , (5) for all x ∈ Ω and r > 0. For simplicity we assume in what follows that μ(B(x, r)) ≥ r α . In what follows we will, furthermore, assume that the measure μ is continuous, i.e. μ satisfies that the map r → μ(B(x, r)) is continuous5 or that μ is doubling, i.e. there exists a constant CD such that, for all x ∈ Ω and for all r > 0, we have that μ(B(x, 2r)) ≤ CD μ(B(x, r)). Notice that in both cases there is a constant c = cμ ≥ 1 such that given t > 0, for all x ∈ Ω, there is a positive number r(x) such that t ≤ μ(B(x, r(x)) ≤ ct. In the doubling case, given x ∈ Ω, consider r0 (x) = sup {r : μ(B(x, r)) < t} and take r such that r < r0 (x) < 2r, then t ≤ μ(B(x, 2r)) ≤ CD μ(B(x, r)) ≤ CD t. In what follows we call these measures c−almost continuous6 . 2.2 Background on Rearrangement Invariant Spaces For measurable functions f : Ω → R, the distribution function of f is given by μf (t) = μ{x ∈ Ω : |f (x)| > t} (t > 0). The decreasing rearrangement fμ∗ of f is the right-continuous non-increasing function from [0, ∞) into [0, ∞] which is equimeasurable with f . Namely, fμ∗ (s) = inf{t ≥ 0 : μf (t) ≤ s}. We will write in what follows f ∗ instead of fμ∗ . In particular, for t > 0 we get (see [6, Prop. 1.7. Chapter 2]) μ(f ∗ (t)) = μ {x ∈ Ω : |f (x)| > f ∗ (t)} ≤ t. (6) It is easy to see that for any measurable set E ⊂ Ω μ(E) |f (x)| dμ ≤ f ∗ (s)ds. (7) E 0 Since f ∗ is decreasing, the function f ∗∗ , defined by 1 t f ∗∗ (t) = f ∗ (s)ds, (8) t 0 5 In the Appendix A we describe measures with this property. 6 An example of an α−lower bounded measure that does not satisfy this condition is given in the Appendix A Sobolev Embeddings for Fractional... is also decreasing and, moreover, f ∗ ≤ f ∗∗ . Remark 3 An elementary computation shows that ∂ ∗∗ f ∗∗ (t) − f ∗ (t) f (t) = − ∂t t and that the function t → t (f ∗∗ (t) − f ∗ (t)) is increasing. Moreover, it is well known and easy to see t f ∗∗ (t) − f ∗ (t) = |f (x)| − f ∗ (t) dt. (9) {x∈Ω:|f (x)|>f ∗ (t)} 2.2.1 Rearrangement Invariant Spaces We recall briefly the basic definitions and conventions we use from the theory of rearrangement-invariant (r.i.) spaces and refer the reader to [6, 25], for a complete treatment. We say that a Banach function space X = X(Ω) on (Ω, d, μ) is rearrangement-invariant (r.i.) space, if g ∈ X implies that all μ− measurable functions f with the same decreasing rearrangement function with respect to the measure μ, i.e. such that f ∗ = g ∗ , also belong to X, and, moreover, f X = g X . For any r.i. space X(Ω) we have L∞ (Ω) ∩ L1 (Ω) ⊂ X(Ω) ⊂ L1 (Ω) + L∞ (Ω), with continuous embedding. A r.i. space X(Ω) can be represented by an r.i. space on the interval (0, μ(Ω)), with Lebesgue measure, X̄ = X̄(0, μ(Ω)), such that f X = f∗ X̄ , for every f ∈ X. A characterization of the norm · X̄ is available (see [6, Theorem 4.10 and subsequent remarks]). Typical examples of r.i. spaces are the Lp -spaces, Lorentz spaces and Orlicz spaces. The associated space X (Ω) of X(Ω) is the r.i. space of all measurable functions h for which the r.i. norm given by |g(x)h(x)| dμ h X (Ω) = sup Ω (10) g=0 g X(Ω) is finite. Note that by the definition (10), the generalized Hölder inequality |g(x)h(x)| dμ ≤ g X(Ω) h X (Ω) (11) Ω holds. Let X(Ω) be an r.i. space. Then, the function φX : [0, ∞) → [0, ∞) given by φX (s) = χ[0,s) X̄ , is called the fundamental function of X(Ω). The fundamental function φX of any r.i. space X(Ω) is quasiconcave, in the sense that it is non-decreasing on [0, ∞), φX (s) = 0 and φX (s)/s is non-increasing on (0, ∞). Moreover, one has that (see [6, Theroem 5.2. Chapter 2.]) φX (s)φX (s) = s, (s > 0). (12) J. Martı́n, W.A. Ortiz Classically conditions on r.i. spaces are given in terms of the Hardy defined by 1 t 1/q 1 ∞ dx 1/q (q) P (q) f (t) = |f (x)|q , Qλ f (t) = |f (x)|q , t > 0, t 0 tλ t x 1−λ here 0 < q < ∞, 0 ≤ λ < 1. The boundedness of these operators on r.i. spaces can be described in terms of the so called Boyd indices7 defined by ln hX (s) ln hX (s) ᾱX = inf and α X = sup , s>1 ln s s<1 ln s where hX (s) denotes the norm of the compression/dilation operator Es on X̄, defined for s > 0, by Es f (t) = f ∗ ( st ). For example if X = Lp with p > 1, then ᾱX = α X = p1 . It is well known that (see [26], and [33]) 1 α X < ⇔ P (q) is bounded on X, (13) q λ (q) αX > ⇔ Qλ is bounded on X. (14) q 3 Symmetrization Inequalities and Embeddings for Fractional Hajłasz-Sobolev Spaces The method of proof of the following theorem follows the ideas of [30, Theorem 2] (see also [31]). Theorem 1 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous ∞ and α−lower bounded. Let s > 0, f ∈ M s,L +L and g ∈ D s (f ). Let 0 < p ≤ 1. Then, 1 for all t > 0, we have ∗∗ ∗ 1/p ∗∗ 1/p |f |p (t) − |f |p (t) ≤ Ct s/α gp (t) , (15) where C = C(c, p) is a constant that just depends on c and p. ∗∗ ∗ Proof Take t > 0 if |f |p (t) − |f |p (t) = 0, then (15) is obvious, otherwise let ∗ A = x ∈ Ω : |f (x)|p > |f |p (t) , notice that by (9) the set A is not empty. Given x ∈ A, since μ is c−almost continuous, there is a radius r(x) such that 2t ≤ μ(B(x, r(x))) ≤ 2ct. Let r = min((2t)1/α , r(x)), and for every x ∈ A, set ∗ Ax = y ∈ B(x, r) : |f (y)|p ≤ |f |p (t) . From B(x, r) = (B(x, r) ∩ A) ∪ Ax 7 Introduced by D.W. Boyd in [7]. Sobolev Embeddings for Fractional... we see that 2t ≤ μ(B(x, r)) ≤ μ(A) + μ(Ax ) ≤ t + μ(Ax ), (by (6)) whence t ≤ μ(Ax ). (16) Then ∗ I = |f (x)|p − |f |p (t) dμ(x) A 1 ≤ |f (x)|p − |f (y)|p dμ(y) dμ(x) A μ (Ax ) Ax 1 = |f (x)|p − |f (y)|p dμ(y) dμ(x) A μ (Ax ) Ax 1 ≤ |f (x)|p − |f (y)|p dμ(y)dμ(x) (by (16)) t A Ax 1 ≤ |f (x)|p − |f (y)|p dμ(y)dμ(x) (sinceAx ⊂ B(x, r)) t A B(x,r) = J. Taking into account that |x|p − |y|p ≤ |x − y|p , we get 1 J ≤ |f (x) − f (y)|p dμ(y)dμ(x) t A B(x,r) 1 ≤ d(x, y)sp (g(x) + g(y))p dμ(y)dμ(x). t A B(x,r) 1 ≤ r sp g(x)p + g(y)p dμ(y)dμ(x) t A B(x,r) sp/α (2t) ≤ g(x)p + g(y)p dμ(y)dμ(x) t A B(x,r) (2t)sp/α ≤ g(x)p dμ(y)dμ(x) + g(y)p dμ(y)dμ(x) t A B(x,r) A B(x,r) (2t)sp/α μ(B(x,r)) ≤ g(x)p μ (B(x, r)) dμ(x)+ g ∗ (z)p dz dμ(x) (by (7)) t A A 0 (2t)sp/α t 2ct ≤ 2ct g ∗ (z)p dz + g ∗ (z)p dz dμ(x) t 0 A 0 (2t)sp/α t 2ct ≤ 2ct g ∗ (z)p dz + t g ∗ (z)p dz (by (8)) t 0 0 t ≤ (2c)2 2sp/α t sp/α g ∗ (z)p dz. 0 Finally, the formula (see Remark 3) ∗∗ ∗ ∗ t |f |p (t) − |f |p (t) = |f (x)|p − |f |p (t) dμ(x) A J. Martı́n, W.A. Ortiz yields ∗∗ ∗ 1/p ∗∗ 1/p |f |p (t) − |f |p (t) ≤ Ct s/α gp (t) , as we wished to show. Theorem 2 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous. Then the following statements are equivalent (i) μ is α−lower bounded. ∞ If 0 < s ≤ 1, then for every f ∈ M s,L +L and g ∈ D s (f ), we have that 1 (ii) f ∗∗ (t) − f ∗ (t) ≤ Ct s/α g ∗∗ (t). (17) Proof If (i) holds, then by Theorem 1 we get (ii). Assume that (17) holds. Fix x0 ∈ Ω and r > 0. Define the function fr ,x0 by (r − d(x0 , x))s if d(x0 , x) ≤ r, fr ,x0 (x) = 0 if d(x0 , x) > r It is easy see that gr ,x0 (x) = χB(x0 ,r) , satisfies that fr ,x0 (x) − fr ,x0 (y) ≤ d(x, y)s gr ,x0 (x) + gr ,x0 (y) . An elementary computation shows that μ B(x0 , r − λ1/s ) if 0 < λ ≤ r s , μfr ,x0 (λ) = (18) 0 if λ > r s . By hypothesis, ∗∗ ∗ ∗∗ fr ,x0 (t) − fr ,x0 (t) ≤ Ct s/α gr ,x0 (t) Thus, ∗∗ ∗∗ fr ,x0 (0) − fr ,x0 (2μ(B(x0 , r))) 2μ(B(x0 ,r)) dt ∗∗ ∗ = fr ,x0 (t) − fr ,x0 (t) 0 t 2μ(B(x0 ,r)) ∗∗ ≤ C t s/α−1 gr ,x0 (t)dt. (19) 0 But 1 t μ(B(x0 , r)) ∗∗ gr ,x0 (t) = χ[0,μ(B(x0 ,r))) (s)ds = min 1, , t 0 t thus 2μ(B(x0 ,r)) ∗∗ I = t s/α−1 gr ,x0 (t)dt (20) 0 μ(B(x0 ,r)) 2μ(B(x0 ,r)) = t s/α−1 dt + μ(B(x0 , r)) t s/α−2 dt 0 μ(B(x0 ,r)) μ(B(x0 , r))s/α . Combining (19) and (20) we get ∗∗ ∗∗ fr ,x0 (0) − fr ,x0 (2μ(B(x0 , r))) μ(B(x0 , r))s/α . (21) On the other hand ∗∗ fr ,x0 (0) = fr ,x0 L∞ = rs (22) Sobolev Embeddings for Fractional... ∗ so from (18) we get that fr ,x0 (t) = 0 if t > μ(B(x0 , r)), therefore 1 2μ(B(x0 ,r)) ∗∗ ∗ fr ,x0 (2μ(B(x0 , r))) = fr ,x0 (t)dt 2μ(B(x0 , r)) 0 1 μ(B(x0 ,r)) ∗ = fr ,x0 (t)dt 2μ(B(x0 , r)) 0 1 = fr ,x0 L1 2μ(B(x0 , r)) 1 rs ≤ r s μ(B(x0 , r) = . (23) 2μ(B(x0 , r)) 2 Thus rs rs ∗∗ ∗∗ = rs − ≤ fr ,x0 (0) − fr ,x0 (2μ(B(x0 , r))) (by (22) and (21)) 2 2 μ(B(x0 , r))s/α (by (21)) which implies that μ is α−lower bounded up to constants. Remark 4 In Kalis’ 2007 PhD thesis at FAU (see also [20]) was proved that Sobolev embed- dings estimates for Hömander vector fields imply a lower bound for the growth of the measure. Theorem 3 provides us the following Sobolev embedding result for Fractional Hajłasz- Sobolev spaces. Theorem 3 Let (Ω, d, μ) be a metric measure space such that μ is c−almost continuous and α−lower bounded. Let X be a r.i. space. Let f ∈ M s,X and g ∈ D s (f ). (i) If s/α < 1, (a) If α X > s/α, then t −s/α f ∗∗ (t) X̄ g X . (b) If ᾱX < s/α, then f L∞ ≤ g X + f L1 +L∞ . (ii) If s/α = 1, then (f ∗∗ (t) − f ∗ (t)) sup φX (t) g X . t>0 t (iii) If s/α > 1, then f L∞ ≤ g X + f L1 +L∞ . Proof Case (i) Assume s/α < 1, then: (a) Condition α X > s/α, implies f ∗∗ (∞) = 0, so ∞ (f ∗∗ (z) − f ∗ (z)) dz ∗∗ ∗ s (1) (f (·) − f (·)) t −s/α f ∗∗ (t) = t −s/α zα s = Qs s (t). t zα z α (·) α J. Martı́n, W.A. Ortiz Hence (f ∗∗ (·) − f ∗ (·)) t −s/α f ∗∗ (t) (1) X̄ = Qs s (t) α (·) α X̄ f ∗∗ (t) − f ∗ (t) ≤ C (since α X > s/α). t s/α X̄ ≤ C t −s/α f ∗∗ (t) X̄ . Therefore, if ᾱX < 1, we have that f ∗∗ (t)−f ∗ (t) t −s/α f ∗∗ (t) X̄ t s/α X̄ ∗∗ g (t) X̄ g X . In case that ᾱX = 1, let 0 < p < 1, and consider the function |f |p . By (15) we have that ∗∗ ∗ 1/p ∗∗ 1/p t −sp/α |f |p (t) − |f |p (t) gp (t) . (24) The formula ∞ dz ∗∗ ∗∗ ∗ |f |p (t) = |f |p (z) − |f |p (z) t z ∞ 1/p p dz sp sp ∗∗ ∗ = zα z− α |f |p (z) − |f |p (z) , t z yields ∞ dz 1/p s ∗∗ 1/p sp sp s − pα ∗∗ ∗ 1/p p t− α |f |p (t) = t− α zα z |f |p (z)− |f |p (z) t z (p) −s ∗∗ ∗ 1/p = Q sp (·) pα |f |p (·) − |f |p (·) (t), α (p) Since α X > αs , the operator Q s is bounded on X (by (14)), thus α − αs ∗∗ − αs ∗∗ 1/p t f t |f |p (t) X̄ X̄ −s (p) p ∗∗ ∗ 1/p = Q s (·) pα |f | (·) − |f |p (·) (t) pα X̄ p ∗∗ p ∗ s − pα 1/p z |f | (z) − |f | (z) X̄ p ∗∗ 1/p g (t) (by (24)) X̄ g X (by (13)). (b) If ᾱX < s/α, by Theorem 2.3 of [8], we have that f ∗∗ (t) − f ∗ (t) f L∞ ≤ + f L1 +L∞ t s/α X̄ g ∗∗ (t) X̄ g X . Case (ii) If s/α = 1, then f ∗∗ (t) − f ∗ (t) 1 t φX (t) ≤ Cg ∗∗ (t) = g ∗ (s)ds ≤ g X (by (11)). t t 0 t Sobolev Embeddings for Fractional... Consequently, (f ∗∗ (t) − f ∗ (t)) φX (t) g X (by (12)). t Case (3) If s/α > 1, then f ∗∗ (t) − f ∗ (t) 1 t φX (t) g ∗ (s)ds ≤ g X . t s/α t 0 t On the other hand 1 dz 1 f ∗∗ (z) − f ∗ (z) f ∗∗ (0) − f ∗∗ (1) = f ∗∗ (z) − f ∗ (z) = zs/α−1 dz 0 z 0 zs/α 1 φX (z) g X zs/α−1 dz 0 z 1 ≤ g X φX (1) zs/α−2 dz 0 g X . Finally, using that f L∞ = f ∗∗ (0), we obtain f L∞ ≤ g X + f L1 +L∞ . As we wished to show. Remark 5 In the classical setting, embedding theorems for W 1,p (Rn ) have different behavior when p < n, p = n,or p > n. The point is that |B(x, r)| r n. (25) In the metric-measure context the counterpart condition (25) is provided by the lower bound for the growth of the measure (5). Notice also the different character on the embeddings for p < n, p = n, or p > n in the r.i. context is done by the role of the Boyd index. Corollary 1 Under conditions of Theorem 3, in the particular case of X = Lp , (0 < p < ∞), we obtain (i) If s/α < 1, 1 (a) If s/α > p, then f Lp ∗ ,p g Lp . where p ∗ = αp α−sp . (b) If 1 p = s/α, then ⎛ ⎛ ⎞p ⎞1/p 1 f ∗∗ (t) ⎝ ⎝ ⎠ dt ⎠ g Lp + f L1 +L∞ . 0 1 + ln t 1 t 1 (c) If p < s/α, then f L∞ ≤ g Lp + f L1 +L∞ . (ii) If s/α = 1, then (f ∗∗ (t) − f ∗ (t)) sup t 1/p g Lp . t>0 t J. Martı́n, W.A. Ortiz In particular, if Lp = L1 , we obtain f L∞ w f M s,Lp , where L∞ w is the weak ∞ L −space 8 (iii) If s/α > 1, then f L∞ ≤ g Lp + f L1 +L∞ . Proof Except (i.b), the remaining statements are a particular case of Theorem 3. To see (i.b), by Theorem 1, we have that t −1/p f ∗∗ (t) − f ∗ (t) ≤ Cg ∗∗ (t). Hence ∞ p 1/p t −1/p f ∗∗ (t) − f ∗ (t) dt g Lp , 0 and by [4, Lemma 5.4] we have that ⎛ ⎛ ⎞p ⎞1/p 1 f ∗∗ (t) ∞ 1/p ⎠ dt ⎠ p ⎝ ⎝ t −1/p f ∗∗ (t) − f ∗ (t) dt + f L1 +L∞ . 0 1 + ln 1 t 0 t Appendix A A.1 Description of Measures μ Satisfying that the Map r → μ(B (x , r )) is Continuous Let (Ω, d) be a metric measure space, we say that a measure μ is metrically continuous with respect to metric d if all x ∈ Ω and all r > 0 it holds that lim μ(B(x, r) B(y, r)) = 0 d(x,y)→0 where A B stands for a symmetric difference of sets A, B ⊂ Ω and is defined as follows: A B := (A\B) ∪ (B\A). The following lemma collects some basic facts about continuity of a measure with respect to the metric (see [9] and [1] for the proof). Lemma 1 Let (Ω, d, μ) be a metric space with a Borel regular measure μ. Then the following hold: (i) If μ is continuous with respect to the metric d, then the map x → μ(B(x, r)) is continuous in d. (ii) If for every x ∈ Ω and every r > 0 it holds that μ(∂B(x, r)) = 0, then μ is continuous with respect to the metric d. (iii) If for every x ∈ Ω the function r → μ(B(x, r)) is continuous, then μ is continuous with respect to the metric d. It is easy to see that if we take Rn with Lebesgue measure (or with an absolutely continuous measure respect to the Lebesgue measure) with the Euclidean distance, then 8 Bennett, DeVore and Sharpley introduced the space weak L∞ defined as f L∞ = supt>0 (f ∗∗ (t) − f ∗ (t)) w in [5] where studied its relationship with functions of bounded mean oscillation Sobolev Embeddings for Fractional... this measure is metrically continuous. In fact we have more (see [9, Proposition 2.1]) if (Ω, d, μ) and (Ω, d, ν) are metric measure spaces then if μ ≺≺ ν and ν is metrically continuous, then μ is metrically continuous too. An important example is the following (see [32]). Lemma 2 Let μ be a nonnegative Radon measure on Rn . Assume that for any point p ∈ Rn , μ({p}) = 0, then we choose the coordinate axes in such a way that μ(∂Q) = 0 for all cubes Q with sides parallel to the axes. In particular the function → μ(Q(x, )) where denotes the length of the edge and x is the center of Q, is continuous. A.2 An Example of an α−lower Bounded Measure which is not c −almost Continuous Consider R2 with the distance d∞ (x, y) = max {|x| , |y|} and the measure μ = Lebesgue measure in the plane + length measure in vertical axis + length measure in vertical straight line passing through (1, 0). It is easy to see that r2 ≤ μ(B(x, r)) 4 however, is not c−almost continuous. Acknowledgements The authors are grateful to professor Xavier Tolsa for providing us the above example and to the referee for his/her useful suggestions to improve the quality of the paper. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. Data availability statement Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Declarations Conflict of Interests The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Adamowicz, T., Gaczkowski, M., Górka, P.: Harmonic functions on metric measure spaces. Rev. Mat. Complut. 32, 141–186 (2019) 2. Alvarado, R., Górka, P., Hajł, a.sz., P.: Sobolev embedding for M 1 , p spaces is equivalent to a lower bound of the measure. J. Funct Anal. 279, 1–39 (2020) 3. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces Oxford Lecture Series in Mathematics and Its Applications, vol. 25. Oxford University Press, Oxford (2004) 4. Bastero, J., Milman, M., Ruiz, F.: A note in L(∞, q) spaces and Sobolev embeddings. Indiana Univ. Math. J. 52, 1215–1230 (2003) 5. Bennett, C., DeVore, R.A., Sharpley, R.: Weak-l ∞ and BMO. Ann. Math. 113, 601–611 (1981) 6. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988) J. Martı́n, W.A. Ortiz 7. Boyd, D.W.: Indices of function spaces and their relationship to interpolation. Canad. J. Math. 21, 1245– 1254 (1969) 8. Feo, F., Martı́n, J., Posteraro, M.R.: Sobolev anisotropic inequalities with monomial weights. J. Math. Anal. Appl. 505(1), 125557 (2022) 9. Gaczkowski, M., Górka, P.: Harmonic functions on metric measure spaces: Convergence and compact- ness. Potential Anal. 31, 203–214 (2009) 10. Górka, P.: In metric-measure spaces Sobolev embedding is equivalent to a lower bound for the measure. Potential Anal. 47, 13–19 (2017) 11. Hajłasz, P.: Sobolev spaces on an arbitrary metric spaces. Potential Anal. 5, 403–415 (1996) 12. Hajłasz, P.: Sobolev spaces on metric-measure spaces. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris 2002). Contemp. Math, Amer. Math. Soc. Providence, RI. 338, 173–218 (2003) 13. Heikkinen, T., Koskela, P., Tuominen, H.: Sobolev-type spaces from generalized poincaré inequalities. Studia Math. 181, 1–16 (2007) 14. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2001) 15. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998) 16. Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev spaces on metric measure spaces an approach based on upper gradients (2015) 17. Hu, J.: A note on hajłasz-sobolev spaces on fractals. J. Math. Anal. Appl. 280, 91–101 (2003) 18. Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254, 1217–1234 (2008) 19. Jiang, R., Shanmugalingam, N., Yang, D., Yuan, W.: Hajłasz gradients are upper gradients. J. Math. Anal. Appl. 422, 397–407 (2015) 20. Kalis̆, J., Milman, M.: Symmetrization and sharp Sobolev inequalities in metric spaces. Rev. Mat. Complut. 22, 499–515 (2009) 21. Karak, N.: Sobolev embedding implies regularity of measure in metric measure spaces. arXiv: 1903.02342v2 (2019) 22. Karak, N.: Lower bound of measure and embeddings of Sobolev, Besov and Triebel-Lizorkin spaces. Math. Nach. 293, 120–128 (2020) 23. Korobenko, L.: Orlicz Sobolev inequalities and the doubling condition. Ann. Fenn Math. 46(1), 153–161 (2021) 24. Koskela, P., Saksman, E.: Pointwise characterizations of Hardy-Sobolev functions. Math. Res. Lett. 15, 727–744 (2008) 25. Krein, S.G., Petunin, Y.I., Semenov, E.M.: Interpolation of linear operators, Transl. Math. Monogr. Amer. Math Soc. 54 Providence (1982) 26. Martı́n, J., Milman, M.: Weighted norm inequalities and indices. J. Funct. Spaces Appl. 4, 43–71 (2006) 27. Martı́n, J., Milman, M.: Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225, 121–199 (2010) 28. Martı́n, J., Milman, M.: Fractional Sobolev inequalities: symmetrization. Isoperimetry and Interpolation, Astérisque 366, x+127 (2014) 29. Martı́n, J., Milman, M.: Isoperimetry and Symmetrization for Logarithmic Sobolev inequalities. J. Funct. Anal. 256, 149–178 (2009) 30. Martı́n, J., Milman, M.: Symmetrization inequalities in and Sobolev embeddings. Proc. Am. Math. Sci. 134, 2335–2347 (2006) 31. Mastyło, M.: The modulus of smoothness in metric spaces and related problems. Potential Anal. 35, 301–328 (2011) 32. Mateu, J., Mattila, P., Nicolau, A., Orobitg, J.: BMO For nondoubling measures. Duke Math. J. 102, 533–565 (2000) 33. Montgomery-Smith, S.: The Hardy Operator and Boyd Indices, in interaction between probability, har- monic analysis and functional analysis. In: Kalton, N., Montgomery-Smith, S.J., Saab, E. (eds.) Lecture Notes in Pure and Applied Mathematics, vol. 175. Marcel Dekker, New York (1995) 34. Ohno, T., Shimomura, T.: Musielak-orlicz-sobolev spaces on metric measure spaces. Czech Math J. 65, 435–474 (2015) 35. Yang, D.: New characterizations of Hajłasz-Sobolev spaces on metric spaces. Sci. China Ser. A-Math. 46, 675–689 (2003) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.