# Generalized Lebesgue Points for Hajłasz Functions

Authors: Toni Heikkinen
Journal: Journal of Function Spaces
 (2018)

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## Abstract

LetX be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.


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## Body

## 1. Introduction and Main Results

LetP=(P,d,μ) be a doubling metric measure space. By the Lebesgue differentiation theorem, almost every point of a locally integrable function is a Lebesgue point. As expected, for smoother functions, the set of non-Lebesgue points is smaller. In [1], Kinnunen and Latvala showed that, for a quasicontinuous Hajłasz–Sobolev function u∈M1,p(P), p>1, there exists a set E of M1,p-capacity zero such that(1)limr→01μBx,r∫Bx,ruydμy=uxfor every x∈P∖E. The case p=1 was studied in [2, 3]. Recently, in [4] and independently in [5], it was shown that if(2)μBy,rμBx,R≥crRQfor every x∈P, 0<r≤R and y∈B(x,R), then for every quasicontinuous u∈M˙s,p with p>Q/(Q+s), (1) holds true outside a set of Ms,p-capacity zero.If we replace integral averages in (1) by medians, then the result holds true also for small p>0. For 0<γ<1, A⊂P, and u∈L0, denote(3)muγA=infa∈R:μx∈A:ux>a<γμA. If p>0 and u∈M˙s,p is quasicontinuous, then by [4, Theorem 1.2], there exists a set E⊂P of Ms,p-capacity zero such that(4)limr→0muγBx,r=ux,for every x∈P∖E and 0<γ≤1/2.In this paper, we will study the existence of (generalized) Lebesgue points for functionsu whose Hajłasz gradient belongs to a general quasi-Banach function space X. This approach allows us to simultaneously cover, for example, Orlicz–Hajłasz, Lorentz–Hajłasz, and variable exponent Hajłasz functions.For0<γ<1, 0<R≤∞, u∈L0(P), and x∈P, denote(5)MRγux=sup0<r<RmuγBx,r. Operator Mγ=M∞γ and its variants have turned out to be useful in harmonic analysis and in the theory of function spaces; see, for example, [4, 6–24].Theorem 1.
Letμ be doubling. Suppose that X has absolutely continuous quasinorm and that, for every 0<γ<1 and for every ball B⊂P, there exists a constant C such that(6)M1γgχBX≤CgXfor every g∈X. Let 0<s≤1. Then, for every quasicontinuous u∈M˙s,X(P), there exists a set E⊂P with CMs,X(E)=0 such that(7)limr→0mu-uxγBx,r=0,limr→0muγBx,r=uxfor every x∈P∖E and 0<γ<1.We say that a pointx∈P satisfying (7) for every 0<γ<1 is ageneralized Lebesgue point of u.The (restricted) Hardy–Littlewood maximal function of a locally integrable functionu is(8)MRux=sup0<r<R1μBx,r∫Bx,ruydμy. As usual, we denote M=M∞.Theorem 2.
Let0<s≤1 and let μ satisfy (2). Denote q=Q/(Q+s). Suppose that X has absolutely continuous quasinorm and that, for every ball B⊂P, there exists a constant C such that(9)M1gq1/qχBX≤CgXfor every 0≤g∈X. Then, for every quasicontinuous u∈M˙s,X(P), quasievery point is a Lebesgue point of u.For any quasi-Banach function spaceX over P, define ΦX:(0,1)→[1,∞];(10)ΦXγ=supfX≤1MγfX. Thegeneralized upper Boyd index of X is(11)αX=limγ→0log⁡ΦXγlog⁡1/γ. The generalized upper Boyd index was introduced in [20], where it was shown that the Hardy–Littlewood maximal operator is bounded on X(Rn) if and only if αX<1. As a corollary of Theorems 1 and 2 we have the following result.Theorem 3.
Suppose thatX has absolutely continuous quasinorm and that μ satisfies (2). Let 0<s≤1 and let u∈M˙s,X(P) be quasicontinuous. (i)
IfαX<∞, then quasievery point is a generalized Lebesgue point of u.(ii)
IfαX<(Q+s)/Q, then quasievery point is a Lebesgue point of u.


## 2. Preliminaries

### 2.1. Basic Assumptions

In this paper,P=(P,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 ball is positive and finite. The doubling property means that there exists a constant cd>0 such that(12)μBx,2r≤cdμBx,r for every ball B(x,r)={y∈P:d(y,x)<r}, where x∈P and r>0.The doubling condition is equivalent to the existence of constantsc and Q such that (2) holds for every 0<r≤R, x∈P, and y∈B(x,R).The integral average of a locally integrable function u over a set A of positive and finite measure is(13)uA=⨍Audμ=1μA∫Audμ.ByχE, we denote the characteristic function of a set E⊂P and by R¯, the extended real numbers [-∞,∞]. L0=L0(P) is the set of all measurable, almost everywhere finite functions u:P→R¯. In general, C and c are positive constants whose values are not necessarily same at each occurrence. When we want to stress that the constant depends on other constants or parameters a,b,…, we write C=C(a,b,…).


### 2.2. Quasi-Banach Function Spaces

A quasinorm on a subspace ofL0(P) is a functional · such that(i)
f=0⟺f=0 a.e.;(ii)
αf=αf for every α∈R;(iii)
there exists a constantcΔ such that f+g≤cΔ(f+g).A quasi-Banach function spaceX over P is a subspace of L0(P) equipped with a complete quasinorm ·X that has the following properties:(i)
g∈X and f≤g a.e. ⇒f∈X and fX≤gX;(ii)
μ(E)<∞⇒χEX<∞;(iii)
0≤fk↑f a.e. ⇒fkX↑fX.By the Aoki–Rolewicz theorem ([25, 26]), there exists a constant 0<ρ≤1 such that(14)∑k=1∞fkX≤41/ρ∑k=1∞fkXρ1/ρfor all f1,f2,…∈X.A quasinorm·X on X is absolutely continuous if fχEkX→0 whenever f∈X and (Ek)k=1∞ is a decreasing sequence of sets such that ⋂k=1∞Ek=∅.


### 2.3. Hajłasz Spaces

Let0<s<∞. A measurable function g:P→[0,∞] is an s-gradient of a function u∈L0(P) if there exists a set E⊂P with μ(E)=0 such that, for all x,y∈X∖E,(15)ux-uy≤dx,ysgx+gy.The collection of all s-gradients of u is denoted by Ds(u).Thehomogeneous Hajłasz spaceM˙s,X=M˙s,X(P) consists of measurable functions u for which(16)uM˙s,X=infg∈DsugX is finite. TheHajłasz spaceMs,X=Ms,X(P) is M˙s,X∩X equipped with the norm(17)uMs,X=uX+uM˙s,X.Hajłasz spacesMs,p(P)=Ms,Lp(P) were introduced in [27] for s=1, p≥1 and in [28] for fractional scales. Recall that, for p>1, M1,p(Rn)=W1,p(Rn) (see [27]), whereas for n/(n+1)<p≤1, M1,p(Rn) coincides with the Hardy–Sobolev space H1,p(Rn) by [29, Thm 1].Next two lemmas fors-gradients follow easily from the definition; see [30, Lemmas 2.4] and [1, Lemma 2.6].Lemma 4.
Letu,v∈L0(P), g∈Ds(u), and h∈Ds(v). Then max{g,h} is an s-gradient of functions max⁡{u,v} and min{u,v}.Lemma 5.
Letui∈L0(P) and gi∈Ds(ui), i∈N. Let u=supi∈Nui and g=supi∈Ngi. If u∈L0(P), then g∈Ds(u).The following lemma is essential [4, Lemma 7.2].Lemma 6.
Let0<s≤1 and let S⊂P be a measurable set. Let u:P→R be a measurable function with g∈Ds(u) and let φ be a bounded L-Lipschitz function supported in S. Then(18)h=φ∞g+2φ∞1-sLsuχS∈Dsuφ. Consequently, there exists a constant C=C(s,φ∞,L) such that(19)uφMs,X≤CuχSX+infg∈DsugχSX for every u∈M˙s,X.Lemma 7.
Let0<s≤1 and suppose that X has absolutely continuous quasinorm. Then s-Hölder continuous functions are dense in Ms,X.Proof.
Letu∈Ms,X,g∈Ds(u)∩X and let E be the exceptional set for (15). Then u is s-Hölder continuous with constant 2λ in the set Eλ={x∈P∖E:g(x)≤λ}. By [31], there is an extension uλ of u∣Eλ to P such that uλ is s-Hölder continuous with constant 2λ. It is easy to see that (g+3λ)χP∖Eλ∈Ds(u-uλ); see [32, Proposition 4.5]. By the absolute continuity of ·X, g+3λχP∖EλX→0 and u-uλX≤2uχP∖EλX→0 as λ→0.


### 2.4.γ-Median

Let0<γ<1. The γ-median of a measurable function u:P→R¯ over a set A of finite measure is(20)muγA=infa∈R:μx∈A:ux>a<γμA. Note that if u∈L0(A) and 0<μ(A)<∞, then muγ(A) is finite.In the following lemma, we list some basic properties of theγ-median. Properties (a), (b), (d), (f), and (g) follow from [21, Propositions 1.1 and 1.2] and (h) and (i) from [21, Theorem 2.1]. The remaining properties (c) and (e) follow immediately from the definition.Lemma 8.
Theγ-median has the following properties: (a)
Ifγ≤γ′, then muγ(A)≥muγ′(A).(b)
Ifu≤v almost everywhere, then muγ(A)≤mvγ(A).(c)
IfA⊂B and μ(B)≤Cμ(A), then muγ(A)≤muγ/C(B).(d)
Ifc∈R, then muγ(A)+c=mu+cγ(A).(e)
Ifc>0, then mcuγ(A)=cmuγ(A).(f)
muγA≤mumin{γ,1-γ}(A).(g)
mu+vγ(A)≤muγ/2(A)+mvγ/2(A).(h)
Ifu is continuous, then for every x∈P,(21)limr→0muγBx,r=ux.(i)
Ifu∈L0(P), then there exists a set E with μ(E)=0 such that(22)limr→0muγBx,r=uxfor every0<γ<1 and x∈P∖E.


### 2.5. Discrete Maximal Functions

In this subsection, we define “discrete” versions of the Hardy–Littlewood maximal function(23)MRux=sup0<r<RuBx,r. and the median maximal function(24)MRγux=sup0<r<RmuγBx,r.We first recall the definition of a discrete convolution. Discrete convolutions are basic tools in harmonic analysis in homogeneous spaces; see, for example, [33, 34]. The following lemma is well known.Lemma 9.
For everyr>0, there exists a collection of balls {Bi=B(xi,r):i∈I}, where I⊂N, and functions φi:P→[0,1], i∈I, with the following properties: (a)
The ballsB(xi,r/2), i∈I, cover P.(b)
∑i∈Iχ2Bi≤C.(c)
For everyi∈I, φi is C/r-Lipschitz, φi≥C-1 on Bi and φi=0 outside 2Bi.(d)
∑i∈Iφi=1.
Here, the constantC depends only on the doubling constant cd.For each scaler>0, we choose a collection of balls {Bi:i∈I} and a collection of functions {φi:i∈I} satisfying conditions (a)–(d) of Lemma 9.Definition 10.
Thediscrete convolution of a locally integrable function u at the scale r is(25)ur=∑i∈IuBiφi,where {Bi:i∈I} and {φi:i∈I} are the chosen collections of balls and functions for the scale r.
Thediscrete maximal function of u is MR∗u:P→R¯,(26)MR∗ux=supq∈Q,0<q<Ruqx.The discrete maximal function, which can be seen as a smooth version of the Hardy–Littlewood maximal function, was introduced in [1].Similarly, we define median versions of a discrete convolution and a discrete maximal function.Definition 11.
Let0<γ<1. Thediscreteγ-median convolution of a function u∈L0 at scale r>0 is(27)urγ=∑i∈ImuγBiφi, where {Bi:i∈I} and {φi:i∈I} are as in Definition 10.
Thediscreteγ-median maximal function of u is MRγ,∗u:P→R¯,(28)MRγ,∗ux=supq∈Q,0<q<Ruqγx.As a supremum of continuous functions, the discrete maximal functions are lower semicontinuous and hence measurable.Lemma 12.
Let0<R≤∞. There exists a constant C=C(cd)≥1 such that(29)C-1MR/2u≤MR∗u≤CM3Rufor every u∈Lloc1 and(30)MR/2γu≤CMRγ/C,∗u,MRγ,∗u≤CM3Rγ/Cufor every u∈L0 and 0<γ<1.Proof.
We will prove (30). The proof of (29) is similar. Let x∈P, r>0, and let urγ=∑i∈Imuγ(Bi)φi be as in Definition 11. If x∈2Bi, then Bi⊂B(x,3r). By the doubling property, μ(B(x,3r))≤Cμ(Bi), and so, by Lemma 8(c), muγ(Bi)≤muγ/C(B(x,3r)). Since ∑i∈Iχ2Bi(x)≤C, it follows that (31)urγx≤Cmγ/CBx,3r.
On the other hand, since the balls(1/2)Bi, i∈I, cover P, there is i∈I such that B(x,r/2)⊂Bi. By the doubling property, μ(Bi)≤Cμ(B(x,r/2)), and so, by Lemma 8(c), muγ(B(x,r/2))≤muγ/C(Bi). Since φi≥C-1 on Bi, we have that (32)muγBx,r2≤Curγ/Cx. The claim (30) follows immediately from these estimates.


## 2.1. Basic Assumptions

In this paper,P=(P,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 ball is positive and finite. The doubling property means that there exists a constant cd>0 such that(12)μBx,2r≤cdμBx,r for every ball B(x,r)={y∈P:d(y,x)<r}, where x∈P and r>0.The doubling condition is equivalent to the existence of constantsc and Q such that (2) holds for every 0<r≤R, x∈P, and y∈B(x,R).The integral average of a locally integrable function u over a set A of positive and finite measure is(13)uA=⨍Audμ=1μA∫Audμ.ByχE, we denote the characteristic function of a set E⊂P and by R¯, the extended real numbers [-∞,∞]. L0=L0(P) is the set of all measurable, almost everywhere finite functions u:P→R¯. In general, C and c are positive constants whose values are not necessarily same at each occurrence. When we want to stress that the constant depends on other constants or parameters a,b,…, we write C=C(a,b,…).


## 2.2. Quasi-Banach Function Spaces

A quasinorm on a subspace ofL0(P) is a functional · such that(i)
f=0⟺f=0 a.e.;(ii)
αf=αf for every α∈R;(iii)
there exists a constantcΔ such that f+g≤cΔ(f+g).A quasi-Banach function spaceX over P is a subspace of L0(P) equipped with a complete quasinorm ·X that has the following properties:(i)
g∈X and f≤g a.e. ⇒f∈X and fX≤gX;(ii)
μ(E)<∞⇒χEX<∞;(iii)
0≤fk↑f a.e. ⇒fkX↑fX.By the Aoki–Rolewicz theorem ([25, 26]), there exists a constant 0<ρ≤1 such that(14)∑k=1∞fkX≤41/ρ∑k=1∞fkXρ1/ρfor all f1,f2,…∈X.A quasinorm·X on X is absolutely continuous if fχEkX→0 whenever f∈X and (Ek)k=1∞ is a decreasing sequence of sets such that ⋂k=1∞Ek=∅.


## 2.3. Hajłasz Spaces

Let0<s<∞. A measurable function g:P→[0,∞] is an s-gradient of a function u∈L0(P) if there exists a set E⊂P with μ(E)=0 such that, for all x,y∈X∖E,(15)ux-uy≤dx,ysgx+gy.The collection of all s-gradients of u is denoted by Ds(u).Thehomogeneous Hajłasz spaceM˙s,X=M˙s,X(P) consists of measurable functions u for which(16)uM˙s,X=infg∈DsugX is finite. TheHajłasz spaceMs,X=Ms,X(P) is M˙s,X∩X equipped with the norm(17)uMs,X=uX+uM˙s,X.Hajłasz spacesMs,p(P)=Ms,Lp(P) were introduced in [27] for s=1, p≥1 and in [28] for fractional scales. Recall that, for p>1, M1,p(Rn)=W1,p(Rn) (see [27]), whereas for n/(n+1)<p≤1, M1,p(Rn) coincides with the Hardy–Sobolev space H1,p(Rn) by [29, Thm 1].Next two lemmas fors-gradients follow easily from the definition; see [30, Lemmas 2.4] and [1, Lemma 2.6].Lemma 4.
Letu,v∈L0(P), g∈Ds(u), and h∈Ds(v). Then max{g,h} is an s-gradient of functions max⁡{u,v} and min{u,v}.Lemma 5.
Letui∈L0(P) and gi∈Ds(ui), i∈N. Let u=supi∈Nui and g=supi∈Ngi. If u∈L0(P), then g∈Ds(u).The following lemma is essential [4, Lemma 7.2].Lemma 6.
Let0<s≤1 and let S⊂P be a measurable set. Let u:P→R be a measurable function with g∈Ds(u) and let φ be a bounded L-Lipschitz function supported in S. Then(18)h=φ∞g+2φ∞1-sLsuχS∈Dsuφ. Consequently, there exists a constant C=C(s,φ∞,L) such that(19)uφMs,X≤CuχSX+infg∈DsugχSX for every u∈M˙s,X.Lemma 7.
Let0<s≤1 and suppose that X has absolutely continuous quasinorm. Then s-Hölder continuous functions are dense in Ms,X.Proof.
Letu∈Ms,X,g∈Ds(u)∩X and let E be the exceptional set for (15). Then u is s-Hölder continuous with constant 2λ in the set Eλ={x∈P∖E:g(x)≤λ}. By [31], there is an extension uλ of u∣Eλ to P such that uλ is s-Hölder continuous with constant 2λ. It is easy to see that (g+3λ)χP∖Eλ∈Ds(u-uλ); see [32, Proposition 4.5]. By the absolute continuity of ·X, g+3λχP∖EλX→0 and u-uλX≤2uχP∖EλX→0 as λ→0.


## 2.4.γ-Median

Let0<γ<1. The γ-median of a measurable function u:P→R¯ over a set A of finite measure is(20)muγA=infa∈R:μx∈A:ux>a<γμA. Note that if u∈L0(A) and 0<μ(A)<∞, then muγ(A) is finite.In the following lemma, we list some basic properties of theγ-median. Properties (a), (b), (d), (f), and (g) follow from [21, Propositions 1.1 and 1.2] and (h) and (i) from [21, Theorem 2.1]. The remaining properties (c) and (e) follow immediately from the definition.Lemma 8.
Theγ-median has the following properties: (a)
Ifγ≤γ′, then muγ(A)≥muγ′(A).(b)
Ifu≤v almost everywhere, then muγ(A)≤mvγ(A).(c)
IfA⊂B and μ(B)≤Cμ(A), then muγ(A)≤muγ/C(B).(d)
Ifc∈R, then muγ(A)+c=mu+cγ(A).(e)
Ifc>0, then mcuγ(A)=cmuγ(A).(f)
muγA≤mumin{γ,1-γ}(A).(g)
mu+vγ(A)≤muγ/2(A)+mvγ/2(A).(h)
Ifu is continuous, then for every x∈P,(21)limr→0muγBx,r=ux.(i)
Ifu∈L0(P), then there exists a set E with μ(E)=0 such that(22)limr→0muγBx,r=uxfor every0<γ<1 and x∈P∖E.


## 2.5. Discrete Maximal Functions

In this subsection, we define “discrete” versions of the Hardy–Littlewood maximal function(23)MRux=sup0<r<RuBx,r. and the median maximal function(24)MRγux=sup0<r<RmuγBx,r.We first recall the definition of a discrete convolution. Discrete convolutions are basic tools in harmonic analysis in homogeneous spaces; see, for example, [33, 34]. The following lemma is well known.Lemma 9.
For everyr>0, there exists a collection of balls {Bi=B(xi,r):i∈I}, where I⊂N, and functions φi:P→[0,1], i∈I, with the following properties: (a)
The ballsB(xi,r/2), i∈I, cover P.(b)
∑i∈Iχ2Bi≤C.(c)
For everyi∈I, φi is C/r-Lipschitz, φi≥C-1 on Bi and φi=0 outside 2Bi.(d)
∑i∈Iφi=1.
Here, the constantC depends only on the doubling constant cd.For each scaler>0, we choose a collection of balls {Bi:i∈I} and a collection of functions {φi:i∈I} satisfying conditions (a)–(d) of Lemma 9.Definition 10.
Thediscrete convolution of a locally integrable function u at the scale r is(25)ur=∑i∈IuBiφi,where {Bi:i∈I} and {φi:i∈I} are the chosen collections of balls and functions for the scale r.
Thediscrete maximal function of u is MR∗u:P→R¯,(26)MR∗ux=supq∈Q,0<q<Ruqx.The discrete maximal function, which can be seen as a smooth version of the Hardy–Littlewood maximal function, was introduced in [1].Similarly, we define median versions of a discrete convolution and a discrete maximal function.Definition 11.
Let0<γ<1. Thediscreteγ-median convolution of a function u∈L0 at scale r>0 is(27)urγ=∑i∈ImuγBiφi, where {Bi:i∈I} and {φi:i∈I} are as in Definition 10.
Thediscreteγ-median maximal function of u is MRγ,∗u:P→R¯,(28)MRγ,∗ux=supq∈Q,0<q<Ruqγx.As a supremum of continuous functions, the discrete maximal functions are lower semicontinuous and hence measurable.Lemma 12.
Let0<R≤∞. There exists a constant C=C(cd)≥1 such that(29)C-1MR/2u≤MR∗u≤CM3Rufor every u∈Lloc1 and(30)MR/2γu≤CMRγ/C,∗u,MRγ,∗u≤CM3Rγ/Cufor every u∈L0 and 0<γ<1.Proof.
We will prove (30). The proof of (29) is similar. Let x∈P, r>0, and let urγ=∑i∈Imuγ(Bi)φi be as in Definition 11. If x∈2Bi, then Bi⊂B(x,3r). By the doubling property, μ(B(x,3r))≤Cμ(Bi), and so, by Lemma 8(c), muγ(Bi)≤muγ/C(B(x,3r)). Since ∑i∈Iχ2Bi(x)≤C, it follows that (31)urγx≤Cmγ/CBx,3r.
On the other hand, since the balls(1/2)Bi, i∈I, cover P, there is i∈I such that B(x,r/2)⊂Bi. By the doubling property, μ(Bi)≤Cμ(B(x,r/2)), and so, by Lemma 8(c), muγ(B(x,r/2))≤muγ/C(Bi). Since φi≥C-1 on Bi, we have that (32)muγBx,r2≤Curγ/Cx. The claim (30) follows immediately from these estimates.


## 3. Capacity

In this section, we define theMs,X-capacity and prove some of its basic properties.Definition 13.
Let0<s<∞. The Ms,X-capacity of a set E⊂P is(33)CMs,XE=infuMs,X:u∈AMs,XE, where(34)AMs,XE=u∈Ms,X:u≥1on  a  neighbourhood  ofEis a set of admissible functions for the capacity. We say that a property holdsquasieverywhere if it holds outside a set of Ms,X-capacity zero.Remark 14.
Lemma4 easily implies that(35)CMs,XE=infuMs,X:u∈AMs,X′E, where AMs,X′(E)={u∈AMs,X(E):0≤u≤1}.Remark 15.
It is easy to see that theMs,X-capacity is an outer capacity, which means that(36)CMs,XE=infCMs,XU:U⊃E,Uisopen.TheMs,X-capacity is not generally subadditive, but for most purposes, it suffices that it satisfies inequality (37) below.Lemma 16.
Let0<s<∞ and let ρ be the constant from (14). Then(37)CMs,X⋃i∈IEiρ≤8∑i∈ICMs,XEiρwhenever Ei⊂P, i∈I⊂N.Proof.
Letε>0. We may assume that ∑i∈ICMs,X(Ei)ρ<∞. There are functions ui∈AMs,X′(Ei) with gi∈Ds(ui) such that(38)uiX+giXρ<CMs,XEiρ+2-iε. By Lemma 5, u=supi∈Iui∈AMs,X′(⋃i∈IEi) and g=supi∈Ngi∈Ds(u). Hence(39)CMs,X⋃i∈IEiρ≤uX+gXρ≤uXρ+gXρ≤4∑uiXρ+giXρ≤8∑uiX+giXρ≤8ε+∑i∈ICMs,XEiρand the claim follows by letting ε→0.A functionu:P→[-∞,∞] isquasicontinuous if for every ε>0, there exists a set E⊂P such that CMs,X(E)<ε and the restriction of u to P∖E is continuous. By Remark 15, the set E can be chosen to be open.The following lemma follows from a result of Kilpeläinen [35].Lemma 17.
Suppose thatu and v are quasicontinuous. If u=v almost everywhere in an open set U, then u=v quasieverywhere in U.Lemma 18.
Suppose thatX has absolutely continuous quasinorm. Then, for every u∈M˙s,X, there exists quasicontinuous u∗ such that u=u∗ almost everywhere.Proof.
Suppose first thatu∈Ms,X. By Lemma 7, there are continuous functions ui∈Ms,X converging to u in Ms,X such that(40)ui-ui+1Ms,X<2-2i for every i∈N. Moreover, by [36, Lemma 3.3], we may assume that ui→u pointwise almost everywhere. Denote Ei={x∈P:uix-ui+1x>2-i} and Fj=⋃i=j∞Ei. Then(41)uj-uk≤∑i=jk-1ui-ui+1≤∑i=jk-12-i=21-j in P∖Fj for every k>j. Hence (ui) converges pointwise in P∖⋂j=1∞Fj and the convergence is uniform in P∖Fj. By continuity, 2iui-ui+1∈AMs,X(Ei) and so(42)CMs,XEi≤2iui-ui+1Ms,X<2-i for every i∈N. Hence, by Lemma 16,(43)CMs,XFjρ≤8∑i=j∞CMs,XEiρ≤C2-jρ, which implies that CMs,X(⋂j=1∞Fj)=0. It follows that the function u∗=limsupi→∞ui is quasicontinuous. Moreover, u∗=limi→∞ui=u almost everywhere.
Suppose then thatu∈M˙s,X. Let x∈P. For k∈N, let φk be a Lipschitz function of bounded support such that φk=1 in B(x,k). Then, by Lemma 6, uk=uφk∈Ms,X. By the first part of the proof, there exists quasicontinuous uk∗ such that uk∗=uk almost everywhere. Since uk+1∗=u=uk∗ almost everywhere in B(x,k), Lemma 17 implies that there exists Ek with CMs,X(Ek)=0 such that uk+1∗=uk∗ in B(x,k)∖Ek. It follows that the limit limk→∞uk∗ exists in P∖E, where, by Lemma 16, E=⋃k=1∞Ek is of Ms,X-capacity zero. Define u∗=limsupk→∞uk∗. Then, clearly u∗=u almost everywhere. Let ε>0. For every k∈N, there exists Uk with CMs,X(Uk)<2-kε such that uk∣P∖Uk∗ is continuous. It follows that u∣P∖⋃k=1∞Uk∪E∗ is continuous and, by Lemma 16, CMs,X(⋃k=1∞Uk∪E)<ε.The following lemma gives a useful characterization of the capacity in terms of quasicontinuous functions. The proof of the lemma is essentially same as the proof of [37, Theorem 3.4]. For E⊂P, denote(44)QAMs,XE=u∈Ms,X:uisquasicontinuousandu≥1quasieverywhereinE,C~Ms,XE=infu∈QAMs,XEuMs,X.Lemma 19.
Suppose thatX has absolutely continuous quasinorm. Then(45)C~Ms,XE≤CMs,XE≤cΔC~Ms,XE for every E⊂P.Proof.
To prove the first inequality, letu∈AMs,X(E) and let u∗ be a quasicontinuous representative of u. Since u≥1 in some open set U containing E and u∗=u almost everywhere, it follows that min{0,u∗-1}=0 almost everywhere in U. Since min{0,u∗-1} is quasicontinuous, the equality actually holds quasieverywhere in U. Hence u∗≥1 quasieverywhere in U, which implies that u∗∈QAMs,X(E).
For the second inequality, letv∈QAMs,X(E). By truncation, we may assume that 0≤u≤1. Fix 0<ε<1, and choose an open set V with CMs,X(V)<ε so that v=1 on E∖V and that v is continuous in P∖V. By continuity, there is an open set U⊂P such that(46)x∈P:vx>1-ε∖V=U∖V. Clearly, E∖V⊂U∖V. Choose u∈AMs,X(V) such that uMs,X<ε and that 0≤u≤1. Define w=v/(1-ε)+u. Then w≥1 on (U∖V)∪V=U∪V which is an open neighbourhood of E. Hence w∈AMs,X(E) and so(47)CMs,XE≤wMs,X≤cΔ11-εvMs,X+uMs,X≤cΔ11-εvMs,X+ε. Since ε>0 and v∈QAMs,X(E) are arbitrary, the desired inequality CMs,X(E)≤cΔC~Ms,X(E) follows.


## 4. Generalized Lebesgue Points

In this section, we prove the first main result of the paper, Theorem1. The main ingredient of the proof of is a capacitary weak type estimate, Theorem 21.Lemma 20.
Let0<s≤1, 0<γ≤1/2, and 0<R<∞. Let u∈L0 and g∈Ds(u)∩L0. Then there exists a constant C≥1 such that CM3Rγ/Cg is an s-gradient of MRγ,∗u.Proof.
Letr>0. By the definition of the discrete γ-median convolution urγ and by the properties of the functions φi,(48)urγ=u+∑i∈ImuγBi-uφi. By Lemma 6, function(49)g+Cr-su-muγBiχ2Bi is an s-gradient of function (muγ(Bi)-u)φi for each i.
Letx∈2Bi∖E, where E is the exceptional set for (15). Using Lemma 8 and the facts that Bi⊂B(x,3r) and μ(B(x,3r))≤Cμ(Bi), we obtain(50)ux-muγBi≤mu-uxγBi≤CrsmgγBi+gx≤Crsmgγ/CBx,3r+gx≤Crsgx+M3rγ/Cgx.Since g(x)≤M3rγ/Cg(x) for almost every x and since the balls 2Bi have bounded overlap, it follows that(51)CM3rγ/Cg∈Dsurγ, for every r>0. Consequently, by Lemma 5,(52)CM3Rγ/Cg∈DsMRγ,∗u as desired.Theorem 21.
Suppose that the assumptions of Theorem1 are in force. Then, for every ball B=B(x0,r) and for every 0<γ<1, there exists a constant C such that(53)CMs,Xx∈B:M1/3γ,∗ux>λ≤Cλ-1uMs,X for every u∈Ms,X and λ>0.Proof.
SinceM1/3γ,∗u≤M1/3γ′,∗u when γ′≤γ, it suffices to prove the claim for 0<γ≤1/2. Let u∈Ms,X and let φ:P→[0,1] be a Lipschitz function such that φ=1 in B(x0,r+4/3) and φ=0 outside B(x0,r+2). Then M1/3γ,∗u=M1/3γ,∗(uφ) in B and so(54)x∈B:M1/3γ,∗ux>λ⊂x∈P:M1/3γ,∗uφx>λ.By (30) and (6),(55)M1/3γ,∗uφX≤CM1γ/CuφX≤CM1γ/CuχBx0,r+3X≤CuX and, by Lemmas 20 and 6 and (6),(56)M1/3γ,∗uφM˙s,X≤Cinfg∈DsuφM1γ/CgX≤Cinfg∈DsuM1γ/Cu+gχBx0,r+2X≤Cinfg∈DsuM1γ/Cu+gχBx0,r+3X≤Cinfg∈Dsuu+gX≤CuMs,X.Hence, M1/3γ,∗(uφ)∈Ms,X. Since M1/3γ,∗(uφ) is lower semicontinuous, λ-1M1/3γ,∗(uφ)∈AMs,X({x∈P:M1/3γ,∗(uφ)>λ}). Thus,(57)CMs,Xx∈P:M1/3γ,∗uφ>λ≤λ-1M1/3γ,∗uφMs,X≤Cλ-1uMs,X and the claim follows.Lemma 22.
Suppose thatX has absolutely continuous quasinorm and that B is a ball such that, for every 0<γ<1 and λ>0,(58)limk→∞CMs,Xx∈B:limsupr→0mukγBx,r>λ=0.whenever limk→∞ukMs,X=0. Then, for every quasicontinuous u∈Ms,X, quasievery point in B is a generalized Lebesgue point of u.Proof.
By Lemma7, continuous functions are dense in Ms,X. Let u∈Ms,X be quasicontinuous and let vk∈Ms,X, k=1,2,…, be continuous such that(59)u-vkMs,X→0 as k→∞. Denote wk=u-vk. Fix 0<γ<1 and λ>0. By Lemma 8,(60)limsupr→0mu-uxγBx,r≤limsupr→0mvk-vkxγ/2Bx,r+limsupr→0mwk-wkxγ/2Bx,r≤limsupr→0mwkγ/2Bx,r+wkx.Hence, by Lemma 16,(61)CMs,Xx∈B:limsupr→0mu-uxγBx,r>λρ≤8CMs,Xx∈B:limsupr→0mwkγ/2Bx,r>λ2ρ+8CMs,Xx∈B:wkx>λ2ρ.By assumption,(62)CMs,Xx∈B:limsupr→0mwkγ/2Bx,r>λ2→0 as k→∞. Since wk is quasicontinuous, Lemma 19 gives(63)CMs,Xx∈B:wkx>λ2≤CC~Ms,Xx∈B:wkx>λ2≤C2λ-1wkMs,X→0as k→∞. It follows that(64)CMs,Xx∈B:limsupr→0mu-uxγBx,r>λ=0 for every 0<γ<1 and λ>0. Denote(65)E=x∈B:limsupr→0mu-uxγBx,r>0for  some0<γ<1. Then(66)E=⋃n,m≥2x∈B:limsupr→0mu-ux1/mBx,r>1n and so, by Lemma 16, CMs,X(E)=0. Since, by Lemma 8,(67)muγBx,r-ux=mu-uxγBx,r≤mu-uxminγ,1-γBx,r, the claim follows.Proof of Theorem1.
Letu∈M˙s,X be quasicontinuous. Fix x∈P and, for k∈N, let φk:P→[0,1] be a Lipschitz function of bounded support such that φk=1 in B(x,k). Then, by Lemma 6, uk=uφk∈Ms,X. By Theorem 21 and Lemmas 12 and 22, for every k, quasievery point is a generalized Lebesgue point of uk. Hence, for every k∈N, quasievery point in B(x,k) is a generalized Lebesgue point of u. Thus, by Lemma 16, quasievery point is a generalized Lebesgue point of u.


## 5. Lebesgue Points

In this section we prove Theorem2.Lemma 23.
Suppose thatμ satisfies (2). Let 0<s≤1, R>0, u∈L0, and g∈Ds(u)∩Llocq where q=Q/(Q+s). Then there exists a constant C such that C(M6Rgq)1/q is an s-gradient of MR∗u.Proof.
By the Sobolev–Poincaré inequality ([8, Lemma 2.2]), u is locally integrable and there exists a constant C such that(68)infc∈Ru-cBx,r≤CrsgqBx,2r1/qfor every x∈P and r>0.
Fixr>0. By the definition of the discrete convolution ur and by the properties of the functions φi,(69)ur=u+∑i∈NuBi-uφi. By Lemma 6, function(70)g+Cr-su-uBiχ2Bi is an s-gradient of the function (uBi-u)φi for each i. A standard chaining argument and (68) imply that, for almost every x∈2Bi,(71)ux-uBi≤CM6rgqx1/q. Since g(x)≤(M6rgq(x))1/q, for almost every x, and since the balls 2Bi have bounded overlap, it follows that(72)CM6rgq1/q∈Dsur. The claim follows by Lemma 5.Lemma 24.
Letu∈M˙s,X be such that u=0 outside B=B(x0,r) and suppose that there exists y0∈P such that d(x0,y0)=3r. Then there exists a constant C such that uMs,X(P)≤Cinfg∈Dsugχ4BX.Proof.
Letg∈Ds(u). Then gχ2B+r-su∈Ds(u). Indeed, if x,y∈2B, then, by definition,(73)ux-uy≤dx,ysgx+gy, and if one of the points, say x, does not belong to 2B, then(74)ux-uy=uy≤dx,ysr-suy. Hence, it suffices to show that uX≤Cgχ4BX. For almost every x∈B,(75)ux≤5srsgx+ess⁡infy∈4B∖2B⁡gy. Since By0,r⊂4B∖2B, we have that μ(4B∖2B)>0. Clearly,(76)ess⁡inf4B∖2B⁡gχ4B∖2BX≤gχ4BX. Hence(77)uX=uχBX≤CgχBX+ess⁡inf4B∖2B⁡gχBX≤CgχBX+χBXχ4B∖2BXgχ4BX≤Cgχ4BX.Theorem 25.
Suppose that the assumptions of Theorem2 are in force. Let B=B(x0,r) be a ball and assume that the sphere {y:d(y,x0)=6r} is nonempty. Then there exist constants C≥1 and R>0 such that(78)CMs,Xx∈B:MR∗ux>λ≤Cλ-1uMs,X for every u∈Ms,X and λ>0.Proof.
LetR=(1/9)min{1,r} and let φ:P→[0,1] be a Lipschitz function such that φ=1 in B(x0,r+4R) and φ=0 outside B(x0,r+5R). Then, by Lemma 6, uφ∈Ms,X and(79)uφMs,X≤CuMs,X.If x∈B, then MR∗u(x)=MR∗(uφ)(x). Hence,(80)x∈B:MR∗ux>λ⊂x∈P:λ-1MR∗uφx>1. If x∈P∖B(x0,r+9R), then Mr∗(uφ)(x)=0. In particular, Mr∗(uφ)=0 outside 2B. Hence, by Lemmas 23 and 24, MR∗(uφ)∈Ms,X and(81)MR∗uφMs,X≤Cinfg∈DsMR∗uφgχ8BX≤Cinfg∈DsuφM6Rgq1/qχ8BX,where q=Q/(Q+s). Since MR∗(uφ) is lower semicontinuous, it follows that(82)CMs,Xx∈B:MR∗ux>λ≤λ-1MR∗uφMs,X≤Cλ-1infg∈DsuφM6Rgq1/qχ8BX.By the fact that 6R<1, assumptions (9) and (79) imply that(83)infg∈DsuφM6Rgq1/qχ8BX≤infg∈DsuφM1gq1/qχ8BX≤infg∈DsuφgX≤uφMs,X≤CuMs,Xand the claim follows.The proof of the next lemma, which is very similar to the proof of Lemma22, will be omitted.Lemma 26.
Suppose that·X is absolutely continuous and that B⊂P is ball such that, for every λ>0,(84)limi→∞CMs,Xx∈B:limsupr→0uiBx,r>λ=0whenever limi→∞uiMs,X=0. Then, for every quasicontinuous u∈Ms,X, quasievery point in B is a Lebesgue point of u.Proof of Theorem2.
We may assume thatP contains at least two points. Then P can be covered by balls Bk=B(xk,rk), k∈I, where I⊂N, such that the spheres {y:d(xk,y)=6rk} are nonempty.
Letu∈M˙s,X be quasicontinuous and, for k∈I, let φk be a Lipschitz function of bounded support such that φk=1 in B(xk,rk). Then, by Lemma 6, uk=uφk∈Ms,X. By Theorem 25 and Lemmas 12 and 26, for every k∈I, quasievery point in Bk is a Lebesgue point of uk and hence of u. Hence, by Lemma 16, quasievery point in P is a Lebesgue point of u.


## 6. Proof of Theorem3

IfαX<∞, then clearly Mγ is bounded on X. Hence, the first part of Theorem 3 follows from Theorem 1. The second part follows from Theorem 2 via the following lemma.Lemma 27.
Letp≤1. If αX<1/p, then the operator u↦(Mup)1/p is bounded on X.Proof.
By assumption, there are constantsα<1/p and C>0 such that(85)MγuX≤Cγ-αuX.for every u∈X and 0<γ<1. Denote by v∗ the decreasing rearrangement of a function v, that is,(86)v∗t=infa≥0:μx∈P:vx>a<t. Then, for every ball B, we have(87)⨍Bupdμ=1μB∫0μBuχB∗tpdt=∫01uχB∗γμBpdγ=∫01muγBpdγ. Hence,(88)Mupx≤∫01Mγuxpdγ≤∑i=1∞2-iM2-iuxp. Let 0<ε<1 be such that α<ε/p. By the Hölder inequality, we obtain(89)Mupx1/p≤C∑i=1∞2-iε/pM2-iux. Thus, by (85) and (14),(90)Mup1/pX≤C∑i=1∞2-iε/pM2-iuX≤C∑i=1∞2-iε/pM2-iuXσ1/σ≤C∑i=1∞2-iε/p-ασ1/σuX≤CuX.


## 7. Examples

### 7.1. Lorentz Spaces

For0<p<∞, 0<q≤∞, and u∈L0, denote(91)uLp,q=∫0∞λqμx∈P:ux>λq/pdλλ1/q when 0<q<∞ and(92)uLp,∞=supλ>0λμx∈P:ux>λ1/p. Then Lp,q={u∈L0:uLp,q<∞} equipped with ·Lp,q is a quasi-Banach function space. If 0<p,q<∞, quasinorm ·Lp,q is absolutely continuous.Lemma 28.
There exists a constantC=C(cd) such that(93)μx∈P:Mγux>λ≤Cγ-1μx∈P:ux>λ for every u∈L0 and λ>0.Proof.
It follows easily from the definitions that, for everyu∈L0, x∈P, 0<γ<1, and λ>0, we have(94)Mγux>λ⟺Mχy∈P:uy>λx>γ.Hence, the claim follows from the well-known weak type inequality for the Hardy–Littlewood maximal operator.The following lemma is an immediate consequence of Lemma28.Lemma 29.
Let0<p<∞ and 0<q≤∞. There exists a constant C=C(cd,p) such that(95)MγuLp,q≤Cγ-1/puLp,q for every u∈Lp,q and 0<γ<1. Consequently, αLp,q≤1/p.Lemma29 and Theorem 3 imply the following result for Lorentz–Hajłasz functions.Theorem 30.
Let0<s≤1, 0<p,q<∞, and let u∈M˙s,Lp,q be quasicontinuous. Then quasievery point is a generalized Lebesgue point of u. Moreover, if p>Q/(Q+s), then quasievery point is a Lebesgue point of u.


### 7.2. Orlicz Spaces

LetΦ:[0,∞)→[0,∞) be a continuous increasing function and suppose that there exists a constant C≥1 such that(96)ΦtC≤Φt2for every t>0. Then(97)LΦ=u∈L0:∫PΦuλdμ<∞for  someλ>0 equipped with a quasinorm(98)uLΦ=infλ>0:∫PΦuλdμ≤1 becomes a quasi-Banach function space.It is easy to see that (96) is equivalent to the existence of constants C0≥1 and β>0 such that(99)Φs≤C0stβΦtwhenever 0<s≤t<∞.A functionΦ:[0,∞)→[0,∞) is doubling, if there exists a constant C such that(100)Φ2t≤CΦtfor every t>0. It is easy to see that if Φ is doubling, then ·Lϕ is absolutely continuous.Lemma 31.
LetΦ satisfy (99). Then there exists a constant C such that(101)MγuLΦ≤Cγ-1/βuLΦ for every u∈LΦ and 0<γ<1. Consequently, αLΦ≤1/β.Proof.
We may assumeuLΦ≤1. Since Φ(Mγu)=MγΦ(u) and, by Lemma 28, MγvL1≤C1γ-1vL1 for all v∈L1, we have that(102)ΦMγuλL1≤C0λ-βΦMγuL1=C0λ-βMγΦuL1≤C0C1λ-βγ-1ΦuL1≤C0C1λ-βγ-1≤1when λ≥(C0C1)1/βγ-1/β.By combining Theorem3 and Lemma 31, we obtain the following result for Orlicz–Hajłasz spaces Ms,Φ≔Ms,LΦ.Theorem 32.
Suppose thatμ satisfies (2) and that Φ is doubling and satisfies (96). Let 0<s≤1 and let u∈M˙s,Φ be quasicontinuous. Then quasievery point is a generalized Lebesgue point of u. Moreover, if Φ satisfies (99) with β>Q/(Q+s), then quasievery point is a Lebesgue point of u.


### 7.3. Variable Exponent Spaces

Letp:P→(0,∞) be a measurable function. The space Lp(·) consisting of functions u for which(103)∫Puxλpxdμx<∞ for some λ>0 equipped with quasinorm(104)uLp·=infλ>0:∫Puxλpxdμx≤1 is a quasi-Banach function space.A measurable functionp:P→(0,∞) islocally log-Hölder continuous if there exists a constant Cp>0 such that(105)px-py≤Cploge+1/dx,yfor all x,y∈PDenote(106)p-=ess⁡infx∈P⁡px,p+=ess⁡supx∈P⁡pxWe need the following result from [19].Lemma 33.
Letp:P→(0,∞) be locally log-Hölder continuous with p->0 and p+<∞. Suppose that there exist constants p∞>0 and 0<a<1 such that(107)∫Pa1/px-p∞dμx<∞.Then there exists a constant C such that(108)MγuLp·≤Cγ-1/p-uLp· for every u∈Lp(·) and 0<γ<1. Consequently, αLp(·)≤1/p-.The next lemma follows from [38, Lemma 2.3], [38, Corollary 3.5] and from the fact that the function t↦1/t is bi-Lipschitz from [a,b] to [1/b,1/a] whenever 0<a<b<∞.Lemma 34.
Suppose thatp:P→(0,∞) is locally log-Hölder continuous and that p->0 and p+<∞. Then, for any ball B⊂P, there exists a locally log-Hölder continuous extension p~ of p∣B such that p~-=p-, p~+=p+ and(109)∫Pa1/p~x-p~∞dμx<∞.for some p~∞>0 and 0<a<1.Theorem 35.
Suppose thatμ satisfies (2) and that p:P→(0,∞) is locally log-Hölder continuous with p+<∞. Let 0<s≤1 and let u∈M˙s,p(·)≔Ms,Lp(·) be quasicontinuous. (1)
Ifp->0, then quasievery point is a generalized Lebesgue point of u.(2)
Ifp->Q/(Q+s), then quasievery point is a Lebesgue point of u.Proof.
( 1 ) Since p+<∞, Lp(·) has absolutely continuous quasinorm. By Theorem 1, it suffices to show that, for every ball B=B(x,r) and 0<γ<1, there exists a constant C such that (110)M1γgχBLp·≤CgLp·for every g∈Lp(·). By Lemma 34, p|B(x,r+1) can be extended to p~ on P such that p~+<∞, p~->0, and (109) holds true for some 0<a<1 and p~∞>0. By Lemma 33, there exists a constant C such that(111)MγvLp~·≤Cγ-1/p~-vLp~· for every v∈Lp~(·). Thus,(112)M1γgχBLp·≤MγgχBx,r+1Lp~·≤Cγ-1/p~-gχBx,r+1Lp~·≤Cγ-1/p~-gLp·for every g∈Lp(·).
( 2 ) Denote q=Q/(Q+s). By Theorem 2, it suffices to show that, for every ball B=B(x,r), there exists a constant C such that(113)M1gq1/qχBLp·≤CgLp· for every 0≤g∈Lp(·). By Lemma 34, p∣B(x,r+1) can be extended to p~ on P such that p~+<∞, p~->q, and (109) holds true for some 0<a<1 and p~∞>0. Hence, by Lemma 33, αLp~(·)≤1/p~-<1/q and so, by Lemma 27, operator g↦(Mgq)1/q is bounded on Lp~(·). Thus,(114)M1gq1/qχBLp·≤MgχBx,r+1q1/qLp~·≤CgχBx,r+1Lp~·≤CgLp·for every 0≤g∈Lp(·).


## 7.1. Lorentz Spaces

For0<p<∞, 0<q≤∞, and u∈L0, denote(91)uLp,q=∫0∞λqμx∈P:ux>λq/pdλλ1/q when 0<q<∞ and(92)uLp,∞=supλ>0λμx∈P:ux>λ1/p. Then Lp,q={u∈L0:uLp,q<∞} equipped with ·Lp,q is a quasi-Banach function space. If 0<p,q<∞, quasinorm ·Lp,q is absolutely continuous.Lemma 28.
There exists a constantC=C(cd) such that(93)μx∈P:Mγux>λ≤Cγ-1μx∈P:ux>λ for every u∈L0 and λ>0.Proof.
It follows easily from the definitions that, for everyu∈L0, x∈P, 0<γ<1, and λ>0, we have(94)Mγux>λ⟺Mχy∈P:uy>λx>γ.Hence, the claim follows from the well-known weak type inequality for the Hardy–Littlewood maximal operator.The following lemma is an immediate consequence of Lemma28.Lemma 29.
Let0<p<∞ and 0<q≤∞. There exists a constant C=C(cd,p) such that(95)MγuLp,q≤Cγ-1/puLp,q for every u∈Lp,q and 0<γ<1. Consequently, αLp,q≤1/p.Lemma29 and Theorem 3 imply the following result for Lorentz–Hajłasz functions.Theorem 30.
Let0<s≤1, 0<p,q<∞, and let u∈M˙s,Lp,q be quasicontinuous. Then quasievery point is a generalized Lebesgue point of u. Moreover, if p>Q/(Q+s), then quasievery point is a Lebesgue point of u.


## 7.2. Orlicz Spaces

LetΦ:[0,∞)→[0,∞) be a continuous increasing function and suppose that there exists a constant C≥1 such that(96)ΦtC≤Φt2for every t>0. Then(97)LΦ=u∈L0:∫PΦuλdμ<∞for  someλ>0 equipped with a quasinorm(98)uLΦ=infλ>0:∫PΦuλdμ≤1 becomes a quasi-Banach function space.It is easy to see that (96) is equivalent to the existence of constants C0≥1 and β>0 such that(99)Φs≤C0stβΦtwhenever 0<s≤t<∞.A functionΦ:[0,∞)→[0,∞) is doubling, if there exists a constant C such that(100)Φ2t≤CΦtfor every t>0. It is easy to see that if Φ is doubling, then ·Lϕ is absolutely continuous.Lemma 31.
LetΦ satisfy (99). Then there exists a constant C such that(101)MγuLΦ≤Cγ-1/βuLΦ for every u∈LΦ and 0<γ<1. Consequently, αLΦ≤1/β.Proof.
We may assumeuLΦ≤1. Since Φ(Mγu)=MγΦ(u) and, by Lemma 28, MγvL1≤C1γ-1vL1 for all v∈L1, we have that(102)ΦMγuλL1≤C0λ-βΦMγuL1=C0λ-βMγΦuL1≤C0C1λ-βγ-1ΦuL1≤C0C1λ-βγ-1≤1when λ≥(C0C1)1/βγ-1/β.By combining Theorem3 and Lemma 31, we obtain the following result for Orlicz–Hajłasz spaces Ms,Φ≔Ms,LΦ.Theorem 32.
Suppose thatμ satisfies (2) and that Φ is doubling and satisfies (96). Let 0<s≤1 and let u∈M˙s,Φ be quasicontinuous. Then quasievery point is a generalized Lebesgue point of u. Moreover, if Φ satisfies (99) with β>Q/(Q+s), then quasievery point is a Lebesgue point of u.


## 7.3. Variable Exponent Spaces

Letp:P→(0,∞) be a measurable function. The space Lp(·) consisting of functions u for which(103)∫Puxλpxdμx<∞ for some λ>0 equipped with quasinorm(104)uLp·=infλ>0:∫Puxλpxdμx≤1 is a quasi-Banach function space.A measurable functionp:P→(0,∞) islocally log-Hölder continuous if there exists a constant Cp>0 such that(105)px-py≤Cploge+1/dx,yfor all x,y∈PDenote(106)p-=ess⁡infx∈P⁡px,p+=ess⁡supx∈P⁡pxWe need the following result from [19].Lemma 33.
Letp:P→(0,∞) be locally log-Hölder continuous with p->0 and p+<∞. Suppose that there exist constants p∞>0 and 0<a<1 such that(107)∫Pa1/px-p∞dμx<∞.Then there exists a constant C such that(108)MγuLp·≤Cγ-1/p-uLp· for every u∈Lp(·) and 0<γ<1. Consequently, αLp(·)≤1/p-.The next lemma follows from [38, Lemma 2.3], [38, Corollary 3.5] and from the fact that the function t↦1/t is bi-Lipschitz from [a,b] to [1/b,1/a] whenever 0<a<b<∞.Lemma 34.
Suppose thatp:P→(0,∞) is locally log-Hölder continuous and that p->0 and p+<∞. Then, for any ball B⊂P, there exists a locally log-Hölder continuous extension p~ of p∣B such that p~-=p-, p~+=p+ and(109)∫Pa1/p~x-p~∞dμx<∞.for some p~∞>0 and 0<a<1.Theorem 35.
Suppose thatμ satisfies (2) and that p:P→(0,∞) is locally log-Hölder continuous with p+<∞. Let 0<s≤1 and let u∈M˙s,p(·)≔Ms,Lp(·) be quasicontinuous. (1)
Ifp->0, then quasievery point is a generalized Lebesgue point of u.(2)
Ifp->Q/(Q+s), then quasievery point is a Lebesgue point of u.Proof.
( 1 ) Since p+<∞, Lp(·) has absolutely continuous quasinorm. By Theorem 1, it suffices to show that, for every ball B=B(x,r) and 0<γ<1, there exists a constant C such that (110)M1γgχBLp·≤CgLp·for every g∈Lp(·). By Lemma 34, p|B(x,r+1) can be extended to p~ on P such that p~+<∞, p~->0, and (109) holds true for some 0<a<1 and p~∞>0. By Lemma 33, there exists a constant C such that(111)MγvLp~·≤Cγ-1/p~-vLp~· for every v∈Lp~(·). Thus,(112)M1γgχBLp·≤MγgχBx,r+1Lp~·≤Cγ-1/p~-gχBx,r+1Lp~·≤Cγ-1/p~-gLp·for every g∈Lp(·).
( 2 ) Denote q=Q/(Q+s). By Theorem 2, it suffices to show that, for every ball B=B(x,r), there exists a constant C such that(113)M1gq1/qχBLp·≤CgLp· for every 0≤g∈Lp(·). By Lemma 34, p∣B(x,r+1) can be extended to p~ on P such that p~+<∞, p~->q, and (109) holds true for some 0<a<1 and p~∞>0. Hence, by Lemma 33, αLp~(·)≤1/p~-<1/q and so, by Lemma 27, operator g↦(Mgq)1/q is bounded on Lp~(·). Thus,(114)M1gq1/qχBLp·≤MgχBx,r+1q1/qLp~·≤CgχBx,r+1Lp~·≤CgLp·for every 0≤g∈Lp(·).



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*Source: 5637042-2018-11-11.xml*