# Lp Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

Authors: Ferit Gürbüz; Ahmed Loulit
Journal: Journal of Mathematics
 (2021)

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## Abstract

It is known, in harmonic analysis theory, that maximal operators measure local smoothness ofLp functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition Mα#f∈Lp,μ, where μ is a lower doubling measure, Mα#f stands for the sharp maximal function of f, and 0≤α≤1 is the degree of smoothness.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the some continuous embeddings on weighted Besov–Triebel–Lizorkin spaces via a general sharp maximal function introduced by Calderón and Scott [6]. Furthermore, we investigate the spaces introduced by Hajłasz [13] that are defined via pointwise inequalities and their connection with the Triebel–Lizorkin spaces. For more details, see [11, 12].Now, let us begin by recalling some definitions and classical results in harmonic analysis on then-dimensional Euclidean space ℝn needed for later sections.(1) A cube onℝn will always mean a cube with sides parallel to the axes and has nonempty interior. For j∈ℤ and k∈ℤn, we denote by Qjk the dyadic cube 2−j0,1n+k, where lQjk=2−j is its side length, xQjk=2−jk is its lower “left-corner,” and cQjk is its center. We set Q=Qjk:j∈ℤ,k∈ℤn and jQ=−log2lQ for all Q∈Q. When the dyadic cube Q appears as an index, such as ∑Q∈Q, it is understood that Q runs over all dyadic cubes in ℝn. For a function ν and dyadic cube Q=Qjk, set(1)νQx=Q−1/2ν2jx−k=Q1/2νjx−xQ,for all x∈ℝn, where νjx=2njν2jx.(2) Throughout the paper,w denotes a weight function, i.e., w is an almost every (a.e.) positive locally integrable function on ℝn. A function f∈Lpw,0<p<∞⟺(2)fp,w=∫ℝnfxpwxdx1/p<∞,and f belongs to the weak-Lp spaces, denoted by Lp,∞w⟺(3)fLp,∞w=supλ>0λwx∈ℝn:fx>λ1/p<∞.Ifw=1, we do not write the subscription w.A weight functionw is said to be in the Muckenhoupt classes Ap,where1≤p<∞, if there exists a constant Cp>0 such that for every cube Q,(4)1Q∫Qwdy1Q∫Qw1−p′dyp−1≤Cp.When 1<p<∞,1/p+1/p′=1; for p=1,(5)1Q∫Qwydy≤C1wx,for a.e. x∈Q, or equivalently Mwx≤C1wx for a.e. x∈ℝn, where M is the Hardy–Littlewood maximal operator.The classAp was introduced by Muckenhoupt [16] in order to characterize the boundedness of the Hardy–Littlewood maximal operator M on the weighted Lebesgue spaces [8, 12]. The pioneering work of Muckenhoupt [16] showed that(6)M:Lpw⟶Lpw,⟺w∈Ap when 1<p<∞ and(7)M:L1w⟶L1,∞w,⟺w∈A1.A weight functionw is in Muckenhoupt’s class Apℝn, 1≤p<∞, of weights if there exists a constant Cp>0 such that for all cubes Q in ℝn,(8)1Q∫Qwydy1Q∫Qwy1−p′dyp−1≤Cp.When 1<p<∞,1/p+1/p′=1; well, for p=1,(9)1Q∫Qwydy≤C1wx,for a.e. x∈B, or equivalently Mwx≤C1wx for a.e. x∈ℝn, where M is the Hardy–Littlewood maximal operator.(3) Note that ifw∈Ap, then w is a doubling measure, i.e., there exists a constant C≥1 such that for all x and all r>0,(10)wBx,2r≤CwBx,r.Another class of functions that plays an important role in harmonic analysis and in partial differential equation theory is the class of functions with bounded mean oscillation denoted byBMOw, i.e., φ∈BMOw, if there is a constant C:(11)supQ1wQ∫Qφy−φQwydy<C,where φQ=1/wQ∫Qφywydy is the average of f on Q with respect to dw. The smallest constant C for which (11) is satisfied is taken to be the norm of φ in the space BMOw and is denoted by ‖φ‖BMOw.(4) The sharp maximal functionM#fx of f is defined by(12)M#fx=supQinfc∈ℝ1Q∫Qfx−cdx,where Q is taken over all cubes in ℝn. Let α≥0. The sharp fractional maximal function Mα#f of f is defined by(13)Mα#fx=supx∈Qinfc∈ℝ1Qα/n∫Qfx−cdx.(5) The space of Schwartz functions: letSℝn be the space of all Schwartz functions on ℝn with the classical topology generated by the family of seminorms:(14)νk,N=supx∈ℝnsupβ≤N1+xk∂βνxk,N∈ℕ0,ν∈Sℝn.The topological dual spaceS′ℝn of Sℝn is the set of all continuous linear functional the space Sℝn is endowed with the weak ∗-topology. We denote by S∞ℝn the topological subspace of functions in Sℝn having all vanishing moments:(15)S∞ℝn=ν∈Sℝn:∫ℝnxβνxdx=0,for everyβ∈ℕn.S∞′ℝn denotes the topological dual space of S∞ℝn, namely, the set of all continuous linear functional on S∞′ℝn. The space S∞′ℝn is also endowed with the weak ∗-topology. It is well known that S∞′ℝn=S′ℝn/Pℝn as topological spaces, where Pℝndenotes the set of all polynomials on ℝn; see, for example, ([21], Proposition 8.1). Similarly, for any R∈ℕ, the space SRℝn is defined to be the set of all Schwartz functions having vanishing moments of order R and SR′ℝn is its topological dual space. We write S−1ℝn=Sℝn.The Fourier transform,ℱν=ν^, of Schwartz function ν is defined by(16)ν^ξ=2π−n∫ℝne−iξ⋅xνxdy.The convolution of two functionsν,μ∈Sℝn is defined by(17)ν∗μx=∫ℝnvx−yμydyand still belongs to Sℝn.The convolution operator can be extended toSℝn×S′ℝn via ν∗fx=f,μx−⋅. It makes sense pointwise and is a C∞ function on ℝn of at most polynomial growth.To simplify notation, we write oftenνf=ν∗f. In some other situations, to avoid confusion, we keep the notation ν∗f. As usual, νt denotes the function defined by νtx=t−nνx/t.(6) In the rest of this paper,C expresses unspecified positive constant, possibly different at each occurrence; the symbol A≤B means that A≤CB. If A≤B and B≤A, then we write A≃B. The Greek letter χS denotes the characteristics function of a sphere S, where S is a measurable subset of ℝn and S represents its Lebesgue measure; p′ and s′ always denote the conjugate index of any p>1 and s>1, that is, 1/p′:=1−1/p and 1/s′:=1−1/s.Function spaces play a crucial role in the genesis of functional analysis and are widely used in the development of the modern analysis of partial differential equations. For instance, the classical Besov–Triebel–Lizorkin spaces are a class of function spaces containing many well-known classical function spaces and are more suitable in the treatment of a large type of partial differential equations (see for instance [5, 10]). A comprehensive treatment of these function spaces and their history can be found in Triebel’s monographs [18, 19] and in the fundamental paper of Frazier and Jawerth [11].In recent years, there has been increasing interest in a new family of function spaces, called new class of Besov–Triebel–Lizorkin spaces. These spaces unify and generalize many classical spaces including Besov spaces, Morrey spaces, and Triebel–Lizorkin spaces (see for instance [20]).In this paper, we study the extent of smoothness on weighted function spaces under the conditionMα#f∈Lp,μ, where μ is a lower doubling measure, Mα#f stands for the sharp maximal function of f, and 0≤α≤1 is the degree of smoothness. When α=0, M0#f=M#f is the classical sharp maximal function. It is well known that the Hardy–Littlewood maximal function Mf is controlled by the sharp maximal function M#f via the celebrated Stein–Fefferman inequality: Mfp≤M#fp and in the case of α=1, it is shown that M1#fp:fHp for some range of p. As a result, we extend the above results to the some general weighted spaces. Embedding results on weighted Besov–Triebel–Lizorkin spaces are obtained. Namely, fF˙p∗,wγ,q≤¯Mα#fp,w (Theorem 1). As a consequence, we obtain fF˙p∗,wγ,q≤fW˙α,pw, where W˙α,pw stands for the fractional Sobolev space.Now, we are ready to present the main theorem of this section.Theorem 1.
Letαandγ be real numbers satisfying 0≤α≤1 and γ<α, and w is the lower d−regular doubling measure. Suppose that wfx>ε<∞ for every ε>0 and Mα#f∈Lpw for d/n+α<p<d/α−γ. Then, for each 0<q≤∞,(18)fF˙p∗,wγ,q≤Mα#fp,w,1p∗=1p−α−γd.Remark 1.
The condition thatwfx>ε<∞ for every ε>0 is necessary. On the other hand, under this condition, Mα#f∈Lpw only if p>d/n+α.Proof.
Iff is not a constant function, then there exists a ball Bx0,R such that(19)infc∈ℝ∫Bx0,Rfy−cdy=c0>0.
Therefore, for allx∈ℝn,(20)Mα#fx±R+x−x0−α−n.
Hence,(21)∫ℝnMα#fpwxdy±∫x−x0>RR+x−x0−α−npwxdx±∑k≥0R+2k+1R−α−np∫x−x0<2k+1Rwxdx±R−α−np+d∑k≥02k−α−np+kd=∞,if p≤d/n+α.Corollary 1.
Under the same conditions in Theorem1, we have, for each 0<q≤∞,(22)fB˙p∗,wγ,q≤Mα#fp,w,1p∗=1p−α−γd,for each 0<p∗<q≤∞.Proof.
By Minkowski’s inequality, we have(23)fB˙p∗,wγ,q≤fF˙p∗,wγ,q,0<p∗<q≤∞.
Then, applying Theorem1, we obtain (22). This completes the proof.


## 2. Preliminaries

In this section, we introduce some necessary and important definitions, notations, lemmas, and results.Definition 1.
Letv be in the Schwartz space with supp ν^ contained in an annulus about the origin and(24)∑j∈ℤν^2−jξ=1for allξ≠0.
Letμ be a doubling measure and 0<p,q≤∞ and γ∈ℝ; the homogeneous Triebel–Lizorkin space F˙p,vγ,q is the set of all distributions f (modulo polynomials) such that(25)fF˙p,μγ,q=∑j∈ℤ2jγqv2−jfq1/qp,μ<∞;0<p,q<∞,fF˙∞,μγ,q=supQ1μQ∫Q∑j=−log2lQ∞2jγqv2−jfqdμx1/q<∞;0<q≤∞,with the interpretation that when q=∞,(26)fF˙∞,wγ,q=supQsupj≥−log2lQ1μQ∫Q2jγv2−jfdμx<∞.
The homogeneous Besov–Lipschitz spaceB˙pγ,q is the set of all distributions f (modulo polynomials) such that(27)fB˙pγ,q=∑j∈ℤ2jγqv2−jfp,wq1/q<∞;0<p,q<∞.
The supremum is taken over all dyadic cubesQ, and lQ denotes the length of sides of the cube Q.Moreover, it is well known that the Besov–Lipschitz spaces and the Triebel–Lizorkin spaces are independent of the choices ofν (see, for example [2–4, 11]). Throughout this paper, ν will be taken as in Definition 1. It is well known that many classical smoothness spaces are covered by the Besov and Triebel–Lizorkin spaces. We recall some examples in the case when dμ=wdx and w∈A∞:(1)
F˙p,w0,2=Hp,w,0<p<∞.(2)
Fp,w0,2=hp,w,0<p<∞, where Hp,w denotes the weighted Hardy spaces of f∈S′ for which(28)fHp,w=supt>0μt∗fp,w<∞andhp,w is the local weighted Hardy space of f∈S′ for which(29)fhp,w=sup0<t<1μt∗fp,w<∞,whereμ is a fixed function in S with ∫ℝnμxdx≠0. By the fundamental work of Fefferman and Stein [9] adapted to the weighted case, Hp,w or hp,w does not depend on the choices of μ. In particular,(30)F˙p,w0,2=Lpw,1<p<∞.(3)
F˙p,wγ,2=Hp,wγ,1<p<∞, where Hp,wγ denotes the weighted Bessel potential space defined by(31)fHp,wγ=ℱ−11+ξ2γ/2ℱfp,w.In particular, when the exponent is a natural number, sayγ=N∈ℕ, then the weighted Bessel potential space can be identified with the classical Sobolev space:(32)Wp,wN=f∈Lp,w:∑|σ|≤N∂σfLp,w<∞,1<p<∞.(4)
F˙∞,w0,2=BMOw.All the above identities have to be understood in the sense of equivalent quasi-norms.Definition 2.
We say that a doubling measureμ is lower d−regular, where d≥n, if there is some constant C>0 such that(33)μBx,t≥Ctdholds for all ball Bx,t⊂ℝn.Remark 2.
An example of measureμ lower d−regular is dμ=wdx,where(34)wx=xαlogβ2+x−1.
In fact,w∈Ap if −n<α<np−1 and β∈ℝ; hence, w is doubling. Moreover, if 0≤α<np−1 and β≥0, then w satisfies wBx,t≥Ctn+α for all 0<t<∞ and all x.Lemma 1.
Letw∈Ap and d−regular. Then, we have(35)Mα#fp,w≤∫ℝn∫ℝnfx−fypx−yαp+dwxwydxdy1/p,1≤p<∞.Proof.
LetQ be a cube and x,y∈Q. Then,(36)fy−fQ=fy−1Q∫Qfzdz≤1Q∫Qfy−fzdz≤fy−fx+1Q∫Qfz−fxdz.
Integrating over the cubeQ with respect to y, we get(37)∫Qfy−fQdy≤∫Qfy−fxdy.
Ifw∈A1, then we have for almost all x∈Q, wx≽1/Q∫Qwydy≽Q−1+d/n. Hence,(38)∫Qfy−fQdy≤Q1−d/n∫Qfy−fxwydy.
The last inequality implies that(39)1Qα/n+1∫Qfy−fQdy≤Q−α−d/n∫Qfy−fxwydy≤∫ℝnfy−fxy−xα+dwydy.
On the other hand, ifw∈Ap, p>1; then using (37) and Hölder’s inequality, we obtain(40)∫Qfy−fQdy≤∫Qfy−fxpwydy1/p∫Qw1−p′ydyp−1/p≤Q1−d/np∫Qfy−fxpwydy1/p≤Q1−d/np∫Qfy−fxpwydy1/p≤Q1+α/n∫ℝnfx−fypx−yαp+dwydy1/p.
Therefore, we conclude that(41)Mα#fx≤∫ℝnfx−fypx−yαp+dwydy1/p,1≤p<∞.
This completes the proof.Lemma 2.
We say thatf is in the fractional Sobolev space W˙α,pw,0<α<1,1≤p<∞, if(42)fW˙α,pw=∫ℝn∫ℝnfx−fypx−yαp+dwxwydxdy1/p<∞.Corollary 2.
Letw∈Ap and lower d−regular, 0<α<1, 1≤p<∞, and f∈W˙α,pw. Then,(43)Mα#fp,w≤fW˙α,pw.One can immediately obtain the following corollary.Corollary 3.
Letαandγ be real numbers satisfying 0<α<1 and γ<α. Assume w∈Ap and f∈W˙α,pw with d/n+α<p<d/α−γ. Then, for each 0<q≤∞,(44)fF˙p∗,wγ,q≤fW˙α,pw,where p∗ is given by 1/p∗=1/p−α−γ/d.Recall that for0<α<1,1<p<∞ and w∈A∞; we have (see [17])(45)fF˙p,wα,∞≈Mα#fp,w.In particular,(46)fF˙p∗,wγ,q≤fF˙p,wα,∞,with 0<α<1,1<p<∞, and p∗ is as before.Lemma 3.
Letf∈Wloc1,1ℝn and 0≤α≤1. Then, for every x∈ℝn, there is a constant Cn such that(47)Mα#fx≤CnM1−α∇fx.Proof.
The proof is an immediate consequence of the well-known Poincaré inequality.
For all ballBx,R and all f∈Wloc1,1ℝn, there is a constant Cn such that(48)∫Bx,Rfy−fBx,Rdy≤CnR∫Bx,R∇fydyholds.Corollary 4.
Letf be a locally integrable function such that ∇f∈Hrw and 0<p∗≤∞ are determined by(49)1p⋆=1r+γ−1d.
Then,f is in F˙p∗,wγ,q. Moreover, we have(50)fF˙p∗,wγ,q≤∇fHrw.Proof.
LetPtx=cnt/t2+x2n+1/2 be the Poisson kernel with the constant Cn such that ∫ℝnPtxdx=1. Then, there exists a constant C=Cn such that Mfx≤Csupt>0Pt∗f. In fact, if Q is a cube with diam Q=t and x∈Q, then we have(51)1Q∫Qfydy=1Q∫Qt2+x−y2n+1/2CntPtx−yfydy≤CntnQ∫QPtx−yfydy≤CnPt∗f.
Thus, we have(52)Mfx≤Csupt>0Pt∗fx.
Using Lemma3 with α=1 and Proposition 2 (see below), we obtain(53)fF˙p∗,wγ,q≤M1#fr,w≤supt>0Pt∗∇fr,w≤∇fHrw.Remark 3.
If we takeγ=0, q=2, r>1, and w=1, in Corollary 4, we obtain the classical Sobolev–Gagliardo–Nirenberg inequality:(54)fp∗≤∇fr,with(55)1p∗=1r−1n.


## 3. Some Useful Lemmas

We start this section with some useful lemmas that will be helpful in proving our main result.Lemma 4 (see [7]).
Providedγ<1, λ>0, and 0<q≤1, there exist Schwartz functions v and μ on ℝn such that(1)
suppv⊂B0,1 and ν^0=0(2)
suppμ^⊂1/2≤ξ≤2 and μ^ξ≥c>0 on 3/5≤ξ≤25/3(3)
∑j∈ℤ2jγqμ2−j∗fq≤C∑j∈ℤ2jγqv2−j∗fqLemma 5.
Assume thatwBx,t≥Cwtd for each x∈ℝn and each t>0, and let v∈S supported on B0,1 such that(56)∫ℝnvxdx=0.
Fix a largeλ>0, and define(57)vt∗fx=supy∈ℝnvt∗fy1+x−y/tλ.
Then,(58)vt∗fx≤minsαMα#fx,sα−d/pMα#fp,w.Proof.
We adapt here the proof given in [7] in the unweighted case. Use the well-known estimate(59)1+x−zs−λ≤∑k=1∞2−kλχx−z2ks,λ>0,where χ denotes the characteristic function of the interval 0,1, to obtain, for any λ>0,(60)νs∗fx≤∑k=1∞2−kλsupz∈ℝnνs∗fzχx−z2ks.
By taking anyz∈Bx,2ks and using the fact that ν is supported in the unit ball and has mean equal zero, we obtain(61)νs∗fz≤∫Bz,1+2ksνsz−yfy−fBx,1+2ksdy≤sα1+2kn+αMα#fx,which holds. Hence,(62)νs∗fx≤sα∑k=1∞2−kλ1+2kn+αMα#fx.
If we chooseλ large enough, we obtain(63)νs∗fx≤sαMα#fx.
On the other hand, by (61), we have for any fixed x∈Bz,s,(64)νs∗fz≤sαMα#fx.
Rising (65) to the pth power and integrating over the ball Bz,s with respect to wxdx, one has that(65)νs∗fz≤wBz,s−1/psαMα#fLpw≤sα−d/pMα#fp,w.
By using (60), we obtain(66)νs∗fx≤sα−d/pMα#fp,w.Proof.
Proof of Theorem1.Proof.
We consider only the case when0<q≤1. In the case when 1<q≤∞, estimate (18) follows from the case q=1 by the embedding(67)F˙p,wγ,q0⊂F˙p,wγ,q1,0<q0≤q1≤∞.
Letk>0 be chosen later and let μ and ν be as in Lemma 4. Assume 0<α−γ<d/pand0<q≤1. Then, using (58), we get(68)∑j∈ℤ2jγqμ2−j∗fxq≤∑j∈ℤ2jγqν2−j∗fxq≤Mα#fx∑j≤k2−jα−γq+Mα#fp,w∑j>k2−jα−γ−d/pq≤2−kα−γqMα#fx+2−kα−γ−d/pqMα#fp,w.
Choose2−k=Mα#fx/Mα#fp,w to deduce that(69)∑j∈ℤ2jγqμ2−j∗fxq1/q≤Mα#fxp/p∗Mα#fp,w1−p/p∗,where p∗ is given by 1/p∗=1/p−α−γ/d. Thus, we have(70)fF˙p∗,wγ,q≤Mα#fp,w.


## 4. Some Extensions

In this section, we will assume thatμ is a nonnegative Borel doubling measure on ℝn; there exists β=βμ>0 such that(71)μB2r≤2βnμBr,for all ball Br. The smallest such β is called a doubling constant of μ.For eachN∈ℕ∪−1,m∈ℕ0, and l∈ℕ, we set(72)A=AN,ml=ν∈SNℝn:νm,N+l+1≤1.Definition 3.
Letγ∈ℝ, 0<p<∞, and 0<q≤∞. The homogeneous grand Tribel–Lizorkin space is the set of all tempered functions f such that when 0<q<∞,(73)fAF˙p,μγ,q=∑j∈ℤ2jγqsupv∈Av2−jfq1/qp,μ<∞and when q=∞,(74)fAF˙p,μγ,∞=supj∈ℤ2jγsupv∈Av2−jfp,μ<∞.Proposition 1.
Letγ∈ℝ, 0<p≤∞, and 0<q≤∞, and μ is the doubling measure with a constant equal to β. Set J=nβmax1,1/p,1/q. If A=AN,ml with l∈ℕ, N+1>maxγ,J−n−γ, and m>maxJ,n+N+1, then (75)fAF˙p,μγ,q=fF˙p,μγ,q.Proof.
Arguing as in the proof in ([15], Theorem 1.2) and using the almost-diagonality theorem (see [1], Theorem 4.2), we obtain the desired result.Proposition 2.
Letα and γ be real numbers satisfying 0≤α≤1 and γ<α and μ be a lower d−regular doubling measure. Assume f is a smooth function and Mα#f∈Lpw with d/n+α<p<d/α−γ. Then, for each 0<q≤∞,(1)
(76)fAF˙p∗,μγ,q≤Mα#fp,μ,wherep∗ is given by 1/p∗=1/p−α−γ/d(2)
For alln/n+α<p≤∞ and 0≤α<∞,(77)fAF˙p,μα,∞≤Mα#fp,μ.(3)
For alln/n+α<p≤∞, 0≤α<∞,(78)fAF˙∞,μα−d/p,∞≤Mα#fp,μ.Proof.
We have from (58) that if μ is a lower d-regular measure, then(79)supν∈Aν2−jf≤min2jαMα#fx,2jα−d/pMα#fp,w.
Arguing as in the proof of Proposition1, we obtain the desired result easily.Definition 4.
Letμ be a doubling measure 0<p<∞ and 0<α≤1. The homogeneous fractional Hajłasz–Sobolev space M˙μα,pℝn is the set of all measurable functions Lμ,locp for which there exists a nonnegative function g∈Lμp such that(80)fx−fy≤x−yαgx+gy,for μ−a.e.x,y∈ℝn.M˙μα,pℝn is equipped with the seminorm(81)fM˙μα,p=infg∈Dfgp,μ,where Df denotes the class of all nonnegative Borel functions g satisfying (80). Thus, Lemma 4.1 in [15] implies the following Sobolev embedding.Lemma 6.
Let0<α≤1, 0<δ<n/α, and p∗ be given by 1/p∗=1/δ−α/n. Then, for all x∈ℝn, 0<r<∞, f∈M˙μα,pℝn, and g∈Df,(82)infc∈ℝ1μBx,r∫Bx,rfy−cp∗dμ1/p∗≤rα1μBx,2r∫Bx,2rgyδdμ1/δ.Remark 4.
Lemma6 is due to Hajłasz ([13], Theorem 8.7) when α=1.Corollary 5.
Letα,γ, and δ be real numbers satisfying 0≤α≤1, γ<α, and n/n+α<p<n/α−γ. Assume f∈M˙μα,pℝn. Then, for each 0<q≤∞,(83)fAF˙p∗,μγ,q≤fM˙μα,p,where 1/p∗=1/p−α−γ/n.Proof.
Fix a ballBx,2r. Then, using Lemma 6 and by taking δ=n/n+α and Hölder’s inequality, we obtain(84)infc∈ℝ1μBx,r∫Bx,rfy−cdμ≤rα1μBx,2r∫Bx,2rgyδdμ1/δ.
Hence,(85)Mα#fx≤Mμgδ1/δxholds, where Mμg is the maximal function with respect to the measure μ. The Lp/δ-boundedness of Mμ when δ<p<∞ and Proposition 2 lead to estimate (83).Also, recall thatM˙μα,pℝn=AF˙p,μα,∞ℝn for 0<α≤1andn/n+α<p<∞ in [15] and M˙μ1,pℝn=F˙p,μ1,2ℝn≈H˙μp for n/n+1<p<∞ in [14]. Here, H˙μp denotes, for p>0, the homogeneous Hardy–Sobolev space, i.e., the space of tempered distributions f on ℝn, such that ∂jf∈Hμp for each j=1,…,n and(86)fH˙μp=∑j=1n∂jfHp,μ.Consequently, iff∈H˙μp with n/n+1<p<n/1−γ, then(87)fF˙p∗γ,q≤fH˙μp,1p∗=1p−1−γn.In particular, we have, forn/n+1<p<n, the following well-known result:(88)fp∗≤fH˙p,1p∗=1p−1n.



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*Source: 8104815-2021-11-24.xml*