# A Capacity Associated with the Weighted Lebesgue Space and Its Applications

Authors: Guoliang Li; Guanglan Wang; Lei Zhang
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we focus on a further study of the weighted Lebesgue capacity associated with the following fractional heat equation:∂t+−Δxαut,x=0,∀α,t,x∈0,1×0,∞×ℝn,u0,x=fx,∀x∈ℝn.. More properties of that capacity are explored, and applications to a trace inequality for the weak solution of the equation are considered.


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## Body

## 1. Introduction

Letα∈0,1, ℝ+1+n=0,∞×ℝn, and −Δxα denote the fractional power of the spatial Laplacian which can be defined by(1)−Δxαu⋅,x=ℱ−1ξ2αℱu⋅,ξx,∀x∈ℝn,for which ℱ is the Fourier transform and ℱ−1 is its inverse. Many research studies have been carried out on the following fractional heat equation with initial data:(2)∂t+−Δxαut,x=0,∀t,x∈ℝ+1+n;u0,x=fx,∀x∈ℝn.The weak solution of (2) can be written as(3)ut,x=Rαft,x=e−t−Δxαfx=∫ℝnKtαx−yfydy,where Ktαx is the fractional heat kernel.(4)Ktαx≔2π−n/2∫ℝneix⋅y−ty2αdy,∀t,x∈ℝ+1+n.It is well known thatKt1x and Kt1/2x are the heat kernel and Poisson kernel, respectively. Although it is hard to obtain an explicit formula for Ktαx when α∈0,1∖1/2 (cf. [1–5]), we are interested to find the following useful estimates (cf. [6, 7]):(5)Ktαx≈mint−n/2α,tx−n+2α≈tt1/2α+xn+2α,∀t,x∈ℝ+1+n;∫ℝnKtαxdx=1,∀t∈0,∞.To study the traces ofu, Chang and Xiao [8] introduced the following Lp capacity for a compact set K⊂ℝ+1+n:(6)CαpK=inffLpℝnp:f≥0&Rαf≥1K,where 1K stands for the characteristic function of K. Using Cαp⋅, they characterized a nonnegative Radon measure μ on R+1+n such that the mapping Rα:Lpℝn⟶Lμqℝ+1+n is continuous for 1<p,q<∞. Based on the theory of the Hedberg–Wolff potential, a noncapacitary characterization of the extension RαLpℝn⊆Lμqℝ+1+n was given in [9]. In [10], the authors considered the fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations. As a complement of [10], when the measure coincided with the Muckenhoupt class, this article addresses a weighted version of that extension.We first recall a nonnegative measurable functionw on ℝn and Ap weight (the Muckenhoupt class, cf. [11, 12]), denoted by w∈Ap, if(7)supB⊂ℝnwBw1−p′Bp−1<∞as1<p<∞,supB⊂ℝnwBinfx∈Bwx−1<∞asp=1,supB⊂ℝnwBexplogw−1B<∞asp=∞.Here and after,B≔Brx=y∈ℝn:x−y<r is the classical Euclidean ball centered at x with radius r for any r,x∈ℝ+1+n, and gB≔1/B∫Bgxdx denotes the average of the function g on B. In the literature, A∞ weights can also be given by A∞=∪1<p<∞Ap or(8)w∈A∞⇔C1EΩξ≤wEwΩ≤C2EΩηasξ≥1,η≤1,andE⊂Ω,for which C1,C2 are constants independent of the sets E,Ω and(9)wE=∫Ewxdx&E=∫Edx.LetλB stand for Bλrx with λ>0. A weight w is said to be a doubling weight if there is a constant C such that w2B≤CwB. It is well known that Ap weights are doubling weights. The weights we considered in this paper are either the Muckenhoupt classes Ap or weights satisfying(10)w1−p′∈A∞,which is a larger class than the Ap class. Condition (10) was first introduced to study weighted nonlinear potential theory by Adams in [13].The plan of this paper is as follows. Section2 presents the definition of a weighted Lp capacity and its properties. Section 3 addresses the application of the weighted capacity to characterize the trace inequalities of weak solutions to (2). Some of the results can be seen as not only the weighted extension of [8, 9] but also the parabolic case of [13].Until further statement, we always assume throughout the paper that0<α<1 and 1<p,q<∞ with 1/p+1/p′=1 and 1/q+1/q′=1. For each natural number k, CkX stands for the space of all functions being k times continuously differentiable in X, and CckX denotes the space of all functions in CkX having compact support. Cc∞X is the subspace of CckX given by Cc∞X≔∩kCckX. U+X denotes the class of all nonnegative Radon measures supported by the set X. LpX is the usual Lebesgue p space, and LμpX is the Lebesgue space with measure μ. The letter C is used to express a positive constant which is independent of the main parameters, but may vary from line to line. The symbol a≲b means a≤Cb; moreover, a≈b whenever a≲b and b≲a.


## 2. A WeightedLp Capacity

Letw be a weight function. The weighted Lp capacity associated with (2) can be defined as(11)CαpwK=inffLwpℝnp:f≥0&Rαf≥1Kfor compact setK⊂ℝ+1+n,CαpwO=supCαpwK:compactK⊂Ofor open setO⊂ℝ+1+n,CαpwE=infCαpwO:openOEfor any setE⊂ℝ+1+n.∀t0,x0∈ℝ+1+n and r>0, the parabolic ball, with center at t0,x0 and radius r, is denoted by(12)Brαt0,x0≔t,x∈ℝ+1+n:r2α<t−t0<2r2α&x−x0<r.It is obvious that its volumeBrαt0,x0≈rn+2α. The following theorem gives the estimates for CαpwBrαt0,x0.Theorem 1.
Letr,t0,x0∈ℝ+×ℝ+1+n. Then,(13)CαpwBrαt0,x0≲∫r∞s−n/p−1wBsx01−p′dss1−p,aswsatisfies10,≈∫r∞s−n/p−1wBsx01−p′dss1−p,asw∈Ap.The proof of Theorem1 needs a potential theory and will be given in the beginning of Section 3. If w≡1, then Cαp1Brαt0,x0≈rn by Theorem 1, which reduces to the well-known estimate for the unweighted Lp capacity of Brαt0,x0 (cf. Proposition 3 of [8]).Firstly, we give some fundamental properties ofCαpw..Proposition 1.
As a nonnegative set function, the mappingK⟶CαpwK enjoys some essential properties as follows:(a)
Cαpw∅=0.(b)
CαpwK1≤CαpwK2asK1,K2are compact subsets ofℝ+1+nK1⊆K2,Cαpw1K≤Cαpw2Kasw1,w2are nonnegative weight functions onℝn,w1≤w2.(c)
IfKi⊂ℝ+1+n is a compact sequence, then Cαpw∪i=1∞Ki≤∑i=1∞CαpwKi.(d)
For any two compact setsK1,K2 of ℝ+1+n, one has(14)CαpwK1∪K2+CαpwK1∩K2≤CαpwK1+CαpwK2.(e)
limi⟶∞CαpwKi=CαpwKas compactKi⊂ℝ+1+nis decreasing andK=∩i=1∞Ki,limi⟶∞CαpwEi=CαpwEas anyEi⊂ℝ+1+nis increasing andE=∪i=1∞Ei.Proof.
Write(15)XαpwK:=f:0≤f∈Lwpℝn&Rαf≥1K.
a–c follow immediately from the special case of Proposition 4.2 of [10] (M=ℝn and dμ=wxdx).
Ford, set(16)f=maxf1,f2g=minf1,f2withfi∈XαpwKi,i=1,2,and(17)E1=x∈ℝn:f1x≤f2xE2=ℝn\E1.
Then,f∈XαpwK1∪K2, g∈XαpwK1∩K2, and(18)∫ℝnfxpwxdx=∫E1f2xpwxdx+∫E2f1xpwxdx,∫ℝngxpwxdx=∫E1f1xpwxdx+∫E2f2xpwxdx.
This clearly forces(19)fLwpℝnp+gLwpℝnp=f1Lwpℝnp+f2Lwpℝnp.
It follows that(20)CαpwK1∪K2+CαpwK1∩K2≤f1Lwpℝnp+f2Lwpℝnp.
Hence,(21)CαpwK1∪K2+CαpwK1∩K2≤CαpwK1+CαpwK2.
To provee, we first note that, for any 0<ε<1, there exists a function f∈XαpwK satisfying(22)fLwpℝnp<CαpwK+ε.
Ifi is big enough, then the compact set Ki⊆Kε=x∈ℝn:Rαfx≥1−ε. We conclude from b that(23)limi⟶∞CαpwKi≤CαpwKε≤1−ε−pfLwpℝnp.
Consequently,(24)CαpwK≤limi⟶∞CαpwKi≤CαpwKasε⟶0.
Next, we proceed to show the second statement. We first conclude fromb that(25)CαpwE≥limi⟶∞CαpwEi.
To prove the reverse inequality, there is no loss of generality in assumingCαpwEi<∞. The definition of CαpwEi yields the existence of an open set Oi⊇Ei with(26)CαpwOi≤CαpwEi+2−iε,∀ε>0.
SinceCαpw∪i=1kEi=CαpwEk<∞, a slight modification of the telescoping inequality (Lemma 23 of [14]) (see also Lemma 2.5 of [15]) yields the following inequality:(27)Cαpw∪i=1kOi−Cαpw∪ikEi<ε.
LetK be a compact subset of ∪i=1∞Oi. Then, there exists a natural number k such that K⊆∪i=1kOi. Therefore,(28)CαpwK≤Cαpw∪i=1kOi≤Cαpw∪ikEi+ε≤limk⟶∞CαpwEk+ε.
This along withb gives(29)CαpwE≤Cαpw∪i=1∞Oi=supK⊂∪i=1∞OiCαpwK≤limk⟶∞CαpwEk+ε,which yields the desired result(30)CαpwE≤limi⟶∞CαpwEiasε⟶0.Secondly, we prove that the infimum for the norm⋅Lwpℝn in the definition of Cαpw⋅ can be achieved in Xαpw⋅. We call this function the capacitary potential.Proposition 2.
LetCαpwK<∞. Then, there is a unique function f0∈XαpwK such that(31)f0Lwpℝnp=CαpwK.Proof.
Assume thatfn⊂XαpwK such that limn⟶∞fnLwpℝnp=CαpwK. Since XαpwK is a closed, convex, and nonempty subset of the reflexive Banach space Lwpℝn, by possibly passing to a subsequence, we may assume that(32)fn⟶f0weakly inLwpℝnf0∈XαpwK.
Moreover, we havef0Lwpℝnp=CαpwK.
The uniform convexity of the spaceLwpℝn implies that f0 is unique.


## 3. Application to a Trace Inequality

As an application ofCαpwK on a given compact set K⊂ℝ+1+n, this section is devoted to some characterizations for the trace inequality of the weak solution to (2). In principle, we investigate nonnegative Radon measures μ∈U+K and weight functions w such that the mapping Rα:Lwpℝn⟶Lμqℝ+1+n is continuous. That is, the following trace inequality(33)RαfLμqℝ+1+n≲fLwpℝnholds for any f∈Lwpℝn. To do so, first we need some preliminaries of the potential theory.Letw satisfy (10) and μ∈U+ℝ+1+n. Then, the energy of μ with respect to w is given by(34)Eαpwμ=∫ℝnRα∗μxp′wx1−p′dx.By a direct calculation, we have thatEαpwμ can be rewritten as(35)Eαpwμ=∫ℝ+1+nRαw−1Rα∗μt,xp′−1dμt,x≔∫ℝ+1+nWαpwμt,xdμt,x,where Wαpwμ is called the weighted nonlinear potential of the measure μ with respect to the weight w. When w≡1, these potentials have been studied extensively (cf. [7–9]).Letμ∈U+ℝ+1+n and Mαμx=supr>0r−nμBrαr2α,x be its fractional parabolic maximal function. Then, the first result of this section is about the Lp−boundedness of Mαμ.Lemma 1.
Assume thatw∈A∞. Then,(36)Rα∗μLwpℝn≈MαμLwpℝn.Proof.
Since(37)Rα∗μx>˜∫Brαr2α,xtt1/2α+x−yn+2αdμt,y>˜μBrαr2α,xrn,∀r>0,we have Rα∗μx>˜Mαμx. Hence, MαμLwpℝn≲Rα∗μLwpℝn.
For the converse inequality, we first note that a slight modification of (1.2) of [16] yields that there exist two constants a>1 and b>0 such that, for any λ>0 and 0<ε≤1, one has the following weighted good-λ inequality:(38)wx∈ℝn:Rα∗μx>aλ≤bεn+2α/nwx∈ℝn:Rα∗μx>λ+wx∈ℝn:Mαμx>ελ.
Next, we proceed the proof following Theorem 3.6.1 of [17]. For any γ>0, multiplying the above inequality by λp−1 and integrating in λ, one has(39)∫0γwx∈ℝn:Rα∗μx>aλλp−1dλ≤bεn+2α/n∫0γwx∈ℝn:Rα∗μx>λλp−1dλ+∫0γwx∈ℝn:Mαμx>ελλp−1dλ.
By changing of variables, we get(40)a−p∫0aγwx∈ℝn:Rα∗μx>λλp−1dλ≤bεn+2α/n∫0γwx∈ℝn:Rα∗μx>λλp−1dλ+ε−p∫0εγwx∈ℝn:Mαμx>λλp−1dλ.
We thus obtain(41)a−p∫ℝnRα∗μxpwxdx≤2ε−p∫ℝnMαμxpwxdx.
By lettingε small enough such that bεn+2α/n≤1/2a−p and γ⟶∞, therefore,(42)Rα∗μLwpℝn≲MαμLwpℝnas desired.Usually, it is difficult to get the estimate forEαpwμ directly. One way around this difficulty is to try to give some simpler equivalent expression first. To do so, we introduce the following weighted parabolic Hedberg–Wolff potential for Rα inspired by the idea due to Hedberg and Wolff [18] for the Riesz potential:(43)Pαpwμt,x=∫0∞μBrαt,xrnp′−1w1−p′Brxdrr,∀t,x∈ℝ+1+n.Lemma 2.
Letw satisfy (10). Then,(44)Eαpwμ≈∫ℝ+1+nPαpwμdμ.Proof.
The forthcoming demonstration is a modification of Theorem 3.2 of [13]. Since(45)μBrαr2α,xrnp′≈∫r2rμBsαs2α,xsnp′dss≲∫0∞μBsαs2α,xsnp′dss,we have(46)Mαμx≲∫0∞μBrαr2α,xrnp′drr1/p′.
We conclude from Lemma1 that(47)Eαpwμ≲∫ℝnMαμxp′wx1−p′dx≲∫ℝn∫0∞μBrαr2α,xrnp′wx1−p′drrdx.
Substituting the following estimate(48)∫ℝnμBrαr2α,xp′wx1−p′dx=∫ℝnμBrαr2α,xp′−1∫Brαr2α,xdμt,ywx1−p′dx≲∫ℝ+1+nμBrα2r2α,yp′−1∫Brywx1−p′dxdμt,yinto (2), one has(49)Eαpwμ≲∫0∞∫ℝ+1+nμBrα2r2α,yp′−1∫Brywx1−p′dxdμt,ydrrnp′+1≲∫ℝ+1+n∫0∞μBrαt,yrnp′−1w1−p′Brydrrdμt,y=∫ℝ+1+nPαpwμdμ.
To prove the reverse, we first note that(50)Wαpwμt,x=Ktα∗w−1Rα∗μp′−1t,x>˜∑m∈ℤ∫B2−mxt−n/2αw−1y∫B2−mαt,xs−n/2αdμs,zp′−1dy>˜∑m∈ℤ∫B2−mx2mnμB2−mαt,x2−mnp′−1w1−p′ydy>˜∫0∞μBrαt,xrnp′−1w1−p′Brxdrr=Pαpwμt,x.
The proof is completed by the fact that(51)Eαpwμ=∫ℝ+1+nWαpwμt,xdμt,x>˜∫ℝ+1+nPαpwμt,xdμt,x.The following capacitary inequality forPαpwμ is a straightforward adaption of Proposition 4.1 of [13].Lemma 3.
Ifw satisfies (10), then for any μ∈U+ℝ+1+n, one has the following inequality:(52)CαpwPαpwμ>λ≲λ1−pμ.On account of the above analysis, we have the following.Lemma 4.
LetK be a compact set in ℝ+1+n, μ∈U+ℝ+1+n, and w satisfy (10). Then,(53)CαpwK>˜μasPαpwμ≤1onK,≲μasPαpwμ≥1onK.Proof.
Due to Proposition 4.1 of [10] and the proof of Lemma 2, there is μK∈U+K such that PαpwμK≲1 on suppμK. Then, combining with Lemma 3, we give the proof.Using Lemma4, we are now in a position to show the proof of Theorem 1 in Section 2. Since(54)Pαpwμt0,x0=∫02rμBsαt0,x0snp′−1w1−p′Bsx0dss+∫2r∞μBsαt0,x0snp′−1w1−p′Bsx0dss≔I+II,applying Lemma 4 to dμ=Cwdxdt on ball Bsαt0,x0 in which constant will be determined later, and we can obtain(55)II≈Cp′−1∫2r∞∫Bsαt0,x0wxdtdxsnp′−1w1−p′Bsx0dss.ChoosingC such that the right side of the above estimate is equal to 1, then Pαpwμ≥1 on Brαt0,x0. It follows from Lemma 4 and the doubling property of A∞ weights that(56)CαpwBrαt0,x0≲‖μ‖≈∫r∞s−np−1w1−p′Bsx0dss1−p.Ifw∈Ap, then for the above C, there is r such that(57)I≲∫02rs2αp′−1wBsx0p′−1w1−p′Bsx0dss≲r2αp′−1≲Cp′−1II≈1.Then, a further application of Lemma4 gives(58)CαpwBrαt0,x0>˜μ≈∫r∞s−n/p−1w1−p′Bsx0dss1−p.


### 3.1. Trace Inequality for1<p≤q<∞

In this section, we characterize (33) under the lower case 1<p≤q<∞ by Cαpw⋅. In Theorem 5.3 of [10], the authors obtained the following result.Theorem 2.
Let1<p≤q<∞, μ∈U+ℝ+1+n, and w satisfy (10). Then, the following two statements are equivalent:(a)
(33) is valid for any nonnegative f∈Lwpℝn(b)
supλ>0λp/q/Φλ,μw<∞, for which Φλ,μw=infCαpwK:compactK⊂ℝ+1+n&μK≥λIn particular, ifp≠q (i.e., 1<p<q<∞), we have the following deep understanding of (33).Theorem 3.
Assume that1<p<q<∞, μ∈U+ℝ+1+n, and w satisfies (10). Then, the following four statements are equivalent:(a)
(33) holds for all nonnegative f∈Lwpℝn(b)
μK≲CαpwKp/q for all compact sets K⊂ℝ+1+n(c)
μBrαt,x≲CαpwBrαt,xp/q for all r,t,x∈ℝ+×ℝ+×ℝn(d)
μBrαt,x≲Ψxr−q/p′ for all r,t,x∈ℝ+×ℝ+×ℝn, where Ψxr=∫r∞w1−p′Bsx/snp′−1ds/sProof.
a⇒b follows directly from the proof of Theorem 2.
b⇒c is trivial.
c⇒d is a consequence of Theorem 1.
d⇒a: let Kλf be defined as in the proof of Theorem 2 and μKλ be μ-restricted to Kλf. Following Lemma 2, we can obtain that(59)λμKλ≤∫KλRαfdμ=∫ℝnfxRα∗μKλxdx≲fLwpℝn∫ℝ+1+nPαpwμKλdμKλ1/p′.
For suitableρ>0 to be determined later, PαpwμKλ can be decomposed as(60)PαpwμKλt,x=∫0ρμKλBrαt,xrnp′−1w1−p′Brxdrr+∫ρ∞μKλBrαt,xrnp′−1w1−p′Brxdrr≔J+JJ.
Used to derive that(61)J≲∫0ρΨxr−q/pw1−p′Brxrnp′−1drr=∫0ρΨxr−q/pdΨxr=Ψxρ−q/p−Ψx0−q/p,and(62)JJ≲μKλp′−1Ψxρ.
Choose(63)ρ=ρx,λso thatΨxρ=μKλ−p′/qasΨx0−q/p<μKλp′−1,0asΨx0−q/p≥μKλp′−1.
Then, a further application ofd and Lemma 2 gives(64)λμKλ≲fLwpℝn∫ℝ+1+nμKλp′−1ΨxρdμKλ1/p′≲fLwpℝnμKλ1/q′asΨx0−q/p<μKλp′−1,fLwpℝn∫ℝ+1+nμKλp′−1Ψx0dμKλ1/p′≲fLwpℝnμKλ1/q′asΨx0−q/p≥μKλp′−1,which implies (33).


### 3.2. Trace Inequality for1<q<p<∞

This section focuses on a further trace inequality under the upper case1<q<p<∞. The main result can be formulated as follows.Theorem 4.
Let1<q<p<∞ and μ,w, and Φλ,μw be as in Theorem 2. Then, the following two statements are equivalent:(a)
(33) holds for all nonnegative f∈Lwpℝn(b)
Pαpwμ∈Lμqp−1/p−qℝ+1+nProof.
The proof can be seen as a weighted fractional heat potential version of the Riesz potential treatment carried out in [19].
To showa⇒b, we first denote by Qlα the α-dyadic cube with side length l≡lQlα and corners in the set l2αℤ+,lℤn with ℤ+=0,1,2,…, namely,(65)Qlα≡k0l2α,k0+1l2α×Q≔k0l2α,k0+1l2α×k1l,k1+1l×⋯×knl,kn+1l,as k0∈ℤ+ and ki∈ℤ for i=1,2,…,n. Next, we introduce the following weighted fractional heat Hedberg–Wolff potential generated by Dα—the family of all the above-defined α-dyadic cubes in ℝ+1+n:(66)Pαpd,wμt,x=∑Qlα∈DαμQlαlnp′−1w1−p′Q1Qlαt,x,and we need to prove(67)a⇒Pαpd,wμ∈Lμqp−1/p−qℝ+1+n.
Sincea is equivalent to the inequality(68)Rα∗gdμLw1−p′p′ℝnp′≲gLμq′ℝ+1+np′,∀g∈Lμq′ℝ+1+n,we notice that Lemma 2 is also true with Pαpd,wμ in place of Pαpwμ and gdμ in place of dμ, and it may be concluded that(69)Rα∗gdμLw1−p′p′ℝnp′>˜∫ℝ+1+nPαpd,wgdμt,xgt,xdμt,x≈∑Qlα∫Qlαgt,xdμt,xlnp′lnw1−p′Q,namely,(70)∑Qlα∫Qlαgt,xdμt,xlnp′lnw1−p′Q≲gLμq′ℝ+1+np′.
For the convenience of notation, (70) can be rewritten as(71)∑QlαΥQlα∫QlαgdμμQlαp′w1−p′Q≲gLμq′ℝ+1+np′,for which(72)ΥQlα=μQlαlnp′ln.
Since the following dyadic Hardy–Littlewood maximal function(73)Mμdht,x=supt,x∈Qα1μQα∫Qαhs,ydμs,y,∀Qα∈Dα,is bounded on Lμpℝ+1+n for 1<p<∞ and(74)∫Qlαgt,xdμt,xμQlαp′>˜∫Qht,xdμt,xμQlα,with the choice of gt,x=Mμdh1/p′t,x under 0≤h∈Lμq′/p′ℝ+1+n, we deduce that(75)gLμq′ℝ+1+np′≲hLμq′/p′ℝ+1+n.
It follows that(76)∑QlαΥQlα∫Qlαht,xdμt,xμQlαw1−p′Q≲hLμq′/p′ℝ+1+n,∀h∈Lμq′/p′ℝ+1+n.
By duality, we can get(77)∑QlαλQlαμQlα1Qlαw1−p′Q∈Lμq′/q′−p′ℝ+1+n,that is, (67).
We proceed the proof by defining(78)Pαpd,τ,wμt,x=∑Qlα∈DταμQlαlnp′−1w1−p′Q1Qlαt,x&Dτα=Dα+τ=Qlα+τQlα′∈Dα,where Qlα+τ=t,x+τ:t,x∈Qlα means the ℝ+1+n∋τ-shift of Qlα. Then, (67) implies(79)supτ∈ℝ+1+n∫ℝ+1+nPαpd,τ,wμt,xqp−1/p−qdμt,x<∞.
The proof will be completed by showing that(80)Pαpwμ∈Lμqp−1/p−qℝ+1+n.
Suppose first thatμ is a doubling measure. Then, (80) is a consequence of (67) and the following observation:(81)Pαpwμt,x≲∑QlαμQlα∗lnp′−1w1−p′Q1Qlαt,x,where Qlα∗ is the cube with the same center as Qlα and side length two times as Qlα. We are left with the task of determining when μ is a possibly nondoubling measure. For any δ>0, we first write(82)Pαp,δwμt,x=∫0δμBrαt,xrnp′−1w1−p′Brxdrr.
Next, we claim that(83)Pαp,δwμt,x≲δ−n+1∫τ≲δPαpd,τ,wμt,xdτ.
In fact, for fixedt,x∈ℝ+1+n and δ>0 such that 2i−1ρ≤δ<2iρ with i∈ℤ and ρ>0 will be determined later, we have(84)Pαp,δwμt,x≲∑j=−∞iμB2jραt,x2jρnp′−1w1−p′B2jρx.
Forj≤i, let Ql,jα be a cube centered at x with 2j−1<l≤2j. Then, B2jραt,x⊆Ql,jα for sufficiently small ρ. Let E be the set of all points τ∈ℝ+1+n enjoying τ≲δ with E being the 1+n-dimensional Lebesgue measure, but also, there exists Qlα,τ∈Dτα satisfying l=2j+1 and Ql,jα⊆Qlα,τ. A geometric consideration produces a dimensional constant cn>0 such that E≥cnδn+1∀j≤i. Therefore,(85)μB2jραt,xp′−1≲E−1∫E∑l=2j+1μQlα,τp′−11Qlα,τt,xdτ≲δ−n+1∫τ≲δ∑l=2j+1μQlα,τp′−11Qlα,τt,xdτ,whence reaching(86)Pαp,δwμt,x≲δ−n+1∫τ≲δ∑j=−∞i∑l=2j+1μQlα,τ2jρnp′−1w1−p′B2jρx1Qlα,τt,xds≲δ−n+1∫τ≲δPαpd,τ,wμt,xdτ,which is our claim. By Hölder’s inequality and Fubini’s theorem, we give(87)∫ℝ+1+nPαp,δwμt,xqp−1/p−qdμt,x≲∫ℝ+1+nδ−n+1∫τ≤CδPαpd,τ,wμqp−1/p−qdτp−q/qp−1∫τ≲δdτ1−p−q/qp−1qp−1/p−qdμ≲δ−n+1∫τ≲δ∫ℝ+1+nPαpd,τ,wμqp−1/p−qdμdτ<∞,which produces(88)Pαpwμ∈Lμqp−1/p−qℝ+1+nasδ⟶∞,by the monotone convergence theorem.
b⇒a: by duality and Lemma 2, we only need to show that(89)∫ℝ+1+nPαpwgdμt,xgt,xdμ≲gLμq′ℝ+1+np′,∀g∈Lμq′ℝ+1+n.
Let(90)Mμgt,x=supr>01μBrαt,x∫Brαt,xgs,ydμs,y.
Denote the centered Hardy–Littlewood maximal function ofg with respect to μ. We note that(91)Pαpwgdμt,x≈∫0∞μBrαt,xrnp′−1∫Brαt,xgt,xdμμBrαt,xp′−1w1−p′Brxdrr≲Mμgt,xp′−1Pαpwμt,x.
Then, the Hölder inequality yields(92)∫ℝ+1+nPαpwgdμt,xdμt,x≲∫ℝ+1+nMμgt,xp′−1Pαpwμt,xgt,xdμt,x≲∫ℝ+1+nMμgt,xq′dμt,xq′/p′−1∫ℝ+1+ngt,xPαpwμt,xq′/q′−p′+1dμt,xq′−p′+1/q′≲gLμq′ℝ+1+np′∫ℝ+1+nPαpwμqp−1/p−qdμp−q/qp−1≲gLμq′ℝ+1+np′,where we have used the boundedness of Mμ on Lμq′ℝ+1+n (cf. [20]).


## 3.1. Trace Inequality for1<p≤q<∞

In this section, we characterize (33) under the lower case 1<p≤q<∞ by Cαpw⋅. In Theorem 5.3 of [10], the authors obtained the following result.Theorem 2.
Let1<p≤q<∞, μ∈U+ℝ+1+n, and w satisfy (10). Then, the following two statements are equivalent:(a)
(33) is valid for any nonnegative f∈Lwpℝn(b)
supλ>0λp/q/Φλ,μw<∞, for which Φλ,μw=infCαpwK:compactK⊂ℝ+1+n&μK≥λIn particular, ifp≠q (i.e., 1<p<q<∞), we have the following deep understanding of (33).Theorem 3.
Assume that1<p<q<∞, μ∈U+ℝ+1+n, and w satisfies (10). Then, the following four statements are equivalent:(a)
(33) holds for all nonnegative f∈Lwpℝn(b)
μK≲CαpwKp/q for all compact sets K⊂ℝ+1+n(c)
μBrαt,x≲CαpwBrαt,xp/q for all r,t,x∈ℝ+×ℝ+×ℝn(d)
μBrαt,x≲Ψxr−q/p′ for all r,t,x∈ℝ+×ℝ+×ℝn, where Ψxr=∫r∞w1−p′Bsx/snp′−1ds/sProof.
a⇒b follows directly from the proof of Theorem 2.
b⇒c is trivial.
c⇒d is a consequence of Theorem 1.
d⇒a: let Kλf be defined as in the proof of Theorem 2 and μKλ be μ-restricted to Kλf. Following Lemma 2, we can obtain that(59)λμKλ≤∫KλRαfdμ=∫ℝnfxRα∗μKλxdx≲fLwpℝn∫ℝ+1+nPαpwμKλdμKλ1/p′.
For suitableρ>0 to be determined later, PαpwμKλ can be decomposed as(60)PαpwμKλt,x=∫0ρμKλBrαt,xrnp′−1w1−p′Brxdrr+∫ρ∞μKλBrαt,xrnp′−1w1−p′Brxdrr≔J+JJ.
Used to derive that(61)J≲∫0ρΨxr−q/pw1−p′Brxrnp′−1drr=∫0ρΨxr−q/pdΨxr=Ψxρ−q/p−Ψx0−q/p,and(62)JJ≲μKλp′−1Ψxρ.
Choose(63)ρ=ρx,λso thatΨxρ=μKλ−p′/qasΨx0−q/p<μKλp′−1,0asΨx0−q/p≥μKλp′−1.
Then, a further application ofd and Lemma 2 gives(64)λμKλ≲fLwpℝn∫ℝ+1+nμKλp′−1ΨxρdμKλ1/p′≲fLwpℝnμKλ1/q′asΨx0−q/p<μKλp′−1,fLwpℝn∫ℝ+1+nμKλp′−1Ψx0dμKλ1/p′≲fLwpℝnμKλ1/q′asΨx0−q/p≥μKλp′−1,which implies (33).


## 3.2. Trace Inequality for1<q<p<∞

This section focuses on a further trace inequality under the upper case1<q<p<∞. The main result can be formulated as follows.Theorem 4.
Let1<q<p<∞ and μ,w, and Φλ,μw be as in Theorem 2. Then, the following two statements are equivalent:(a)
(33) holds for all nonnegative f∈Lwpℝn(b)
Pαpwμ∈Lμqp−1/p−qℝ+1+nProof.
The proof can be seen as a weighted fractional heat potential version of the Riesz potential treatment carried out in [19].
To showa⇒b, we first denote by Qlα the α-dyadic cube with side length l≡lQlα and corners in the set l2αℤ+,lℤn with ℤ+=0,1,2,…, namely,(65)Qlα≡k0l2α,k0+1l2α×Q≔k0l2α,k0+1l2α×k1l,k1+1l×⋯×knl,kn+1l,as k0∈ℤ+ and ki∈ℤ for i=1,2,…,n. Next, we introduce the following weighted fractional heat Hedberg–Wolff potential generated by Dα—the family of all the above-defined α-dyadic cubes in ℝ+1+n:(66)Pαpd,wμt,x=∑Qlα∈DαμQlαlnp′−1w1−p′Q1Qlαt,x,and we need to prove(67)a⇒Pαpd,wμ∈Lμqp−1/p−qℝ+1+n.
Sincea is equivalent to the inequality(68)Rα∗gdμLw1−p′p′ℝnp′≲gLμq′ℝ+1+np′,∀g∈Lμq′ℝ+1+n,we notice that Lemma 2 is also true with Pαpd,wμ in place of Pαpwμ and gdμ in place of dμ, and it may be concluded that(69)Rα∗gdμLw1−p′p′ℝnp′>˜∫ℝ+1+nPαpd,wgdμt,xgt,xdμt,x≈∑Qlα∫Qlαgt,xdμt,xlnp′lnw1−p′Q,namely,(70)∑Qlα∫Qlαgt,xdμt,xlnp′lnw1−p′Q≲gLμq′ℝ+1+np′.
For the convenience of notation, (70) can be rewritten as(71)∑QlαΥQlα∫QlαgdμμQlαp′w1−p′Q≲gLμq′ℝ+1+np′,for which(72)ΥQlα=μQlαlnp′ln.
Since the following dyadic Hardy–Littlewood maximal function(73)Mμdht,x=supt,x∈Qα1μQα∫Qαhs,ydμs,y,∀Qα∈Dα,is bounded on Lμpℝ+1+n for 1<p<∞ and(74)∫Qlαgt,xdμt,xμQlαp′>˜∫Qht,xdμt,xμQlα,with the choice of gt,x=Mμdh1/p′t,x under 0≤h∈Lμq′/p′ℝ+1+n, we deduce that(75)gLμq′ℝ+1+np′≲hLμq′/p′ℝ+1+n.
It follows that(76)∑QlαΥQlα∫Qlαht,xdμt,xμQlαw1−p′Q≲hLμq′/p′ℝ+1+n,∀h∈Lμq′/p′ℝ+1+n.
By duality, we can get(77)∑QlαλQlαμQlα1Qlαw1−p′Q∈Lμq′/q′−p′ℝ+1+n,that is, (67).
We proceed the proof by defining(78)Pαpd,τ,wμt,x=∑Qlα∈DταμQlαlnp′−1w1−p′Q1Qlαt,x&Dτα=Dα+τ=Qlα+τQlα′∈Dα,where Qlα+τ=t,x+τ:t,x∈Qlα means the ℝ+1+n∋τ-shift of Qlα. Then, (67) implies(79)supτ∈ℝ+1+n∫ℝ+1+nPαpd,τ,wμt,xqp−1/p−qdμt,x<∞.
The proof will be completed by showing that(80)Pαpwμ∈Lμqp−1/p−qℝ+1+n.
Suppose first thatμ is a doubling measure. Then, (80) is a consequence of (67) and the following observation:(81)Pαpwμt,x≲∑QlαμQlα∗lnp′−1w1−p′Q1Qlαt,x,where Qlα∗ is the cube with the same center as Qlα and side length two times as Qlα. We are left with the task of determining when μ is a possibly nondoubling measure. For any δ>0, we first write(82)Pαp,δwμt,x=∫0δμBrαt,xrnp′−1w1−p′Brxdrr.
Next, we claim that(83)Pαp,δwμt,x≲δ−n+1∫τ≲δPαpd,τ,wμt,xdτ.
In fact, for fixedt,x∈ℝ+1+n and δ>0 such that 2i−1ρ≤δ<2iρ with i∈ℤ and ρ>0 will be determined later, we have(84)Pαp,δwμt,x≲∑j=−∞iμB2jραt,x2jρnp′−1w1−p′B2jρx.
Forj≤i, let Ql,jα be a cube centered at x with 2j−1<l≤2j. Then, B2jραt,x⊆Ql,jα for sufficiently small ρ. Let E be the set of all points τ∈ℝ+1+n enjoying τ≲δ with E being the 1+n-dimensional Lebesgue measure, but also, there exists Qlα,τ∈Dτα satisfying l=2j+1 and Ql,jα⊆Qlα,τ. A geometric consideration produces a dimensional constant cn>0 such that E≥cnδn+1∀j≤i. Therefore,(85)μB2jραt,xp′−1≲E−1∫E∑l=2j+1μQlα,τp′−11Qlα,τt,xdτ≲δ−n+1∫τ≲δ∑l=2j+1μQlα,τp′−11Qlα,τt,xdτ,whence reaching(86)Pαp,δwμt,x≲δ−n+1∫τ≲δ∑j=−∞i∑l=2j+1μQlα,τ2jρnp′−1w1−p′B2jρx1Qlα,τt,xds≲δ−n+1∫τ≲δPαpd,τ,wμt,xdτ,which is our claim. By Hölder’s inequality and Fubini’s theorem, we give(87)∫ℝ+1+nPαp,δwμt,xqp−1/p−qdμt,x≲∫ℝ+1+nδ−n+1∫τ≤CδPαpd,τ,wμqp−1/p−qdτp−q/qp−1∫τ≲δdτ1−p−q/qp−1qp−1/p−qdμ≲δ−n+1∫τ≲δ∫ℝ+1+nPαpd,τ,wμqp−1/p−qdμdτ<∞,which produces(88)Pαpwμ∈Lμqp−1/p−qℝ+1+nasδ⟶∞,by the monotone convergence theorem.
b⇒a: by duality and Lemma 2, we only need to show that(89)∫ℝ+1+nPαpwgdμt,xgt,xdμ≲gLμq′ℝ+1+np′,∀g∈Lμq′ℝ+1+n.
Let(90)Mμgt,x=supr>01μBrαt,x∫Brαt,xgs,ydμs,y.
Denote the centered Hardy–Littlewood maximal function ofg with respect to μ. We note that(91)Pαpwgdμt,x≈∫0∞μBrαt,xrnp′−1∫Brαt,xgt,xdμμBrαt,xp′−1w1−p′Brxdrr≲Mμgt,xp′−1Pαpwμt,x.
Then, the Hölder inequality yields(92)∫ℝ+1+nPαpwgdμt,xdμt,x≲∫ℝ+1+nMμgt,xp′−1Pαpwμt,xgt,xdμt,x≲∫ℝ+1+nMμgt,xq′dμt,xq′/p′−1∫ℝ+1+ngt,xPαpwμt,xq′/q′−p′+1dμt,xq′−p′+1/q′≲gLμq′ℝ+1+np′∫ℝ+1+nPαpwμqp−1/p−qdμp−q/qp−1≲gLμq′ℝ+1+np′,where we have used the boundedness of Mμ on Lμq′ℝ+1+n (cf. [20]).



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*Source: 1257963-2022-02-28.xml*