# Optimal Weak Type Estimates for the P-ADIC Hardy Type Operators on Higher-Dimensional Product Spaces

Authors: Junchao Wei
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we introduce the fractional p-adic Hardy operators and its conjugate operators and obtain its optimal weak type estimates on the p-adic Lebesgue product spaces.


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## Body

## 1. Introduction

In recent years,p-adic analysis has been widely used in quantum mechanics, the probability theory, and the dynamical systems [1, 2]. Meanwhile, there is an increasing attention in pseudo-differential equations, wavelet theory, and harmonic analysis (see [3–8]).For a prime numberp, let ℚp be the field of p-adic numbers, a nonzero rational number x is represented as x=pγm/n, where γ is an integer and the integers m,n are not divisible by p. Then, the norm is defined as xp=p−γ, and it is easy to see that the norm satisfies the following properties:(i)
xp≥0,∀x∈ℚp,xp=0⇔x=0(ii)
xyp=xpyp,∀x,y∈ℚp(iii)
x+yp≤maxxp,yp,∀x,y∈ℚp, in the case when xp≠yp, we have x+yp=maxxp,ypIt is well known thatℚp is a typical model of non-Archimedean local fields. From the standard p-adic analysis, any x∈ℚp/0 can be uniquely represented as a canonical form(1)x=pγ∑k=0∞αkpk,where αk,γ∈ℤ,α0≠0≤αk<p, note that the series (1) converges with respect to the norm xp because one has pγαkpkp=p−γ−k. The space ℚpn consists of elements x=x1,x2,…,xn, where xi∈ℚp, i=1,2,…,n. The norm in this space is(2)xp≔max1≤i≤nxip,x∈ℚpn.The symbolsBγa and Sγa represent, respectively, the ball and the sphere with center at a∈ℚpn and radius pγ, defined by(3)Bγa=x∈ℚpn:x−ap≤pγ,Sγa=x∈ℚpn:x−ap=pγ.It is clear thatSγa=Bγa/Bγ−1a, and we set Bγ0=Bγ and Sγ0=Sγ.Asℚpn is a locally compact commutative group with respect to addition, there exists a Harr measure dx on ℚpn, which is unique up to a positive constant factor and is translation invariant, that is, dx+a=dx. We normalize the measure dx such that(4)∫B00dx=B00H=1,where BH denotes the Harr measure of a measure subset B of ℚpn. By simple calculation, we can obtain that(5)BγaH=pγn,SγaH=pγn1−p−n.The classical Hardy operator(6)ℋfx≔1x∫0xftdt,x>0was introduced by Hardy in [9], and a celebrated integral inequality states that(7)ℋfLqℝ+≤qq−1fLqℝ+,1<q<∞.It was also shown that the constant factorq/q−1 is optimal, knowing its importance in analysis.Faris in [10] and Christ and Grafakos in [11] proposed an extension of (1) and its adjoint to the n-dimensional Euclidian spaces ℝn of which the equivalent forms are(8)Hfx=1xn∫y|≤|xfydy,H∗fx=∫y>xfyyndy,x∈ℝn0.The norm ofH and H∗ on Lqℝn was evaluated and found to be equal to that of the classical Hardy operator. For more details about the boundedness of the Hardy operator and its adjoint, we included some references [12–14].On the other hand, the fractional Hardy operator and its adjoint are obtained by merely interchanging⋅n with ⋅n+α0≤α<n in (8). The weak and strong type optimal bounds for the fractional Hardy and adjoint Hardy operator have also spotlighted many researchers in the past, see [15–18].The n-dimensional fractional p-adic Hardy and adjoint Hardy operator are defined and studied in [19], which, for f∈Llocℚpn and 0≤β<∞, are given as(9)Hβpfx=1xpn−β∫yp≤xpfydy,Hβp,∗fx=∫yp≥xpfyypn−βdy,x∈ℚpn0.whenβ=0, the fractional p-adic Hardy and adjoint Hardy operator reduces to p-adic Hardy and adjoint Hardy operator. Some other papers showing the boundedness of p-adic Hardy-type operators are included [19–26].In 2020, Li et al. [27] introduced the definition of the fractional Hardy operator on higher-dimensional product spaces as follows:(10)Hβ1,…βmfx≔∏i=1m1B0,xi1−βi/ni∫y1<x1⋯∫ym<xmfy1,…,ymdy1⋯dym,where f be a nonnegative integrable function on ℝn1×ℝn2×⋯×ℝnm, xi∈ℝni, m∈ℕ,0≤βi<ni, i=1,…,m, x=x1,x2,…,xm∈ℝn1×ℝn2×⋯×ℝnm with ∏i=1mxi≠0. Furthermore, the corresponding operator norm on the weak Lebesgue product spaces was obtained.Next, we will introduce the definition of the fractional Hardy operator on higher-dimensional p-adic product spaces and obtain sharp weak bounds.Definition 1.
Letm,ni∈ℕ,xi∈ℚpni, 0≤βi<ni,i=1,…,m, and f be a nonnegative integrable function on ℚpn1×ℚpn2×⋯×ℚpnm. Define the fractional p-adic Hardy operator on higher-dimensional product spaces by(11)Hβ1,…βmpfx=9∏i=1m1B0,xipH1−βi/ni00∫y1p<x1p⋯∫ymp<xmpfy1,⋯ymdy1⋯dym,where x=x1,x2,…,xm∈ℚpn1×ℚpn2×⋯×ℚpnm with ∏i=1mxip≠0.
In 2020, Wang et al. [28] gave the definition of fractional conjugate Hardy operator on higher-dimensional product spaces as follows:(12)Hβ1,…βm∗fx≔∫y1>x1⋯∫ym>xm∏i=1mfy1,…,ymB0,yi1−βi/nidy1…dym,where f be a nonnegative integrable function on ℝn1×ℝn2×⋯×ℝnm, xi∈ℝni, m∈ℕ, 0≤βi<ni,i=1,…,m, x=x1,x2,…,xm∈ℝn1×ℝn2×⋯×ℝnm with ∏i=1mxi≠0, and they also got the corresponding operator norm on the weak Lebesgue product spaces.
Next, we will give a higher-dimensional version of the fractional p-adic conjugate Hardy operator and obtain sharp weak bounds.Definition 2.
Letm∈ℕ,ni∈ℕ,xi∈ℚpni, 0≤βi<ni,i=1,…,m, and f be a nonnegative integrable function on ℚpn1×ℚpn2×⋯×ℚpnm. Define the fractional p-adic conjugate Hardy operator on higher-dimensional product spaces by(13)Hβ1,…βmp,∗fx=∫y1p>x1p⋯∫ymp>xmp∏i=1mfy1,…,ymB0,yipH1−βi/nidy1…dym,where x=x1,x2,…,xm∈ℚpn1×ℚpn2×⋯×ℚpnm with ∏i=1mxi≠0.
In this article, we will obtain sharp weak bounds for the fractional p-adic Hardy operators and its conjugate operators on the p-adic Lebesgue product spaces. Our method of proving the main results involves a frequent use of the following formula:(14)∫ℚpnfxdx=∑γ∈ℤ∫Sγfxdx.


## 2. Sharp Weak Bounds for Fractional Hardy Operators

This section considers the problem of obtaining optimal weak bounds forHβ1,…βmp and our results as follows.Theorem 1.
Set0<βi<ni, let Q=n1/n1−β1,…,nm/nm−βm, i=1,…,m. If f∈L1ℚpn1×ℚpn2×⋯×ℚpnm, then we have(15)Hβ1,…,βmpfwLQℚpn1×ℚpn2×⋯×ℚpnm≤1⋅fL1ℚpn1×ℚpn2×⋯×ℚpnm.
Furthermore,(16)Hβ1,⋯βmpL1ℚpn1×ℚpn2×⋯×ℚpnm⟶wLQℚpn1×ℚpn2×⋯×ℚpnm=1.
To obtain the desired result, we need the following lemma.Lemma 1.
Suppose that0≤β<n, if f∈L1ℚpn, then for any λ>0,(17)HβpfLn/n−β,∞ℚpn≤1⋅fL1ℚpn.
Moreover,(18)HβpL1ℚpn⟶Ln/n−β,∞ℚpn=1.Proof.
Since(19)Hβpfx≤1xpn−β∫yp≤xpfydy≤xp−n−βfL1ℚpn.
Next, we letL=fL1ℚpn, then, for any λ>0, we get(20)x∈ℚpn:Hβpfx>λ⊂x∈ℚpn:xp<L/λ1/n−β.
Thus,(21)HβpfLn/n−β,∞ℚpn=supλ>0λ∫ℚpnχx∈ℚpn:Hβpfx>λxdxn−β/n≤supλ>0λ∫ℚpnχx∈ℚpn:xp<L/λ1/n−βxdxn−β/n=supλ>0λ∫|x|p<L/λ1/n−βdxn−β/n=supλ>0λ∑j=−∞logpL/λ1n−β∫Sjdxn−β/n=1−p−nn−β/nsupλ>0λ∑j=−∞logpL/λ1/n−βpjndxn−β/n=supλ>0λLλ=fL1ℚpn.
On the other hand, we letf0x=χx∈ℚpn:xp≤1x, then(22)f0L1ℚpn=∫ℚpnχx∈ℚpn:xp≤1xdx=1.
Also,(23)Hβpf0x=1xpn−β∫yp≤xpf0ydy=1xpn−β∫yp≤xpχx∈ℚpn:xp≤1dy=1xpn−β∫yp≤xpdy,xp≤1,∫yp<1dy,xp>1.=xpβ,xp≤1,xpβ−n,xp>1.
Now,(24)x∈ℚpn:Hβpf0x>λ=xp≤1:xpβ>λ∪xp>1:xpβ−n>λ.
Since0<β<n, when λ≥1, then(25)x∈ℚpn:Hβpf0x>λ=∅.
Also, when0<λ<1, then(26)x∈ℚpn:Hβpf0x>λ=x∈ℚpn:λ1/β<xp<1λ1/n−β.
Therefore,(27)Hβpf0Ln/n−β,∞ℚpn=sup0<λ<1λ∫ℚpnχx∈ℚpn:λ1/β<xp<1/λ1/β−nxdxn−β/n=sup0<λ<1λ∫λ1/β<xp<1/λ1/n−βdxn−β/n=sup0<λ<1λ∑j=logpλ1/βlogp1/λ1/n−β∫Sjdxn−β/n=1−p−nn−β/nsup0<λ<1λplogpλ1/β+1n−plogpλ1/β−n+1n1−pnn−β/n=1−p−nn−β/nsup0<λ<11−λn/βλn/n−β1−p−nn−β/n=f0L1ℚpn.
Thus, as above, we get(28)HβpL1ℚpn⟶Ln/n−β,∞ℚpn=1.
Now let us prove Theorem1.Proof.
Without loss of generality, we consider only the situation whenm=2, and then, the case m≥3 is just a repetition of the case m=2. For m=2, the operator Hβ1,β2p can be written as(29)Hβ1,β2pfx1,x2=1B0,x1pH1−β1/n11B0,x2pH1−β2/n2∫y1p≤x1p∫y2p≤x2pfy1,y2dy1dy2.
Whenf∈L1ℚpn1×ℚpn2, we get(30)1B0,|x2|pH1−β2/n2∫y2p≤x2pf⋅,y2dy2∈L1ℚpn1,for allx2∈ℚpn2.
Then by Lemma 1,(31)Hβ1,β2pf⋅,x2Ln1/n1−β1,∞ℚpn1=supλ1>0λ1x1:Hβ1,β2px1,x2>λ1n1−β1/n1≤1⋅1B0,x2pH1−β2/n2∫y2p<x2pf⋅,y2dy2L1ℚpn1.
Using Fubini theorem, we obtain that(32)1|B0,|x2|p|H1−β2/n2∫|y2|p<|x2|pf⋅,y2dy2L1ℚpn1=1|B0,|x2|p|H1−β2/n2∫ℚpn1∫|y2|p<|x2|pfy1,y2dy2dy1≤1|B0,|x2|p|H1−β2/n2∫|y2|p<|x2|p∫ℚpn1fy1,y2dy1dy2.
Obviously,∫ℚpn1fy1,y2dy1∈L1ℚpn2, if f∈L1ℚpn1×ℚpn2. Then, applying the lemma again, we get(33)1|B0,|x2|p|H1−β2/n2∫|y2|p≤|x2|p∫ℚpn1fy1,y2dy1dy2Ln2/n2−β2,∞ℚpn2=supλ2>0λ2x2:1|B0,|x2|p|H1−β2/n2∫|y2|p≤|x2|p∫ℚpn1fy1,y2dy1dy2>λ2n2−β2/n2≤1⋅∫ℚpn2∫ℚpn1fy1,y2dy1dy2=fL1ℚpn1×ℚpn2.
Combining (31)–(33), we get(34)Hβ1,β2pfwLn1/n1−β1,n2/n2−β2ℚpn1×ℚpn2≤1⋅‖f‖L1ℚpn1×ℚpn2,for all f∈L1ℚpn1×ℚpn2.
Conversely, to prove that the constant 1 is optimal, we took(35)f0x=χx∈ℚpn:xp≤1x.
And choosingFx1,x2=f0x1f0x2, where x1∈ℚpn1,x2∈ℚpn2, we get from the definition of Hβ1,β2p that(36)Hβ1,β2pFx1,x2=∏i=121B0,xipH1−βi/ni∫y1p<x1p∫y2p<x2pf0y1f0y2dy1dy2=Hβ1pf0x1Hβ2pf0x2.
Also,(37)Hβ1pf0x1=1B0,x1pH1−β1/n1∫y1p≤x1pf0y1dy=1x1pn1−β1∫y1p≤x1pdy,x1p<1,∫y1p<1dy,x1p≥1,=x1pβ1x1p<1,x1pβ1−n1,x1p≥1.
We now that letL=Hβ2pf0x2, for 0<λ1<L, then combining both the cases, we get(38)x1∈ℚpn1:Hβ1,β2pFx1,x2>λ1=x1p<1:x1pβ1L>λ1+x1p≥1:x1pβ1−n1L>λ1=x1∈ℚpn1:λ1L1/β1<x1p<1+x1∈ℚpn1:1≤x1p<Lλ11/n1−β1=x1∈ℚpn1:λ1L1/β1<x1p<Lλ11/n1−β1.
Therefore, we have(39)Hβ1,β2pFx1,x2Ln1n1−β1,∞ℚpn1=sup0<λ1<Lλ1∫ℚpn1χλ1/L1/β1<|x1|p<L/λ11n1−β1x1dx1n1−β1/n1=sup0<λ1<Lλ1∫λ1/L1/β1<|x1|p<L/λ11/n1−β1dx1n1−β1/n1=sup0<λ1<Lλ1∑j=−logpλ1/Lβ1+1logpL/λ11/n1−β1∫Sjdx1n1−β1/n1=1−p−n1n1−β1/n1sup0<λ1<Lλ1∑j=−logpλ1/Lβ1+1logpL/λ11/n1−β1pjn1n1−β1/n1=1−p−n1n1−β1/n1sup0<λ1<Lλ1p−logpλ1/Lβ1+1n1−plogpL/λ11/n1−β1+1n11−pn1n1−β1/n1=1−p−n1n1−β1/n1sup0<λ1<Lλ1λ1/Ln1/β1−L/λ1n1/n1−β1p−n1−1n1−β1/n1=1−p−n1n1−β1/n1sup0<λ1<Lλ1Ln1/n1−β1−λ1/Ln1/β1λ1n1/n1−β11−p−n1n1−β1/n1=1⋅L.
For0<λ2<1, we also divide x2 into two cases x2p<1 and x2p≥1. As above, we get(40)sup0<λ2<1λ2x2∈ℚpn2:1⋅Hβ2pf0x2>λ2n2−β2/n2=1×1.
SinceFL1ℚpn1×ℚpn2=1×1, by combining (37) with (38),(41)Hβ1,β2pFwLn1/n1−β1,n2/n2−β2ℚpn1×ℚpn2=1⋅FL1ℚpn1×ℚpn2.
This completes the proof.


## 3. Sharp Weak Bounds for Fractional Conjugate Hardy Operators

Likewise, this section contains the results having sharp weak bounds for fractional p-adic conjugate Hardy operators, and our results are as follows.Theorem 2.
Set0<βi<ni, let Q=n1/n1−β1,…,nm/nm−βm,i=1,…,m. If f∈L1ℚpn1×ℚpn2×⋯×ℚpnm, then we have(42)Hβ1,…,βmp,∗fwLQℚpn1×ℚpn2×⋯×ℚpnm≤1⋅fL1ℚpn1×ℚpn2×⋯×ℚpnm.
Furthermore,(43)Hβ1,⋯βmp,∗L1ℚpn1×ℚpn2×⋯×ℚpnm⟶wLQℚpn1×ℚpn2×⋯×ℚpnm=1.
In order to prove our theorem, we need the following lemma.Lemma 2.
Suppose that0≤β<n, if f∈L1ℚpn, then for any λ>0,(44)Hβp,∗fLn/n−β,∞ℚpn≤1⋅fL1ℚpn.
Moreover,(45)Hβp,∗L1ℚpn⟶Ln/n−β,∞ℚpn=1.
The proof of this result is almost the same as Lemma1; here, we omit the proof details. Next, we give the proof of Theorem 2.Proof.
Without loss of generality, we only discuss the casem=2, and then, the case m≥3 is just a repetition of the case m=2. When m=2, the operator Hβ1,β2p,∗ can be written as(46)Hβ1,β2p,∗fx1,x2=∫y1p>x1p∫y2p>x2pfy1,y2B0,y1pH1−β1/n1B0,y2pH1−β2/n2dy1dy2.
Using Lemma2 and Fubini theorem, it implies that(47)Hβ1,β2p,∗f⋅,x2Ln1/n1−β1,∞ℚpn1=supλ1>0λ1|x1:Hβ1,β2p,∗fX1,x2>λ1|n1/n1−β1≤1⋅∫|y2|p>|x2|pf⋅,y2|B0,|y2|p|1−β2/n2L1ℚpn1≤1⋅∫|y2|p>|x2|p1|B0,|yp||1−β2/n2∫ℚpn1fy1,y2dy1dy2.
We conclude that(48)∫|y2|p>|x2|p1|B0,|yp||1−β2/n2∫ℚpn1fy1,y2dy1dy2Ln2/n2−β2,∞ℚpn2=supλ2>0λ2x2:∫|y2|p>|x2|p1|B0,|yp||1−β2/n2∫ℚpn1fy1,y2dy1dy2>λ2n2/n2−β2≤1⋅∫ℚpn2∫ℚpn1fy1,y2dy1dy2=1⋅fℚpn1×ℚpn2.
Consequently, combining (45) and (46), we get(49)Hβ1,β2p,∗fwLn1/n1−β1,n2/n2−β2ℚpn1×ℚpn2≤1⋅fL1ℚpn1×ℚpn2.
On the other hand, for any0<ε<1, we took(50)fεx=xp−β+n/ε,xp≥1,0,|x|p<1.
LetFx1,x2=fϵ1x1fϵ2x2, where x1∈ℚpn1,x2∈ℚpn2, then(51)FL1ℚpn1×ℚpn2=fϵ1L1ℚpn1fϵ2L1ℚpn2=1−p−n1pβ1+n1/ϵ1−n1−11−p−n2pβ2+n2/ϵ2−n2−1.
We have(52)Hβip,∗fεixi=∫|yi|p>|xi|p|yi|p−βi+ni/εi−ni−βiχ|yi|p≥1yidyi=1−p−ni|xi|βi−βi+ni/εipβi+ni/εi−βi−1,|xi|p≥1,1−p−nipβi+ni/εi−βi−1,|xi|p<1.
SetCϵi=1−p−ni/pβi+ni/ϵi−βi−1 and M=Cϵ1Hβ2p,∗f2x2/λ11/β1+n1/ϵ1−β1, we obtain that(53)x1∈ℚpn1:|Hβ1,β2p,∗Fx1,x2|>λ1=|x1|p<1:Cϵ1Hβ2p,∗f2x2>λ1+|x1|p≥1:Cϵ1|x1|β1−β1+n1/ϵ1Hβ2p,∗f2x2>λ1=x1∈ℚpn1:|x1|pβ1+n1/ϵ1−β1<Cϵ1Hβ2p,∗f2x2/λ1.
Notice that whenλi>Cϵi,xi∈ℚpn1:Hβip,∗fixi>λi=∅, if ϵi is small enough, Cϵi tends to zero; therefore, when εi is small enough, we get(54)I0≔sup0<λ1<Hβ2p,∗f2x2λ1x1∈ℚpn1:|Hβ1,β2p,∗|>λ1n1/n1−β1=sup0<λ1<Hβ2p,∗f2x2λ1∫|x1|pβ1+n1/ε1−β1<Cε1Hβ2p,∗f2x2/λ1dx1n1/n1−β1=sup0<λ1<Hβ2p,∗f2x2λ1∑j=−∞logpM∫Sjdx1n1/n1−β1=1−p−n1n1/n1−β1sup0<λ1<Hβ2p,∗f2x2λ1∑j=−∞logpMpjn1n1/n1−β1=sup0<λ1<Hβ2p,∗f2x2λ1Mn1n1/n1−β1=sup0<λ1<Hβ2p,∗f2x2λ1Cε1Hβ2p,∗f2x2/λ11/β1+n1/ε1−β1n1n1/n1−β1=Cε1Hβ2p,∗f2x21/n1−β1β1+n1/ε1−β1×sup0<λ1<minHβ2p,∗f2x2,Cε1λ11−1/n1−β1β1+n1/ε1−β1=Cε1Hβ2p,∗f2x21/n1−β1β1+n1/ε1−β1=Hβ2p,∗f2x21/n1−β1β1+n1/ε1−β1pβ1+n1/ε1−n1−1pβ1+n1/ε1−β1−1‖fε1‖L1ℚpn1.
Using the same method forx2, we obtain that(55)supλ2>0λ2x2∈ℚpn2:I0>λ2n2/n2−β2=Cε2pβ1+n1/ε1−n1−1pβ1+n1/ε1−β1−1fε1L1ℚpn1n1−β1/n2−β2pβ1+n1/ε1−n1−1/pβ1+n1/ε1−β1−1×sup0<λ2<Cε2λ21−n1−β1/n2−β2pβ1+n1/ε1−n1−1/pβ1+n1/ε1−β1−1=Cε2pβ1+n1/ε1−n1−1pβ1+n1/ε1−β1−1fε1L1ℚpn1n1−β1/n2−β2pβ1+n1/ε1−n1−1/pβ1+n1/ε1−β1−1=pβ1+n1/ε1−n1−1pβ1+n1/ε1−β1−1fε1L1ℚpn1n1−β1/n2−β2pβ1+n1/ε1−n1−1/pβ1+n1/ε1−β1−1×pβ2+n2/ε2−n2−1pβ2+n2/ε2−β2−1fε2L1ℚpn2.
Letε1⟶0+ and ε2⟶0+, it implies that(56)Hβ1,β2p,∗FwLn1/n1−β1,n2/n2−β2ℚpn1×ℚpn2≥1⋅FL1ℚpn1×ℚpn2.
This finishes the proof of Theorem2.



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