# A Note about Young’s Inequality with Different Measures

Authors: Saba Mehmood; Eridani Eridani; Fatmawati Fatmawati
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

The key purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators defined with doubling measures in Lebesgue spaces with different measures. Relating Bessel decaying the kernel of the operators is satisfying some elementary properties. Doubling measure, Young's inequality, and Minköwski’s inequality will be used in proofs of boundedness of integral operators. In addition, we also explore the relation between the parameters of the kernel and generalized integral operators and see the norm of these generalized operators which will also be bounded by the norm of their kernel with different measures.


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## Body

## 1. Introduction

Suppose we are givenKα:0,∞⟶0,∞, with Kαt:=tα−n,0<α<n. Fractional integral operator, one of integral operators, is often studied since early last century. This operator is defined by(1)Tαfx:=∫ℝnKαx−yfydy,for every f∈Llocpℝn, with 1≤p<∞. Here, Kα is called fractional integral kernel or Riesz kernel [1].Studies about Riesz potentialsTα were started since 1920’s. Hardy-Littlewood [2,3] proved the boundedness of Riesz potentials on Lebesgue spaces for n=1. After 50’s, Hardy-Littlewood and Sobolev [4] proved the boundedness of Tα for n∈N. [5] (see also D. Edmunds [[6], Chapter 6]) Kokilashvili had a complete description of nondoubling measure μ guaranteeing the boundedness of fractional integral operator Tα from Lpμ,X to Lqμ,X, 1<p<q<1. We notice that this result was derived in [7] for potentials on Euclidean spaces. In [4], theorems of Sobolev and Adams type for fractional integrals defined on quasimetric measure spaces were established.Some two-weight norm inequalities for fractional operators onℝn with nondoubling measure were studied in [8]. The boundedness of the Riesz potential in Lebesgue and Morrey spaces defined on Euclidean spaces was studied in Peter and Adams’s paper [9,10]. The same problem for fractional integrals on ℝn with nondoubling measure was investigated by Sawano in [11]. Eridani ([12], Theorem 4, Theorem 3.1, Theorem 3.3) established the boundedness of fractional integral operators and mention the necessary and sufficient conditions for the boundedness of maximal operators.Since 1930s, some researchers [2,3] have studied the boundedness of Tα on some function spaces.Theorem 1 (Hardy-Littlewood-Sobolev) (see [4]).
Iff∈Lpℝn, with 1<p<n/α, then there exists Cp,q>0 such that(2)Tαf:Lqℝn≤Cp,qf:Lpℝn,1q=1p−αn.
From now on,C,Cp,q,Cp,q,s>0 will be serve as a positive constant, not necessarily the same one.
The purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators defined in Lebesgue spaces with different measures. Role of Young’s inequality and Minköwski’s inequality will be used in proofs of boundedness of integral operators.
Here, we define(3)f:Lp:=∫ℝnftpdt1/p,1≤p<∞,and f∈Lp=Lpℝn, as a collection of f such that f:Lp<∞. Next, for a given Kα,γ:0,∞⟶0,∞, with(4)Kα,γt:=tα−n1+tγ,0<α<n,0≤γ.
We define Bessel-Riesz operator as(5)Tαfx:=∫ℝnKα,γx−yfydy,
For everyf∈Llocpℝn, with 1≤p<∞. Here, Kα,γ is called Bessel–Riesz kernel.
The origin of Bessel–Riesz operators is Schröndinger equation. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. In quantum mechanics, the analogue of Newton’s law is Schrödinger’s equation. In 1999, Kazuhiro Kurata, Seiichi Nishigaki, and Satoko Sugano [13] studied boundedness of integral operators on Lebesgue and generalized Morrey spaces and its application to estimate in Morrey spaces for the Schrödinger operator L2=−Δ+Vx+Wx with nonnegative V∈RH∞ (reverse Hölders class) and small perturbed potentials W.
In a recent previous year 2016, Idris et al. ([14], Theorem 6, pp. 3) presented the boundedness of Bessel–Riesz operators. They obtained results that were similar to Chiarenza–Frasca’s result [15] for the boundedness of fractional integral operators.
Eridani et al. [12] presented the boundedness of fractional integral operators defined on quasimetric measure spaces. Moreover, Idris et al. [14] have investigated the boundedness of generalized Morrey spaces with weight and presented the boundedness of these operators on Lebesgue spaces and Morrey spaces for Euclidean spaces.
Since Euclidean spaces are the simplest example of measure metric spaces. Kurata et al. [16] have investigated the boundedness of Bessel–Riesz operators on generalized Morrey spaces with weight. The boundedness of these operators on Lebesgue spaces in Euclidean settings will be proved using Young’s inequality and Minköwski’s inequality.
Moreover, we will also find the norm of the generalized Bessel–Riesz operators bounded by the norm of the kernels. Saba et al. [17] used Young’s inequality, to prove the boundedness of Bessel–Riesz operators on Lebesgue spaces in measure metric spaces, which are easy consequences of Young's inequality. The second consequence after the fact Bessel–Riesz operator is bounded on Lebesgue spaces; we entered to next phenomena of Morrey spaces. In Young's inequality, we have the best constant known as 1. But at this point, we still have no information about the best constant in Morrey spaces. So in this paper, we move towards generalized Bessel–Riesz operators in Lebesgue spaces.
We will also mention the case when the measure satisfies the doubling condition. The derived conditions are simultaneously necessary and sufficient for appropriate inequalities that were derived in [18,19]. We also have the following result [14] about the boundedness of such an operator as follows:Theorem 2.
(Idris–Gunawan–Lindiarni–Eridani). Iff∈Lp and Kα,γ∈Lq, then there exists Cp,q,s>0 such that(6)Tα,γf:Ls≤Cp,q,sKα,γ:Lq⋅f:Lp,1s=1p+1q−1.
For some functionsf:ℝn⟶ℝ and g:ℝn⟶ℝ, we define(7)f⋆gx:=∫ℝngx−yfydy,x∈ℝn.
We know thatf⋆g is a generalization of Tα and Tα,γ. Moreover, the above result is a particular case of the following [20].Theorem 3.
(Young’s inequality). Iff∈Lp and g∈Lq, then there exists Cp,q,s∗>0 such that(8)f⋆g:Ls≤Cp,q,s∗g:Lq⋅f:Lp,1s=1p+1q−1.
Kurata et al. [16] have shown that W⋅Tα,γ is bounded on generalized Morrey spaces where W is any real functions. For applications of the above operators in Euclidean spaces setting, see [16].


## 2. The Kernel

For1≤s<∞,0<γ<∞, and ℝ+:=0,∞, we define functions ρ:ℝ+⟶ℝ+, with the following conditions:(9)ρI.12≤ab≤2⇒1C1≤ρaρb≤C1;ρII.∫0rρtstns−1+1dt+∫r∞ρtstns−1+sγ+1dt<∞,r>0;ρIII.∫01ρtstns−1+1dt+∫1∞ρtstns−1+sγ+1dt≤C2<∞.FromρI we will have(10)∫r2rρttdt∼ρr.That is, there exists0<c1≤c2 such that(11)c1ρr≤∫r2rρttdt≤c2ρr,r>0.Since(12)1rsγ∫01ρtstns−1+1dt≥∫r1ρtstns−1+sγ+1dt,0<r<1,∫1rρtstns−1+1dt≤rsγ∫1∞ρtstns−1+sγ+1dt,1<r<∞,thenρII and ρIII are equivalent, with the following explanations.It is easy to see thatρII⇒ρIII. Suppose ρIII is true. Then, for 0<r<1, we have(13)∫0rρtstns−1+1dt≤∫01ρtstns−1+1dt<∞,∫r∞ρtstns−1+sγ+1dt=∫r1+∫1∞≤C2+∫r1≤C2+1rsγ∫01ρtstns−1+1dt<∞.On the other hand, for1<r<∞, then(14)∫r∞ρtstns−1+sγ+1dt<∫1∞ρtstns−1+sγ+1dt<C2<∞,∫0rρtstns−1+1dt=∫01+∫1r≤C2+rsγ∫1∞ρtstns−1+sγ+1dt<∞,and we already prove that both of the conditions are equivalent.We define(15)Kρ,γx:=ρxxn1+xγ,x∈ℝn.UsingρII with 1≤s<∞, and any R>0, then(16)∫ℝnKρ,γxsdx=∫0≤xρxsxsn1+xsγdx=∑k∈Z∫2kR≤x<2k+1Rρxsxsn1+xsγdx∼∑k=−∞−1ρ2kRs2kRns−11+2kRsγ+∑k=0∞ρ2kRs2kRns−11+2kRsγ≤∑k=−∞−1ρ2kRs2kRns−1+∑k=0∞ρ2kRs2kRns−1+sγ∼∑k=−∞−1∫2kR2k+1Rρtstns−1+1dt+∑k=0∞∫2kR2k+1Rρtst1+ns−1+sγdt=∫0Rρtstns−1+1dt+∫R∞ρtst1+ns−1+sγdt<∞.ByρII, then Kρ,γ∈Ls, for 1≤s<∞.Ifk∈−1,−2,−3,…, then 0<2k<1 and 1+2knγ<2nγ. On the other hand, we can also have for k∈0,1,2,3,…, then 1≤2k, and 2k<1+2k<2k+1.Finally, we will have(17)∫01ρtstns−1+1dt+∫1∞ρtst1+ns−1+sγdt=∑k=−∞−1∫2k2k+1ρtstns−1+1dt+∑k=0∞∫2k2k+1ρtst1+ns−1+sγdt∼∑k=−∞−1ρ2ks2kns−1+∑k=0∞ρ2ks2kns−1+sγ≤C∑k=−∞−1ρ2ks2kn2kns1+2knγ+∑k=0∞ρ2ks2kn2kns1+2knγ=C∑k∈Z∫2k≤x<2k+1ρxsxsn1+xsγdx=C∫0≤xρxsxsn1+xsγdx=∫ℝnKρ,γxsdx.After the above estimates, we conclude.Lemma 1.
For1≤s<∞, we will have(18)∫ℝnKρ,γxsdx∼∫01ρtstns−1+1dt+∫1∞ρtst1+ns−1+sγdt.
With the same technique as used to estimate the kernel defined by equation (15) or everyy∈ℝn, we also have(19)∫ℝnKρ,γx−ysdx∼∫01ρtstns−1+1dt+∫1∞ρtst1+ns−1+sγdt.
As a classical example, we can consider the following:(20)ρt:=tα+ns−1/s,1≤α+1γ<s<∞.
If we consider(21)Tρ,γfx:=∫ℝnKρ,γx−yfydy,x∈ℝn,then as a consequences of Young’s inequality, we also have the following.Lemma 2.
If1/q=1/p+1/s−1 and f∈Lp, then there exists C>0 such that(22)Tρ,γf:Lq≤CKρ,γ:Ls⋅f:Lp,1<p,q,r<∞.
From the above lemma, there existsC>0 such that(23)Tρ,γf:Lp≤CKρ,γ:Lp⋅f:L1,orTρ,γf:Lp≤CKρ,γ:L1⋅f:Lp,1≤p<∞.
Supposeμ is an arbitrary measure on ℝn. We define μ∈GC (growth condition), if and only if there exists C2>0 such that(24)μBa,R≤C2Rn,ror every open balls Ba,R:=x∈ℝn:x−a<R in ℝn. For more information about this kind of measure, see [21].
Based on the above definitions, we try to estimate(25)Kρ,γ:Lsμ:=∫ℝnKρ,γxsdμx1/s,1≤s<∞.
For1≤s<∞, and R>0, we consider the following:(26)∫x<Rρxsxsn1+xsγdμx=∑k=−∞−1∫2kR≤x<2k+1Rρxsxsn1+xsγdμx≤C∑k=−∞−1ρ2kRsμB0,2kR2kRsn1+2kRsγ≤C∑k=−∞−1ρ2kRs2kRs−1n≤C∑k=−∞−1∫2kR2k+1Rρtsts−1n+1dt=C∫0Rρtsts−1n+1dt,
and also(27)∫R≤xρxsxsn1+xsγdμx=∑k=1∞∫2kR≤x<2k+1Rρxsxsn1+xsγdμx≤C∑k=0∞ρ2kRsμB0,2kR2kRsn1+2kRsγ≤C∑k=0∞ρ2kRs2kRs−1n+sγ≤C∑k=0∞∫2kR2k+1Rρtsts−1n+sγ+1dt=C∫R∞ρtsts−1n+sγ+1dt,where letter C>0 shall always denote a constant, not necessarily the same one.
At this point, for1≤s<∞, and μ∈GC, we define(28)Kρ,γ:Lsμ=supR>0∫0Rρtsts−1n+1dt+∫R∞ρtsts−1n+sγ+1dt1/s.


## 3. Main Results

For any measureν on ℝn, any measurable functions f:ℝn⟶ℝ, and x∈ℝn, we define(29)Tρ,γfx:=∫ℝnKρ,γx−yfydνy.Before we state our main results, about the boundedness ofTρ,γ, we consider the following simple result [13].Lemma 3 (Minköwski’s inequality).
Suppose1≤p<∞, and we are given F:ℝn×ℝn⟶ℝ. For any measure ν and μ on ℝn, then(30)∫ℝn∫ℝnFx,ydνypdμx1/p≤∫ℝn∫ℝnFx,ypdμx1/pdνy.
Now, we state the following:Theorem 4.
Supposeμ∈GC and ν is any measure on ℝn.
Iff∈L1ν and Kρ,γ∈Lsμ, then there exists Cs>0 such that(31)Tρ,γf:Lsμ≤CsKρ,γ:Lsμ⋅f:L1ν,1≤s<∞.Proof.
By Minköwski’s inequality, with1≤s<∞, then(32)∫ℝnTρ,γfxsdμx1/s=∫ℝn∫ℝnKρ,γx−yfydνysdμx1/s≤∫ℝn∫ℝnKρ,γx−yfysdμx1/sdνy≤∫ℝn∫ℝnKρ,γx−ysdμx1/sfydνy≤CKρ,γ:Lsμ⋅f:L1ν.
As a corollary of the above result, we also haveCorollary 1.
Supposeμ∈GC and ν is any measure on ℝn.
Iff∈L1ν and Kρ,γ∈L1μ, then there exists C>0 such that(33)Tρ,γf:L1μ≤CKρ,γ:L1μ⋅f:L1ν.
Suppose we have the(34)1s=1p+1q−1,or1=1s+1−1p+1−1q=1s+1p′+1q′.
It is noted that(35)p=q′1−ps,q=p′1−qs.
After the above simple calculations, we haveTheorem 5.
Supposeμ∈GC and ν any measure on ℝn such that(36)νB≤C∗μB,for some C∗>0 and for every open balls B on ℝn.
Iff∈Lpν and Kρ,γ∈Lqμ, then there exists C>0 such that(37)Tρ,γf:Lsμ≤CKρ,γ:Lqμ⋅f:Lpν,1s=1p+1q−1.Proof.
With Hölder’s inequality, we start with the following:(38)Tρ,γfx≤∫ℝnfyp/s+1−p/sKρ,γx−yq/s+1−p/sdνy≤∫ℝnKρ,γx−yqfypdνy1/s∫ℝnfyq′1−p/sdνy1/q′×∫ℝnKρ,γx−yp′1−q/sdνy1/p′=∫ℝnKρ,γx−yqfypdνy1/s∫ℝnfypdνy1/q′×∫ℝnKρ,γx−yqdνy1/p′.
If we define(39)Tρ,γp,qfx:=∫ℝnKρ,γx−yqfypdνy,then for every x, we come to(40)Tρ,γfxs≤f:Lpνsp/q′⋅Tρ,γp,qfx∫ℝnKρ,γx−yqdνys/p′.
Now, we want to estimate the right hand side, especially, forx∈ℝn and R>0, we will have(41)∫ℝnKρ,γx−yqdνy=∫x−y<R+∫x−y≥R.
We start with(42)∫x−y<R=∑k=−∞−1∫2kR≤x−y<2k+1R∼C∑k=−∞−1Kρ,γ2kRqνBx,2k+1R≤C∑k=−∞−1ρ2kRq2kRq−1n≤C∫0Rρtqtq−1n+1dt≤CKρ,γ:Lqμq,and also(43)∫x−y≥R∼C∑k=0∞Kρ,γ2kRqνBx,2k+1R≤C∑k=0∞ρ2kRq2kRnq−n+qγ≤C∫R∞ρtqtnq−1+qγ+1dt≤CKρ,γ:Lqμq.
Up to now, for everyx∈ℝn, we already have(44)Tρ,γfxs≤CKρ,γ:Lqμqs/p′⋅f:Lpνqs/p′⋅Tρ,γp,qfx.
By the previous fact, we know that(45)Tρ,γf:L1μ≤C1Kρ,γ:L1μ⋅f:L1ν,and after this, we come to(46)∫ℝnTρ,γp,qfxdμx≤C1Kρ,γq:L1μ⋅fp:L1ν=C1Kρ,γ:Lqμq⋅f:Lpνp.
And finally,(47)Tρ,γf:Lsμs≤CKρ,γ:Lqμqs/p′⋅f:Lpνsp/q′∫ℝnTρ,γp,qfxdμx≤CKρ,γ:Lqμq+qs/p′⋅f:Lpνp+sp/q′.
Sinceq+qs/p′=s=p+sp/q′, then we are done.
As a corollary, we also have:Corollary 2.
Supposeμ∈GC and ν any measure on ℝn such that(48)νB≤C∗μB,for some C∗>0 and for every open balls B on ℝn.
Iff∈Lsν and Kρ,γ∈L1μ, then there exists C>0 such that(49)Tρ,γf:Lsμ≤CKρ,γ:L1μ⋅f:Lsν,1≤s<∞.


## 4. Pointwise Multiplier Operators

Supposef∈Lpμ and g∈Lp′μ, with 1<p<∞ and 1=1/p+1/p′. Then, by Hölder inequality, we will have f⋅g:L1μ≤f:Lpμ⋅g:Lp′μ.Next, we consider a pointwise multipliers operatorW by(50)W:f↦W⋅f,withW⋅fx≔Wx⋅fx,x∈ℝn.So, from Hölder inequality, ifW∈Lp′μ, then W is a bounded operators from Lpμ to L1μ, with W⋅f:L1μ≤W:Lp′μ⋅f:Lpμ,1<p<∞.Another example is the following equation. Suppose, we consider a fractional integral operatorsTα, and we define(51)W⋅Tα:f↦W⋅Tαf,withW⋅Tαfx≔Wx⋅Tαfx,x∈ℝn.Again, since1<p<n/α and 1/q+α/n=1/p, in view of Hölder inequality, then(52)W⋅Tαf:Lp≤W:Ln/α⋅Tαf:Lq≤CW:Ln/α⋅f:Lp.So, ifW∈Ln/α, then W⋅Tα:Lp⟶Lp is a bounded operator.From our main results, we also have:Corollary 3.
Supposeμ∈GC and ν is any measure on ℝn.
Iff∈L1ν,W∈Ls′μ, and Kρ,γ∈Lsμ, then(53)W⋅Tρ,γ:L1ν⟶L1μ,is a bounded operator. That is, for every s∈1,∞, there exists Cs>0 such that(54)W⋅Tρ,γf:L1μ≤CsW:Ls′μ⋅Kρ,γ:Lsμ⋅f:L1ν.Corollary 4.
Supposeμ∈GC and ν any measure on ℝn such that(55)νB≤C∗μB,for some C∗>0 and for every open balls B on ℝn.
Iff∈Lsν,W∈Ls′μ, and Kρ,γ∈L1μ, then(56)W⋅Tρ,γ:Lsν⟶L1μ,
is a bounded operator. That is, for everys∈1,∞, there exists Cs>0 such that(57)W⋅Tρ,γf:L1μ≤CKρ,γ:L1μ⋅W:Ls′μ⋅f:Lsν.
Next, this will be served as our last corollary.Corollary 5.
Supposeμ∈GC and ν any measure on ℝn such that(58)νB≤C∗μB,for some C∗>0 and for every open balls B on ℝn.
Iff∈Lpν,W∈Lq′μ, and Kρ,γ∈Lqμ, then(59)W⋅Tρ,γ:Lpν⟶Lpμ,1s+1q′=1p,is a bounded operator. That is, there exists C>0 such that(60)W⋅Tρ,γf:Lpμ≤CKρ,γ:Lqμ⋅W:Lq′μ⋅f:Lpν.


## 5. Conclusions

From the result of this study, we have seen the boundedness of the generalized Bessel–Riesz operators in Lebesgue spaces, in the framework of different measures (Theorem 3.1, Theorem 5) by using, as tools, an estimate of the Lebesgue norm of the Bessel–Riesz kernel (Theorem2), the Young inequality (Theorem 3), and functions with doubling conditions. In the future, we shall continue this study to prove the boundedness of generalized Bessel–Riesz operators on Morrey spaces and generalized Morrey spaces.



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