Pure Mathematics
Vol. 09  No. 08 ( 2019 ), Article ID: 32571 , 8 pages
10.12677/PM.2019.98117

The Action of the Bessel Potentials on ${L}^{p}\left({R}^{n}\right)\left(0 Space

Renkun Shi

Department of Mathematics, College of Science, Hohai University, Nanjing Jiangsu

Received: Sep. 23rd, 2019; accepted: Oct. 10th, 2019; published: Oct. 17th, 2019

ABSTRACT

In this paper, we concern the action of the Bessel potentials on ${L}^{p}\left({R}^{n}\right)$ space with $0. By the method of contradiction, we prove that for any $r>0$, the Bessel potential cannot be a bounded linear operator from ${L}^{p}\left({R}^{n}\right)$ to ${L}^{r}\left({R}^{n}\right)$, and obtain other related results.

Keywords:Bessel Potentials, ${L}^{p}\left({R}^{n}\right)$ Space, Unbounded Operators

Bessel位势在空间 ${L}^{p}\left({R}^{n}\right)\left(0 上的作用

1. 引言

${‖uv‖}_{{L}^{r}\left(\Omega \right)}\le {‖u‖}_{{L}^{p}\left(\Omega \right)}{‖v‖}_{{L}^{q}\left(\Omega \right)},$ (1)

$p,q,r>0,\text{\hspace{0.17em}}\frac{1}{r}=\frac{1}{p}+\frac{1}{q},$ (2)

$\Omega \subset {\text{R}}^{n}$ 是一区域。值得指出的是，只要 $p,q,r$ 满足(2)式，则(1)式对任意的 $u\in {L}^{p}\left(\Omega \right)$$v\in {L}^{q}\left(\Omega \right)$ 都成立。特别地，当 $u\in {L}^{p}\left(\Omega \right)$,$v\in {L}^{q}\left(\Omega \right)$

$p,q>0,\text{\hspace{0.17em}}\frac{1}{p}+\frac{1}{q}>1$ (3)

${‖A\left(uv\right)‖}_{{L}^{r}\left(\Omega \right)}\le C\left({‖u‖}_{{L}^{p}\left(\Omega \right)},{‖v‖}_{{L}^{q}\left(\Omega \right)}\right)$ ? (P)

Bessel位势是形如

${\left(1-\Delta \right)}^{-\alpha /2},\text{\hspace{0.17em}}\alpha >0$

${K}_{\alpha }\left(x\right)={\left(4\pi \right)}^{-\frac{n}{2}}\Gamma {\left(\frac{\alpha }{2}\right)}^{-1}{\int }_{0}^{\infty }{\text{e}}^{-\frac{|x{|}^{2}}{4s}-s}{s}^{\frac{\alpha -\left(n+2\right)}{2}}\text{d}s.$ (4)

${\stackrel{^}{K}}_{\alpha }\left(\xi \right)={\left(1+|\xi |\right)}^{-\alpha /2}.$

${\left(1-\Delta \right)}^{-\alpha /2}\phi \left(x\right)={\int }_{{\text{R}}^{n}}{K}_{\alpha }\left(x-y\right)\phi \left(y\right)\text{d}y.$ (5)

${‖{\left(1-\Delta \right)}^{-\alpha /2}u‖}_{{W}^{k,p}\left({\text{R}}^{n}\right)}\le C{‖u‖}_{{L}^{1}\left({\text{R}}^{n}\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall u\in {L}^{1}\left({\text{R}}^{n}\right).$ (6)

${‖{\left(1-\Delta \right)}^{-\alpha /2}\left(uv\right)‖}_{{L}^{r}\left({\text{R}}^{n}\right)}\le {C}_{0}\left({‖u‖}_{{L}^{p}\left({\text{R}}^{n}\right)},{‖v‖}_{{L}^{q}\left({\text{R}}^{n}\right)}\right)$ (7)

${‖u‖}_{p}={\left({\int }_{{\text{R}}^{n}}{|u|}^{p}\text{d}x\right)}^{\frac{1}{p}},\text{\hspace{0.17em}}\forall u\in {L}^{p}\left({\text{R}}^{n}\right).$

${‖{\left(1-\Delta \right)}^{-\alpha /2}u‖}_{r}\le {C}_{0}\left({‖u‖}_{p}\right),\text{\hspace{0.17em}}\forall u\in {L}^{p}\left({\text{R}}^{n}\right),\text{\hspace{0.17em}}u\ge 0.$ (8)

${‖{\left(1-\Delta \right)}^{-\alpha /2}u‖}_{r}\le {C}_{0}\left({‖u‖}_{{p}_{1}},{‖u‖}_{{p}_{2}}\right),\text{\hspace{0.17em}}\forall u\in {L}^{{p}_{1}}\left({\text{R}}^{n}\right)\cap {L}^{{p}_{2}}\left({\text{R}}^{n}\right),\text{\hspace{0.17em}}u\ge 0.$ (9)

${‖{\left(1-\Delta \right)}^{-\alpha /2}\left(uv\right)‖}_{r}\le {C}_{0}\left({‖u‖}_{p},{‖v‖}_{q}\right),\text{\hspace{0.17em}}\forall u\in {L}^{p}\left({\text{R}}^{n}\right),\text{\hspace{0.17em}}v\in {L}^{q}\left({\text{R}}^{n}\right).$ (10)

${‖{\left(1-\Delta \right)}^{-\alpha /2}\left(uv\right)‖}_{r}\le {C}_{0}\left({‖u‖}_{\gamma },{‖u‖}_{p},{‖v‖}_{q}\right),\text{\hspace{0.17em}}\forall u\in {L}^{\gamma }\left({\text{R}}^{n}\right)\cap {L}^{p}\left({\text{R}}^{n}\right),\text{\hspace{0.17em}}\forall v\in {L}^{q}\left({\text{R}}^{n}\right).$ (11)

$v\in {L}^{q}\left({\text{R}}^{n}\right)$时，(7)式不可能成立，从而给出了问题的否定回答。定理1.2和定理1.4则说明，

2. 定理的证明

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

，则

(23)

(24)

，则。从而由(10)式知

(25)

，则。从而由(11)式知

(26)

(27)

The Action of the Bessel Potentials on Lp(Rn)(0 < p < 1) Space[J]. 理论数学, 2019, 09(08): 908-915. https://doi.org/10.12677/PM.2019.98117

1. 1. Adams, R.A. and Fournier, J.J.F. (2003) Sobolev Spaces. 2nd Edition, Academic Press, New York.

2. 2. Stein, E.M. (1970) Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton. https://doi.org/10.1515/9781400883882