On a $p$-Adic Integral Operator Induced by a Homogeneous Kernel and Its Applications

Jin Jianjun,1, Tang Shuan,2, Feng Xiaogao,3

 基金资助: 国家自然科学基金.  11501157国家自然科学基金.  12061022国家自然科学基金.  11701459

 Fund supported: the NSFC.  11501157the NSFC.  12061022the NSFC.  11701459

Abstract

In this paper, we introduce and study a $p$-adic integral operator induced by a homogeneous kernel of degree $-λ$ and obtain its sharp norm estimates. As applications, we establish some new $p$-adic inequalities with the best constant factors and their equivalent forms, which extend some known results in the literature.

Keywords： p-Adic integral operator ; p-Adic inequalities ; Homogeneous kernel ; Norm of operator

Jin Jianjun, Tang Shuan, Feng Xiaogao. On a $p$-Adic Integral Operator Induced by a Homogeneous Kernel and Its Applications. Acta Mathematica Scientia[J], 2021, 41(4): 968-977 doi:

1 引言与主要结果

$q>1$, 记${{\mathbb R}} _{+} = (0, +\infty)$, 则有如下著名的Hardy-Littlewood-Pólya不等式

$$$\left[\int_{0}^\infty \left|\int_{0}^\infty\frac{f(y)}{\max\{x, y\}}{\rm d}y\right|^q {\rm d}x\right]^{\frac{1}{q}}\leq \frac{q^2}{q-1} \left[\int_{0}^\infty |f(x)|^q{\rm d}x\right]^{\frac{1}{q}}$$$

$$$x = p^{\gamma} \sum\limits_{j = 0}^{\infty} a_{j}p^j, \quad \gamma = \gamma(x) \in {\mathbb Z},$$$

2 定理1.1的证明

$$$W_1(\lambda_1, \lambda_2; x): = \int_{{\mathbb Q}_p^{*}} {\cal H}(x, y) \cdot \frac{|x|_{p}^{(q-1)\lambda_1}}{|y|_{p}^{\lambda_2}}\, {\rm d}y, \: x \in {\mathbb Q}_p^{*}.$$$

$\begin{eqnarray} W_1(\lambda_1, \lambda_2; x)& = &\int_{{\mathbb Q}_p^{*}} {\cal H}(x, xt) \cdot \frac{|x|_{p}^{(q-1)\lambda_1}}{|xt|_{p}^{\lambda_2}}\, {\rm d}t \\ & = & |x|_{p}^{q\lambda_1-1} \int_{{\mathbb Q}_p^{*}} H_{\lambda}(1, |t|_p) \cdot \frac{1}{|t|_{p}^{\lambda_2}}{\rm d}t \nonumber\\ & = &|x|_{p}^{q\lambda_1-1} \sum\limits_{-\infty <\gamma <\infty}\int_{S_\gamma}H_{\lambda}(1, |t|_p) \cdot \frac{1}{|t|_{p}^{\lambda_2}}\, {\rm d}t \\ & = & |x|_{p}^{q\lambda_1-1}(1-p^{-1})\sum\limits_{-\infty <\gamma <\infty} H_{\lambda}(1, p^{\gamma})\cdot {p}^{-\lambda_2\gamma} \cdot p^{\gamma} \\ & = & C_{p}(\lambda_1, \lambda_2)|x|_{p}^{q\lambda_1-1}. \end{eqnarray}$

$$$W_2(\lambda_1, \lambda_2;y) = C_{p}(\lambda_1, \lambda_2)|y|_{p}^{q'\lambda_2-1}.$$$

$\begin{eqnarray} \|{\bf H}^p\| \geq \frac{||{\bf H}^pf_\varepsilon ||_{q, \overline{\theta}_1}}{\|f_\varepsilon\|_{q, \theta_1}}\geq(\varepsilon^{\varepsilon})^{\frac{1}{q}} \int_{|t|_p \geq \frac{1}{|\varepsilon|_p}}H_{\lambda}(1, |t|_p) |t|_{p}^{-\lambda_1-\frac{\varepsilon}{q}}{\rm d}t. \end{eqnarray}$

$(\varepsilon^{\varepsilon})^{\frac{1}{q}}\rightarrow 1, \: N\rightarrow \infty.$所以, 由Fatou引理及(2.4)式, 有

3 定理1.2的证明

$\begin{eqnarray} &&\int_{{\mathbb Q}_p^{*}} \int_{{\mathbb Q}_p^{*}} {\cal H}(x, y) f(x)g(y){\rm d}x{\rm d}y\\ & = &\int_{{\mathbb Q}_p^{*}} \int_{{\mathbb Q}_p^{*}}{\cal H}(x, y) \left[ \frac{|x|_p^{\frac{\lambda_1}{q'}}} {|y|_p^{\frac{\lambda_2}{q}}}\cdot f(x)\right] \left[ \frac{|x|_p^{{\frac{\lambda_2}{q}}}} {|y|_p^\frac{\lambda_1}{q'}}\cdot g(y)\right]{\rm d}x{\rm d}y \\ &\leq &\left\{ \int_{{\mathbb Q}_p^{*}} {\cal H}(x, y) \cdot \frac{|x|_p^{(q-1)\lambda_1}} {|y|_p^{\lambda_2}}\cdot f^q(x) \, {\rm d}x{\rm d}y\right\}^{\frac{1}{q}} \\ & & \times \left\{ \int_{{\mathbb Q}_p^{*}}{\cal H}(x, y)\cdot \frac{|y|_p^{(q'-1)\lambda_2}} {|x|_p^{\lambda_1}}\cdot g^{q'}(y) \, {\rm d}x{\rm d}y\right\}^{\frac{1}{q'}} \\ & = &\left\{ \int_{{\mathbb Q}_p^{*}} W_{1}(\lambda_1, \lambda_2; x)f^q(x) \, {\rm d}x\right\}^{\frac{1}{q}} \left\{ \int_{{\mathbb Q}_p^{*}} W_{2}(\lambda_1, \lambda_2; y)g^{q'}(y) \, {\rm d}y\right\}^{\frac{1}{q'}}. \end{eqnarray}$

我们先证明充分性. 当$\lambda_1+\lambda_2 = 2-\lambda$时, 容易看到这时有

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