Asymptotic Properties of the p-Adic Fractional Integration Operator

TL;DR

This paper investigates the long-term behavior of a p-adic fractional integration operator, extending previous work on non-Archimedean pseudo-differential equations to understand its asymptotic properties.

Contribution

It provides a detailed analysis of the asymptotic properties of the p-adic fractional integration operator, a novel extension of Kochubei's earlier work.

Findings

Characterization of the asymptotic behavior of the operator

Identification of conditions for specific asymptotic regimes

Extension of classical fractional integration results to p-adic context

Abstract

We study asymptotic properties of the p-adic version of a fractional integration operator introduced in the paper by A. N. Kochubei, Radial solutions of non-Archimedean pseudo-differential equations, Pacif. J. Math. 269 (2014), 355-369.

Equations

$$|x{|}_p=p^{-\nu }\text{mbox}ifx=p^{\nu }\frac{m}{n},$$

$$|x{|}_p=0\text{mbox}ifandonlyifx=0;$$

$$|xy{|}_p=|x{|}_p\cdot |y{|}_p;$$

$$|x+y{|}_p\le \max (|x{|}_p,|y{|}_p).$$

$$|x+y{|}_p=\max (|x{|}_p,|y{|}_p)\text{mbox}if|x{|}_p\ne |y{|}_p.$$

$$\underset{|x{|}_p\le p^n}{\int }|x{|}_p^{\alpha -1}dx=\frac{1-p^{-1}}{1-p^{-\alpha }}{p}^{\alpha n};\text{ here and below }n\in \mathbb{Z},\alpha >0;$$

$$\underset{|x{|}_p\le p^n}{\int }dx=p^n;$$

$$\underset{|x{|}_p=p^n}{\int }dx=(1-\frac{1}{p})p^n;$$

$$\underset{|x{|}_p=1}{\int }|1-x{|}_p^{\alpha -1}=\frac{p-2+p^{-\alpha }}{p(1-p^{-\alpha })}.$$

$$(I^{\alpha }f(x))=\frac{1-p^{-\alpha }}{1-p^{\alpha -1}}\underset{|y{|}_p\le |x{|}_p}{\int }\left(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)f(y)dy,$$

$$f\sim \sum _{n=0}^{\infty }{a}_n|x{|}_p^{M_n},|x{|}_p\to 0,a_n\in \mathbb{C}.$$

$$(I^{\alpha }f(x))\sim \frac{1-p^{-\alpha }}{1-p^{\alpha -1}}\sum _{n=0}^{\infty }{a}_n{b}_n|x{|}_p^{M_n+\alpha },|x{|}_p\to 0,$$

$$b_n=\frac{p^{-\alpha +1}-1}{(1-p^{-\alpha })p}+(1-p^{-1})\sum _{k=1}^{\infty }(1-p^{-k(\alpha -1)})p^{-k(M_n+1)}.$$

$$f=\sum _{n=0}^N{a}_n|x{|}_p^{M_n}+R_N(x),R_N(x)=o(|x{|}_p^{M_N})|x{|}_p\to 0.$$

$$I^{\alpha}_{(1)}=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}\int\limits_{|y|_{p}\leq|x|_{p}}(|x-y|_{p}^{\alpha-1}-|y|_{p}^{\alpha-1})\Bigl{(}\sum_{n=0}^{N}a_{n}|y|_{p}^{M_{n}}\Bigr{)}\,dy,$$

$$I_{(2)}^{\alpha }=\frac{1-p^{-\alpha }}{1-p^{\alpha -1}}\underset{|y{|}_p\le |x{|}_p}{\int }(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1})R_N(y)dy.$$

$$I^{\alpha}_{(1)}=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}|x|_{p}^{\alpha}\int\limits_{|s|_{p}\leq 1}(|1-s|_{p}^{\alpha-1}-|s|_{p}^{\alpha-1})\Bigl{(}\sum_{n=0}^{N}a_{n}|x|_{p}^{M_{n}}|s|_{p}^{M_{n}}\Bigr{)}ds=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}|x|_{p}^{\alpha}(A+B)$$

$$A=\int\limits_{|s|_{p}<1}\left(1-|s|_{p}^{\alpha-1}\right)\Bigl{(}\sum_{n=0}^{N}a_{n}|x|_{p}^{M_{n}}|s|_{p}^{M_{n}}\Bigr{)}\,ds\\ =\sum_{n=0}^{N}a_{n}|x|_{p}^{M_{n}}\sum_{k=1}^{\infty}\left(1-p^{-k(\alpha-1)}\right)p^{-kM_{n}}\int\limits_{|s|_{p}=p^{-k}}\,ds\\ =(1-p^{-1})\sum_{n=0}^{N}a_{n}|x|_{p}^{M_{n}}\sum_{k=1}^{\infty}\left(1-p^{-k(\alpha-1)}\right)p^{-k(M_{n}+1)},$$

$$B=\sum _{n=0}^N{a}_n|x{|}_p^{M_n}\underset{|s{|}_p=1}{\int }\left(|1-s{|}_p^{\alpha -1}-1\right)ds=\frac{p^{-\alpha +1}-1}{(1-p^{-\alpha })p}\sum _{n=0}^N{a}_n|x{|}_p^{M_n}.$$

$$\left|I_{(2)}^{\alpha }\right|\le C_1|x{|}_p^{\alpha +M_{N+1}}\underset{|s{|}_p\le 1}{\int }(|1-s{|}_p^{\alpha -1}-|s{|}_p^{\alpha -1})|s{|}_p^{M_{N+1}}ds=O(|x{|}_p^{\alpha +M_{N+1}}).$$

$$(I^{\alpha }f)(x)\asymp |x{|}_p^{\alpha -1},|x{|}_p\to \infty .$$

$$\left(J_{(1)}^{\alpha }f\right)(x)=\frac{1-p^{-\alpha }}{1-p^{\alpha -1}}\underset{|y{|}_p<1}{\int }\left(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)f(y)dy,$$

$$\left(J_{(2)}^{\alpha }f\right)(x)=\frac{1-p^{-\alpha }}{1-p^{\alpha -1}}\underset{1\le |y{|}_p\le |x{|}_p}{\int }\left(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)f(y)dy$$

$$\left(J_{(1)}^{\alpha }f\right)(x)\asymp \underset{|y{|}_p<1}{\int }\left(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)dy\asymp |x{|}_p^{\alpha -1}.$$

$$\begin{gathered} \left|\left(J_{(2)}^{\alpha }f\right)(x)\right|\le C\underset{1\le |y{|}_p\le |x{|}_p}{\int }\left(|x-y{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)|y{|}_p^{-M}dy \\ =C\left\{\sum _{j=0}^{N-1}\underset{|y{|}_p=p^j}{\int }\left(|x{|}_p^{\alpha -1}-|y{|}_p^{\alpha -1}\right)|y{|}_p^{-M}dy+\underset{|y{|}_p=p^N}{\int }\left(|x-y{|}_p^{\alpha -1}-p^{N(\alpha -1)}\right)p^{-MN}dy\right\} \\ =C\{(1-\frac{1}{p})\sum _{j=0}^{N-1}{p}^j\left(p^{N(\alpha -1)}-p^{j(\alpha -1)}\right)p^{-Mj} \\ +p^{-MN}\underset{|y{|}_p=p^N}{\int }|x-y{|}_p^{\alpha -1}dy-(1-\frac{1}{p})p^{\alpha N-MN}\}. \end{gathered}$$

$$f(x)\sim |x{|}_p^{-\beta }\sum _{n=0}^{\infty }{a}_n(\log |x{|}_p{)}^{\gamma -n},|x{|}_p\to \infty ,$$

$$G_1(r)\stackrel{\text{def}}{=}\underset{|y{|}_p\le r}{\int }f(y)dy=o(r^{1-\lambda }),r\to \infty .$$

$$\underset{|y{|}_p\le p^{\nu }}{\int }|y{|}_p^{-\lambda }dy=\frac{1-p^{-1}}{1-p^{\lambda -1}}{p}^{(1-\lambda )\nu },\nu \in \mathbb{Z},$$

$$G_2(r)\stackrel{\text{def}}{=}\underset{|y{|}_p\le r}{\int }|y{|}_p^{-\lambda }dy=O(r^{1-\lambda }),r\to \infty .$$

$$\frac{G_1(r)}{G_2(r)}=\frac{G_1(r_0(n))+(G_1(r)-G_1(r_0(n)))}{G_2(r_0(n))+(G_2(r)-G_2(r_0(n)))}\le \frac{G_1(r_0(n))+\frac{1}{n}{G}_3(n,r)}{G_2(r_0(n))+G_3(n,r)}$$

$$G_3(n,r)=\underset{r_0\le |y{|}_p\le r}{\int }|y{|}_p^{-\lambda }dy.$$

Full text

**Key words: ** $p$-adic numbers; Vladimirov’s $p$-adic fractional differentiation operator; $p$-adic fractional integration operator; asymptotic expansion

MSC 2010. Primary: 11S80. Secondary: 26A33.

1 Introduction

1.1. In analysis of complex-valued functions on the field $\mathbb{Q}_{p}$ of $p$-adic numbers (or, more generally, on a non-Archimedean local field), the basic operator is Vladimirov’s fractional differentiation operator $D^{\alpha}$, $\alpha>0$, defined via the Fourier transform or, for wider classes of functions, as a hypersingular integral operator [1, 5]. Properties of this $p$-adic pseudo-differential operator were studied by Vladimirov (see [5]) and found to be more complicated than those of its classical counterparts. For example, as an operator on $L^{2}(\mathbb{Q}_{p})$, it has a point spectrum of infinite multiplicity. However, it was shown in [2] to behave much simpler on radial functions $x\to f(|x|_{p})$.

In particular, in [2] the first author introduced a right inverse $I^{\alpha}$ to the operator $D^{\alpha}$ on radial functions, which can be seen as a $p$-adic analog of the Riemann-Liouville fractional integral of real analysis (including the case $\alpha=1$ of the usual antiderivative). Just as the Riemann-Liouville fractional integral is a source of many problems of analysis, that must be true for the operator $I^{\alpha}$.

In this paper we study asymptotic properties of the function $I^{\alpha}f$ for a given asymptotic expansion of $f$; for the asymptotic properties of Riemann-Liouville fractional integral see [3, 4, 7].

1.2. Let us recall the main definitions and notation used below.

Let $p$ be a prime number. The field of $p$-adic numbers is the completion $\mathbb{Q}_{p}$ of the field $\mathbb{Q}$ of rational numbers, with respect to the absolute value $|x|_{p}$ defined by setting $|0|_{p}=0$,

[TABLE]

where $\nu,m,n\in\mathbb{Z}$, and $m,n$ are prime to $p$. It is well known that $\mathbb{Q}_{p}$ is a locally compact topological field with the topology determined by the metric $|x-y|_{p}$, and that there are no absolute values on $\mathbb{Q}$, which are not equivalent to the “Euclidean” one, or one of $|\cdot|_{p}$. We will denote by $dx$ the Haar measure on the additive group of $\mathbb{Q}_{p}$ normalized by the condition $\int_{|x|_{p}\leq 1}dx=1$.

The absolute value $|x|_{p}$, $x\in\mathbb{Q}_{p}$, has the following properties:

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The latter property called the ultrametric inequality (or the non-Archimedean property) implies the total disconnectedness of $\mathbb{Q}_{p}$ and unusual geometric properties. Note also the following consequence of the ultrametric inequality:

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We will often use the integration formulas (see [1, 5, 6]):

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in particular,

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See [1, 5] for further details of analysis of complex-valued functions on $\mathbb{Q}_{p}$.

From now on, we consider the case $\alpha>1$. The integral operator $I^{\alpha}$ introduced in [2] has the form

[TABLE]

where $f$ is a locally integrable function on $\mathbb{Q}_{p}$. See [2] for its connection to the Vladimirov operator $D^{\alpha}$ and applications to non-Archimedean counterparts of ordinary differential equations. Note that our results can be generalized easily to the case of general non-Archimedean local fields.

2 Asymptotics at the origin

Let $0<M_{0}<M_{1}<M_{2}<\ldots$, $M_{n}\to\infty$. Then the sequence $f_{n}(x)=|x|_{p}^{M_{n}}$ is an asymptotic scale for $x\to 0$ (see, for example, §16 of [4] for the main notions regarding asymptotic expansions).

Theorem 1**.**

Suppose that a function $f$ admits an asymptotic series expansion

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Then

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where

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Proof. We have

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Then $I^{\alpha}f=I^{\alpha}_{(1)}+I^{\alpha}_{(2)}$,

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After the change of variables $y=sx$ we get

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where

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On the other hand, since $|R_{N}(x)|\leq C|x|_{p}^{M_{N+1}}$, we find that for some constant $C_{1}>0$,

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The above calculations result in the asymptotic relation (2). $\qquad\blacksquare$

3 Asymptotics at infinity

For positive functions $\varphi,\psi$, we write $\varphi(x)\asymp\psi(x)$, $|x|_{p}\to\infty$, if $c\psi(x)\leq\varphi(x)\leq d\psi(x)$, for large values of $|x|_{p}$, $x\in\mathbb{Q}_{p}$, for some positive constants $c,d$.

Theorem 2**.**

Suppose that $a\leq f(x)\leq b$ ($a,b>0$) for $|x|_{p}<1$, $|f(x)|\leq C|x|_{p}^{-M}$, $M>1$, $C>0$, for $|x|_{p}\geq 1$. Then

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Proof. Let us rewrite (1) with $|x|_{p}\geq 1$ in the form $I^{\alpha}f=J^{\alpha}_{(1)}f+J^{\alpha}_{(2)}f$ where

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[TABLE]

Then

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Next, if $|x|_{p}=p^{N}$, $N\geq 0$, then

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Calculating the integral as above and finding the sums of geometric progressions we see that $\left|\left(J^{\alpha}_{(2)}f\right)(x)\right|\leq\operatorname{const}\cdot|x|_{p}^{\alpha-1}$, which proves (3). $\qquad\blacksquare$

4 Logarithmic asymptotics

If a function $f$ decays slower than it did under the assumptions of Theorem 2, then a richer asymptotic behavior is possible. Let us consider the case where $f(t)\geq 0$,

[TABLE]

where $0\leq\beta<1$, $\gamma\geq 0$, $a_{n}\in\mathbb{R}$.

First we need some auxiliary results.

Lemma 1**.**

Let $0\leq f(x)=o\left(|x|_{p}^{-\lambda}\right)$, $|x|_{p}\to\infty$, where $0<\lambda<1$. Then

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Proof. Let $n_{0}=[\log_{p}r]$. Then $p^{n_{0}}\leq r\leq p^{n_{0}+1}$. It is known (see Section 1) that

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so that

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By our assumption, for any $n\in\mathbb{N}$, there exists such $r_{0}=r_{0}(n)$ that $f(x)<\frac{1}{n}|x|_{p}^{-\lambda}$ for $|x|_{p}>r_{0}$. Then we can write

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where

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It follows from (6) that $G_{3}(n,r)\to\infty$, so that

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where $n$ is arbitrary. Therefore

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which gives, together with (7), the required asymptotic relation (5). $\qquad\blacksquare$

Lemma 2**.**

Let $0\leq\beta<1$, $k\in\mathbb{N}$. For any $\varepsilon>0$, such that $\beta+\varepsilon<1$,

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Proof. Assuming that $r>2$, we have $|t|_{p}<\frac{1}{2}$, so that $|1-t|_{p}^{\alpha-1}-|t|_{p}^{\alpha-1}=1-|t|_{p}^{\alpha-1}\leq 1$, and we find that

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if $r$ is large enough, and the relation (8) follows from the integration formula (6). $\qquad\blacksquare$

Now we are ready to consider the asymptotics of $I^{\alpha}f$ for a function $f$ satisfying (4). Below we use the notation

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for any real positive number $\gamma$ and $n\in\mathbb{N}$.

Theorem 3**.**

If a function $f\geq 0$ satisfies the asymptotic relation (4), then

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where

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[TABLE]

Proof. Let us write $(I^{\alpha}f)(x)$ for $|x|_{p}\geq 1$ as the sum of two integrals $I_{1}$ and $I_{2}$, with the integration over $\{y:\ |y|_{p}<|x|_{p}^{1/2}\}$ and $\{y:\ |x|_{p}^{1/2}\leq|y|_{p}\leq|x|_{p}\}$ respectively.

Denote $\mathcal{K}(x,y)=|x-y|_{p}^{\alpha-1}-|y|_{p}^{\alpha-1}$. Considering $I_{1}$, for $|y|_{p}\leq|x|_{p}$, we have

[TABLE]

Indeed, if $|x|_{p}>1$, then $|y|_{p}<|x|_{p}$, $\mathcal{K}(x,y)=|x|_{p}^{\alpha-1}-|y|_{p}^{\alpha-1}$, and we get (10). If $|x|_{p}=1$, $|y|_{p}<1$, then $0<\mathcal{K}(x,y)=1-|y|_{p}^{\alpha-1}<|x|_{p}^{\alpha-1}$.

It follows from (10) that

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and by (4) and Lemma 1, for any small $\varepsilon>0$,

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Considering $I_{2}$ we write

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Denote

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where on the domain of integration,

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and we may write, for a non-integer $\gamma$, the convergent binomial series

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Note that we can use the Taylor formula with the integral form of the remainder

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where

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If $-\frac{1}{2}<s<\frac{1}{2}$, $0<\tau<1$, then $\frac{1}{2}\leq 1+s(1-\tau)\leq\frac{3}{2}$. Therefore

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[TABLE]

and this asymptotics is uniform with respect to $t,|t|_{p}\in[|x|_{p}^{-1/2},1]$.

Substituting and using Lemma 2 we obtain the expansion

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We have

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where

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The last estimate is a consequence of (12).

Now the asymptotic relations (11) and (12) imply the required relation (9). $\qquad\blacksquare$

In our final result, we give a modification of Theorem 3 for the case where $\beta=1$.

Theorem 4**.**

Suppose that $f$ is nonnegative,

[TABLE]

Then

[TABLE]

where

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[TABLE]

Proof. Let us write $I^{\alpha}f=\frac{1-p^{-\alpha}}{1-p^{\alpha-1}}(J_{1}+J_{2}+J_{3})$ where

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[TABLE]

[TABLE]

Choosing $\varepsilon>0$, such that $1+\varepsilon<\alpha$, we see that $f(x)=o\left(|x|_{p}^{-1+\varepsilon}\right)$, $|x|_{p}\to\infty$. By Lemma 1,

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For the kernel of the above integral operator we get, considering various cases, the estimate

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It follows from Lemma 1 that

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By our assumption,

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Let us consider the expression

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It follows from the first integration formula from Section 1 that

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This implies (just as in the proof of Theorem 3) the expansion

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Taking into account (14), we come to (13). $\qquad\blacksquare$

Acknowledgments

The work of the first author was supported in part by Grant 23/16-18 “Statistical dynamics, generalized Fokker-Planck equations, and their applications in the theory of complex systems” of the Ministry of Education and Science of Ukraine.