---
abstract: 'We give an appropriate version of the result in the paper by Dipierro, Savin, and Valdinoci for different, not necessarily fractional, powers of the Laplacian.'
address: '127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455'
author:
- 'N.V. Krylov'
title: 'On the paper “All functions are locally s-harmonic up to a small error" by Dipierro, Savin, and Valdinoci'
---

Introduction
============

Let ${\mathbb{R}}^{d}$ be a $d$-dimensional Euclidean space. Denote $B_{r}=\{x\in{\mathbb{R}}^{d}
:|x|<r\}$, and for integers $k=0,1,...$ let $C^{k}(\bar B_{r})$ be the set of $k$-times continuously differentiable functions on $\bar B_{r}$ provided with the usual norm $\|\cdot\|_{C^{k}(B_{r})}$ equal to the sum of maximal magnitudes of all derivatives up to the $k$-th order.

Here is Theorem 1.1 of [@DSV_17] in a slightly different form.

\[theorem 10.10.1\] For any $k\in\{0,1,...\}$, $s\in(0,1)$, $f\in C^{k}(\bar B_{r})$, and $\varepsilon>0$, there exists $u\in C^{\infty}_{0}({\mathbb{R}}^{d})$ such that $(-\Delta)^{s}u=0$ in $B_{1}$ and $$\|f-u\|_{C^{k}(  B_{1})}\leq\varepsilon.$$

Actually, Theorem \[theorem 10.10.1\] may look like a generalization of Theorem 1.1 of [@DSV_17] because the smoothness class of $u$’s in the latter is much wider. However, by using mollifications one easily deduces Theorem \[theorem 10.10.1\] from Theorem 1.1 of [@DSV_17].

Many interesting consequences are extracted in [@DSV_17], in particular, that $s$-harmonic functions in $B_{1}$ do not satisfy the Harnack inequality.

One can rewrite Theorem \[theorem 10.10.1\] in different terms. For real $n$ introduce $$K_{n}(x,y)=|x-y|^{n},\quad K_{n}g(y)=\int_{{\mathbb{R}}^{d}}K_{n}(x-y)g(x)\,dx.$$ Since the function $u$ from Theorem \[theorem 10.10.1\] has a representation $u=c_{d,s}K_{2s-d}g$, where $g=(-\Delta)^{s}u$ and $c_{d,s}$ is an appropriate constant, Theorem \[theorem 10.10.1\] is equivalent to the following.

\[theorem 10.10.2\] For any $k\in\{0,1,...\}$, $s\in(0,1)$, $f\in C^{k}(\bar B_{1})$, and $\varepsilon>0$, there exists $g\in C^{\infty} ({\mathbb{R}}^{d})$ having all bounded derivatives of any order such that $g=0$ in $B_{1}$, $K_{2s-d}g\in  C^{\infty}_{0}({\mathbb{R}}^{d})$, and $$\|f-K_{2s-d}g\|_{C^{k}(  B_{1})}\leq\varepsilon.$$

The goal of this paper is to almost extend this result from $n=2s-d$, $s\in(0,1)$, to arbitrary $n\in(-d,0)$ such that $n+d \not \in\{2,4,...\}$. By “almost” we mean that we will have $K_{n}g\in  C^{\infty} ({\mathbb{R}}^{d})$ with each derivative of any order tending to zero as $|x|\to\infty$ instead of $K_{2s-d}g\in  C^{\infty}_{0}({\mathbb{R}}^{d})$, but on the other hand we will have $g\in C^{\infty}_{0} (B_{4}\setminus\bar B_{3})$. Wider range of $n$ than $(-d,0)$ is also treated.

Main result
===========

Fix a real number $n$ such that $n+d \not \in\{2,4,...\}$ and fix a $k\in\{0,1,...\}$. Define $S $ as the set of functions $f$ on $B_{1}$ each of which has a representation on $B_{1}$ as $f=K_{n}g$, where $g\in   C^{\infty}_{0}(B_{4}\setminus\bar B_{3})$. Observe right away that if $n\in(-d,0)$, then $K_{n}g$ is well defined in ${\mathbb{R}}^{d}$ and is bounded along with each of its derivatives of any order, which, in addition, tend to zero at infinity. This justifies the last paragraph in the Introduction.

Here is our result.

\[theorem 9.10.1\] The set $S$ is everywhere dense in $C^{k}(\bar B_{1})$.

To prove the theorem we need rather rough estimates from the following lemma proved in Section \[section 10.9.1\].

\[lemma 9.10.1\] Take $r\in\{0,1,...\}$ such that $2r>k$. Then there exist a constant $N=N(d,k,r)$ such that, for any $x\in B_{4}\setminus\bar B_{3}$ and real $m$, we have $$\label{9.10.1}
\|K_{m}(x,\cdot)\|_{C^{ k}(B_{1})}
\leq N^{r}(m^{2r}+N^{r}).$$

[**Proof of Theorem \[theorem 9.10.1\]**]{}. Let $\bar S$ be the closure of $S$ in $C^{k}(\bar B_{1})$. Then $K(x,\cdot)\in \bar S$ for any $x\in B_{4}\setminus\bar  B_{3}$, which is seen from the formula $$K_{n}(x,y)=\lim_{m\to\infty}\int_{{\mathbb{R}}^{d}}K_{n}(x+z/n,y)\zeta 
(z )\,dz$$ $$=\lim_{m\to\infty}\int_{B_{4}\setminus\bar  B_{3}}
K_{n}(z,y)\zeta_{m}
(z-x)\,dz,$$ where $\zeta_{m}(z)=m^{d}\zeta(mz)$ and $\zeta$ is any $C^{\infty}_{0}({\mathbb{R}}^{d})$ function which integrates to one.

Now, since partial derivatives are limits of finite differences, $\Delta^{m}_{x}K_{n}(x,\cdot)\in \bar S$ for any $m=0,1,...$ and $x\in B_{4}\setminus \bar B_{3}$, where $\Delta_{x}$ is the Laplacian applied with respect to the $x$ variable. Observe that $K_{n}(x-y)$ is a radial function if the origin is taken to be $y$. We know how the Laplacian looks on radial functions, so that, for $m\geq1$, $$\label{9.10.2}
\Delta _{x}K_{n} =n(n-1)  K_{n-2}+(d-1)n
 K_{n-2}=
n(n+d-2) K_{n-2}.$$ It follows that $\Delta^{m}_{x}K_{n}(x,\cdot)=c_{m}K_{n-2m}(x,\cdot)$, where $c_{m}$ are some constants, and our assumption that $n+d \not \in\{2,4,...\}$ implies that $c_{m}\ne0$. Thus $$K_{n-2m}(x,\cdot)\in \bar S\quad m=0,1,....$$

Next, owing to Lemma \[lemma 9.10.1\], for any $x\in B_{4}\setminus \bar B_{3}$, $r\in\{0,1,...\}$, and real $t$, the series $$\sum_{m=0}^{\infty}\frac{1}{m!}t^{m}K_{n-2r-2m}(x,\cdot)$$ converges in $C^{k}(\bar B_{1})$. Thus, for any $t$ and $x\in B_{4}\setminus \bar B_{3}$, $$K_{n-2r}(x-\cdot)\exp(-t/|x-\cdot|^{2})\in \bar S.$$ Obviously, the integral $$\int_{0}^{\infty}t^{m}K_{n-2r}(x-\cdot)\exp(-t/|x-\cdot|^{2})\,dt$$ converges in $C^{k}(\bar B_{1})$ for any $x\in B_{4}\setminus \bar B_{3}$ and $m>-1$. Thus, $$K_{n-2r+2m-2}(x-\cdot)\in \bar S.$$ For any real $j $ it is possible to find (many) $r\in\{0,1,...\}$ and real $m>-1$ such that $n-2r+2m-2=j$. Hence, for any $x\in B_{4}\setminus \bar B_{3}$ $$K_{j}(x-\cdot)\in \bar S\quad \forall j\in{\mathbb{R}}.$$ For $j\in\{0,1,...\}$ the functions $K_{j}(x-y)$ are polynomials with respect to $x$ with the coefficients depending on $y$. These coefficients can be found by differentiating $K_{j}(x-y)$ with respect to $x\in B_{4}\setminus\bar B_{3}$ and it follows that they belong to $\bar S$. But then, for any $x$, $K_{j}(x-\cdot)$ is in $\bar S$ being just a linear combination of elements of $\bar S$. Thus, $$K_{j}(x-\cdot)\in \bar S\quad \forall j=0,1,..., x\in{\mathbb{R}}^{d}.$$

Furthermore, clearly the series of polynomials $$\sum_{m=0}^{\infty}\frac{1}{m!}t^{-m}K_{2m}(x-\cdot)$$ converges in $C^{k}(\bar B_{1})$ for any $x\in {\mathbb{R}}^{d}$ and $t>0$. It follows that $$\exp(-|x-\cdot|^{2}/t)\in \bar S.$$

The rest is routine. Take an $f\in C^{k}(\bar B_{1})$, continue it to the whole ${\mathbb{R}}^{d}$ preserving its class and making it vanish outside $B_{2}$, and for $t>0$ define $$f_{t}(y)= t^{-d/2}\int_{{\mathbb{R}}^{d}}
f(x)\exp(-|x-y|^{2}/t)\,dx.$$ Obviously the integral converges in $C^{k}(\bar B_{1})$, so that $f_{t}\in \bar S$. We know that, if a constant $c$ is chosen appropriately, then $cf_{t}\to f$ in $C^{k}(\bar B_{1})$ as $t\downarrow 0$. Since $cf_{t}\in \bar S$, $f \in \bar S$ and the theorem is proved.

Proof of Lemma \[lemma 9.10.1\]
===============================

\[section 10.9.1\]

Take $p\in(1,\infty)$ and $u\in W^{2}_{p}(B_{2})$. By representing it as $h+v$, where $h$ is harmonic and $v=0$ on $\partial B_{2}$ it is easy to conclude that $\|u\|_{W^{2}_{p}(B_{1})}\leq N(\|u\|_{L_{p}(B_{2})}
+\|\Delta u\|_{L_{p}(B_{2})})$, where and below by $N$ we denote generic constants $\geq1$ depending only on $d$, $p$, and $r$ (which is coming later). This fact also holds for $B_{\rho_{1}}\subset \bar B_{\rho_{1}}\subset   B_{\rho_{2}}$ in place of $B_{1}$ and $B_{2}$ with a constant, this time, also depending on $\rho_{1}$ and $\rho_{2}$. By applying these estimates to the derivatives of $u$, we see that, for any $r=0,1,...$, $$\|u\|_{W^{2r}_{p}(B_{1})}
\leq N\sum_{i=0}^{r}\|\Delta^{i} u\|_{L_{p}(B_{2})}.$$

Then and the estimate $|n(n+d-2)|\leq N(n^{2}+1)$ show that, for $|y|\leq2$ and $3\leq|x|\leq4$, $$|\Delta^{i} K_{m}(x,y)|\leq c^{i}(m^{2}+1)((m-2)^{2}+1)
\cdot...\cdot((m-2i+2)^{2}+1)K_{m-2i}(x,y)$$ $$\leq c^{i}(m^{2}+1)((m-2)^{2}+1)
\cdot...\cdot((m-2i+2)^{2}+1).$$ Since, for $j=0,...,r$, we have $(m-2j)^{2}+1\leq 2m^{2}+N$, it holds that $$\|K_{m}(x,\cdot)\|_{W^{2r}_{p}(B_{1})}
\leq N^{r}(2m^{2}+N)^{r}.$$

Finally, by embedding theorems $W^{2r}_{p}(B_{1})\subset C^{k}(\bar B_{1})$ if $(2r-k)p>d$, which holds indeed for large enough $p$. This proves the lemma.

We conclude the paper with the following.

In [@DSV_17] the authors prove that the set of polynomials is dense in $C^{k}(\bar B_{1})$. Actually, this fact follows from Lemma 3 and Theorem 3 in Section 3 of an old paper [@Gr_41]. By using Bernstein’s polynomials this fact is also follows from Section 3.8 of [@Kr_95].

[10]{}

S. Dipierro, O. Savin, and E.Valdinoci, [*All functions are locally s-harmonic up to a small error*]{}, J. Eur. Math. Soc., Vol. 19 (2017), 957–955.

L.M. Graves, [*Some general approximation theorems*]{}, Annals of Mathematics, Second Series, Vol. 42, No. 1 (Jan., 1941), 281–292.

N.V. Krylov, “Introduction to the theory of diffusion processes”, Amer. Math. Soc., Providence, RI, 1995.
