On $L_{p}-$ Theory for Integro-Differential Operators with Spatially Dependent Coefficients

Abstract

The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by Lévy measures with O-regularly varying profile. The coefficients are assumed to be bounded and Hölder continuous in the spatial variable. Our results can cover interesting classes of Lévy measures that go beyond those comparable to $dy/\left| y\right| ^{d+\alpha }.$

Appendix

We collect some results on O-RV functions which are used frequently in this paper. Some results are taken from [3, 6] and slightly modified for our need.

Lemma 14

([6, Lemma 1]) Assume $w\left( r\right) ,r>0,$ is an O-RV function at zero and infinity with indices $p_{1},q_{1},p_{2},q_{2}$ defined in (2.2), (2.3), and $p_{1},p_{2}>0.$ Then for any $\alpha _{1}>q_{1}\vee q_{2}$, $0<\alpha _{2}<p_{1}\wedge p_{2}$, there exist $c_{1}=c_{1}\left( \alpha _{1}\right) ,c_{2}=c_{2}\left( \alpha _{2}\right) >0$ such that

$$\begin{aligned} c_{1}\left( \frac{y}{x}\right) ^{\alpha _{2}}\le \frac{w\left( y\right) }{w\left( x\right) }\le c_{2}\left( \frac{y}{x}\right) ^{\alpha _{1}},0<x\le y<\infty . \end{aligned}$$

Lemma 15

([3, Lemma 8]) Assume $w\left( r\right) ,r>0,$ is an O-RV function at zero with lower and upper indices $p_{1},q_{1},$ that is,

$$\begin{aligned} r_{1}\left( x\right) =\overline{\lim _{\varepsilon \rightarrow 0}}\frac{w\left( \epsilon x\right) }{w\left( \epsilon \right) }<\infty ,x>0, \end{aligned}$$

and

$$\begin{aligned} p_{1}=\lim _{\epsilon \rightarrow 0}\frac{\log r_{1}\left( \epsilon \right) }{\log \epsilon }\le q_{1}=\lim _{\epsilon \rightarrow \infty }\frac{\log r_{1}\left( \epsilon \right) }{\log \left( \epsilon \right) }. \end{aligned}$$

• (i) Let $\beta >0$ and $\tau >-\beta p_{1}.$ There is $C>0$ so that

  $$\begin{aligned} \int _{0}^{x}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in (0,1], \end{aligned}$$

  and $\lim _{x\rightarrow 0}x^{\tau }w\left( x\right) ^{\beta }=0.$

• (ii) Let $\beta >0$ and $\tau <-\beta q_{1}$. There is $C>0$ so that

  $$\begin{aligned} \int _{x}^{1}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in (0,1],\text { } \end{aligned}$$

  and $\lim _{x\rightarrow 0}x^{\tau }w\left( x\right) ^{\beta }=\infty .$

• (iii) Let $\beta <0$ and $\tau >-\beta q_{1}$. There is $C>0$ so that

  $$\begin{aligned} \int _{0}^{x}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in (0,1], \end{aligned}$$

  and $\lim _{x\rightarrow 0}x^{\tau }w\left( x\right) ^{\beta }=0.$

• (iv) Let $\beta <0$ and $\tau <-\beta p_{1}$. There is $C>0$ so that

  $$\begin{aligned} \int _{x}^{1}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}=\int _{1}^{x^{-1}}t^{-\tau }w\left( \frac{1}{t}\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in (0,1], \end{aligned}$$

  and $\lim _{x\rightarrow 0}x^{\tau }w\left( x\right) ^{\beta }=\infty .$

Similar statement holds for O-RV functions at infinity.

Lemma 16

([6, Lemma 3]) Assume $w\left( r\right) ,r>0,$ is an O-RV function at infinity with lower and upper indices $p_{2},q_{2},$ that is,

$$\begin{aligned} r_{2}\left( x\right) =\overline{\lim _{\varepsilon \rightarrow \infty }}\frac{w\left( \epsilon x\right) }{w\left( \epsilon \right) }<\infty ,x>0, \end{aligned}$$

and

$$\begin{aligned} p_{2}=\lim _{\epsilon \rightarrow 0}\frac{\log r_{2}\left( \epsilon \right) }{\log \epsilon }\le q_{2}=\lim _{\epsilon \rightarrow \infty }\frac{\log r_{2}\left( \epsilon \right) }{\log \left( \epsilon \right) }. \end{aligned}$$

• (i) Let $\beta >0$ and $-\tau >\beta q_{2}.$ There is $C>0$ so that

  $$\begin{aligned} \int _{x}^{\infty }t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in [1,\infty ), \end{aligned}$$

  and $\lim _{x\rightarrow \infty }x^{\tau }w\left( x\right) ^{\beta }=0.$

• (ii) Let $\beta >0$ and $-\tau <\beta p_{2}$. There is $C>0$ so that

  $$\begin{aligned} \int _{1}^{x}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in [1,\infty ),\text { } \end{aligned}$$

  and $\lim _{x\rightarrow \infty }x^{\tau }w\left( x\right) ^{\beta }=\infty .$

• (iii) Let $\beta <0$ and $\tau <-\beta p_{2}$. There is $C>0$ so that

  $$\begin{aligned} \int _{x}^{\infty }t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in [1,\infty ), \end{aligned}$$

  and $\lim _{x\rightarrow \infty }x^{\tau }w\left( x\right) ^{\beta }=0.$

• (iv) Let $\beta <0$ and $\tau >-\beta q_{2}$. There is $C>0$ so that

  $$\begin{aligned} \int _{1}^{x}t^{\tau }w\left( t\right) ^{\beta }\frac{dt}{t}\le Cx^{\tau }w\left( x\right) ^{\beta },x\in [1,\infty ), \end{aligned}$$

  and $\lim _{x\rightarrow \infty }x^{\tau }w\left( x\right) ^{\beta }=\infty .$

Lemma 17

([6, Lemma 5 ]) Let $w\left( r\right) ,r>0,$ be a continuous, non-decreasing O-RV function at zero and infinity with indices $p_{1},q_{1},p_{2},q_{2}$ defined in (2.2), (2.3), and $p_{1},p_{2}>0.$ Let

$$\begin{aligned} a_{1}\left( r\right) =\inf \left\{ t>0:w\left( t\right) \ge r\right\} ,r>0, \end{aligned}$$

$$\begin{aligned} a_{2}\left( r\right) =\inf \left\{ t>0:w\left( t\right)>r\right\} ,r>0. \end{aligned}$$

If a is either $a_{1}$ or $a_{2}$, then

• (i) $w\left( a\left( t\right) \right) =t,t>0$ and

  $$\begin{aligned} a_{1}\left( w\left( t\right) \right) \le t\le a_{1}\left( w\left( t\right) +\right) ,t>0, \end{aligned}$$
  $$\begin{aligned} a_{2}\left( w\left( t\right) -\right) \le t\le a_{2}\left( w\left( t\right) \right) ,t>0. \end{aligned}$$

• (ii) a is O-RV at zero and infinity with lower indices $p,\bar{p}$ and upper indices $q,\bar{q}$ respectively such that

  $$\begin{aligned} \frac{1}{q_{1}}\le p\le q\le \frac{1}{p_{1}},\frac{1}{q_{2}}\le \bar{p}\le \bar{q}\le \frac{1}{p_{2}}. \end{aligned}$$

• (iii) $a\left( w\left( t\right) \right) \asymp t,t>0.$

Proof

(i) is straightforward, (ii) is proved in [6, Lemma 5 ] and (iii) easily follows from (ii) and Lemma 14. $\square$

Corollary 2

([6, Corollary 1]) Let $w\left( r\right) ,r>0,$ be a continuous, non-decreasing O-RV function at zero and infinity with indices $p_{1},q_{1},p_{2},q_{2}$ defined in (2.2), (2.3), and $p_{1},p_{2}>0.$ Let

$$\begin{aligned} a_{1}\left( r\right) =\inf \left\{ t>0:w\left( t\right) \ge r\right\} ,r>0, \end{aligned}$$

$$\begin{aligned} a_{2}\left( r\right) =\inf \left\{ t>0:w\left( t\right)>r\right\} ,r>0. \end{aligned}$$

If a is either $a_{1}$ or $a_{2}$, then

• (i) For any $\beta >0$ and $\tau <\frac{\beta }{q_{1}}\wedge \frac{\beta }{q_{2}}$ there is $C>0$ such that

  $$\begin{aligned} \int _{0}^{r}t^{-\tau }a\left( t\right) ^{\beta }\frac{dt}{t}\le & {} Cr^{-\tau }a\left( r\right) ^{\beta },r>0,\\ \lim _{r\rightarrow 0}r^{-\tau }a\left( r\right) ^{\beta }= & {} 0,\lim _{r\rightarrow \infty }r^{-\tau }a\left( r\right) ^{\beta }=\infty , \end{aligned}$$

  and for any $\beta <0,\tau >\left( -\frac{\beta }{p_{1}}\right) \vee \left( -\frac{\beta }{p_{2}}\right)$ there is $C>0$ such that

  $$\begin{aligned} \int _{0}^{r}t^{\tau }a\left( t\right) ^{\beta }\frac{dt}{t}\le & {} Cr^{\tau }a\left( r\right) ^{\beta },r>0,\\ \lim _{r\rightarrow 0}r^{\tau }a\left( r\right) ^{\beta }= & {} 0,\lim _{r\rightarrow \infty }r^{\tau }a\left( r\right) ^{\beta }=\infty . \end{aligned}$$

• (ii) For any $\gamma >0$ and $\tau >\frac{\gamma }{p_{1}}\vee \frac{\gamma }{p_{2}}$ there is $C>0$ such that

  $$\begin{aligned} \int _{r}^{\infty }t^{-\tau }a\left( t\right) ^{\gamma }\frac{dt}{t}\le & {} Cr^{-\tau }a\left( r\right) ^{\gamma },r>0,\\ \lim _{r\rightarrow 0}r^{-\tau }a\left( r\right) ^{\gamma }= & {} \infty ,\lim _{r\rightarrow \infty }r^{-\tau }a\left( r\right) ^{\gamma }=0, \end{aligned}$$

  and for any $\gamma <0$ and $\tau <\left( -\frac{\gamma }{q_{1}}\right) \wedge \left( -\frac{\gamma }{q_{2}}\right)$ there is $C>0$ such that

  $$\begin{aligned} \int _{r}^{\infty }t^{\tau }a\left( t\right) ^{\gamma }\frac{dt}{t}{} & {} \le Cr^{\tau }a\left( r\right) ^{\gamma },r>0,\\ \lim _{r\rightarrow 0}r^{\tau }a\left( r\right) ^{\gamma }{} & {} = \infty ,\lim _{r\rightarrow \infty }r^{\tau }a\left( r\right) ^{\gamma }=0. \end{aligned}$$

Lemma 18

Let $\nu \in \mathfrak {A}^{\sigma }$,$\sigma \in \left[ 1,2\right)$, $w=w_{\nu }$ be an O-RV function at zero and infinity with indices $p_{1},q_{1},p_{2},q_{2}$ defined in (2.2), (2.3) and $p_{1},p_{2}>0$. If $\frac{\sigma -1}{\left( p_{1}\wedge p_{2}\right) }<\beta <1$ then there exists $N=N\left( \beta ,\nu \right) >0$ such that

$$\begin{aligned} \int _{\left| y\right| \le 1}\left| y\right| w\left( \left| y\right| \right) ^{\beta }\nu \left( dy\right) \le N. \end{aligned}$$

Proof

Since $1+\left( p_{1}\wedge p_{2}\right) \beta >\sigma$, we may choose $0<\alpha <\left( p_{1}\wedge p_{2}\right)$ so that $1+\alpha \beta >\sigma .$ By Lemma 14 and definition of $\sigma$,

$$\begin{aligned} \int _{\left| y\right| \le 1}\left| y\right| w\left( \left| y\right| \right) ^{\beta }\nu \left( dy\right) \le N\int _{\left| y\right| \le 1}\left| y\right| ^{1+\alpha \beta }\nu \left( dy\right) \le N. \end{aligned}$$

$\square$

Lemma 19

Let $\nu \left( dy\right) =j_{d}\left( \left| y\right| \right) dy$ and D hold then

$$\begin{aligned} w_{\nu }\left( r\right) \asymp \left( \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\right) ^{-1}\asymp \gamma \left( r\right) ,r>0. \end{aligned}$$

Proof

The first estimate follows from the definition of $w_{\nu }.$ The second estimate follows from basic properties of O-RV functions. With definition of O-RV functions in mind, we consider the function $\gamma ^{-1}$,

$$\begin{aligned} \limsup _{\epsilon \rightarrow \infty }\frac{\gamma \left( \epsilon x\right) ^{-1}}{\gamma \left( \epsilon \right) ^{-1}}=\limsup _{\epsilon \rightarrow \infty }\frac{\gamma \left( \epsilon \right) }{\gamma \left( \epsilon x\right) }=\limsup _{\epsilon \rightarrow \infty }\frac{\gamma \left( x^{-1}\epsilon x\right) }{\gamma \left( \epsilon x\right) }=r_{2}\left( x^{-1}\right) . \end{aligned}$$

Hence, $\gamma {}^{-1}$ is an O-RV function at infinity with the lower index of

$$\begin{aligned} p=\lim _{\epsilon \rightarrow 0}\frac{\log r_{2}\left( \epsilon ^{-1}\right) }{\log \epsilon }=-\lim _{\epsilon \rightarrow \infty }\frac{\log r_{2}\left( \epsilon \right) }{\log \epsilon }=-q_{2} \end{aligned}$$

and by a similar calculation the upper index is

$$\begin{aligned} q=-p_{2}. \end{aligned}$$

Similarly, we consider the function $\gamma \left( x^{-1}\right) ^{-1},x>0,$

$$\begin{aligned} \limsup _{\epsilon \rightarrow \infty }\frac{\gamma ^{-1}\left( \frac{1}{\epsilon x}\right) }{\gamma ^{-1}\left( \frac{1}{\epsilon }\right) }=\limsup _{\epsilon \rightarrow 0}\frac{\gamma \left( \epsilon \right) }{\gamma \left( \epsilon x^{-1}\right) }=\limsup _{\epsilon \rightarrow 0}\frac{\gamma \left( \epsilon x^{-1}x\right) }{\gamma \left( \epsilon x^{-1}\right) }=r_{2}\left( x\right) . \end{aligned}$$

Therefore, the function $\gamma \left( x^{-1}\right) ^{-1},x>0$ is an O-RV function at infinity with

$$\begin{aligned} p=p_{2},q=q_{2}. \end{aligned}$$

By applying [1, Theorem 3] with $\gamma {}^{-1}$, there exists $N>1$ so that for $r\ge N$,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\asymp \gamma \left( r\right) ^{-1}. \end{aligned}$$

(8.1)

Now we extend the estimate to $r\in \left[ 1,N\right)$. On one hand, by (8.1) and Lemma 14,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds= & {} \int _{r}^{N}\frac{1}{\gamma \left( s\right) s}ds+\int _{N}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\\\le & {} C\gamma \left( r\right) ^{-1}\ln \left( N\right) +C\gamma \left( N\right) ^{-1}\\\le & {} C\gamma \left( r\right) ^{-1}. \end{aligned}$$

On the other hand, by Lemma 14,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\ge & {} \int _{N}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\ge C\gamma \left( N\right) ^{-1}\\\ge & {} C\gamma \left( r\right) ^{-1}. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\asymp \gamma \left( r\right) ^{-1},r\ge 1. \end{aligned}$$

(8.2)

For $r<1$, we first note that by applying [1, Theorem 3] with $\gamma \left( x^{-1}\right) ^{-1},x>0$, there exists $N>1$ so that for $x\ge N$.

$$\begin{aligned} \int _{1}^{x}\frac{1}{\gamma \left( s^{-1}\right) s}ds\asymp \gamma \left( x^{-1}\right) ^{-1}. \end{aligned}$$

(8.3)

For $0<r\le \frac{1}{N}$, we change the variable of integration then apply (8.3), (8.2) and Lemma 14,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds= & {} \int _{r}^{1}\frac{1}{\gamma \left( s\right) s}ds+\int _{1}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\\= & {} \int _{1}^{1/r}\frac{1}{\gamma \left( s^{-1}\right) s}ds+\int _{1}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\\\le & {} C\gamma \left( r\right) ^{-1}. \end{aligned}$$

Moreover,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\ge \int _{1}^{1/r}\frac{1}{\gamma \left( s^{-1}\right) s}ds\ge C\gamma \left( r\right) ^{-1}. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{r}^{\infty }\frac{1}{\gamma \left( s\right) s}ds\asymp \gamma \left( r\right) ^{-1},r\in \left( 0,\frac{1}{N}\right] . \end{aligned}$$

(8.4)

Finally, the estimate for $r\in \left[ \frac{1}{N},1\right)$ follows from (8.2), (8.4) and Lemma 14. $\square$