An \(L_q(L_p)\)-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps

We investigate an \(L_{q}(L_{p})\)-regularity (\(1<p,q<\infty \)) theory for space-time nonlocal equations of the type \(\partial ^{\alpha }_{t}u = \mathcal {L}u +f\). Here, \(\partial ^{\alpha }_{t}\) is the Caputo fractional derivative of order \(\alpha \in (0,1)\) and \(\mathcal {L}\) is an integro-differential operator

which is the infinitesimal generator of an isotropic unimodal Lévy process. We assume that the jumping kernel \(j_{d}(r)\) is comparable to \(r^{-d} \ell (r^{-1})\), where \(\ell \) is a continuous function satisfying

where \(0\le \delta _{1}\le \delta _{2}<2\). Hence, \(\ell \) can be slowly varying at infinity. Our result covers \(\mathcal {L}\) whose Fourier multiplier \(\Psi (\xi )\) satisfies \(\Psi (\xi )\asymp -\log {(1+|\xi |^{\beta })}\) for \(\beta \in (0,2]\) and \(\Psi (\xi ) \asymp -(\log (1+|\xi |^{\beta /4}))^{2}\) for \(\beta \in (0,2)\) by taking \(\ell (r) \asymp 1\) and \(\ell (r) \asymp \log {(1+r^{\beta })}\) for \(r\ge 1\) respectively. In this article, we use the Calderón–Zygmund approach and function space theory for operators having slowly varying symbols.

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

The authors are sincerely grateful to the anonymous referee for careful reading, valuable comments, finding errors. The paper could be considerably improved by the referee’s comments.

Authors and Affiliations

1. Department of Mathematical Sciences, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Republic of Korea

   Jaehoon Kang

2. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02445, Republic of Korea

   Daehan Park

Corresponding author

Correspondence to Daehan Park.

Conflict of interest

The authors declare that they have no conflict of interest.

Publisher's Note

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The first author was supported by BK21 SNU Mathematical Sciences Division. The second author was supported by Samsung Science & Technology Foundation (SSTF)’s grants (No. SSTF-BA1401-51).

Appendix

Lemma A.1

1. (i)

   Suppsose that \(\ell _{1}\in \mathcal {G}\) and let \(\ell _{2}\) be a function which satisfies \(\ell _{2} \in (a,b)\) for some \(0<a<b<\infty \). Then \(\ell _{1}\ell _{2}\in \mathcal {G}\).

2. (ii)

   Suppose \(\ell \in \mathcal {G}\). Then for any \(b>0\), \(r\mapsto \ell (r^b)\) belongs to \(\mathcal {G}\).

3. (iii)

   Let \(\ell _1\in \mathcal {G}\) and \(\ell _2\) be an increasing function satisfying \(\ell _2\ge c\) on \([1,\infty )\) for some \(c>0\). Then, \(\ell _1/\ell _2\in \mathcal {G}\).

4. (iv)

   Let \(\ell _{1}(r)=\log {(1+r)}\) and let \(\ell _{k+1}=\ell _{k}\circ \ell _{1}(r)\) for \(k\in \mathbb {N}\). Then for any \(n\in \mathbb {N}\) \(k_{1},\dots ,k_{n}\in \mathbb {N}\) and \(b_{1},\dots ,b_{n}>0\) we have

   $$\begin{aligned} \Lambda (r) = \prod _{i=1}^{n} (\ell _{k_{i}}(r))^{b_{i}} \in \mathcal {G}. \end{aligned}$$
5. (v)

   For \(b\in (0,1/2)\), \(\ell (r)=(e^{(\log {(1+r)})^{b}}-1) \in \mathcal {G}\).

Proof

(i) & (ii) Trivial.

(iii) Let \(\ell =\ell _1/\ell _2\). Using that \(\ell _2\ge c\), \(\ell _2\) is increasing and \(\ell _1\in \mathcal {G}\), we see that for any \(a>0\)

(iv) Let \({\tilde{\Lambda }}=(\ell _{1})^{2}\Lambda \). Then \({\tilde{\Lambda }}(r) = \prod ^{n+1}_{i=1} (\ell _{k_{i}}(r))^{b_{i}}\), and we may set \(k_{1}=1\), \(b_{1}=2\). If we show that \({\tilde{\Lambda }} \in \mathcal {G}\), then due to (iii), it follows that \(\Lambda \in \mathcal {G}\). Thus we will show that \({\tilde{\Lambda }} \in \mathcal {G}\). Set \(\ell _{0}(s)=s\). By using the change of variable,

It is easy to see that

Moreover, by the integration by parts and \(((\ell _{k_{i}-1})^{b_{i}})'(s) \le C(b_{i},k_{i})\),

Since \(k_{1}=1\) and \(b_{1}=2\) (for \({\tilde{\Lambda }}\)), there exists \(M>1\) such that

Thus, we have

The above inequality implies that \(\Lambda \in \mathcal {G}\) since it is bounded on (1, M].

(v) By the change of variable, we have

Also, by the integration by parts, we obtain

This shows that there exists \(M>1\) such that for \(r>M\)

Hence, we have (recall that \(b\in (0,1/2))\)

Thus \(\ell \in \mathcal {G}\). The lemma is proved. \(\square \)

Lemma A.2

Let \(f:(0,\infty ) \rightarrow (0,\infty )\) be an increasing continuous function which satisfies \(\sup _{0<r<1}f(r)<\infty \), \(\lim _{r\rightarrow \infty }f(r)=\infty \) and

for some \(c_{1}, \delta >0\). Then there is a strictly increasing continuous function \({\tilde{f}}:(0,\infty ) \rightarrow (0,\infty )\) satisfying

where the constant C does not depend on r.

Proof

We prove the lemma by constructing \({\tilde{f}}\). Extend f to \(\mathbb {R}_{+} \cup \{0\}\) by

Now let \(A= \{ r\ge 0: \exists \, s\ge 0, s\ne r ~ \text {such that} ~ f(r)=f(s) \}\). Then, we can check that \(A=\cup _{k=1}^{\infty }[r_{k},l_{k}]\), where \([r_{k},l_{k}]\) are pairwise disjoint closed intervals.

Case 1. Assume that \(f(0)=0\) or f(r) is strictly increasing for \(r\le 1\). Then, there is a positive number \(a>0\) such that \(A\subset [0,a)^{c}\). Note that for \([r_{k},l_{k}]\) we can choose \(\varepsilon _{k}\in (0,1)\) such that \((l_{k},l_{k}+\varepsilon _{k}]\subset A^{c}\). Now define \({\tilde{f}}_{k}\) on \([r_{k},r_{l}+\varepsilon _{k}]\) as

Then \({\tilde{f}}_{k}\) is continuous, strictly increasing on \([r_{k},l_{k}+\varepsilon _{k}]\) and it satisfies \({\tilde{f}}_{k}(r_{k})=f(r_{k})\), and \({\tilde{f}}_{k}(l_{k}+\varepsilon _{k}) = f(l_{k}+\varepsilon _{k})\). Moreover, on \([r_{k},l_{k}+\varepsilon _{k}]\), \({\tilde{f}}_{k}\) satisfies

since \(f(r_{k})=f(l_{k})\), \(l_{k}>r_{k}\ge a\). Now define \({\tilde{A}} = \cup _{k=1}^{\infty }[r_{k},l_{k}+\varepsilon _{k}]\) and

Then \({\tilde{f}}\) is a desired function.

Case 2. Now assume that \(f(0)\ne 0\) and f(r) is not strictly increasing for \(r\le 1\). Then there exists \(b\ge 1\) such that \([0,b]=[r_{1},l_{1}]\). Take \(\varepsilon _{1}\) and \({\tilde{f}}_{1}\) for \([r_{1},l_{1}]\) corresponding to \(\varepsilon _{k}\) and \({\tilde{f}}_{k}\) in above case. Then on \([r_{1},l_{1}]\), we have

since \(f(0)=f(b)>0\). For other k, we have the same result by following Case 1. Hence, by taking \({\tilde{f}}\) as in (A.1), the lemma is proved. \(\square \)

Lemma A.3

Let \(f:(0,\infty )\rightarrow (0,\infty )\) be a strictly increasing continuous function and \(f^{-1}\) be its inverse. Suppose that there exist \(c,\gamma >0\) such that \((f(R)/f(r))\le c(R/r)^\gamma \) for \(0<r\le R<\infty \). Then, for any \(k>0\), there exists \(C>0\) such that for any \(b>0\)

Proof

By the scaling property of f with \(R=b^{-1}\) and \(r=f^{-1}(s^{-1})\), and the fact that \(f(f^{-1}(s^{-1}))= s^{-1}\), we see that

Thus,

\(\square \)

We use the following lemma with \(f(r) = h(r^{-1})\). Note that by using (2.12), one can check that \(h(r^{-1})\) is a strictly increasing function satisfying \(h(R^{-1})\le c (R/r)^{2} h(r^{-1})\) for any \(0<r<R\).

Lemma A.4

Let \(f:(0,\infty )\rightarrow (0,\infty )\) be a strictly increasing function and \(f^{-1}\) be its inverse. Suppose that there exist \(c,\gamma >0\) such that \((f(R)/f(r))\le c(R/r)^\gamma \) for \(0<r\le R<\infty \). Then, there exists \(C>0\) such that for any \(b>0\)

Proof

By the change of variable and Fubini’s theorem and Lemma A.3,

\(\square \)

Lemma A.5

Let \(\alpha \in (0,1)\). Suppose the function \(\ell \) satisfies Assumption 2.3 (i) let \(\kappa (b) = (h(b))^{-1/\alpha }\) and let \(t_{1}>0\) be taken from Lemma 3.2. Then, there exists \(C>0\) depending only on \(\alpha ,\kappa _{1},\kappa _{2},d,\ell ,C_{0},C_{1},C_{2}\), and \({\varvec{\delta }}\) such that for any \(b>0\)

Proof

By (2.15), (2.16), Proposition 3.2 (i), (4.2) and (4.3),

where we used the relations \(se^{-s} \le Ce^{-s/2}\) (\(s>0\)) and \(K(\rho )\le h(\rho )\) for the last inequality. By Fubini’s theorem we have

which shows (A.2).

Now we prove (A.3). Using Proposition 3.2 (i), (4.2) and (4.3), we see that

where for the third inequality we use Lemma 3.1 (i). \(\square \)

The following lemma is counter part of Lemma A.5. The proof is more delicate than that of Lemma A.5 due to the fact that h(r) and \(\ell (r^{-1})\) may not be comparable for \(0<r\le 1\).

Lemma A.6

Let \(\alpha \in (0,1)\). Suppose the function \(\ell \) satisfies Assumption 2.3 (ii)–(2). Let \(\kappa (b)=(h(b))^{-1/\alpha }\) and let \(t_{1}>0\) be taken from Lemma 3.2. Then, there exists \(C>0\) depending only on \(\alpha ,\kappa _{1},\kappa _{2},d,\ell ,C_{0},C_{1},C_{2}\), and \({\varvec{\delta }}\) such that for any \(b>0\)

Proof

Note that p(t, 0) is well-defined on \((0,t_1]\). We first show (A.7). We split the integral into two parts.

We can obtain \(I\le C\) by using Proposition 3.2 (ii) and the same argument in the proof of (A.3) (see (A.5)). Thus, we will show \(II\le C\) for some constant C for the rest of the proof of (A.7). Observe that

Since \(r\mapsto h(r)\) is decreasing, we see that \(h((\ell ^{-1}(a_0/r))^{-1})\ge h(4b)\) for \(r\le a_0(\ell ^*((4b)^{-1}))^{-1}\). Hence, by Proposition 3.2 (ii) and Fubini’s theorem

Also for \(II_{2}\), by using Proposition 3.2 (ii) and relation \(K\le h\) and \(se^{-s} \le c e^{-s/2}\) we have

Thus, we obtain \(II\le C\).

Now, we show (A.6). First, we see that

where

Like (A.4), we have

Hence, we only need to control IV. Observe that by Remarks 3.1 (i), 2.3 and (2.21)

Thus, we obtain (A.6). The lemma is proved. \(\square \)

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