Abstract
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.
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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.
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Appendix
Appendix
Lemma A.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 \(\ell _{1}\ell _{2}\in \mathcal {G}\).
-
(ii)
Suppose \(\ell \in \mathcal {G}\). Then for any \(b>0\), \(r\mapsto \ell (r^b)\) belongs to \(\mathcal {G}\).
-
(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, \(\ell _1/\ell _2\in \mathcal {G}\).
-
(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
$$\begin{aligned} \Lambda (r) = \prod _{i=1}^{n} (\ell _{k_{i}}(r))^{b_{i}} \in \mathcal {G}. \end{aligned}$$ -
(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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Kang, J., Park, D. An \(L_q(L_p)\)-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps. Stoch PDE: Anal Comp 12, 1439–1491 (2024). https://doi.org/10.1007/s40072-023-00309-6
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DOI: https://doi.org/10.1007/s40072-023-00309-6
Keywords
- Space-time nonlocal equations
- Maximal \(L_q(L_p)\)-regularity theory
- Caputo fractional derivative
- Slowly varying symbols
- Heat kernel estimation