Regularity for the Supercritical Fractional Laplacian with Drift

Abstract

We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, wherein the dominant term is given by the drift instead of the diffusion component, we prove local regularity of solutions in Sobolev spaces employing tools from the theory of pseudo-differential operators. The regularity of solutions in the supercritical case is as expected from the subcritical case. In the subcritical case the diffusion is at least as strong as the drift, and the operator is an elliptic pseudo-differential operator, which is not the case in the supercritical regime. We also compute the leading part of the singularity of the Green’s kernel for the supercritical case, which displays some rather unusual behavior.

The Analysis of \(p^0_s(x';x_n)\)

The fact that the integrand in (4.15) has only a single pole in the upper half plane shows that the behavior of the $\tau$-integral is quite different for $x_n>0$ and $x_n<0$. As noted this echoes the behavior of the kernel, $p^0_s(x';x_n)$, for the heat operator $e^{-x_n(-\Delta _{\mathbb {R}^{n-1}})^s}$, which vanishes identically in the set $\{(x',x_n):\, x_n<0\}$. In this appendix we give a precise description of this heat kernel. Our analysis is suggested by a similar treatment given by Michael Taylor in a unpublished note Remarks on Fractional Diffusion Equations (available at http://www.unc.edu/math/Faculty/met/fdif).

An elementary calculation shows that where $x_n>0$ we have:

$$\begin{aligned} \begin{aligned} p^0_s(x';x_n)&=\frac{2\pi c_n\omega _{n-2}}{|x'|^{\frac{n-3}{2}}}\int \nolimits _{0}^{\infty } J_{\frac{n-3}{2}}(r|x'|) r^{\frac{n-1}{2}}e^{-x_nr^{2s}}dr\\&= \frac{2\pi c_n\omega _{n-2}}{|x'|^{n-1}}\int \nolimits _{0}^{\infty } J_{\frac{n-3}{2}}(r) r^{\frac{n-1}{2}}e^{-\frac{x_n}{|x'|^{2s}}r^{2s}}dr. \end{aligned} \end{aligned}$$

(5.1)

This kernel is an-isotropically homogeneous:

$$\begin{aligned} p^0_s(\mu x';\mu ^{2s}x_n)=\mu ^{-(n-1)}p^0_s(x';x_n). \end{aligned}$$

(5.2)

When $x_n/|x'|^{2s}$ is bounded away from zero, we can simply expand the Bessel function in a power series to obtain that:

$$\begin{aligned} p^0_s(x';x_n)=\frac{1}{|x_n|^{\frac{n-1}{2s}}}\sum _{j=0}^{\infty }b_j\left( \frac{|x'|^2}{x_n^{\frac{1}{s}}}\right) ^{j}, \end{aligned}$$

(5.3)

which is a smooth function of $|x'|^2/x_n^{\frac{1}{s}}$, where $x_n>0$.

A priori, it appears much subtler to deduce the behavior of $p^0_s(x';x_n)$ as $x_n/|x'|^{2s}$ tends to zero. We could use an argument similar to that used in the proof of Proposition 4.9, but there is an alternate approach, using the notion of subordination, which we have decided to employ. Using it we derive a second formula for $p^0_s$ from which these asymptotics are more transparent. Indeed, this second formula gives a method to compute the asymptotics of the integrals

$$\begin{aligned} \int \limits _{0}^{\infty } J_{\frac{n-3}{2}}(r) r^{\frac{n-1}{2}}e^{-\lambda r^{2s}}dr \end{aligned}$$

(5.4)

as $\lambda \rightarrow 0^+$, for $0<s<1$. These asymptotics are, by no means obvious, as they result from global cancellations taking place within the integral. As this analysis does not appear to be in the literature we pause to consider this question. Our result is:

Proposition 5.1

For $\nu \ge 0$ and $0<s<1$, there exists a sequence $\{\mu _{sk}\}$ so that

$$\begin{aligned} \int \limits _{0}^{\infty } J_{\nu }(r) r^{\nu +1}e^{-\lambda r^{2s}}dr\sim \sum _{k=1}^{\infty }\mu _{sk}\lambda ^{k}, \end{aligned}$$

(5.5)

as $\lambda \rightarrow 0^+$.

Proof

Let $\Phi _s(x)$ be the function defined by the Laplace transform formula:

$$\begin{aligned} e^{-\lambda ^s}=\int \limits _{0}^{\infty }\Phi _s(x)e^{-\lambda x}dx. \end{aligned}$$

(5.6)

The Laplace inversion formula shows that, for any $\sigma \in [0,\infty )$, we have

$$\begin{aligned} \Phi _s(x)=\frac{1}{2\pi i}\int \limits _{\sigma -i\infty }^{\sigma +i\infty }e^{xz-z^s}dz, \end{aligned}$$

(5.7)

where $z^s$, defined on $\mathbb {C}\setminus (-\infty ,0]$, is normalized to be a positive real number for $z\in (0,\infty )$.

If $L$ is a positive operator generating a positivity preserving semigroup and $0<s<1$, then

$$\begin{aligned} e^{-tL^s}=\int \limits _{0}^{\infty }\frac{e^{-x L}}{t^{\frac{1}{s}}}\Phi _s\left( \frac{x}{t^{\frac{1}{s}}}\right) dx, \end{aligned}$$

(5.8)

see [22, § IX.11]. Using (5.6) and [9], Formula6.631.4],

$$\begin{aligned} \int \limits _{0}^{\infty }x^{\nu +1}J_{\nu }(x)e^{-\alpha x^2}dx=\frac{1}{(2\alpha )^{\nu +1}} \exp \left( \frac{-1}{4\alpha }\right) , \end{aligned}$$

(5.9)

we easily prove:$\square$

Lemma 5.2

For $0<s<1$ and $0\le \nu$, there is a constant $C_{\nu }$ so that the identity

$$\begin{aligned} \int \limits _{0}^{\infty }\frac{\Phi _s(\sigma )e^{-\frac{1}{4\sigma \lambda ^{\frac{1}{s}}}}d\sigma }{(\lambda ^{\frac{1}{s}}\sigma )^{\nu +1}}= C_{\nu }\int \limits _{0}^{\infty }J_{\nu }(\tau )e^{-\lambda \tau ^{2s}}\tau ^{\nu +1}d\tau \end{aligned}$$

(5.10)

holds for $\lambda$ with $\mathfrak {R}\lambda >0$.

From formula (5.7) it is clear that $\Phi _s(x)$ is in ${\mathcal {C}}^{\infty }(\mathbb {R})$ and vanishes where $x<0$. For real, positive $x$, we can replace this formula with

$$\begin{aligned} \Phi _s(x)=\mathfrak {R}\left[ \frac{1}{\pi i}\int \nolimits _{0}^{\infty }e^{xe^{i\theta }r-(e^{i\theta }r)^s}e^{i\theta } dr\right] \end{aligned}$$

(5.11)

for $\theta \in [\frac{\pi }{2},\pi ]$. Taking $\theta =\pi$, and using the Taylor series for $e^{-e^{is\pi }r^s}$ gives, for $x>0$, that

$$\begin{aligned} \Phi _s(x)=\mathfrak {R}\left[ \frac{i}{\pi } \int \nolimits _{0}^{\infty }e^{-xr-e^{is\pi }r^s}dr\right] =\frac{1}{\pi }\sum _{j=1}^{\infty } \frac{(-1)^{j-1}\Gamma (sj+1)\sin \pi s j}{j!}\frac{1}{x^{sj+1}}.\nonumber \\ \end{aligned}$$

(5.12)

The asymptotic expansion in (5.5) follows from (5.10) and this expansion.

Remark 5.3

The identity in (5.10) trades the $\lambda$ in the exponent on the right-hand side for $1/\lambda ^{\frac{1}{s}}$ on the left-hand side. This in turn allows us to trade the very subtle, globally determined asymptotic behavior of the integral on the right-hand side as $\lambda \rightarrow 0^+$, for the much simpler asymptotic analysis, as $1/\lambda ^{\frac{1}{s}}\rightarrow \infty$ on the left. Paying somewhat closer attention to the coefficients in the expansion shows that if $2s<1$, then the infinite sum on the right-hand side of (5.5), given by,

$$\begin{aligned} \frac{1}{\pi }\sum _{j=1}^{\infty }\frac{\Gamma (sj+1)\Gamma (sj+\nu +1)\sin (s\pi j)}{j!}(-1)^{j-1}(4^s\lambda )^{j}, \end{aligned}$$

(5.13)

is an entire function. If $s=1/2$, then this series has a finite radius of convergence, and if $1/2<s<1$, then it is an asymptotic series.

Using this result we see from (5.1) and (5.5) that, as $x_n/|x'|^{2s}\rightarrow 0$, we have:

$$\begin{aligned} p^0_s(x';x_n)\sim \frac{\chi _{[0,\infty )}(x_n)}{|x'|^{n-1}}\left( \frac{c_n x_n}{|x'|^{2s}}+O\left[ \frac{x_n}{|x'|^{2s}}\right] ^2\right) . \end{aligned}$$

(5.14)

The asymptotic result can also be obtained using standard microlocal techniques. Note, however, that our formula is valid whether or not both $|x'|$ and $x_n$ tend to zero, and shows that away from $x'=0$ this kernel vanishes to order 1 as $x_n\rightarrow 0^+$. It also shows that for fixed $x_n>0$, this kernel behaves like $1/|x'|^{n-1+2s}$ as $|x'|$ tends to infinity. Combining this formula with (5.3) establishes the following result.

Proposition 5.4

The kernel $p^0_s(x';x_n)$ for the operator $e^{-x_n(-\Delta _{\mathbb {R}^{n-1}})^s}$, has the following representation

$$\begin{aligned} p^0_s(x';x_n)= \chi _{[0,\infty )}(x_n)\frac{H^+\left( \frac{|x'|^2}{x_n^{\frac{1}{s}}}\right) }{(x_n^{\frac{1}{s}}+|x'|^2)^{\frac{n-1}{2}}}, \end{aligned}$$

(5.15)

where $H^+(\lambda )$ is a smooth positive function in $[0,\infty )$ with an expansion of the form

$$\begin{aligned} H^+(\lambda )=\frac{c_n}{\lambda ^s}\left( 1+O(1/\lambda ^s)\right) , \end{aligned}$$

(5.16)

as $\lambda \rightarrow \infty$.