# Fractional Laplacian

### From Mwiki

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All the definitions below are equivalent. | All the definitions below are equivalent. | ||

- | === As a | + | === As a Fourier multiplier === |

- | The fractional Laplacian is the | + | The fractional Laplacian is the operator with symbol $|\xi|^{2s}$. In other words, the following formula holds |

\[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] | \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] | ||

for any function (or tempered distribution) for which the right hand side makes sense. | for any function (or tempered distribution) for which the right hand side makes sense. | ||

This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it. | This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it. | ||

+ | |||

+ | === Via the heat semigroup === | ||

+ | Recall the numerical formula | ||

+ | $$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$ | ||

+ | valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=|\xi|^2$ and using the Fourier transform definition of the fractional Laplacian we get | ||

+ | \begin{equation}\label{semigroup} | ||

+ | (-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}. | ||

+ | \end{equation} | ||

+ | Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely, | ||

+ | $$\begin{cases} | ||

+ | \partial_tv=\Delta v,&\hbox{for}~(x,t)\in\mathbb{R}^n\times(0,\infty)\\ | ||

+ | v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n. | ||

+ | \end{cases}$$ | ||

+ | This semigroup language approach was introduced in <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>. | ||

=== From functional calculus === | === From functional calculus === | ||

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Where $c_{n,s}$ is a constant depending on dimension and $s$. | Where $c_{n,s}$ is a constant depending on dimension and $s$. | ||

+ | |||

+ | One way of deducing this formula is the following: write down the heat kernel in the semigroup formula \eqref{semigroup}, use that $e^{t\Delta}1\equiv1$ and apply Fubini's Theorem. See <ref name="Stinga"/> and <ref name="Stinga-Torrea"/> for the details of this computation. Notice that this derivation gives the explicit value of the constant: | ||

+ | $$c_{n,s}=\frac{4^s\Gamma(n/2+s)}{\pi^{n/2}|\Gamma(-s)|}.$$ | ||

This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems. | This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems. | ||

Line 38: | Line 55: | ||

The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'': | The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'': | ||

\[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\] | \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\] | ||

- | which holds as long as $u$ is | + | which holds as long as $u$ is integrable enough for the right hand side to make sense. |

== Heat kernel == | == Heat kernel == | ||

Line 84: | Line 101: | ||

== Regularity issues == | == Regularity issues == | ||

Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls. | Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls. | ||

+ | |||

+ | More generally, one has the estimate | ||

+ | \[\|u\|_{C^{\alpha+2s}(B_{1/2})}\leq C\left( | ||

+ | \|(-\Delta)^s u\|_{C^{\alpha}(B_1)}+\|u\|_{L^{\infty}(B_1)}+\int_{\R^n\setminus B_1}|u(y)|\frac{dy}{|y|^{n+2s}}\right)\] | ||

+ | for any $\alpha\geq0$ such that $\alpha+2s$ is not an integer. | ||

=== Full space regularization of the Riesz potential === | === Full space regularization of the Riesz potential === | ||

- | If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$. | + | If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein<ref name="Stein"/>. |

=== Boundary regularity === | === Boundary regularity === | ||

From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel). | From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel). | ||

+ | |||

+ | Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem | ||

+ | \[\left\{ \begin{array}{rcll} | ||

+ | (-\Delta)^s u &=&g&\textrm{in }\Omega \\ | ||

+ | u&=&0&\textrm{in }\R^n\backslash \Omega. | ||

+ | \end{array}\right.\] | ||

+ | If $\Omega$ is $C^\infty$, then | ||

+ | \[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\] | ||

+ | where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see <ref name="Grubb"/> and also <ref name="RS"/>. | ||

+ | |||

+ | If $\Omega$ is $C^{2,\alpha}$ and $g$ is $C^\alpha$, then $u/d^s$ is $C^{\alpha+s}$ up to the boundary <ref name="RS-K"/>. | ||

+ | |||

+ | Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in <ref name="RS2"/>. | ||

== References == | == References == | ||

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<ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref> | <ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref> | ||

<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref> | <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref> | ||

- | <ref name="RS">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[preprint]] }}</ref> | + | <ref name="RS">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[J. Math. Pures Appl.]] | volume=101 | pages=275-302 }}</ref> |

+ | <ref name="Stein">{{Citation | last1=Stein | first1=E. | title=Singular Integrals And Differentiability Properties Of Functions | publisher=[[Princeton Mathematical Series]] | year=1970}}</ref> | ||

+ | <ref name="RS2">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The extremal solution for the fractional Laplacian | url=http://arxiv.org/abs/1305.2489 | year=2013 | journal=[[Calc. Var. Partial Differential Equations]] | pages=to appear }}</ref> | ||

+ | <ref name="Grubb">{{Citation | last1=Grubb | first1=G. | title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators | url=http://arxiv.org/abs/1310.0951 | year=2014 | journal=[[arXiv]] | pages=1-43 }}</ref> | ||

+ | <ref name="RS-K">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=Boundary regularity for fully nonlinear integro-differential equations | year=2014 | journal=[[preprint arXiv]] }}</ref> | ||

+ | <ref name="Stinga">{{Citation | last1=Stinga | first1=P. R.| title=Fractional powers of second order partial differential operators: extension problem and regularity theory | url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf | year=2010 | journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}} | ||

+ | </ref><ref name="Stinga-Torrea">{{Citation | last1=Stinga | first1=P. R.| last2=Torrea | first2=J. L. | title=Extension problem and Harnack's inequality for some fractional operators | url=http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q | year=2010 | journal=[[Comm. Partial Differential Equations]] | volume=35 | issue=11 | pages=2092-2122}}</ref> | ||

}} | }} |

## Latest revision as of 19:04, 5 August 2015

The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic elliptic linear integro-differential operator of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^n)$ we can write the operator as

\[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]

where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a comparison principle can be derived for equations involving the fractional Laplacian.

## Contents |

## Definitions

All the definitions below are equivalent.

### As a Fourier multiplier

The fractional Laplacian is the operator with symbol $|\xi|^{2s}$. In other words, the following formula holds \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] for any function (or tempered distribution) for which the right hand side makes sense.

This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.

### Via the heat semigroup

Recall the numerical formula
$$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$
valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=|\xi|^2$ and using the Fourier transform definition of the fractional Laplacian we get
\begin{equation}\tag{1}
(-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.
\end{equation}
Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely,
$$\begin{cases}
\partial_tv=\Delta v,&\hbox{for}~(x,t)\in\mathbb{R}^n\times(0,\infty)\\
v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n.
\end{cases}$$
This semigroup language approach was introduced in ^{[1]} and ^{[2]}.

### From functional calculus

Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.

This definition is not as useful for practical applications, since it does not provide any explicit formula.

### As a singular integral

If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]

Where $c_{n,s}$ is a constant depending on dimension and $s$.

One way of deducing this formula is the following: write down the heat kernel in the semigroup formula (1), use that $e^{t\Delta}1\equiv1$ and apply Fubini's Theorem. See ^{[1]} and ^{[2]} for the details of this computation. Notice that this derivation gives the explicit value of the constant:
$$c_{n,s}=\frac{4^s\Gamma(n/2+s)}{\pi^{n/2}|\Gamma(-s)|}.$$

This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.

### As a generator of a Levy process

The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]

This definition is important for applications to probability.

## Inverse operator

The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the *Riesz potential*:
\[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
which holds as long as $u$ is integrable enough for the right hand side to make sense.

## Heat kernel

The fractional heat kernel $p(t,x)$ is the fundamental solution to the fractional heat equation. It is the function which solves the equation \begin{align*} p(0,x) &= \delta_0 \\ p_t(t,x) + (-\Delta)^s p &= 0 \end{align*}

The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$ \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]

Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \]

## Poisson kernel

Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem \begin{align*} u(x) &= g(x) \qquad \text{if } x \notin B_1 \\ (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1. \end{align*}

The solution can be computed explicitly using the Poisson kernel
\[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
where^{[3]}
\[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]

The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')^{[4]}.

## Green's function for the ball

For a function $g \in L^2(B_1)$, there exists a unique function $u \in H^s(\R^n)$ such that \begin{align*} u(x) &= 0 && \text{if } x \notin B_1 \\ (-\Delta)^s u &= g(x) && \text{if } x \in B_1. \end{align*}

The solution is given explicitly using the Green's function,
\[ u(x) = \int_{B_1} G_{B_1}(x, y) g(y) \mathrm d y, \]
where^{[3]}
\[ G_{B_1}(x, y) = C_{n,s} |x - y|^{2 s - n} \int_0^{r_0(x, y)} \frac{r^{s-1}}{(r+1)^{n/2}} \, \mathrm d r \]
with
\[ r_0(x, y) = \frac{(1 - |x|^2) (1 - |y|^2)}{|x - y|^2} . \]
The above formula holds for all $s \in (0, 1)$ also for $n = 1$.^{[5]}

## Regularity issues

Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls.

More generally, one has the estimate \[\|u\|_{C^{\alpha+2s}(B_{1/2})}\leq C\left( \|(-\Delta)^s u\|_{C^{\alpha}(B_1)}+\|u\|_{L^{\infty}(B_1)}+\int_{\R^n\setminus B_1}|u(y)|\frac{dy}{|y|^{n+2s}}\right)\] for any $\alpha\geq0$ such that $\alpha+2s$ is not an integer.

### Full space regularization of the Riesz potential

If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions *up to $2s$ derivatives*. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of ^{[6]}, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein^{[7]}.

### Boundary regularity

From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).

Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem
\[\left\{ \begin{array}{rcll}
(-\Delta)^s u &=&g&\textrm{in }\Omega \\
u&=&0&\textrm{in }\R^n\backslash \Omega.
\end{array}\right.\]
If $\Omega$ is $C^\infty$, then
\[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\]
where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see ^{[8]} and also ^{[9]}.

If $\Omega$ is $C^{2,\alpha}$ and $g$ is $C^\alpha$, then $u/d^s$ is $C^{\alpha+s}$ up to the boundary ^{[10]}.

Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in ^{[11]}.

## References

- ↑
^{1.0}^{1.1}Stinga, P. R. (2010), "Fractional powers of second order partial differential operators: extension problem and regularity theory",*PhD. Thesis - Universidad Aut\'onoma de Madrid*, https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf - ↑
^{2.0}^{2.1}Stinga, P. R.; Torrea, J. L. (2010), "Extension problem and Harnack's inequality for some fractional operators",*Comm. Partial Differential Equations***35**(11): 2092-2122, http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q - ↑
^{3.0}^{3.1}Riesz, M. (1938), "Intégrales de Riemann-Liouville et potentiels",*Acta Sci. Math. Szeged***9**(1): 1–42, ISSN 0001-6969, http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article - ↑ Landkof, N. S. (1972),
*Foundations of modern potential theory*, Berlin, New York: Springer-Verlag - ↑ Blumenthal, R. M.; Getoor, R. K.; Ray, D. B. (1961), "On the distribution of first hits for the symmetric stable processes",
*Trans. Amer. Math. Soc.***99**: 540–554, ISSN 0002-9947, http://www.jstor.org/stable/1993561 - ↑ Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator",
*Communications on Pure and Applied Mathematics***60**(1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153 - ↑ Stein, E. (1970),
*Singular Integrals And Differentiability Properties Of Functions*, Princeton Mathematical Series - ↑ Grubb, G. (2014), "Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators",
*arXiv*: 1-43, http://arxiv.org/abs/1310.0951 - ↑ Ros-Oton, X.; Serra, J. (2012), "The Dirichlet problem for the fractional Laplacian: regularity up to the boundary",
*J. Math. Pures Appl.***101**: 275-302, http://arxiv.org/abs/1207.5985 - ↑ Ros-Oton, X.; Serra, J. (2014), "Boundary regularity for fully nonlinear integro-differential equations",
*preprint arXiv* - ↑ Ros-Oton, X.; Serra, J. (2013), "The extremal solution for the fractional Laplacian",
*Calc. Var. Partial Differential Equations*: to appear, http://arxiv.org/abs/1305.2489