Bessel potential

In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity.

If s is a complex number with positive real part then the Bessel potential of order s is the operator

$(I-\Delta)^{-s/2}$

where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.

Yukawa potentials are particular cases of Bessel potentials for $s=2$ in the 3-dimensional space.

Representation in Fourier space

The Bessel potential acts by multiplication on the Fourier transforms: for each $\xi \in \mathbb{R}^d$

$\mathcal{F}((I-\Delta)^{-s/2} u) (\xi)= \frac{\mathcal{F}u (\xi)}{(1 + 4 \pi^2 \vert \xi \vert^2)^{s/2}}.$

Integral representations

When $s > 0$, the Bessel potential on $\mathbb{R}^d$ can be represented by

$(I - \Delta)^{-s/2} = G_s \ast u,$

where the Bessel kernel $G_s$ is defined for $x \in \mathbb{R}^d \setminus \{0\}$ by the integral formula ^[1]

$G_s (x) u = \frac{1}{(4 \pi)^{s/2}\Gamma (s/2)} \int_0^\infty \frac{e^{-\frac{\pi \vert x \vert^2}{\delta}-\frac{\delta}{4 \pi}}}{\delta^{1 + \frac{d - s}{2}}}\,\mathrm{d}\delta.$

Here $\Gamma$ denotes the Gamma function. The Bessel kernel can also be represented for $x \in \mathbb{R}^d \setminus \{0\}$ by^[2]

$G_s (x) = \frac{e^{-\vert x \vert}}{(2\pi)^\frac{d-1}{2} 2^\frac{s}{2} \Gamma (\frac{s}{2}) \Gamma (\frac{d - s + 1}{2})} \int_0^\infty e^{-\vert x \vert t} \Big(t + \frac{t^2}{2}\Big)^\frac{d - s - 1}{2} \,\mathrm{d}t.$

Asymptotics

At the origin, one has as $\vert x\vert \to 0$,^[3]

$G_s (x) = \frac{\Gamma (\frac{d - s}{2})}{2^s \pi^{s/2} \vert x\vert^{n - s}}(1 + o (1)) \quad \text{ if } 0 < s < d,$

$G_s (x) = \frac{1}{2^{d - 1} \pi^{d/2} }\ln \frac{1}{\vert x \vert}(1 + o (1)) ,$

$G_s (x) = \frac{\Gamma (\frac{s - d}{2})}{2^s \pi^{s/2} }(1 + o (1)) \quad \text{ if }s > d.$

In particular, when $0 < s < d$ the Bessel potential behaves asymptotically as the Riesz potential.

At infinity, one has, as $\vert x\vert \to \infty$, ^[4]

$G_s (x) = \frac{e^{-\vert x \vert}}{2^\frac{d + s - 1}{2} \pi^\frac{d - 1}{2} \Gamma (\frac{s}{2}) \vert x \vert^\frac{n + 1 - s}{2}}(1 + o (1)).$

See also

• Riesz potential
• Fractional integration
• Sobolev space
• Fractional Schrödinger equation

References

1. Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University Press. Chapter V eq. (26). ISBN 0-691-08079-8.
2. N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier 11. 385–475, (4,2).
3. N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier 11. 385–475, (4,3).
4. N. Aronszajn; K. T. Smith (1961). "Theory of Bessel potentials I". Ann. Inst. Fourier 11: 385–475.

• Duduchava, R. (2001), "Bessel potential operator", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics 250 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316
• Hedberg, L.I. (2001), "Bessel potential space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Solomentsev, E.D. (2001), "B/b015870", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8