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}}}.

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.

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)).

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

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