Matematikada **Bessel potentsiali** **Riesz potentsialiga** o'xshash potentsialdir (Fridrix Vilgelm Bessel nomi bilan atalgan), lekin cheksizlikda yaxshiroq parchalanish xususiyatlariga ega.

Agar *s* musbat haqiqiy qismga ega kompleks son bo'lsa, *s* tartibli Bessel potensiali operatori:

{\displaystyle (I-\Delta )^{-s/2}}
bu yerda Δ - Laplas operatori va kasr quvvati Furye transformlari yordamida aniqlanadi.

Yukava potensiali Bessel potentsiallarining {\displaystyle s=2} 3 o'lchovli fazoda xususiy holatlaridir

Furye fazosida ko'rinishi
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Bessel potentsiali Furye o'zgarishlariga ko'paytirish orqali ta'sir qiladi: har bir {\displaystyle \xi \in \mathbb {R} ^{d}}uchun:

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

Integral ko'rinishlari
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{\displaystyle s>0} bo'lganda, Bessel potensiali {\displaystyle \mathbb {R} ^{d}}da quyidagicha ifodalanishi mumkin:

{\displaystyle (I-\Delta )^{-s/2}u=G\_{s}\ast u,}
bu yerda Bessel yadrosi {\displaystyle G\_{s}} {\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}} uchun integral formula bo'yicha ifodalanadi

{\displaystyle G\_{s}(x)={\frac {1}{(4\pi )^{s/2}\Gamma (s/2)}}\int \_{0}^{\infty }{\frac {e^{-{\frac {\pi \vert x\vert ^{2}}{y}}-{\frac {y}{4\pi }}}}{y^{1+{\frac {d-s}{2}}}}}\,\mathrm {d} y.}
Bu yerda {\displaystyle \Gamma } Gamma funksiyasini bildiradi. Bessel yadrosi orqali {\displaystyle x\in \mathbb {R} ^{d}\setminus \{0\}} ham ifodalanishi mumkin:

{\displaystyle 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.}
Ushbu oxirgi ifodani o'zgartirilgan Bessel funksiyasi nuqtai nazaridan qisqaroq yozish mumkin, shu sababli potensial o'z nomini oladi:

{\displaystyle G\_{s}(x)={\frac {1}{2^{(s-2)/2}(2\pi )^{d/2}\Gamma ({\frac {s}{2}})}}K\_{(d-s)/2}(\vert x\vert )\vert x\vert ^{(s-d)/2}.}

Asimptotiklar
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Kelib chiqishida birida {\displaystyle \vert x\vert \to 0},

{\displaystyle G\_{s}(x)={\frac {\Gamma ({\frac {d-s}{2}})}{2^{s}\pi ^{s/2}\vert x\vert ^{d-s}}}(1+o(1))\quad {\text{ if }}0<s<d,}
{\displaystyle G\_{d}(x)={\frac {1}{2^{d-1}\pi ^{d/2}}}\ln {\frac {1}{\vert x\vert }}(1+o(1)),}
{\displaystyle G\_{s}(x)={\frac {\Gamma ({\frac {s-d}{2}})}{2^{s}\pi ^{s/2}}}(1+o(1))\quad {\text{ if }}s>d.}
Xususan, {\displaystyle 0<s<d} bo'lganda Bessel potensiali Riesz potensiali kabi asimptotik tarzda o'zini tutadi.

Cheksizlikda, {\displaystyle \vert x\vert \to \infty },

{\displaystyle 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 {d+1-s}{2}}}}(1+o(1)).}

Shuningdek qarang:
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* Riesz potensiali
* Sobolev fazosi
* Shredinger kasr tenglamasi
* Yukava potensiali

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

* 
* Grafakos, Loukas (2009), *Modern Fourier analysis*, Graduate Texts in Mathematics, 250-jild (2nd-nashr), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316
* Stein, Elias (1970), *Singular integrals and differentiability properties of functions*, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8