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Riesz Potential

The Riesz potential of order 0<alpha<n is the integral operator

I_alphaf(x)=c_(n,alpha)int_(R^n)(f(y))/(||x-y||^(n-alpha))dy,

defined whenever the integral converges, where ||·|| is the Euclidean norm and c_(n,alpha)>0 is the normalization constant. It is a convolution with a power-law kernel. With the Fourier transform convention using e^(-ix·xi), its Fourier multiplier is ||xi||^(-alpha).

For alpha=2s with 0<s<1 and 2s<n, this is the inverse potential associated with the fractional Laplacian (Mei and Wei 2026).

See also

Convolution, Fourier Multiplier, Fourier Transform, Fractional Laplacian, Riesz Transform

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References

Mei, L. and Wei, J. "A Proof of the Fractional Lane-Emden Conjecture." 7 Oct 2026. https://arxiv.org/abs/2610.10188.Stein, E. M. and Weiss, G. "Fractional Integrals on n-Dimensional Euclidean Space." J. Math. Mech. 7, 503-514, 1958.

Cite this as:

Weisstein, Eric W. "Riesz Potential." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RieszPotential.html

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