In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines a fractional inverse power ( − Δ ) − α / 2 {\displaystyle (-\Delta )^{-\alpha /2}} of the Laplace operator on Euclidean space. It generalizes to several variables the Riemann–Liouville integrals of one variable.
Definition If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by
where the constant is given by
c α = π n / 2 2 α Γ ( α / 2 ) Γ ( ( n − α ) / 2 ) . {\displaystyle c_{\alpha }=\pi ^{n/2}2^{\alpha }{\frac {\Gamma (\alpha /2)}{\Gamma ((n-\alpha )/2)}}.}
This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α. The classical result due to Sobolev states that the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)
‖ I α f ‖ p ∗ ≤ C p ‖ f ‖ p , p ∗ = n p n − α p , ∀ 1 < p < n α {\displaystyle \|I_{\alpha }f\|_{p^{*}}\leq C_{p}\|f\|_{p},\quad p^{*}={\frac {np}{n-\alpha p}},\quad \forall 1<p<{\frac {n}{\alpha }}}
For p=1 the result was extended by (Schikorra, Spector & Van Schaftingen 2014),
‖ I α f ‖ 1 ∗ ≤ C p ‖ R f ‖ 1 . {\displaystyle \|I_{\alpha }f\|_{1^{*}}\leq C_{p}\|Rf\|_{1}.}
where R f = D I 1 f {\displaystyle Rf=DI_{1}f} is the vector-valued Riesz transform. More generally, the operators Iα are well-defined for complex α such that 0 < Re α < n. The Riesz potential can be defined more generally in a weak sense as the convolution
I α f = f ∗ K α {\displaystyle I_{\alpha }f=f*K_{\alpha }}
where Kα is the locally integrable function:
K α ( x ) = 1 c α 1 | x | n − α . {\displaystyle K_{\alpha }(x)={\frac {1}{c_{\alpha }}}{\frac {1}{|x|^{n-\alpha }}}.}
The Riesz potential can therefore be defined whenever f is a compactly supported distribution. In this connection, the Riesz potential of a positive Borel measure μ with compact support is chiefly of interest in potential theory because Iαμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rn. Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier. In fact, one has
K α ^ ( ξ ) = ∫ R n K α ( x ) e − 2 π i x ξ d x = | 2 π ξ | − α {\displaystyle {\widehat {K_{\alpha }}}(\xi )=\int _{\mathbb {R} ^{n}}K_{\alpha }(x)e^{-2\pi ix\xi }\,\mathrm {d} x=|2\pi \xi |^{-\alpha }}
and so, by the convolution theorem,
… excerpt ends here. Continue reading the full article.
