# Riesz potential

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$\alpha$-potential

A potential of the form

$$V _ \alpha ( x) = V( x; \alpha , \mu ) = \ \int\limits \frac{d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0,$$

where $\mu$ is a positive Borel measure of compact support on the Euclidean space $\mathbf R ^ {n}$, $n \geq 2$, and $| x- y |$ is the distance between the points $x, y \in \mathbf R ^ {n}$. When $n \geq 3$ and $\alpha = n- 2$, the Riesz potential coincides with the classical Newton potential; when $n= 2$ and $\alpha \rightarrow 0$, the limit case of the Riesz potential is in some sense the logarithmic potential. When $n \geq 3$ and $0 < \alpha \leq n- 2$, the Riesz potential is a superharmonic function on the entire space $\mathbf R ^ {n}$; moreover, in the classical case $\alpha = n- 2$, outside the support $S( \mu )$ of $\mu$, the potential $V( x) = V _ {n- 2} ( x)$ is a harmonic function. When $\alpha > n- 2$, the Riesz potential $V _ \alpha ( x)$ is a subharmonic function outside $S( \mu )$. For all $\alpha > 0$ the Riesz potential $V _ \alpha ( x)$ is a lower semi-continuous function on $\mathbf R ^ {n}$, continuous outside $S( \mu )$.

Among the general properties of Riesz potentials the following are the most important. The continuity principle: If $x _ {0} \in S( \mu )$ and if the restriction $V _ \alpha ( x) \mid _ {S( \mu ) }$ is continuous at the point $x _ {0}$, then $V _ \alpha ( x)$ is continuous at $x _ {0}$ as a function on $\mathbf R ^ {n}$. The restricted maximum principle: If $V _ \alpha ( x) \mid _ {S( \mu ) } \leq M$, then $V _ \alpha ( x) \leq 2 ^ \alpha M$ everywhere on $\mathbf R ^ {n}$. When $n- 2 \leq \alpha < n$, a more precise maximum principle is valid: If $V _ \alpha ( x) \mid _ {S( \mu ) } \leq M$, then $V _ \alpha ( x) \leq M$ everywhere on $\mathbf R ^ {n}$ (this statement remains valid also when $n= 2$ and $\alpha \rightarrow 0$, that is, for the logarithmic potential).

The capacity theory for Riesz potentials can be constructed, for example, on the basis of the concept of the $\alpha$-energy of a measures $\mu$:

$$E _ \alpha ( \mu ) = \int\limits \int\limits \frac{d \mu ( x) d \mu ( y) }{| x- y | ^ \alpha } ,\ \alpha > 0.$$

One may assume that for a compact set $K$,

$$V _ \alpha ( K) = \inf \{ E _ \alpha ( \mu ) \} ,$$

where the infimum is taken over all measures $\mu$ concentrated on $K$ and such that $\mu ( K) = 1$; then the $\alpha$-capacity is equal to

$$C _ \alpha ( K) = [ V _ \alpha ( K)] ^ {- 1/ \alpha } .$$

If $V _ \alpha ( K) < + \infty$, then the infimum is attained on the capacitary measure $\lambda$ (also called equilibrium measure), which is concentrated on $K$, $\lambda ( K) = 1$, generating the corresponding capacitary $\alpha$-potential $V( x; \alpha , \lambda )$ (cf. also Capacity potential). The further construction of $\alpha$-capacities of arbitrary sets is carried out in the same way as for the classical capacities.

The Riesz potential is called after M. Riesz (see [2]), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see [1]).

#### References

 [1] O. Frostman, "Potentiel d'equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions" Medd. Lunds Univ. Mat. Sem. , 3 (1935) pp. 1–118 [2] M. Riesz, "Intégrales de Riemann–Liouville et potentiels" Acata Sci. Math. Szeged , 9 (1938) pp. 1–42 [3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) [4] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)

For $n$ even and $\alpha = n- 2 m \leq 0$, $| x - y | ^ {2m- n} \mathop{\rm log} | x- y |$ is a fundamental solution of the polyharmonic equation $\Delta ^ {m} u = 0$, otherwise $| x- y | ^ {2m- n}$ is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order $> 2$, see [a2]. A treatment of Riesz potentials in the framework of balayage spaces is given in [a1].
The Riesz kernels $| x- y | ^ {- \alpha }$ are the standard examples of convolution kernels. Thus, Riesz potentials may be regarded as special singular integrals. For more details on this interesting point of view see [a3].