# Riesz potential

-potential

A potential of the form

where is a positive Borel measure of compact support on the Euclidean space , , and is the distance between the points . When and , the Riesz potential coincides with the classical Newton potential; when and , the limit case of the Riesz potential is in some sense the logarithmic potential. When and , the Riesz potential is a superharmonic function on the entire space ; moreover, in the classical case , outside the support of , the potential is a harmonic function. When , the Riesz potential is a subharmonic function outside . For all the Riesz potential is a lower semi-continuous function on , continuous outside .

Among the general properties of Riesz potentials the following are the most important. The continuity principle: If and if the restriction is continuous at the point , then is continuous at as a function on . The restricted maximum principle: If , then everywhere on . When , a more precise maximum principle is valid: If , then everywhere on (this statement remains valid also when and , 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 -energy of a measures :

One may assume that for a compact set ,

where the infimum is taken over all measures concentrated on and such that ; then the -capacity is equal to

If , then the infimum is attained on the capacitary measure (also called equilibrium measure), which is concentrated on , , generating the corresponding capacitary -potential (cf. also Capacity potential). The further construction of -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)