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 ), who obtained a number of important properties of Riesz potentials; for the first time such potentials were studied by O. Frostman (see ).
|||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|
|||M. Riesz, "Intégrales de Riemann–Liouville et potentiels" Acata Sci. Math. Szeged , 9 (1938) pp. 1–42|
|||N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)|
|||W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)|
For even and , is a fundamental solution of the polyharmonic equation , otherwise is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order , see [a2]. A treatment of Riesz potentials in the framework of balayage spaces is given in [a1].
The Riesz kernels 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].
|[a1]||J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)|
|[a2]||B.W. Schulze, G. Wildenhain, "Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung" , Birkhäuser (1977)|
|[a3]||E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970)|
|[a4]||L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)|
Riesz potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_potential&oldid=14705