Riesz potential

$ \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'équilibre 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)

Comments

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].

References

[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)