Harmonic morphisms applied to classical potential theory

TY - JOUR

T1 - Harmonic morphisms applied to classical potential theory

AU - Fuglede, Bent

PY - 2011

Y1 - 2011

N2 - It is shown that if φ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces X and Y, then a function f, defined on an open set V ⊂ Y, is superharmonic if and only if f {ring operator}φ is superharmonic on φ-1(V) ⊂ X. The "only if" part is due to Constantinescu and Cornea, with φ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where φ is the projection from R{double strock}N to R{double strock}n (N >n≥ 1) or where φ is the radial projection from R{double strock}N \ {0} to the unit sphere in R{double strock}N (N ≥ 2).

AB - It is shown that if φ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces X and Y, then a function f, defined on an open set V ⊂ Y, is superharmonic if and only if f {ring operator}φ is superharmonic on φ-1(V) ⊂ X. The "only if" part is due to Constantinescu and Cornea, with φ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where φ is the projection from R{double strock}N to R{double strock}n (N >n≥ 1) or where φ is the radial projection from R{double strock}N \ {0} to the unit sphere in R{double strock}N (N ≥ 2).

M3 - Journal article

SN - 0027-7630

VL - 202

SP - 107

EP - 126

JO - Nagoya Mathematical Journal

JF - Nagoya Mathematical Journal

ER -