Circle Means of Green’s Functions

Nakai, Mitsuru

doi:10.1017/s0027763000015245

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Consider a Function W P>n = W N E H(ω P π R N -R Q ) π C(ώ P1

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1973

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“…3. As direct consequences of (11) we first obtain the circle mean formula of Green's function ( [18]) which will be convenient for our later calculations: (12) f G(z, ζ)dθ(ζ) = -2π max (log p, log r(ζ)) .…”

Section: Consider a Function W P>n = W N E H(ω P π R N -R Q ) π C(ώ Pmentioning

confidence: 99%

Uniform densities on hyperbolic Riemannl surfaces

Nakai

1973

Nagoya Mathematical Journal

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We are interested in the question how the spaces of solutions of elliptic equations vary according to the variations of underlying regions and coefficients of the equation. We will discuss this question for the case of equations Δu = Pu considered on noncompact Riemann surfaces R. Typically we ask the properties of mappings τx: (R,P) → dim PX(R) from the space Φ of pairs (R, P) of noncompact Riemann surfaces R and densities P on R, i.e. P(z)dxdy are 2-forms on R such that P(z)dxdy ≢ 0 and P(z) ≥ 0 are Hölder continuous with respect to local parameters z = x + iy, into cardinals, where PX(R) are the linear spaces of solutions of Δu = Pu on R with certain boundedness properties X.

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Section: Consider a Function W P>n = W N E H(ω P π R N -R Q ) π C(ώ Pmentioning

confidence: 99%