オーリッツ関数のリースポテンシャルの連続性

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In this thesis we study continuity properties or Riesz potentials or order α, 0<α<n, of a nonnegative measurable function f on R^n, which is defined by[numerical formula]. Here it is natural to assume that [numerical formula], which is equiva1ent to (1)[numerical formula] We are concerned with the behavior of Riesz potentials U_αf near a given point, which may be assumed, without loss of generality, to be the origin. To obtain genera1 results, we treat functions f satisfying an Orlicz condition with weight of the form [numerical formula]. Here Φ_p(γ)and ω(γ) are positive monotone functions on the interval (0,∞) with the fo11owing properties : (ψ1)Φ_p(γ)is of the form r^pψ(γ), where 1≤p<∞ and ψ is a positive monotone function on the interva1 (0,∞); set ψ (0)=lim__<γ⇾0>ψ(γ).(ψ2) ψ is of logarithmic type, that is, there exists A_1>0 such that [numerical formula] whenever γ>0. (ω1) ω satisfies the doubling condition; that is, there exists A_2>0 such that[numerical formula] whenever γ >0. Riesz potentials may not, in general, be continuous at any point of R^n. But, it is known that if ρ>1 and (2) [numerical formula], then U_αf is continuous everywhere on R^n; in case αρ>n, (2) holds by condition (ψ2) and the continuity also fo1lows from well-known Sobolev's theorem. For simplicity, let ω(γ)=γ^β, where-n<β≤αρ-n, and ℓ be the nonnegative integer such that ℓ≤α-(n+β)/ρ<ℓ+1. In this case, we treat functions f satisfying (3)[numerical formula]. In Chapter 2,we shall show that if (2) holds, then there exists a polynomial P_ <ℓ>such that (4)[numerical formula] for any function f satisfying (1) and (3), where [numerical formula] Since lim_<γ⇾0>γ^<-ℓ>K(γ)=0,(4) implies that U_αf is ℓ times differentiable at the origin. Let R_α(χ)=|χ|^<α-n> and consider the remainder term of Taylor's expansion : [numerical formula]. Then U_αf(χ)-P_<ℓ>(χ) will be written as [numerical formula], provided [numerical formula], where B(χ, γ) denotes the open ball centered at χ with radius γ<0. In Chapter 3,we study the Holder continuity or Riesz potentials or functions f satisfying (5)[numerical formula], by applying the results in Chapter 2. Recently Edmunds and Krbec studied almost Lipschitz continuity for Bessel potentials of order n/ρ+1 of functions f satisfying [numerical formula] for some α<0. In this chapter, we deal with Riesz potentials of order α, n/ρ≤α≤n/p+1,and give extensions of those results. 0ur aim in this direction is to find κ such that [numerical formula] when fsatisfies (5), and our result implies the above mentioned result by Edmunds and Krbec. If (2) does not hold, then the potential may not be continuous anywhere, and we study the fine limits of U_αf with respect to the relative Orlicz capacity [numerical formula] where κ is a nonnegative Borel measurable function on R^n, G is an open set in R^n and the infimum is taken over all nonnegative measurable functions g on G such that [numerical formula]. In case κ(γ)=γ^<α-n>, we write C_<α, φ_ρ> for C_<κ, φ_p>. For simplicity, we write C_<κ, φ_ρ>(E)=O if C_<k, φ_p>(E∩G;G)=0 for every bounded open set G. If a property holds except for a set E with C_<κ, φ_ρ> (E) =0,then we say that the property holds C_<κ, φ_ρ>-quasi everywhere. In Chapter 4,we show that if f satisfies (1) and (3), then there exist a set E⊂R^n and a polynomia1 P_<ℓ> such that [numerical formula] and [numerical formula] where [numerical formula] and [numerical formula] If in addition (2) holds, then the exceptiona1 set E is empty and the above fine limit is seen to be replaced by the usual limit similar to (4). In Chapter 5,we are concerned with the existence of radial limits.
広島大学の論文
1996-12-28

オーリッツ関数のリースポテンシャルの連続性

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