For λ = 4m(n+m), a function $f(z)=(1-\mid{z}\mid^2)^mg(z)$ with g ∈ $X_\lambda$, the eigenspace of the invariant Laplacian $\tilde{\Delta}$ in the unit ball $B_n$ of $\mathbb{C}^n$, satisfies an elliptic differential equation $\Delta_mf$ = 0. We make a study of the operator $\Delta_m$ as another way to study $\tilde{\Delta}$ - 4m(n + m). For example, if $Z_m$ denotes the class of all solutions f in $C^2(B_n)$ of $\Delta_mf$ = 0, we obtain an $L^2$-growth condition for the projection of a function in $Z_m$ onto H(p,q), the space of all harmonic homogeneous polynomials on $\mathbb{C}^n$ of degree p in z and of degree q in $\overline{z}$, to be 0 unless either p ≤ m or q ≤ m. This corresponds and gives another way to obtain the $L^2$-growth condition for a function in $X_\lambda$ to be in the $\mathcal{M}$-subspace $Y_4$ of $X_\lambda$ in [1,3,7]. $Y_4$ is the space of pluriharmonic functions in case λ = 0.
본 논문에서는 새로이 정의된 편미분 작용소 $\Delta_m$을 가지고 [3]의 결과들을 조사하였다. 그러면 이 작용소 $\Delta_m$ 은 불변 Laplacian보다 다루기 쉬우며 결과 또한 더욱 간단한 형태로 쓰여진다."