In this paper we make up some extensions for the theorems concerning the potential integrals. The results are the following: Let v(x) be the function defined by (1) where the function f(x) vanishes outside a bounded and measurable domain GEn". ∑s be a measurable subset in a S-dimensional hyperplane section of G. Theorem 1 Suppose that P>1

f(x)∈Lp(G)

λ=n(1-1/p) and s>0. Then there exist constants C

K>0 indenpent of f such that integral from ∑s to ∞ exp(|v(x)|/K‖f‖_(Lp(G)))p/(p-1)dx≤Cmes∑s. Theorem 2 Suppose that f(x)∈L1: λ(G)

where 0λ. Then there exist constants C

K>0 independent of f such that integral from ∑s to ∞ exp(|v(x)|/K‖f‖_(L1

λ(G)))dx≤C(diamG) Theorem 3 Suppose thut f(x)∈L1 _μ(G)

0λ

then v∈Lq|v(∑s)

1μ and there exists constant C>0 independent of f such that ‖v‖Lq v(∑s)≤C‖f‖L1:λ(G)) Thcorem 4 Suppose that 1p _μ(G) 0λ

thenr v∈Lq

v(∑s)

(s-μ)/q>(n-μ)/p-(n-λ)

v/q=s/q-[(n-μ)/p-(n-λ)[>μ/q and there exists a constant C>0 independent of f such that ‖v‖_(Lq|v(∑s))≤C‖f‖_(L(p|μ)(G)). KeyWords：