(I) A subset H of C n is a-minimally thin at infinity on C n.
A subset H of C n is a-minimally thin at infinity on C n.
Theorem C If a subset H of C n is a-minimally thin at infinity on C n , then we have ∫ H d P ( 1 + | P | ) n < ∞. (6).
Theorem 2 If a subset H of C n is a-minimally thin at infinity on C n , then we have ∫ H d P ( 1 + | P | ) n < ∞.
diam W k ≤ dist ( W k, R n ∖ C n ≤ 4 diam W k. Theorem 1 If H is a union of cubes from the Whitney cubes of C n , then equation (7) is also sufficient for H to be a-minimally thin at infinity with respect to C n.
Now we obtain u ( P ) ≥ ( c 3 − c ( ∞, v ) ) M Ω a ( P, ∞ ) for any P ∈ H. Hence by a result of [[12], p.69], H is a-minimally thin at infinity on C n with respect to the Schrödinger operator, which is the statement of (I).
Theorem A A subset H of C n is a-minimally thin at infinity on C n if and only if ∑ j = 0 ∞ γ Ω a ( H j ) W ( 2 j ) V − 1 ( 2 j ) < ∞, where H j = H ∩ C n ( Ω ; [ 2 j, 2 j + 1 ) ) and j = 0, 1, 2, … .
Theorem A A subset H of C n is a-minimally thin at infinity on C n if and only if ∑ j = 0 ∞ γ Ω a ( H j ) W ( 2 j ) V − 1 ( 2 j ) < ∞, where H j = H ∩ C n ( Ω ; [ 2 j, 2 j + 1 ) ) and j = 0, 1, 2, … . In recent work, Zhao (see [[2], Theorems 1 and 2]) proved the following results.
In this paper, we shall obtain a series of new criteria for a-minimally thin sets at infinity on C n , which complemented Theorem A by the way completely different from theirs.
This paper gives some new criteria for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by Long-Gao-Deng.
