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8 JUNICHI ARAMAKI
Thus we have
J(t; |x|, λy) = λ −m (2πλ −2 t) −m/2 E e −λ−2 t R 1
0 c|λ(1+q)/p x| 2p |y+ √ λ −2 tYs| 2q ds
= λ −m J(λ −2 t; |λ (1+q)/p x|, y).
Now, we give the proof of Theorem 2.1.
Proof of Theorem 2.1.
Since p(t; x, y, x, y) is a real valued function, using (3.1) and (3.2), we can
write
I(t; x, y) ≡ |p(t; x, y, x, y) − p0(t; x, y, x, y)|
= (2πt) −d/2 E (cos F t (x, y) − 1)e −t R 1
0 V (x+√ tXs,y+ √ tYs)ds .
Since 0 ≤ 1 − cos θ ≤ θ 2 /2 for θ ∈ R, I(t; x, y) is estimated by
1
2 (2πt)−d/2 E F t (x, y) 2 e −t R 1
0 V (x+√ tXs,y+ √ tYs)ds .
By the Schwartz inequality, Lemma 3.2 and hypothesis (A.2) and (V.2), we
have
I(t; x, y) ≤ C1t −d/2 E[F t (x, y) 4 ] 1/2 E e −2t R 1
0 V (x+√ tXs,y+ √ tYs)ds 1/2
≤ C2t 2−d/2 (1 + |x| 2 ) 2a |y| 4b E e −C3t R 1
0 (1+|x+√ tXs| 2 ) p |y+ √ tYs| 2q ds 1/2 .
Define the function ξ by (3.3) and let χ be the characteristic function of the
sets {ξ ≥ |x|/2 √ t}.
Now we decompose
(3.7)
K(t; x, y) ≡ E e −C3t R 1
0 (1+|x+√ tXs| 2 ) p |y+ √ tYs| 2q ds
into the form K(t; x, y) = 2
j=1 Kj(t; x, y) where
K1(t; x, y) = E e −C3t R 1
0 (1+|x+√ tXs| 2 ) p |y+ √ tYs| 2q ds χ
K2(t; x, y) = E e −C3t R 1
0 (1+|x+√ tXs| 2 ) p |y+ √ tYs| 2q ds (1 − χ) .
Then we note that K(t; x, y) 1/2 ≤ 2
j=1 Kj(t; x, y) 1/2 . At first, we consider
K1(t; x, y). Since (1 + |x + √ tXs| 2 ) p ≥ 1, we see that
K1(t; x, y) ≤ E e −C3t R 1
0 |y+√ tYs| 2q )ds E[χ]
= E e −C3t R 1
0 |y+√ tYs| 2q )ds P ({ξ ≥ |x|/2 √ t}).
By Lemma 3.1 and Lemma 3.3 (i), we have
−C5t|y| 2q
K1(t; x, y) ≤ C4(e + e −C6|y| 2 /t −C7|x|
)e 2 /t
.