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N n In conclusion, we have established n T (t, ξ1 , η1 ) ≤ ε 2 −m+1 t m−1 cn (m − 1)! φ L1 + φ n L1 f0 m−1 L1 f0 L1 . 33) This concludes the proof of Proposition 2, and ends this section. 7. Right Non-ordered Graphs The previous analysis readily establishes the estimate T (t, ξ1 , η1 ) ≤ O(εn/2−(m−1) ) in full generality. In the particular case when m − 1 = n/2, this reduces to T (t, ξ1 , η1 ) ≤ O(1). An additional argument is needed to prove that right non-ordered graphs, or wrong graphs with m − 1 > n/2, are actually vanishing as ε → 0.

4) 42 D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti Hence the whole integration procedure over variables tz 2 , tz 3 , . . , tz p , ti , tz p+1 , . . , tz m eventually gives the estimate n t n−(m−1)−1 (n − (m − 1) − 1)! |φ(h j )| |φ(h j )| |h j | |T (ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m ε−d(m−1)− 2 +(m−1)+1 × dh 1 dh 2 · · · dh n j∈Z , j =z p Rnd × |φ(h i )| |φ(h z p )| |area(h z p , h i )| m 1+ k=2 j ∈Z / , j =i 2 Hk ε − α2 . 5) The role of indices i and p is clearly put apart here. Here and below, the reader may keep in mind the relation m − 1 = n/2.

Indeed, changing variables in this way we arrive at the estimate |T (ξ1 , η1 , t)| ≤ cn Nα ( f 0 )m Nα (φ)m−2+n−(m−1)−2 ε × R2d dh i dk |φ(h i )| |φ(k)| |φ(h )| . |area(k, h i )| t m−2 (m − 2)! 9) Term-by-Term Convergence 43 Here h is a function of the H j ’s with j ∈ C p , k, and some h j ’s with j ∈ E p . 10) where c(d) is some universal constant, depending on the dimension d only. Here we used the fact that d ≥ 3. Eventually we have proved |T (ξ1 , η1 , t)| ≤ c(d)n Nα ( f 0 )m Nα (φ)n ε t m−2 .