LECTURE 2
Analytical Tools
In this chapter we introduce some of the analysis necessary for the applications
given later. A comprehensive treatment of the analytical aspects of the pseudo-
holomorphic curve theory is contained in the forth-coming book [1].
2.1. A priori estimates
An important part in dealing with holomorphic curves are the apriori estimates.
The closed t:-disk in C will be denoted in the following by De. We shall also
write D instead of D 1. Denote for p E (2,oo) and k EN by Wk,P(D,C^2 ) the
standard Sobolev space equipped with the norm
llwllk,p = ( L llD<>ull1f,P)^11 P.
1 <>1:5kWe write V for the space of test functions r.p: C -+ C^2 with support in B = { z E
CI lzl < l}. The following estimates are well-known for 8 = 08 + i&t
ll8r.pllk,p;::: Ck,pll'Pllk+l,p, 'PE V
k E N,p E (2, oo) (they also hold for 1 < p ~ 2, but that will not be important for
us). Ifwe replace i by complex multiplications J (i.e. J E .L'.R(C^2 ) with J^2 =-Id)
in a compact set .J, i.e. J E .J and .J is compact, then we find a universal constant
ck,p > 0, so that the above holds for the associated 8-operators. Assume now J is
an almost complex structure on C^2. Our first result is the following.
Theorem 2.1. Let p E (2, oo), E > i:' > 0. Given a constant c > 0 there exists a
constant c' > 0 such that for a solution w: D , -+ C^2 of
the estimate
holds.
W 8 + J(w)wt = 0
lw(O)I ~ 155k::::: 1,