LECTURE 2. FREDHOLM THEORY 163for ( E W^1 ·P([-l, 2] x S^1 ). Moreover, if ( E W^1 •^2 and D( E W 1 ~;,1', then ( E
w,k+l,p
Joe ·
The inequality is proved in three stages. The first is the same inequality with
the L^2 -norm on the right replaced by the LP-norm. This follows directly from
the Calderon-Zygmund inequality. The second stage uses the Sobolev embedding
W^1 ·^2 '---> LP with corresponding estimate ll(llLv :::; c ll(llw1,2· The third stage is
the elliptic estimate for W^1 •^2. In both the first and third stage the domain has to
be increased. Details are easy and are left to the reader. Finally, that D( E Lf 0 c
implies ( E W 1 ~·: is the standard elliptic regularity result.
Step 3: Consider the norm
11(112,p = (1-: ll((s,·)lli 2cs1) ds) l/p
There exist constants c 2 ,c 3 > 0 such that, if ( E W^1 •^2 (JR x S^1 ,JR^2 n) and D( E
LP(JR x S^1 ,JR^2 n), then ( E Wl,P(JR x S^1 ,JR^2 n) and
11(112,p :::; c2 llD(llLP' ll(llw1,p :::; C3 (llD(llLP + 11~11 2 ,p).
It follows from Step 2 that ( E W 1 ~':. Hence, to establish the first assertion, it
only remains to prove that ll(llw1.v < oo and this will follow from the two estimates.
The first of these is just Young's convolution inequality. Namely, by assumption,
we have 'T/ = D( E L^2 (JR, H) n LP(JR, H). Step 1 shows that ( = Q'T/ = K * 'T/· Hence,
by Young's inequality,
2
llQ'T/ll2,p = llK * TJllLP(IR,H) :::; llKllu(JR,.C(H)) ll'T/llLP(IR,H) :::; 5 llTJllLP.
The last inequality uses (18) and llTJ(s)llL2(s1) :::; ll'T/(s)llLv(s1) for P 2 2.
To prove the second inequality in Step 3, we use Step 2 and (a+b)P :::; 2P(aP+bP):
11<11:'.v'·'([k.k+l[xS') < 2' c,' ( t:' llD<ll\'.,(S') ds + (t:' ll<lli'(S') ds t ')
< 2Pc1P (1~:
2llD(llivcs1) ds + 3P/
2
-
1
1 ~:2ll(lli2cs1 l ds)
rk+2
< 3P/
2
-
1
2Pc1P l k-l (llD(llivcs1) + ll(lli2cs1)) ds.
Take the sum over all k to obtain
ll(llfin.v:::; 3P/
2
2Pc1P 1-: (llD(llivcs1) + ll(lli2cs1i) ds,
and hence
ll(llw1.v :::; C3 (llD(lb + 11(112,p) ·
This proves the inequalities and Step 3.
Step 4: We prove the lemma.
Putting the two estimates of Step 3 together we obtain