HARDY TYPE INEQUALITIES
27
sidered in Section 4, because then the appropriate function i p ( if ip1 is
not bounded) is somewhat more complicated. In Section 3 we consider the
3 . 'v,
case c p G C . Remark that we are interested in taking ty very small, hence
c p very large . Since tp! must be dominated by cpT, clearly cpf = cpf is the
best possible choice from this point of view. This will be the choice of
Section 3. We shall leave out some possible generalizations. For example
we make for p the simplest, natural, choice . We also assume the simplest
2 2 2
necessary condition on (2a)P + Q , namely (2a)P + Q ^ 3 + 31P
with constant $ , $1 0. In some cases 8 can be replaced by an increa
sing function of t (e.g. if H = ( p and Q = ytm with u, m G JR , then
Q = (a+m)Q, hence for (a+m)signu 0 and m 0 the function Q grows
as t ).We leave such generalizations to Section 4.
The inequality (2.36) given in Proposition 2.10 becomes simpler if
we assume that c p = c p and q = q. We denote by C(q,p) the set of all func
tions c p : I •+ • ]R satisfying
(2.39) c p G C3(I) , cp » 0 , cp'(t) ^ qp(t) and
tcp"(t) + tcp"'(t) q(l+(p»(t)) for t a + 1,
and we then have
Corollary 2.11: Let q £ 1, a € 3R and assume that S and R are small at
infinity in the sense of Proposition 2.10. Then there are constants
v G (0,1) and c G (1,~) such that for all c p G C(q,p), for all v G (0,v )
and all v G t?(r(v)):
(2.40) £ti^L(cp)vp (v,[(2a)P*+Qa3v) + (Pv, (tcp vc)Pv) +
+ ( v , [ a c p
 2
+
2tcpTcp"

MtpfM
 v c ( l + c p '
2
+ i £ l  ) ] v ) .
Proof: This is a simple consequence of (2.36) if one makes use of the fol
lowing estimates (notice the identity (2.15) and the fact that now
P = ip):