2. CALDERON WEIGHT S 15
has been used. D
We conclude this section showing an application of the reverse Holder type
inequality which shows the relation between the classes Cp for different values of p
PROPOSITION 2.9.
i) If the weight w satisfies the condition Mp then it also satisfies Mq, for all q p
and\\w\\"MqAnwrMp.
ii) If the weight w satisfies the condition Mp then it also satisfies Mq, for all q p
and \\W\\M« 8^K;MP
Hi) If the weight w satisfies the condition Mp then for all q p, the weight w(t)tq~p
satisfies Mq with constant
\\wtq~p\\M
8Q\\W\\M
•
PROOF.
Let us remark that the assertion i) is nothing but Holder's inequality
and ii) and iii) are consequences of reverse Holder type inequalities satisfied by
these classes of weights.
i) Let / G Lq(w). Since q/p 1 we have
/»00 /00 /»00
/ \P{f)\«w P(\f\"/p)Pwnw\\PMp \f\"w.
Jo Jo Jo
ii) Since w G Mp, the reverse Holder inequality for this class of weights derived
in the Proposition 2.4 implies that for all e 0 such that epwAfp 1? the weight
w(t)t~ep G Mp, with constant
\\wt ep\\Mp
lep\\w\
MP
Let / G
Lq(w),
with q p 1, then applying Holder's inequality with exponents
p/q and (q — p)/q, we have
iWoi(f^*)*,(f,'««*)(",/'
C
with C = ( ^—  ) . Thus,
ep
Q{\f(x)\q/pxcq)(tj\P/9
*
r
\Qf\qw °q
r
\Q{\f\q/pxe){t)\pt~€pw{t)dt
Jo Jo L J
C f (\f\q/px€Y x~epw(x)dx = C f \f{x)\qw(x)dx,
with
c
,
=
( q  p \
q P
( 4^Mp
ep J V
i

e
Pll^llMp
Let 2pet/;Afp = 1, then we obtain
\W\M
&\\w\\M*(qp)Xp'q.