2. CALDERON WEIGHT S

13

PROPOSITION

2.7'. IfT is bounded on L°°{v), for any weight v G C^, then T

is bounded on Lp(w), for all w G Cp and for all 1 p oo.

It is important for our purposes to compare the classes Cp with other classes

of weights that have appeared in the literature in connection with interpolation

theory. According to the factorization theorem we shall restrict ourselves to the

case p = 1.

i) We recall that a weight is a quasipower weight (see [BK]) if there exists a

positive constant C such that

C~1w

Sw Cw, (briefly Sw ~ w). We say

that a weight w is a functional parameter or a Kalugina weight if there exist

a positive constant C and a

C^1

positive function p such that

C~l

wp C

and aip(t) t(ff(t) j3ip(t), for some 0 a [3 1 and for all t 0. An

integration by parts shows that a Kalugina weight is a quasipower weight.

The class C\ is strictly larger than the others. Indeed, let

1/y/t, if 0 t 1

Wit) = , .

v ; ' l/Vt^i, if i t

It is very easy to compute that

Sw(t) =

2 + ^ , if 0 t 1;

2 , 2yt-i

t "^ t

2 arctan - 7 = , if 1 t

and therefore w G C\ but w is not a quasipower weight and consequently it is

not a Kalugina weight either (this example is suggested in [HS]).

ii) Using the formulas S = P + Q = PoQ = QoP it is easy to see that Sw ~ w

if and only if Pw ~ Qw ~ w. Moreover, in this case, Pw, Qw and even P^w,

Q(k'w

are quasipower weights.

If w is a quasipower weight and k is a natural number, then w(x) — Q^w(x)p(x),

where p(x) ~ C and Q^w is a nonincreasing

C(fc_1)-function.

Therefore we

can suppose that w is a nonincreasing very regular function whenever w is a

quasipower weight.

hi) Note that Sf(t) — P{Qf){t)1 therefore, since Qf is a nonincreasing function

(if / 0), we see that only the boundedness of P for nonincreasing functions

should be considered. Thus, we have that Cp — Mp D Bp (see Definition 4.1

below).

Furthermore, observe that Mp (1 M1 C Cp. The result will follow from the

following,

PROPOSITION

2.8. A weight w satisfies Mp and M1 if and only if there exists

C 0 such that for all t 0

- J w{x)dx +

tP'1

r ^^dx Cw(t). (2.12)

Furthermore any of these two equivalent conditions implies Mp and Mp.