ABSTRACT
Chapter 4
H-TRANSFORM ON THE SPACE L
;r
In Chapter 3 we constructed so-called L
;2
-theory of the H-transform (3.1.1), where we char-
acterize the existence, the boundedness and representation properties of the transformsH on
the space L
;2
given in (3.1.3). The present chapter is devoted to extending the above results
from r = 2 to any r = 1: Moreover, we shall deal with the study of properties such as the
range and the invertibility of the H-transform on the space L
;r
with any 1 5 r < 1. The
results will be dierent in nine cases:
1) a
= = Re() = 0; 2) a
= = 0; Re() < 0; 3) a
= 0; > 0;
4) a
= 0; < 0; 5) a
> 0; a
> 0; 6) a
> 0; a
= 0;
7) a
= 0; a
> 0; 8) a
> 0; a
> 0; a
< 0; 9) a
> 0; a
< 0; a
> 0:
Here a
, , , a
and a
are given in (1.1.7), (1.1.8), (1.1.10), (1.1.11) and (1.1.12), respec-
tively. We shall also use the constants and dened by (3.4.1) and (3.4.2), respectively.
