A mapping F : [OMEGA] [right arrow] E is called integrably bounded if there exists an integrable function
k such that [absolute value of x] [less than or equal to] k(t) for all x [member of] [F.sub.0](t).
A locally integrable function
is a function which is integrable on every compact subset of its domain of definition.
Suppose that each [[omega].sub.i] (i = 1, ..., m) is a positive locally integrable function
defined on H.
For an arbitrary integrable function
f(t), the definition of the fractional integrals of order [alpha] > 0 is defined as
If f(t) is an integrable function
defined for all t [greater than or equal to] 0, its generalized integral transform G is the integral of f(t) times [u.sup.][alpha]] x [e.sup.-t/u] from t = 0 to [infinity].
Assume that p : [a, b] [right arrow] R is a nonnegative integrable function
with [[integral].sup.b.sub.a] [rho] (t)dt > 0 and (p, q) is a pair of conjugate exponents, that is 1 [less than or equal to] p,q [less than or equal to] [infinity], 1/p + 1/q = 1.
For any square integrable function
x on [-1, 1], we define the function
where x([theta]) is an integrable function
on [0, 2[pi]), called the measurement function.
Now, define the GD of a nonsmooth integrable function
f: (0,1) [right arrow] R.
An integrable function
f over [a, b] is necessarily bounded on that interval.
Let [phi](t) be a square integrable function
, that is, [phi](t) [member of] [L.sup.2](R).
Since (1.5) is not true for every nonnegative integrable function
[phi], the proof of (i) is not correct.