A

locally integrable function is a function which is integrable on every compact subset of its domain of definition.

For any

locally integrable function f and z [member of] H, it follows that

Let w be a weight (i.e., a nonnegative, locally integrable function) on [R.sup.d] and let M stand for the usual Hardy-Littlewood maximal operator.

for any locally integrable function / on Rd and any [lambda] > 0.

Let p: [OMEGA] [right arrow] ]1, +[infinity]] be a

locally integrable function such that 1 < [p.sup.-] = [inf.sub.x [member of] [OMEGA]]p(x) and [p.sup.+] = [sup.sub.x [member of] [OMEGA]]p(x) [less than or equal to] [infinity].

Let g be a nonnegative,

locally integrable function such that [[intergral].sup.t] g [not equal to] 0, and the functions q and r continuous functions with r > 0.

The integral average of a

locally integrable function u over a set A of positive and finite measure is

We say w a weight if w is a non-negative and

locally integrable function. For a subset E [subset] [R.sup.n], [[chi].sub.E] means the characteristic function of E and [absolute value of E] denotes the volume of E.

By a weight, we shall mean a

locally integrable function v on [R.sup.n] such that v(x) > 0 for a.e.

A

locally integrable function f is said to be in [BMO.sup.[theta].sub.p]([phi]) with p [greater than or equal to] 1 and [theta] [greater than or equal to] 0, if there exists a positive constant C such that for any cube Q

A nonnegative

locally integrable function [omega] on [R.sup.n] is said to belong to [A.sub.p] (1 < p < [infinity]), if

Let b be a

locally integrable function; the commutator of homogeneous fractional integral operator [b, [T.sub.[OMEGA],[sigma]]] is defined by