ABSTRACT
The left (right) derivative <p(t) = &_(t) (= $'+(*)) exists and is left (right) continuous, nondecreasing on (0, oo), satisfies 0 < <p(t) < oo for 0 < t < oo, <£>(0) = 0 and lim <p(t) — oo. [Similar description is valid
t->-oo for the right derivative, but we concentrate on the left one.] The left inverse -0 of (p is, by definition, ijj(s) = inf{£ > 0 : (p(t) > s} for s > 0. Then $, ^ given by
Jo JQ are called a pair of complementary N-functions which satisfy the Young inequality:
(2) with equality iff (p(\x\) < \y\ < (p+(\x\) when x is given, or tjj(\y\) < \x < ip+(\y\) when y is given (?/>+ = ^'+ ; ^ + — ^ +)- The TV-function \£ complementary to $ can equally be defined by:
#(y) = sup{x|?/| - $(rc) : re > 0}, y € R. (3) A classification of TV-functions based on their growth rates is facili-
tated by the following: Definition 1. An TV-function $ is said to obey the A2-condition for large x (for small £, or for all #), written often as $ € A2(oo)($ e A2(0), or $ € A2), if there exist constants XQ > Q,K > 2 such that 3>(2x) < K3>(x) for x > XQ (for 0 < x < XQ, or for all x > 0); and it obeys the V2-condition for large x (for small x, or for all x), denoted symbolically as $ € V2(oo) ($ e V2(0), or $ G V2) if there are constants XQ > 0 and c > 1 such that $(rc) < -^<fr(cx) for x > x0 (for 0 < a; < XQ, or for all re > 0).
