On a Hardy-Littlewood Type Integral Inequality with a Monotonic Weight Function

Abstract

The Hardy-Littlewood type integral inequality, which is the subject of study n this paper, is

$${\left( {{{\int_a^\infty {w\left( x \right)\left| {f'\left( x \right)} \right|} }^2}dx} \right)^2} \leqslant 4{\int_a^\infty {w\left( x \right)\left| {f\left( x \right)} \right|} ^2}dx{\int_a^\infty {w\left( x \right)\left| {f''\left( x \right)} \right|} ^2}dx$$

where - ∞ ≤ a < ∞, and w is a positive, monotonic increasing function on (a, ∞). The inequality is valid, with the number 4, for all complex-valued f such that f and f″ ε L ²_W (a, ∞). In certain cases the number 4 is best possible and all cases of equality can be described.

The example w(x) = x on (0, ∞) is considered in detail and it is shown the best possible number in the inequality in this case is strictly less than 4. It is known, from simple examples, that the inequality may not hold if the monotonic increasing condition on the weight w is removed.

The proofs given in this paper are essentially in classical analysis and depend, in a curious way, on the original proof of Hardy and Littlewood from 1932, where w(x) = 1 on (0, ∞). These proofs are compared with the later operator theoretic proof of the inequality above, first given by Kwong and Zettl in 1979, and later in 1981. Both types of proof offer an explanation of the fact that 4 is a global number for the inequality, for all intervals (a, ∞) and all weights w of the kind prescribed above.