On Some Inequalities with Weights, Examples and Counterexamples

Inequalities (1) and (2) below are investigated. We show that (1) holds true (for 0 < p < q < ∞) with || ||X = || ||L1(0, A, dx, Lq/p(0, B, dy)), but does not hold (in general) with the weaker norm || ||X = || ||Lq/p(0, B, dy, L1(0, A, dx)). We also show that the weighted Hardy inequality (2) holds true with r = 1 if and only if the positive reals p, q satisfy q ≤ p (cp. Section 2).