Hardy inequalities

Abstract

In this book we dealt so far several times with Hardy inequalities. But first we wish to mention that the whole story began with Hardy’s note [Had28] and the famous Theorem 330 in [HLP52], p. 245 (in small print). As a consequence (ignoring constants) one gets the following assertion: Let 1 < p < ∞ and m ∈ ℕ. There is a number c > 0 such that

$$\int\limits_\mathbb{R} {\left| t \right|^{ - mp} \left| {u(t)} \right|^p dt \leqslant c} \int\limits_\mathbb{R} {\left| {\frac{{d^m u(t)}} {{dt^m }}} \right|^p dt} $$

((16.1))

for all

$$u \in S(\mathbb{R}) with \frac{{d^j u}} {{dt^j }}(0) = 0 for j = 0, \ldots ,m - 1.$$

.