*Oh! The little more, and how much it is!*

And the little less, and what worlds away!

This quote from Robert Browning’s works adorns the beginning of the archetypal technique book *Inequalities*, written by the potent triumvirate of G.H. Hardy, J.E. Littlewood and G. Pólya. I’m not sure if budding analysts read it anymore; if not, they should. This book is the epitome of classical analysis and has been a staple of those who have wished to learn that art since Cambridge University Press published it in 1934. Reading it one can almost see the glistening ivory towers of English Academia…heady days for those who yearn for the atmosphere of between-wars England (the “long weekend”) when dons in academic garb sat at high table and lived the life of the mind. *The G.H. Hardy Reader*, recently published by the MAA, is a wonderful collection of supporting documents for the full visualization of this dreamy landscape (see the recent review by Michael Berg). *Inequalities* is very much a product of this atmosphere. Names such as Bliss, Courant, Jessen, Schur, Young and Zygmund are all credited with assistance by Hardy in the preface to his book and my own vision of Shangri-La features them all discussing such brilliant mathematics over sherry.

The inequality referenced in the title of this book under review is no. 326 and no. 327 in the ninth chapter of *Inequalities* (“Hilbert’s Inequality and its analogues and extensions”). The focus of this chapter is the “remarkable bilinear form which was first studied by Hilbert….viz. the form \(\sum\sum\frac{a_n b_m}{m+n}\) where \(m\) and \(n\) run from \(1\) to \(\infty\)” (Hardy). The next few pages show the relevance of this result to \(p\)-norm estimates of the form \[ \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{a_n b_m}{m+n} \leq C(p) \left(\sum_{m=1}^\infty a_m^p\right)^{1/p} \left( \sum_{n=1}^\infty b_n^q\right)^{1/q}, \qquad\qquad\qquad \frac{1}{p}+\frac{1}{q}=1.\]

Hardy also establishes the integral form of this inequality and much of the focus of the chapter is devoted to finding the form of the best possible constant \(C(p)\). The inequalities thus established have the right form to say quite a bit about Fourier series and about the extension of analytic functions in the open unit disc to the entire disc. In his own attempts to prove Hilbert’s inequality Hardy produced an interesting pair of his own:

326. If \(p>1\), \(a_n\geq 0\), and \(A_n=a_1+a_2+\dots+a_n\), then \[ \tag{9.8.1} \sum \left(\frac{A_n}{n}\right)^p < \left(\frac{p}{p-1}\right)^p \sum a_n^p\] unless all the \(a\) are zero. The constant is the best possible.

327. If \(p>1\), \(f(x)\geq 0\), and \(F(x)=\int_0^x f(t)\,dt\), then \[\tag{9.8.2} \int_0^\infty \left(\frac{F}{x}\right)^p\, dx < \left(\frac{p}{p-1}\right)^p \int_0^\infty f^p\, dx\] unless \(f\equiv 0\). The constant is the best possible.

Hardy goes on to mention that “These theorems were first proved by Hardy (2), except that Hardy was unable to fix the constant in Theorem 326. This imperfection was removed by Landau (4). A great many alternative proofs of the theorems have been given by various writers, for example by Broadbent (1), Elliott (1), Grandjot (1), Hardy (4), Kaluza and Szego (1), Knopp (1).We begin by giving Elliott’s proof of Theorem 326 and Hardy’s proof of Theorem 327.” So there it is, the birth of this famous inequality described in concise form by its creator.

Is Hardy’s Inequality an important result? The above form(s) certainly make(s) it useful for proving continuity of an averaging operator on \(L^p\) spaces and it certainly fits in nicely with Hilbert’s double-series inequality and its connection with kernels of integral equations. The book under review takes things several large steps further by utilizing multidimensional forms of the inequality. Sobolev spaces are introduced in the first chapter and the relationship between Hardy’s inequality and versions of the Sobolev embedding theorem are fleshed out in detail. The Fourier transform is given a very short introduction and put to use immediately in showing that Hardy’s inequality implies several forms of “uncertainty principles” (including Heisenberg’s inequality for the product of uncertainty in momentum and uncertainty in position for a particle whose location is governed by a probability distribution).

Perhaps the most interesting application of Hardy’s Inequality is to generalized “curvatures” of domains. A multidimensional form of Hardy’s inequality can be written which relates the \(p\)-norm of the gradient of \(f\) over a domain \(\Omega\) to the \(p\)-norm of a quantity involving the quotient of \(|f(x)|\) by the distance \(\delta(x)\) from \(x\) to the boundary \(\partial\Omega\). The authors then introduce the term “skeleton” for the points in \(\Omega\) where \(\delta(x)\) is non-differentiable. The conversation continues on to ridgepoints and cutpoints of the domain and the investigation of regularity of \(\delta(x)\) up to the boundary is analyzed for convex sets. The development from first principles of these ideas is brief but accessible but could have been made even more accessible with a series of pictures illustrating the definitions. The issue of graphics recurs later in the book on page 186 where we are treated to a graph of a lemniscate that look likes it was generated on a dot matrix printer! Still, such breadth of application from such a “simple” inequality!

The largest part of the book involves applications of Hardy’s inequality to mathematical physics of the Operator Theory type. Eigenvalues of the Schrodinger operator on \(L^2(\mathbb{R}^n\), magnetic fields and potentials and Pauli operators all make appearances. The physical background is not developed (only referenced) and I doubt that anyone without some background in Operator Theory and Partial Differential Equations would make much of all this. The terseness of the development throughout make this book more suitable for graduate students.

All in all, the book under review is a lovely compendium of the utility and power of Hardy’s Inequality. It is not without a certain sense of irony that one comes to realize that this would have come as quite a shock to G.H. Hardy! His famous claim not to have generated any “useful” mathematics is clearly wrong as this book shows in great detail. Apparently the safety of the Ivory Tower in its serene isolation from practical matters is no more. It’s a different world now and perhaps we’re better off that way.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.