When mathematicians solve a differential equation, they are usually converting unbounded operators (such as differentiation) which are represented in the equation into bounded operators (such as integration) which represent the solutions. It is rarely possible to give a solution explicitly, but general theory can often show whether a solution exists, whether it is unique, and what properties it has. For this, one often needs to apply suitable (bounded) functions $f$ to unbounded operators $A$ and obtain bounded operators $f(A)$ with good properties. This is the rationale behind the theory of (bounded) functional calculus of (unbounded) operators. Applications include the ability to find the precise rate of decay of energy of damped waves and many systems of similar type.

Oxford Mathematician Charles Batty and collaborators have recently developed a bounded functional calculus which provides a unified and direct approach to various general results. They extend the scope of functional calculus to more functions and provide improved estimates for some functions which have already been considered. To apply the results, one only has to check whether a given function $f$ lies in the appropriate class of functions by checking a simple condition on the first derivative.

The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.