Quantitative continuity estimates

Suppose is a continuous (but nonlinear) map from one normed vector space to another . The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or , particularly if is not known to be uniformly continuous). If is known to be differentiable (in, say, the Fréchet sense), then we in fact have a linear bound of the form

for some depending on , if is small enough; one can of course make independent of (and drop the smallness condition) if is known instead to be Lipschitz continuous.

In many applications in analysis, one would like more explicit and quantitative bounds that estimate quantities like in terms of quantities like . There are a number of ways to do this. First of all, there is of course the trivial estimate arising from the triangle inequality:

This estimate is usually not very good when and are close together. However, when and are far apart, this estimate can be more or less sharp. For instance, if the magnitude of varies so much from to that is more than (say) twice that of , or vice versa, then (1) is sharp up to a multiplicative constant. Also, if is oscillatory in nature, and the distance between and exceeds the “wavelength” of the oscillation of at (or at ), then one also typically expects (1) to be close to sharp. Conversely, if does not vary much in magnitude from to , and the distance between and is less than the wavelength of any oscillation present in , one expects to be able to improve upon (1).

When is relatively simple in form, one can sometimes proceed simply by substituting . For instance, if is the squaring function in a commutative ring , one has

and thus

or in terms of the original variables one has

If the ring is not commutative, one has to modify this to

Thus, for instance, if are matrices and denotes the operator norm, one sees from the triangle inequality and the sub-multiplicativity of operator norm that

If involves (or various components of ) in several places, one can sometimes get a good estimate by “swapping” with at each of the places in turn, using a telescoping series. For instance, if we again use the squaring function in a non-commutative ring, we have

which for instance leads to a slight improvement of (2):

More generally, for any natural number , one has the identity

in a commutative ring, while in a non-commutative ring one must modify this to

and for matrices one has

Exercise 1 If and are unitary matrices, show that the commutator obeys the inequality

(Hint: first control .)

Now suppose (for simplicity) that is a map between Euclidean spaces. If is continuously differentiable, then one can use the fundamental theorem of calculus to write

where is any continuously differentiable path from to . For instance, if one uses the straight line path , one has

In the one-dimensional case , this simplifies to

Among other things, this immediately implies the factor theorem for functions: if is a function for some that vanishes at some point , then factors as the product of and some function . Another basic consequence is that if is uniformly bounded in magnitude by some constant , then is Lipschitz continuous with the same constant .

Applying (4) to the power function , we obtain the identity

which can be compared with (3). Indeed, for and close to , one can use logarithms and Taylor expansion to arrive at the approximation , so (3) behaves a little like a Riemann sum approximation to (5).

Exercise 2 For each , let and be random variables taking values in a measurable space , and let be a bounded measurable function.

• (i) (Lindeberg exchange identity) Show that

  

  

  

• (ii) (Knowles-Yin exchange identity) Show that

  

  

  

  where is a mixture of and , with uniformly drawn from independently of each other and of the .

• (iii) Discuss the relationship between the identities in parts (i), (ii) with the identities (3), (5).

(The identity in (i) is the starting point for the Lindeberg exchange method in probability theory, discussed for instance in this previous post. The identity in (ii) can also be used in the Lindeberg exchange method; the terms in the right-hand side are slightly more symmetric in the indices , which can be a technical advantage in some applications; see this paper of Knowles and Yin for an instance of this.)

Exercise 3 If is continuously times differentiable, establish Taylor’s theorem with remainder

If is bounded, conclude that

For real scalar functions , the average value of the continuous real-valued function must be attained at some point in the interval . We thus conclude the mean-value theorem

for some (that can depend on , , and ). This can for instance give a second proof of fact that continuously differentiable functions with bounded derivative are Lipschitz continuous. However it is worth stressing that the mean-value theorem is only available for real scalar functions; it is false for instance for complex scalar functions. A basic counterexample is given by the function ; there is no for which . On the other hand, as has magnitude , we still know from (4) that is Lipschitz of constant , and when combined with (1) we obtain the basic bounds

which are already very useful for many applications.

Exercise 4 Let be matrices, and let be a non-negative real.

• (i) Establish the Duhamel formula

  

  

  where denotes the matrix exponential of . (Hint: Differentiate or in .)

• (ii) Establish the iterated Duhamel formula

  

  

  

  

  for any .

• (iii) Establish the infinitely iterated Duhamel formula

  

  

• (iv) If is an matrix depending in a continuously differentiable fashion on , establish the variation formula

  

  where is the adjoint representation applied to , and is the function

  

  (thus for non-zero ), with defined using functional calculus.

We remark that further manipulation of (iv) of the above exercise using the fundamental theorem of calculus eventually leads to the Baker-Campbell-Hausdorff-Dynkin formula, as discussed in this previous blog post.

Exercise 5 Let be positive definite matrices, and let be an matrix. Show that there is a unique solution to the Sylvester equation

which is given by the formula

In the above examples we had applied the fundamental theorem of calculus along linear curves . However, it is sometimes better to use other curves. For instance, the circular arc can be useful, particularly if and are “orthogonal” or “independent” in some sense; a good example of this is the proof by Maurey and Pisier of the gaussian concentration inequality, given in Theorem 8 of this previous blog post. In a similar vein, if one wishes to compare a scalar random variable of mean zero and variance one with a Gaussian random variable of mean zero and variance one, it can be useful to introduce the intermediate random variables (where and are independent); note that these variables have mean zero and variance one, and after coupling them together appropriately they evolve by the Ornstein-Uhlenbeck process, which has many useful properties. For instance, one can use these ideas to establish monotonicity formulae for entropy; see e.g. this paper of Courtade for an example of this and further references. More generally, one can exploit curves that flow according to some geometrically natural ODE or PDE; several examples of this occur famously in Perelman’s proof of the Poincaré conjecture via Ricci flow, discussed for instance in this previous set of lecture notes.

In some cases, it is difficult to compute or the derivative directly, but one can instead proceed by implicit differentiation, or some variant thereof. Consider for instance the matrix inversion map (defined on the open dense subset of matrices consisting of invertible matrices). If one wants to compute for close to , one can write temporarily write , thus

Multiplying both sides on the left by to eliminate the term, and on the right by to eliminate the term, one obtains

and thus on reversing these steps we arrive at the basic identity

For instance, if are matrices, and we consider the resolvents

then we have the resolvent identity

as long as does not lie in the spectrum of or (for instance, if , are self-adjoint then one can take to be any strictly complex number). One can iterate this identity to obtain

for any natural number ; in particular, if has operator norm less than one, one has the Neumann series

Similarly, if is a family of invertible matrices that depends in a continuously differentiable fashion on a time variable , then by implicitly differentiating the identity

in using the product rule, we obtain

and hence

(this identity may also be easily derived from (6)). One can then use the fundamental theorem of calculus to obtain variants of (6), for instance by using the curve we arrive at

assuming that the curve stays entirely within the set of invertible matrices. While this identity may seem more complicated than (6), it is more symmetric, which conveys some advantages. For instance, using this identity it is easy to see that if are positive definite with in the sense of positive definite matrices (that is, is positive definite), then . (Try to prove this using (6) instead!)

Exercise 6 If is an invertible matrix and are vectors, establish the Sherman-Morrison formula

whenever is a scalar such that is non-zero. (See also this previous blog post for more discussion of these sorts of identities.)

One can use the Cauchy integral formula to extend these identities to other functions of matrices. For instance, if is an entire function, and is a counterclockwise contour that goes around the spectrum of both and , then we have

and similarly

and hence by (7) one has

similarly, if depends on in a continuously differentiable fashion, then

as long as goes around the spectrum of .

Exercise 7 If is an matrix depending continuously differentiably on , and is an entire function, establish the tracial chain rule

In a similar vein, given that the logarithm function is the antiderivative of the reciprocal, one can express the matrix logarithm of a positive definite matrix by the fundamental theorem of calculus identity

(with the constant term needed to prevent a logarithmic divergence in the integral). Differentiating, we see that if is a family of positive definite matrices depending continuously on , that

This can be used for instance to show that is a monotone increasing function, in the sense that whenever in the sense of positive definite matrices. One can of course integrate this formula to obtain some formulae for the difference of the logarithm of two positive definite matrices .

To compare the square root of two positive definite matrices is trickier; there are multiple ways to proceed. One approach is to use contour integration as before (but one has to take some care to avoid branch cuts of the square root). Another to express the square root in terms of exponentials via the formula

where is the gamma function; this formula can be verified by first diagonalising to reduce to the scalar case and using the definition of the Gamma function. Then one has

and one can use some of the previous identities to control . This is pretty messy though. A third way to proceed is via implicit differentiation. If for instance is a family of positive definite matrices depending continuously differentiably on , we can differentiate the identity

to obtain

This can for instance be solved using Exercise 5 to obtain

and this can in turn be integrated to obtain a formula for . This is again a rather messy formula, but it does at least demonstrate that the square root is a monotone increasing function on positive definite matrices: implies .

Several of the above identities for matrices can be (carefully) extended to operators on Hilbert spaces provided that they are sufficiently well behaved (in particular, if they have a good functional calculus, and if various spectral hypotheses are obeyed). We will not attempt to do so here, however.

Why are solitons stable?

Last month, at the joint AMS/MAA meeting in San Diego, I spoke at the AMS “Current Events” Bulletin on the topic “Why are solitons stable?“. This talk was supposed to be a survey of many of the developments on the rigorous stability theory of solitary waves in dispersive wave models (e.g. the Kortweg-de Vries equation and its generalisations, nonlinear Schrödinger equations, etc.), although my actual talk (which was the usual 50 minutes in length) only managed to cover about half of the material I had planned.

More recently, I completed the article that accompanies the talk, and which will be submitted to the Bulletin of the American Mathematical Society. In this paper I describe the key conflict in these wave models between dispersion (the tendency of waves of differing frequency to move at different speeds, thus causing any localised wave to disperse in space over time) and nonlinearity (which can cause any concentrated portion of the wave to self-amplify). Solitons seem to lie at the exact balancing point between these two forces, neither dispersing nor amplifying, but instead simply traveling at a constant velocity or oscillating in phase at a constant rate. In some cases, this balancing point is unstable; remove even a tiny amount of mass from the soliton and it eventually disperses completely into radiation, or one can add a tiny amount and cause the soliton to concentrate into a point and thence exhibit blowup in finite time. In other cases, the balancing point is stable; small perturbations to a soliton may end up changing the amplitude, position, and/or velocity of the soliton slightly, but the bulk of the solution still closely resembles a soliton in size, shape, and behaviour. Stability is sometimes enforced by linear properties, such as dispersive estimates or spectral properties of the linearised dynamics, but is also often enforced by nonlinear properties, such as nonlinear conservation laws, monotonicity formulae, and local propagation estimates for mass and energy (such as those provided by virial identities). The interplay between all these properties can be remarkably subtle, especially in the critical case when a key conserved quantity is scale-invariant (thus leading to an additional degeneracy in the soliton manifold). This is particularly evident in the remarkable series of papers by Martel and Merle establishing various stability and blowup properties near the ground state soliton of the critical generalised KdV equation, which I spend some time discussing (without going into too many of the (quite numerous) technical details). The focus in my paper is primarily on the non-integrable case, in which the techniques are primarily analytic rather than algebraic or geometric.