SYMMETRIES, SCALING, AND DIMENSIONAL ANALYSIS

TERENCE TAO

Abstract. Dimensional analysis is a powerful tool with which one can quickly and easily check for the “obvious” ways in which an estimate can be patently false, thus guarding against errors and preventing one from wasting too much time trying to prove estimates which are not true. Here We give some rigorous justification to the dimensional analysis of estimates, placing it in the more general context of Newton polytopes associated to various symmetries.

1. Prelude: one-dimensional and multi-dimensional polynomials

Let R^{+} := {x ∈ R : x > 0} be the positive real axis and N := {0, 1, …} be the natural numbers. Define a totally positive one-dimensional polynomial to be a function P : R^{+} → R^{+} of the form

P (x) := ∑_{j∈N} c_{j}x^{j}

where c_{j} are non-negative coefficients with at most finitely many coefficients non-zero, and which is not identically zero. Let us say that one totally positive polynomial Q majorises another P, and write P ≺ Q, if there exists a constant C > 0 for which we have P (x) ≤ CQ(x) for all x ∈ R^{+}. This is clearly a partial ordering on the totally positive polynomials.

It is a natural question to try to characterise the pairs of totally positive polynomials P, Q for which P ≺ Q. Fortunately this is easy.

Definition 1.1 (One-dimensional Newton polytope). Let P (x) = ∑_{j∈N} c_{j}x^{j} be a totally positive polynomial. We define the (upper) degree \overline{deg}(P) of P to be the largest integer j for which c_{j} is non-zero, and the lower degree \underline{deg}(P) to be the smallest integer j for which c_{j} is non-zero. We define the Newton polytope N(P) ⊂ R of P to be the closed interval N(P) := [\underline{deg}(P), \overline{deg}(P)].

Examples 1.2. If P (x) = x + 2x^{2} + 3x^{3}, then N(P) = [1, 3], while if P (x) = x^{n}, then N(P) = {n}.

1991 Mathematics Subject Classification. 42B15, 35L05.

Proposition 1.3. Let P, Q be totally positive one-dimensional polynomials. Then P ≺ Q if and only if N(P) ⊆ N(Q).

Proof. Since P, Q are strictly positive and continuous on R^{+}, we can compare P and Q on any compact subset of R^{+} without difficulty; the only issue is when x → 0 or when x → ∞. Observe that P (x) grows like x^{\overline{deg}(P)} as x → ∞ and decays like x^{\underline{deg}(P)} as x → 0, and similarly for Q. The claim easily follows. □

We also observe that the Newton polytope behaves well under the addition and multiplication operations: N(P + Q) = \overline{N(P) ∪ N(Q)} (where \overline{X} denotes the convex hull of X) and N(P Q) = N(P)+N(Q) := {n + m : n ∈ N(P), m ∈ N(Q)}.

Now we generalise to higher dimensions. Fix a dimension d ≥ 1. A totally positive d-dimensional polynomial is a function P : (R^{+})^{d} → R^{+} of the form

P (x) := ∑_{j∈N^{d}} c_{j}x^{j}

where c_{j} are non-negative coefficients with at most finitely many coefficients non-zero, and which is not identically zero, and x^{j} = x_{1}^{j_{1}} … x_{d}^{j_{d}} is the usual multi-index notation. As before we can define a partial order by setting P ≺ Q whenever there exists C > 0 for which P (x) ≤ CQ(x) for all x ∈ (R^{+})^{d}.

Definition 1.4 (Newton polytope). Let P (x) = ∑_{j∈N^{d}} c_{j}x^{j} be a totally positive polynomial in d dimensions. . We define the (upper) degree \overline{deg}(P) of P to be the largest integer j for which c_{j} is non-zero, and the lower degree \underline{deg}(P) to be the smallest integer j for which c_{j} is non-zero. We define the Newton polytope N(P) ⊂ R^{d} of P to be the convex hull of the set {j ∈ N^{d} : c_{j} > 0}.

Examples 1.5. If P (x, y) = x^{2} + xy + y^{2}, then N(P) is the closed line segment connecting (2, 0) with (0, 2), while if P (x, y) = 1 + x^{2} + y^{2}, then N(P) is the closed triangle with vertices (0, 0), (2, 0), and (0, 2). It is easy to see that this definition generalises the one-dimensional one. Note that in some texts, the Newton polytope is defined to be the larger set N(P) + [0, ∞)^{d}.

Proposition 1.3 generalises:

Proposition 1.6. Let P, Q be totally positive d-dimensional polynomials. Then P ≺ Q if and only if N(P) ⊆ N(Q).

We invite the reader to prove this proposition; we shall prove a more general statement (at least of the “only if” part) in the next section. It has the following corollary. Say that a totally positive d-dimensional polynomial is homogeneous of degree n if we have P (λx) = λ^{n}P (x) for all x ∈ (R^{+})^{n} and λ > 0. This is equivalent to requiring N(P) to lie in the hyperplane {(n_{1}, …, n_{d}) : n_{1} + … + n_{d} = n}. Then if P and Q are homogeneous totally positive d-dimensional polynomials, a necessary (but not sufficient) condition for P ≺ Q is that P and Q have the same degree of homogeneity. Thus for instance we can instantly assert that

x^{2} + 2xy + 3y^{2} ⊀ x^{3} + 2x^{2}y + 3xy^{2} + 4y^{3}

without any additional effort, because both sides are homogeneous and the degrees do not match.

More generally, given a totally positive d-dimensional polynomial x ↦ P (x), we can (for any fixed x ∈ (R^{+})^{d}) consider the totally positive one-dimensional polynomial λ ↦ P (λx). One verifies that the one-dimensional Newton polytope N(P (·x)) of this polynomial is independent of the choice of x; indeed N(P (·x)) is simply the image of N(P) under the map (n_{1}, …, n_{d}) ↦ n. Thus a necessary (but not sufficient) condition for P ≺ Q is that N(P (·x)) ⊆ N(Q(·x)). Thus for instance we can instantly assert that

1 + xy ⊀ x^{2} + y^{2} + x^{3} + 2x^{2}y + 3xy^{2} + 4y^{3}

because the left-hand side has one-dimensional Newton polytope [0, 2], while the right-hand side has one-dimensional Newton polytope [2, 3].

One can also verify the identities N(P + Q) = \overline{N(P) ∪ N(Q)} and N(P Q) = N(P) + N(Q) as before; we leave this to the reader (and shall prove a more general statement below).

2. More general growth functions

One can achieve a similar analysis for more general functions F : (R^{+})^{d} → R^{+} that are not necessarily polynomial in nature. Note that we again have the partial order F ≺ G on such functions, defined as before.

Definition 2.1 (Generalised Newton polytope). Let F : (R^{+})^{d} → R^{+} be a function. A closed half-space

H = {(n_{1}, …, n_{d}) ∈ R^{d} : a_{1}n_{1} + … + a_{d}n_{d} ≤ b}

with (a_{1}, …, a_{d}) ∈ R^{d} and b ∈ R is said to be a supporting half-space for F if for any x = (x_{1}, …, x_{d}) ∈ (R^{+})^{d} there exists c_{x} > 0 such that we have the upper bound

F (λ^{a_{1}}x_{1}, …, λ^{a_{d}}x_{d}) ≤ c_{x}λ^{b}

for all λ ≥ 1. We define the (generalised) Newton polytope N(F) of F to be the intersection of all of its supporting half-spaces; thus N(F) is a closed convex subset of R^{d}.

Remark 2.2. In one dimension, the Newton polytope is easier to define: we have N(F) = [a, b], where b is the best exponent for which F (x) ≤ C_{ε}x^{b+ε} for all ε > 0 and x ≥ 1, and a is the best exponent for which F (x) ≤ C_{ε}x^{a−ε} for all ε > 0 and 0 ≤ x < 1. Thus the Newton polytope in the one-dimensional case captures the growth or decay rate at zero and at infinity. In higher dimensions, the Newton polytope is exploring a sort of directional growth rate (at least if one views a log-log plot of F).

We leave it to the reader to verify that this definition generalises the one for polynomials (which are automatically normal). However, it is applicable to arbitrary functions F : (R^{+})^{d} → R^{+}. For instance, in one dimension we have

N(\sqrt{x}) = \{1/2\}
N(x + \frac{1}{x}) = [−1, 1]
N(max(x, \frac{1}{x})) = [−1, 1]
N((1 + x^{2})^{1/2}) = [0, 1]
N(e^{x}) = [0, +∞)
N(e^{1/x}) = (−∞, 0]
N(e^{x} + e^{1/x}) = R
N(e^{x+ \frac{1}{x}}) = ∅.

In higher dimensions, we easily check that monomials with fractional exponents have a Newton exponent which is a point:

N(x_{1}^{α_{1}} … x_{d}^{α_{d}}) = (α_{1}, …, α_{d}) for any α_{1}, …, α_{d} ∈ R.

In particular monomials are normal. Also we have the following easy rules:

Proposition 2.3 (Newton polytope rules). Let F, G : (R^{+})^{d} → R^{+} be functions with N(F), N(G) non-empty. Then we have N(F + G) = N(max(F, G)) = \overline{N(F) ∪ N(G)}, N(F G) ⊆ N(F)+N(G), and N(F^{α}) = α · N(F) := {αn : n ∈ N(F)} for any α ≥ 0.

Example 2.4. In one dimension, if we start with N(x^{α}) = {α} and apply these rules, we can obtain that N((1 + x^{2})^{s/2}) = [0, s] for s ≥ 0.

Proof. The first identity N(F + G) = N(max(F, G)) = N(F) ∩ N(G) follows from the observation that a half-space H is a supporting hyperplane for F + G if and only if it is a supporting hyperplane for F and G separately (and using the dual characterisation of convex sets). The inclusion N(F G) ⊆ N(F) + N(G) follows from the observation that if

{(n_{1}, …, n_{d}) ∈ R^{d} : a_{1}n_{1} + … + a_{d}n_{d} ≤ b}

is a supporting hyperplane for F, and

{(n_{1}, …, n_{d}) ∈ R^{d} : a_{1}n_{1} + … + a_{d}n_{d} ≤ b′}

is a supporting hyperplane for G then

{(n_{1}, …, n_{d}) ∈ R^{d} : a_{1}n_{1} + … + a_{d}n_{d} ≤ b + b′}

is a supporting hyperplane for F G. The identity N(F^{α}) = α · N(F) follows from the observation that H is a supporting half-space for F if and only if α · H is a supporting half-space for F^{α}. □

Remark 2.5. Under mild conditions on F and G (one needs to prevent F, G from oscillating too much), one can in fact obtain the equality N(F G) = N(F) + N(G), but we will not need this stronger statement here.

We have an easy and useful necessary condition for majorisation:

Proposition 2.6 (Majorisation condition). Let F, G : (R^{+})^{d} → R^{+} be functions such that F ≺ G. Then N(F) ⊆ N(G). (In particular, if F and G are comparable, then N(F) = N(G).)

Proof. Observe that every supporting hyperplane for G is automatically a supporting hyperplane for F. The claim follows. □

Example 2.7. In order to have

xy ≺ x^{a} + y^{b}

it is necessary that the interval spanned by (a, 0) and (0, b) contain (1, 1) (which forces a and b to be dual exponents between 1 and infinity); in this case, this necessary condition is also sufficient. (Indeed, we can reverse the above proposition whenever F and G are finite positive linear combinations of monomials with fractional exponents; we leave this as an exercise to the reader.)

3. Dimensional analysis of actions on functions

The preceding discussion concerned estimates P (x) ≤ CQ(x) that involved functions of finitely many non-negative variables x_{1}, …, x_{d}. However, one can easily apply this machinery to tackle more interesting estimates, particularly those involving one or more functions f_{1}, …, f_{k} that lie in various function spaces X_{1}, …, X_{k}. A typical task is to establish a k-linear estimate such as

∥T (f_{1}, …, f_{k})∥_{Y} ≤ C∥f_{1}∥_{X_{1}} … ∥f_{k}∥_{X_{k}}

or perhaps a more nonlinear estimate such as

∥F (f)∥_{Y} ≤ C(1 + ∥f∥_{Y}^{n}).

Let us work very abstractly, and consider very general estimates of the form

F (f_{1}, …, f_{k}) ≤ CG(f_{1}, …, f_{k})

for all “suitably nice” f_{1} ∈ X_{1}, …, f_{k} ∈ X_{k}, where F and G are some positive quantities that depend (possibly in a nonlinear manner) on f_{1}, …, f_{k}, and C > 0 is a constant. Let us write F ≺ G if an estimate of the above form is true for all suitable f_{1}, …, f_{k}.

Of course, at such a level of generality it is hopeless to try to find usable necessary and sufficient conditions for which F ≺ G is true. However, dimensional analysis tools do allow us to cheaply provide useful necessary conditions for such estimates. This is useful because for randomly chosen F, G, the estimate F ≺ G is almost surely false; using dimensional analysis one can winnow out many of the false estimates, thus leaving one with estimates that are more likely to be true. This can save a lot of time by not expending effort on proving false estimates, and can also guard against errors.

Here’s how dimensional analysis works. One needs a number of scaling parameters λ = (λ_{1}, …, λ_{d}) ∈ (R^{+})^{d}. These scaling parameters need to act on the input functions f⃗:= (f_{1}, …, f_{k}), leading to a rescaled version f⃗_{λ} = (f_{1,λ}, …, f_{k,λ}) of the input functions. This action does not need to be a group action of any sort (though it often is); the only thing that is needed is that every acceptable input f⃗ rescales to another acceptable input f⃗_{λ}. Then if we have F ≺ G, then for any fixed f⃗ we clearly have

F (f⃗_{λ}) ≤ CG(f⃗_{λ}) for all λ ∈ (R^{+})^{d}.

In other words, the function λ ↦ F (f⃗_{λ}) must be majorised by the function λ ↦ G(f⃗_{λ}). Applying Proposition 2.6, we conclude a necessary condition for F ≺ G, namely that

N(λ ↦ F (f⃗_{λ})) ⊆ N(λ ↦ G(f⃗_{λ}))

for arbitrary^{1} inputs f⃗. This may seem like a rather strange condition, but in practice (at least for very simple actions λ : f⃗ ↦ f⃗_{λ}) the Newton polytopes N(λ ↦ F (f⃗_{λ})) are easy to compute (and in fact often do not depend on the initial choice of function f⃗) - these polytopes are essentially the dimensions or units of the left and right-hand sides of the estimates with respect to the scaling action. In most cases, even just the one-dimensional Newton polytopes convey quite a bit of information.

Let’s give some concrete examples. The simplest scaling is the homogeneity scaling, where d = k and f⃗_{λ} = (λ_{1}f_{1}, …, λ_{k}f_{k}) is formed by scaling each input function f_{j} by a different scalar λ_{j} > 0. Consider for instance an estimate of the form

∥T (f_{1}, f_{2})∥_{Y} ≤ C∥f_{1}∥_{X_{1}}∥f_{2}∥_{X_{2}},

where T is a bilinear operator. Applying the scaling we obtain

λ_{1}λ_{2}∥T (f_{1}, f_{2})∥_{Y} ≤ Cλ_{1}∥f_{1}∥_{X_{1}}λ_{2}∥f_{2}∥_{X_{2}}.

1Actually, one needs to assume that f⃗ is sufficiently non-degenerate that the quantities F (f⃗_{λ}), G(f⃗_{λ}) are non-vanishing, but this is easy to establish in practice.

The Newton polytopes of both sides are just {(1, 1)}, so dimensional analysis does not prevent estimates like this from being true. On the other hand, an estimate such as

∥T (f_{1}, f_{2})∥_{Y} ≤ C∥f_{1}∥_{X_{1}}∥f_{2}∥_{X_{2}}^{2}

cannot be true, because the Newton polytopes with respect to homogeneity are {(1, 1)} and {(1, 2)} respectively. On the other hand, an estimate such as

∥T (f_{1}, f_{2})∥_{Y} ≤ C(∥f_{1}∥_{X_{1}}^{2} + ∥f_{2}∥_{X_{2}}^{2})

has a chance of being true, because the left-hand side has Newton polytope {(1, 1)}, and the right-hand side has as Newton polytope the interval between (2, 0) and (0, 2).

Homogeneity explains why in estimates such as the Gagliardo-Nirenberg inequality

∥f∥_{L^{q}(R^{d})} ≤ C∥f∥_{L^{p}(R^{d})}^{1−θ}∥∇f∥_{L^{p}(R^{d})}^{θ}

(for suitable values of p, q, θ and d) the exponents 1 − θ and θ have to add up to 1. Roughly speaking, the “number of f’s” on the LHS has to match (or at least be contained in) the “number of f’s on the RHS”.

If the functions f_{1}, …, f_{k} are lying in a Euclidean space R^{d}, then there is a one-parameter dilation scaling available, by setting^{2} f_{i,λ}(x) := f_{i}(\frac{x}{λ}) for all λ ∈ R^{+}. To see how dimensional analysis works, let us apply it to the Sobolev embedding inequality

∥f∥_{L^{q}(R^{d})} ≤ C(∥f∥_{L^{p}(R^{d})} + ∥∇^{k}f∥_{L^{p}(R^{d})})

for certain values of q, p, k, d. Suppose this estimate were true. We fix some test function f and apply the above inequality to the rescaled function f_{λ} to obtain

∥f_{λ}∥_{L^{q}(R^{d})} ≤ C(∥f_{λ}∥_{L^{p}(R^{d})} + ∥∇^{k}f_{λ}∥_{L^{p}(R^{d})}).

The Newton polytopes of both sides can be computed to be \{\frac{d}{q}\} and [\frac{d}{p} − k, \frac{d}{p}]. So we arrive at the scaling condition

\frac{d}{p} − k ≤ \frac{d}{q} ≤ \frac{d}{p}

which is indeed the basic necessary (and sufficient, when 1 < p < q < ∞) condition for Sobolev embedding. Similarly, if we drop the lower order term and consider the endpoint Sobolev embedding

∥f∥_{L^{q}(R^{d})} ≤ C∥∇^{k}f∥_{L^{p}(R^{d})}

then scaling forces \frac{d}{p} − k = \frac{d}{q}.

2This is the scaling which preserves the L^{∞} norm. In some cases it is more convenient computationally to mix this scaling with the homogeneity scaling, and introduce scalings such as f_{i,λ}(x) := \frac{1}{λ^{d/p_{i}}} f_{i}(\frac{x}{λ}), which preserves the L^{p_{i}}(R^{d}) norm of f_{i}. However, if the estimate was already homogeneous (with the same degrees on both sides), this does not affect the conclusions of dimensional analysis.

These scaling computations can also be performed by more traditional dimensional analysis, giving space the units of length L (one can think of L as a length unit, such as metres or feet), and letting f(x) be dimensionless (because the estimate is already consistent with respect to homogeneity, the choice of dimension for f is not actually relevant). Then ∥f∥_{L^{q}(R^{d})} = (∫_{R^{d}} |f(x)|^{q} dx)^{1/q} has units L^{d/q}, ∥f∥_{L^{p}(R^{d})} has units L^{d/p}, and ∥∇^{k}f∥_{L^{p}(R^{d})} has units L^{d/p−k}; this can be used to recover the preceding necessary conditions. With a little practice, these dimensional computations can be done quite quickly. For example, when applied to the Gagliardo-Nirenberg inequality they yield the condition \frac{d}{q} = (1 − θ)\frac{d}{p} + θ(\frac{d}{p} − 1).

In some cases it is useful to consider anisotropic scalings, in which certain directions are treated differently than others; for instance, when dealing with solutions of the Schrödinger equation iu_{t} + Δu = 0, it is common to consider the scaling u_{λ}(t, x) := u(\frac{t}{λ^{2}} , \frac{x}{λ}) as this preserves the property of solving the equation. If we give space the units of L, this amounts to giving time the units of L^{2}. An alternative is to generalise the problem by introducing some additional parameters, for instance studying the more general Schrödinger equation iu_{t} + \frac{ℏ}{2m}Δu = 0 with ℏ, m constants. This leads to a more general (multiparameter) family of scalings, in which time, space, and the constants ℏ, m can be rescaled in a number of ways; this corresponds more closely with the units that physicists assign to various quantities associated with the Schrödinger equation. In this particular example, adding the ℏ and m parameters doesn’t significantly change the analysis, but if ℏ and m also occur at several other places in the hypotheses and conclusions of the problem then the multiparameter scaling analysis can become more useful. (It also brings into view certain asymptotic limits of the problem, in this case the classical limit ℏ → 0 or the massless limit m → 0, which can be instructive to analyse in their own right.)

Now we turn to some less well-known instances of dimensional analysis. Let us start with the analysis of the translation action f_{i,λ}(x) = f_{i}(x + λe_{0}), where e_{0} ∈ R^{d} is some fixed non-zero direction vector. Many estimates, such as the Sobolev and Gagliardo-Nirenberg estimates, are translation-invariant, and as such a dimensional analysis of translation action yields no useful conclusion. However, one can obtain nontrivial necessary conditions once one deals with estimates with weights (such as power weights). Consider for instance the class of Hardy inequalities

∥f∥_{L^{p}(R^{d};|x|^{−s} dx)} ≤ C∥∇f∥_{L^{q}(R^{d})}  (1)

for suitable values of p, d, s, q. We take a nontrivial test function f and apply the above inequality to f_{λ}(x) := f(x + λe_{0}). Then the left-hand side behaves like λ^{−s} as λ → ∞, while the right-hand side stays constant; thus this estimate can only be valid if s ≥ 0. More generally, we see that the net |x| weight on the left-hand side must always be “more negative” than or equal to that on the right-hand side. Note that this analysis breaks down if one imposes restrictions on the functions which prohibit one from using the translation action. For instance, the radial Sobolev inequality

∥|x|^{1/2}f∥_{L^{∞}(R^{3})} ≤ C∥∇f∥_{L^{2}(R^{3})}  (2)

is valid for radial functions f, even though the above translation analysis shows that it cannot possibly be true for general functions. The point here is that the translation of a radial function is not radial.

The frequency space counterpart to the translation action is the modulation action f_{i,λ}(x) = f_{i}(x)e^{2πiλx·e_{0}}, where e_{0} is some fixed non-zero direction vector. This modulation action does not influence L^{p} spaces but can detect the presence of derivatives on the right or left-hand side of an inequality. For instance, it shows that there is no such thing as a “reverse Sobolev inequality”

∥∇f∥_{L^{p}(R^{d})} ≤ C∥f∥_{L^{q}(R^{d})}

for any choice of p, q, d, because if one replaces f(x) by f_{λ}(x) := f(x)e^{2πiλx·e_{0}} and sends λ → ∞ then the left-hand side goes to infinity while the right-hand side stays constant. A similar example shows that one can only hope to establish a product estimate such as

∥fg∥_{W^{s,p}(R^{d})} ≤ C∥f∥_{W^{s_{1},p_{1}}(R^{d})}∥g∥_{W^{s_{2},p_{2}}(R^{d})}

if we have s ≤ min(s_{1}, s_{2}). Roughly speaking, the modulation analysis asserts that the number of derivatives on the left-hand side cannot exceed the number of derivatives on the right-hand side. A variant of the above analysis (in which two functions are modulated in equal and opposite directions) shows that the product of two functions in negative Sobolev spaces cannot be estimated in any reasonable norm.

We now turn to a somewhat stranger example of an action which can lead to some useful necessary conditions. Given any integer λ > 1 and a function f(x) on R^{d}, consider the replication f_{λ}(x) of f defined as

f_{λ}(x) = f(x − x_{1}) + f(x − x_{2}) + … + f(x − x_{λ})

where x_{1}, x_{2}, …, x_{λ} are extremely widely separated points in R^{d}; we leave the exact description of these x_{i} somewhat vague as it will not be important for us. This is not a group action, but nevertheless can be used in similar frameworks to above. For instance, it quickly yields the principle “the higher exponents are always on the left” (possibly due to Littlewood?); more precisely, if T is any non-trivial translationinvariant linear operator on R^{d}, then the only way that an estimate of the form

∥T (f)∥_{L^{q}(R^{d})} ≤ C∥f∥_{L^{p}(R^{d})}

can be true is if q ≥ p. To see this, take an f for which T f is not identically zero, and apply the above estimate to the replication f_{λ} of f. If the x_{1}, …, x_{λ} are chosen sufficiently far apart, then the left-hand side grows like λ^{1/q} and the right-hand side like λ^{1/p}; setting λ → ∞ yields the result. A similar argument applies for bilinear operators (f, g) ↦ T (f, g) which are translation invariant, thus T (f, g)(x−x_{0}) = T (f(· − x_{0}), g(· − x_{0}))(x); it implies that a bilinear estimate

∥T (f, g)∥_{L^{r}(R^{d})} ≤ C∥f∥_{L^{p}(R^{d})}∥g∥_{L^{q}(R^{d})}

can only be true if \frac{1}{r} ≤ \frac{1}{p} + \frac{1}{q}; compare this with the conditions for Hölder’s inequality.

4. Uses of dimensional analysis

There are several ways in which dimensional analysis can be useful. Firstly, as emphasised above, they give a necessary condition for certain types of estimates to be true. This can save time by eliminating false conjectures before one spends any nontrivial amount of effort trying to prove them. Secondly, they can help guard against errors in a proof; if an estimate halfway through a proof fails to be consistent with dimensional analysis, one can quickly recheck key junctures in earlier parts of the proof and locate the first place where the estimates became inconsistent with dimensional analysis; this will locate the error rather quickly. (In some cases, it also allows for the author to unconsciously correct for things like sign errors or other arithmetic errors which have somehow corrupted various exponents, as the author could later “fill in” those exponents via dimensional analysis rather than work them out by a long sequence of computations.)

There are however more sophisticated ways to use dimensional analysis. Firstly, as briefly mentioned above, it allows one to isolate various scaling limits of a desired estimate, where one replaces the input functions f with f_{λ} and then investigates the limit as λ → ∞ (possibly after dividing out a power of λ from both sides in order to make the limit converge). These limits are logical consequences of the original estimate, and are often simpler to prove (because certain terms in the original estimate will asymptotically vanish). They are thus excellent model problems to study. For instance, suppose one wants to prove the inhomogeneous Sobolev space product estimate

∥fg∥_{W^{s,p}(R^{d})} ≤ C∥f∥_{W^{s_{1},p_{1}}(R^{d})}∥g∥_{W^{s_{2},p_{2}}(R^{d})}

where s, s_{1}, s_{2} ≥ 0. Scaling considerations force

\frac{d}{p} − s ≥ (\frac{d}{p_{1}} − s_{1}) + (\frac{d}{p_{1}} − s_{2}).

Now suppose that we are critical with respect to scaling:

\frac{d}{p} − s = (\frac{d}{p_{1}} − s_{1}) + (\frac{d}{p_{1}} − s_{2}).

Now replace f, g by their scaled counterparts f_{λ}(x) := f(x/λ), g_{λ} := g(x/λ). After dividing out by the common power λ^{d/p−s} and taking limits as λ → ∞, we infer that the inhomogeneous Sobolev space estimate implies the homogeneous counterpart:

∥fg∥_{Ẇ^{s,p}(R^{d})} ≤ C∥f∥_{Ẇ^{s_{1},p_{1}}(R^{d})}∥g∥_{Ẇ^{s_{2},p_{2}}(R^{d})}.

This suggests that one should try to prove the homogeneous estimate first.

If the original estimate was scale-invariant, then of course the scaling limit of the estimate will be equal to itself, which is rather uninteresting. More interesting things happen if the original estimate had a critical (scale-invariant) component as well as a subcritical (nonscale-invariant) component; then in the scaling limit the subcritical components disappear leaving only the critical components (as we saw above). In some cases this makes the scaling limit appear, somewhat paradoxically, to be stronger than the original estimate. Suppose for instance one is trying to prove the endpoint Sobolev embedding

∥f∥_{L^{6}(R^{3})} ≤ C(∥f∥_{L^{2}(R^{3})} + ∥∇f∥_{L^{2}(R^{3})}).  (3)

Taking the scaling limit, we obtain the apparently stronger estimate

∥f∥_{L^{6}(R^{3})} ≤ C∥∇f∥_{L^{2}(R^{3})}.  (4)

Thus these two estimates are equivalent. Now one might think naively that in trying to prove this estimate, one should focus on the weakerlooking statement (3); there are more terms on the right-hand side (one gets to control f as well as ∇f in L^{2}) and so the estimate should be “easier to prove”. This is however incorrect. The scaling analysis shows that the subcritical term ∥f∥_{L^{2}(R^{3})} is useless for the task of controlling the left-hand side; any purported proof of (3) that tries to involve both ∥f∥_{L^{2}} and ∥∇f∥_{L^{2}} can be rescaled to produce a simpler proof which only involves ∥∇f∥_{L^{2}}, which basically means that the ∥f∥_{L^{2}} term cannot actually be non-trivially used for this problem - it is a “red herring”. Leaving this term in, then, will slow down the task of proving the estimate, as the prover may be tempted to try a number of attacks on the estimate which non-trivially involve the ∥f∥_{L^{2}} norm, being unaware that the scaling analysis conclusively demonstrates that such attacks are doomed to failure from the start.

In short, scaling limits can be used as a “razor” to weed out useless hypotheses from a problem. Dually, they can also point towards which conclusions are likely to be harder to prove; as a general rule, a term which survives in the scaling limit is going to be harder to estimate than a term which went to zero in the scaling limit.

Here is another example. Consider the Carleson maximal operator

Cf(x) := sup_{N} | ∫_{−∞}^{N} f̂(ξ)e^{2πixξ} dξ|.

It is known (basically from the work of Kolmogorov) that this operator is not of weak-type (1, 1):

∥Cf∥_{L^{1,∞}(R)} ≰ C∥f∥_{L^{1}(R)}.  (5)

Now someone familiar with Calderón-Zygmund theory might venture the conjecture that perhaps the situation improves if one replaces L^{1} with the stronger Hardy space H^{1}, thus one might hope to establish

∥Cf∥_{L^{1,∞}(R)} ≤ C∥f∥_{H^{1}(R)}.  (6)

At first glance this looks promising; for instance, the Riesz projection P_{−}f(x) := ∫_{−∞}^{0} f̂(ξ)e^{2πixξ} dξ is known to map H^{1} to L^{1}, even though it is not bounded from L^{1} to itself. However, this overlooks the fact that the Carleson operator (in marked contrast to Calderón-Zygmund operators such as P_{−}) is modulation-invariant: if f_{λ}(x) := e^{2πiλx}f(x), then we have

(Cf)_{λ} = C(f_{λ}).

Now suppose the estimate (6) were true, then of course we would have

∥Cf_{λ}∥_{L^{1,∞}(R)} ≤ C∥f_{λ}∥_{H^{1}(R)}

for all λ. The modulation invariance of C (and of L^{1,∞}(R)) shows that the left-hand side is independent of λ. What about the right-hand side? One can compute that lim_{λ→∞} ∥f_{λ}∥_{H^{1}(R)} is essentially (up to constants) the L^{1} norm of f; the Hardy space H^{1} degenerates to the Lebesgue space L^{1} in the high frequency limit. Thus we see that (6) cannot be true, because the high frequency limit of this estimate contradicts (5). Thus the modulation symmetry shows us that there is no advantage to be gained by refining the Lebesgue space control on f to the Hardy space control. (However, the situation changes if one breaks or spends the modulation symmetry of the problem; indeed, in the modern proofs of Carleson’s theorem (essentially, the L^{p} boundedness of C), one decomposes f into various “trees”, each of which has a certain “centre frequency”; when considering the contribution of a single tree, one can spend the modulation symmetry to move that centre frequency to the origin, at which point Hardy spaces become useful again.)

We have seen how dimensional analysis can prune away useless hypotheses and highlight the most difficult conclusions of a proposed estimate. It can also prune away certain proof strategies. One basic principle is that if the Newton polytopes of the left-hand and right-hand sides of an estimate agree, then all intermediate sides must also agree as well. For instance, suppose one wants to prove a linear estimate such as

∥T f∥_{Y} ≤ C∥f∥_{X}.

Presumably the proof will consist of a long series of intermediate inequalities,

∥T f∥_{Y} ≤ … ≤ … ≤ ∥f∥_{X},

or perhaps the inequalities will be arranged in a tree structure rather than in linear order (e.g. one may split into several cases, etc.) Now note that with respect to the homogeneity scaling f ↦ λf, both the left and right-hand side are homogeneous of degree 1 (i.e. f appears exactly once). Unless one breaks the homogeneity scaling (see below), then this forces all intermediate steps in the proof to also be homogeneous of degree 1. For instance, we can rule out trying to begin the proof by using the estimate

∥T f∥_{Y} ≤ \frac{1}{2}(1 + ∥T f∥_{Y}^{2} ).  (7)

This inequality is true, but there is no chance that one can continue this argument and hope to bound \frac{1}{2}(1 + ∥T f∥_{Y}^{2} ) by C∥f∥_{X}, as this would be inconsistent with homogeneity dimensional analysis. So one should only use estimates and techniques which preserve homogeneity. Similarly, one cannot hope to derive Sobolev embedding inequalities from Hardy’s inequalities, basically because the former are translationinvariant while the latter are not.

The above arguments are only valid so long as one maintains the required symmetry. However it is sometimes quite beneficial to break the symmetry (or more precisely, to spend it), most commonly by achieving some sort of useful normalisation. For instance, if one wanted to prove the inequality ∥T f∥_{Y} ≤ C∥f∥_{X}, then one could spend the homogeneity symmetry to normalise ∥f∥_{X} = 1. Once one does so, it becomes legitimate to use estimates such as (7), as the objection to that estimate has disappeared. Thus we see that by spending a symmetry, new proof techniques become available^{3}. This is not always an unmitigated blessing - it may suggest more ways to correctly prove a result, but it may also suggest more ways to incorrectly prove the result. The spending of symmetries should be done carefully - the normalisation should be natural, and the benefit tangible, in order to compensate for the loss of symmetry.

Here is an example in which symmetry is mis-spent. Suppose one wants to prove the Sobolev inequality (4). One might hope to use a dyadic decomposition,

3Of course, one could reverse the normalisation and recover the same proof techniques in the un-normalised setting; for instance, the un-normalised analogue of (7) is ∥T f∥_{Y} ≤ \frac{1}{2}(∥f∥_{X} + \frac{∥T f∥_{Y}^{2}}{∥f∥_{X}} ). But such un-normalised estimates might not be immediately obvious to the person trying to prove the original estimate without first passing to the normalised setting.

∥f∥_{L^{6}(R^{3})} = (∑_{n∈Z} ∥f∥_{L^{6}({x:2^{n}<|x|≤2^{n+1}})}^{6})^{1/6},

estimate each of the dyadic shell components separately, and then sum. This might then lead to an ostensibly easier-looking model problem of showing that

∥f∥_{L^{6}({x:2^{n}<|x|≤2^{n+1}})} ≤ C∥∇f∥_{L^{2}(R^{3})}  (8)

uniformly in n. At this point one might be cheered by the fact that scale invariance allows one to normalise n = 0, thus the model problem becomes a local Sobolev inequality rather than a global one:

∥f∥_{L^{6}({x:1<|x|≤2})} ≤ C∥∇f∥_{L^{2}(R^{3})}.  (9)

One might hope that this inequality (9), which appears to be strictly easier than (4), would be thus easier to prove, and then one could hope to rescale back the n parameter and somehow sum in n to recover the full estimate. However, this is a false simplification, as (9) is just as hard to prove as the original estimate (4), due to the scaling and translation invariances of the Sobolev embedding estimate. Indeed, we already observed that (9) is equivalent to (8). If we then replace f by f_{n}(x) := f(x − 2^{n+0.5}e_{1}) in (8), where e_{1} is a standard basis vector, and take limits as n → ∞, we quickly see that in the asymptotic limit (8) reverts back to (4). The basic problem here is that because the original Sobolev embedding estimate was translation invariant, there is no preferred spatial origin; one point in space is just as important as any other. So performing a dyadic decomposition around an arbitrarily chosen coordinate origin 0 is artificial and pointless. The situation should be contrasted with that of establishing Hardy’s inequality (1) or the radial Sobolev inequality, where there is no translation symmetry, the origin already plays a distinguished role, and dyadic decomposition around that origin is a very good idea. Thus one can see that the absence or presence of symmetries can guide one towards the correct method of proof for various estimates and eliminate incorrect method of proofs.

For another example, any treatment of the L^{p} theory of the Carleson maximal operator or the bilinear Hilbert transform should not proceed by exploiting any tools that give the frequency origin a privileged role, such as the use of Littlewood-Paley theory, H^{1} and BMO, Calderón-Zygmund theory, etc., because these operators are modulation invariant while the tools are not. Indeed the theory that has been set up to treat these operators can be viewed as a modulation-invariant generalisation of Calderón-Zygmund theory; though after a while one identifies certain “centre frequencies” of “trees” in f and spends the modulation invariance to reduce matters to classical Calderón-Zygmund theory again.

The decision on when to spend symmetries is analogous to the debate in geometry as to whether to use coordinates; by introducing coordinates, one allows for more explicit computational power, but at the cost of giving up the coordinate transformation symmetry. If both the hypotheses and conclusion of the problem are coordinate-invariant, then by using coordinates one can run into the danger of losing the coordinate invariance halfway through the proof, from which one cannot recover to finish the proof. However, using coordinates is unquestionably useful if one actually spends the coordinate symmetry to establish some additional normalisation, such as a gauge fixing. My personal opinion is that one should not be dogmatic either about using coordinates or avoiding them; like any other mathematical tool, they are useful if applied correctly and with definite purpose, and are useless or worse when applied incorrectly.

Department of Mathematics, UCLA, Los Angeles, CA 90024

  E-mail address: tao@@math.ucla.edu