LECTURE NOTES 2 FOR 254A

TERENCE TAO

1. Littlewood-Paley projections

This week we will introduce the Littlewood-Paley decomposition, which is a very basic way to carve up the phase plane. After describing its basic properties, we shall apply it to a class of function spaces known as Sobolev spaces. During this week and the next we shall prove such results as the Sobolev embedding and trace lemmas, the Gagliardo-Nirenberg inequality, and the multiplication laws for Sobolev spaces.

There is a certain amount of flexibility in how one sets up the Littlewood-Paley decomposition^{1} on R^{n}, but one standard way is as follows. Let ϕ(ξ) be a real radial bump function supported on {ξ ∈ R^{n} : |ξ| ≤ 2} which equals 1 on {ξ ∈ R^{n} : |ξ| ≤ 1. Let ψ(t) be the function

ψ(ξ) := ϕ(ξ) − ϕ(2ξ).

Thus ψ is a bump function supported on the annulus {1/2 ≤ |ξ| ≤ 2}. By construction we have

∑_{k∈Z} ψ(ξ/2^{k}) = 1

for all ξ ≠ 0. Thus we can partition unity into the functions ψ(ξ/2^{k}) for integers k, each of which is supported on an annulus of the form |ξ| ∼ 2^{k}.

We now define the Littlewood-Paley projection operators^{2} P_{k}, P_{≤k} by

P̂_{k}f(ξ) = ψ(ξ/2^{k})f̂(ξ)
{ \atop P_{≤k}f}(ξ) = ϕ(ξ/2^{k})f̂(ξ).

Informally, P_{k} is a frequency projection to the annulus {|ξ| ∼ 2^{k}}, while P_{≤k} is a frequency projection to the ball {|ξ| ≲ 2^{k}}. We shall sometimes write P_{<k} for P_{≤k−1}.

Observe that P_{k} = P_{≤k} − P_{<k}. Also, if f is an L^{2} function then P_{≤k}f → 0 in L^{2} as k → −∞, and P_{≤k}f → f in L^{2} as k → +∞ (this is an easy consequence of Parseval’s theorem). By telescoping the series, we thus have the Littlewood-Paley decomposition

1The reader may take n = 1 if desired (which is convenient for e.g. drawing phase planes); there is almost no difference between one-dimensional Littlewood-Paley theory and the higherdimensional theory.

2Strictly speaking, these are not quite projections; even though they are self adjoint, they do not quite square to themselves, because we chose ϕ to be a smooth cutoff rather than a rough one. However, the operator P_{k}P_{k} is of the same form as P_{k}, and similarly for P_{≤k}, and so it is still quite reasonable to think of these operators as (smoothed out) projection operators.

f = ∑_{k∈Z} P_{k}f

for all f ∈ L^{2}, where the summation is in the L^{2} sense^{3}

This decomposition takes a single function and writes it as a superposition of a countably infinite family of functions P_{k}f, each one of which has frequency of magnitude roughly 2^{k}. Lower values of k represent low frequency components of f; higher values represent high frequency components.

Indeed, from the basic facts about the Fourier transform we have the representation

P_{≤k}f(x) = f ∗ (2^{nk}ϕ̂(2^{k}·)) = ∫ f(x + 2^{−k}y)ϕ̂(y) dy.  (1)

Note that ϕ̂(y) is a Schwartz function with total mass ∫ ϕ̂(y) dy = ϕ(0) = 1. Thus the function P_{≤k}f is an average of f localized to physical scales ≲ 2^{−k}. In particular, we expect P_{≤k}f to be essentially constant at scales ≪ 2^{−k}; this is consistent with the uncertainty principle.

What does a single function P_{k}f look like? Since P_{k}f = P_{≤k+2}P_{k}f, we see from (1) that we have the self-reproducing formula

P_{k}f(x) = ∫ P_{k}f(x + 2^{−k−2}y)ϕ̂(y) dy.  (2)

Thus P_{k}f is essentially constant at physical scales ≪ 2^{−k}. On the other hand we have P_{≤k−2}P_{k}f = 0, so from (1) we have

∫ P_{k}f(x + 2^{−k+2}y)ϕ̂(y) dy = 0 for all x ∈ R^{n}.

This roughly asserts that P_{k}f has mean zero at scales ≲ 2^{−k+2}: To reconcile these two properties, we see that on each ball of radius O(2^{−k}), the function P_{k}f is smooth at scales ≪ 2^{−k} and contains about O(1) oscillations.

Another indication of oscillation is given by the mean zero property

∫ P_{k}f(x) dx = P̂_{k}f(0) = 0.

In fact all moments of P_{k}f vanish:

∫ (2πix)^{α}P_{k}f(x) dx = (∂_{ξ}^{α}P̂_{k}f)(0) = 0.

Later on we shall compare this informal vision of affairs with the rigorous situation of the Haar system, which is kind of a Walsh analogue of the Littlewood-Paley decomposition.

3Actually, this decomposition works for just about any locally integrable function which has some decay at infinity, and one usually has all the convergence properties of the summation that one needs. In many applications one can make the a priori assumption that f is Schwartz, in which case the convergence is (for instance) uniform. However, if f does not decay, then this formula fails. For instance, if f ≡ 1, then all the projections P_{k}f vanish.

The Littlewood-Paley pieces P_{k}f of f react very well with derivatives. Roughly speaking, we have

∇P_{k}f ≈ 2^{k}P_{k}f.  (3)

Informally this makes sense, since ∇ is multiplication by 2πiξ on the frequency side, and |ξ| ∼ 2^{k} on the support of P_{k}f.

A more precise statement is as follows.

Lemma 1.1. Let k be an integer, and let f be a function with Fourier support in the annulus {2^{k−1} ≤ |ξ| ≤ 2^{k+1}}. Then we have

∥∇f∥_{p} ∼ 2^{k}∥f∥_{p}

for all 1 ≤ p ≤ ∞. In particular, we have ∥∇P_{k}f∥_{p} ∼ 2^{k}∥P_{k}f∥_{p} for arbitrary f.

Proof We have the analogue of (2):

f(x) = ∫ f(x + 2^{−k−2}y)ϕ̂(y) dy.

Differentiating this we have

∇f(x) = ∫ ∇_{x}f(x + 2^{−k−2}y)ϕ̂(y) dy.

Integrating by parts, this becomes

∇f(x) = 2^{k+2} ∫ f(x + 2^{−k−2}y)∇ϕ̂(y) dy.

Since ∇ϕ̂ is a bump function, we see from Minkowski’s inequality that

∥∇f∥_{p} ≲ 2^{k}∥f∥_{p}

To obtain the reverse implication, we need to invert ∇. Observe that

ϕ(ξ/2^{k+2}){ \atop ∂_{x_{j}}f}(ξ) = 2πiξ_{j}f̂(ξ)

and hence that

f̂(ξ) = ∑_{j=1}^{n} ϕ(ξ/2^{k+2})\frac{ξ_{j}}{2πi|ξ|^{2}} { \atop ∂_{x_{j}}f}(ξ).

Inverting the Fourier transform, we obtain a way of recovering f from ∇f:

f = 2^{−k} ∑_{j=1}^{n} K_{k,j} ∗ ∂_{x_{j}}f  (4)

where

K_{k,j}(x) := 2^{k} ∫ ϕ(ξ/2^{k+2})\frac{ξ_{j}}{2πi|ξ|^{2}} e^{2πix·ξ} dξ

= 2^{nk} ∫ ϕ(ξ/4)\frac{ξ_{j}}{2πi|ξ|^{2}} e^{2πi2^{k}x·ξ} dξ.

Replacing everything by absolute values we obtain the bound

|K_{k,j}(x)| ≲ 2^{nk}.

By repeated integration by parts we also obtain the bounds

|K_{k,j}(x)| ≲ 2^{nk}|2^{k}x|^{−s}

for all s > 0, with the constants depending on s (cf. last week’s notes). Thus K_{k,j} is an approximation to the identity at scales 2^{−k}. Thus by Minkowski’s inequality again we may obtain

∥∇f∥_{p} ≳ 2^{k}∥f∥_{p}

and we are done.

From (3) we have the heuristic relationship

∇ ∼ ∑_{k} 2^{k}P_{k};  (5)

in other words, one can morally split a derivative as a linear combination of Littlewood-Paley operators. This heuristic asserts that the effect of a derivative on a function f is to accentuate the high frequencies and diminish the low frequencies, since each frequency component P_{k}f is essentially multiplied by its own frequency 2^{k}. One can apply similar heuristics to other differentiation operators such as the Laplacian Δ, as well as integration operators such as the inverse Laplacian Δ^{−1}^{4}.

We now study how the Littlewood-Paley pieces P_{k}f (or P_{≤k}f) of a function are related to the function itself. Specifically, we are interested in how the L^{p} behavior of the P_{k}f relate to the L^{p} behavior of f.

From (1) and Minkowski’s inequality we have (for 1 ≤ p ≤ ∞)

∥P_{≤k}f∥_{p} ≤ ∫ ∥f(x + 2^{−k}y)∥_{p}|ϕ̂(y)| dy ≲ ∥f∥_{p}.

Thus P_{≤k}f does not get any bigger than f itself as measured in L^{p} (or indeed in any translation invariant Banach space). This makes sense, since P_{≤k}f is an average of f. Similarly ∥P_{k}f∥_{p} ≲ ∥f∥_{p}. On the other hand, we have f = ∑_{k} P_{k}f. So by the triangle inequality we thus have the cheap Littlewood-Paley inequality

sup_{k} ∥P_{k}f∥_{p} ≲ ∥f∥_{p} ≲ ∑_{k} ∥P_{k}f∥_{p}.  (6)

This inequality was ridiculously easy to prove, yet it is already quite powerful. Generally it is adequate for one’s purposes as long as one has an epsilon number of derivatives to spare somewhere^{5}. This comment will seem less cryptic as time goes by, but I’ll illustrate one application right away:

4However, to get both sides of the analogue of (3) for such operators one needs to assume that the symbol of these operators are elliptic. For instance, for a second-order constant co-efficient operator such as ∑_{j,j′} c_{j,j′}∂_{x_{j}} ∂_{x_{j′}}, one can obtain both sides of the analogue of (3) provided one assumes the ellipticity assumption | ∑_{j,j′} c_{j,j′}ξ_{j}ξ_{j′}| ∼ |ξ|^{2} for all ξ ∈ R^{n}.

5It is often said that an inequality such as the cheap Littlewood-Paley inequality is “sharp up to a logarithm”. To explain this, suppose that for some reason we only needed to consider frequencies in the range a ≲ |ξ| ≲ A for some 0 < a < A < ∞. For instance a and A could be the lowest and highest physically attainable frequency in the universe. Then the number of dyadic scales k which are relevant is only O(log_{2}(A/a)). Thus the left and right sides of (6) differ by at most a factor of O(log(A/a)). In many applications in real life, or those in which one can lose an epsilon of derivatives (which corresponds to something like (A/a)^{ε}, this logarithmic loss is quite acceptable. This principle has been stated rather succinctly by Coifman as “As a first approximation, log ∞ = O(1)”.

Lemma 1.2 (Non-endpoint Sobolev embedding). Let 1 ≤ p < q ≤ ∞ be such that

\frac{1}{p} − \frac{1}{n} > \frac{1}{q}.

Then we have

∥f∥_{L^{q}(R^{n})} ≲ ∥f∥_{L^{p}(R^{n})} + ∥∇f∥_{L^{p}(R^{n})}  (7)

for all functions f in R^{n} for which the right-hand side is finite. (The implicit constant depends on p, q, n).

Proof By the usual limiting arguments we may assume that f is Schwartz. Let’s denote the right-hand side of (7) by X. From (6) we have

∥P_{k}f∥_{p} ≲ X

for all k. We also have

∥∇P_{k}f∥_{p} ≲ X.

By Lemma 1.1 we thus have

∥P_{k}f∥_{p} ≲ 2^{−k}X.

Combining the two we thus have

∥P_{k}f∥_{p} ≲ min(1, 2^{−k})X.

Note how the first component of X is good at controlling low frequencies k ≤ 0, while the second component of X is good at controlling high frequencies k ≥ 0.

Now P_{k}f has Fourier support in |ξ| ∼ 2^{k}. Since q > p, we can then apply Bernstein’s inequality (from last week’s notes) to obtain

∥P_{k}f∥_{q} ≲ 2^{( \frac{1}{p} − \frac{1}{q})nk} min(1, 2^{−k})X.

Applying the cheap Littlewood-Paley inequality (6) again we obtain

∥f∥_{q} ≲ ∑_{k} 2^{( \frac{1}{p} − \frac{1}{q})nk} min(1, 2^{−k})X.

If you take a plot of 2^{( \frac{1}{p} − \frac{1}{q})nk} min(1, 2^{−k}) as a function of k, it reaches a maximum of 1 at k = 0 and decays exponentially as k → ±∞ because of our assumptions on p, q. So we can sum this and obtain the desired conclusion.

The above proof uses a very useful strategy in proving any sort of estimates involving derivatives: first prove it for a single Littlewood-Paley piece P_{k}f (i.e. assume all frequencies ξ are of magnitude ∼ 2^{k}). If you can’t even prove it for a single piece, there is very little chance you can prove it for the whole function f. Once you see what’s going on for a single piece, add things up (either by (6), or by more sophisticated estimates). This method works surprisingly well in a large number of situations (see e.g. the exercises for this week).

The Sobolev embedding theorem also holds at the endpoint \frac{1}{p} − \frac{1}{n} = \frac{1}{q} (as long as p ≠ 1 and q ≠ ∞), and in this case we don’t need the ∥f∥_{L^{p}} component:

Lemma 1.3 (Endpoint Sobolev embedding). Let 1 < p < q < ∞ be such that

\frac{1}{p} − \frac{1}{n} = \frac{1}{q}.

Then we have

∥f∥_{L^{q}(R^{n})} ≲ ∥∇f∥_{L^{p}(R^{n})}  (8)

for all functions f in R^{n} for which the right-hand side is finite. (The implicit constant depends on p, q, n).

Proof We first prove the weak-type estimate

∥f∥_{L^{q,∞}(R^{n})} ≲ ∥∇f∥_{L^{p}(R^{n})}  (9)

and show how to bootstrap this to the strong-type estimate at the end. By scaling we may take ∥∇f∥_{p} = 1.

We wish to prove that

|{|f| > λ}| ≲ λ^{−q}

for all λ > 0. We follow the Littlewood-Paley philosophy, and begin with the problem of estimating

|{|P_{k}f| > λ}|

for some integer k. First of all, by Chebyshev we have

|\{|P_{k}f| > λ\}| ≲ \frac{∥P_{k}f∥_{p}^{p}}{λ^{p}} ∼ \frac{(2^{−k}∥∇P_{k}f∥_{p})^{p}}{λ^{p}} ≲ 2^{−kp}λ^{−p}.

On the other hand, by Bernstein’s inequality we also have

∥P_{k}f∥_{∞} ≲ 2^{nk/p}∥P_{k}f∥_{p} ≲ 2^{nk/p}2^{−k}.  (10)

Thus {|P_{k}f| > λ} vanishes unless λ ≲ 2^{nk/p}2^{−k}. But then we have

2^{−kp}λ^{−p} ≲ 2^{−kp}λ^{−q}(2^{nk/p}2^{−k})^{q−p}
= λ^{−q}

by the definition^{6} of q. Thus we can prove the weak-type inequality for a single Littlewood-Paley piece P_{k}f.

To prove the weak-type inequality for the entire function f, we note in the above argument that for a fixed λ there is a critical value k_{λ} of k, defined (up to an O(1) error) by

λ ∼ 2^{nk_{λ}/p}2^{−k_{λ}}.

From our hypotheses we have p < n, so that the right-hand side of (10) is increasing in k. We can therefore sum and obtain

∥P_{≤k_{λ}−C}f∥_{∞} < λ/2  (11)

for some sufficiently large constant C. On the other hand, we have

∥P_{k}f∥_{p} ≲ 2^{−k}

6It may seem like magic that the powers of k magically cancelled, but this is to be expected by scaling or dimensional analysis considerations. If one gives physical space the units of length, so that frequency space has the units of length^{−1}, then the relationship 1/q = 1/p − 1/n is forced by dimensional analysis.

so by summing we obtain

∥P_{>k_{λ}−C}f∥_{p} ≲ 2^{−k_{λ}}

and so by Chebyshev and the previous computation

|{|P_{>k_{λ}−C}f| ≥ λ/2}| ≲ 2^{−k_{λ}p}λ^{−p} ∼ λ^{−q}.

Combining this with (11) we obtain the weak-type estimate.

Now to get the strong-type estimate from the weak-type estimate. In principle this is immediate from the Marcinkiewicz interpolation theorem, but the ∇ on the right makes things a little tricky. Morally one should be able to rewrite (9) as

∥∇^{−1}g∥_{L^{q,∞}(R^{n})} ≲ ∥g∥_{L^{p}(R^{n})}

where g = ∇f, in which case the Marcinkiewicz interpolation theorem does apply, however there is a catch because ∇ is not surjective. To get around this we need to extend the ∇^{−1} operator to all n-tuples of functions (not just those which are gradients). This can be done e.g. by the operator Δ^{−1}∇·, which inverts ∇. By repeating the above arguments we see that

∥Δ^{−1}∇ · g∥_{L^{q,∞}(R^{n})} ≲ ∥g∥_{L^{p}(R^{n})}

for all n-tuples of Schwartz functions g, and so by interpolation we can bootstrap the weak-type estimate to a strong type estimate, and we are done.

As the name suggests, the cheap Littlewood-Paley inequality is not the sharpest statement one can make connecting the L^{p} norms of P_{k}f with those of f. One can already see this when p = 2, in which case we have

∥f∥_{2} ∼ (∑_{k} ∥P_{k}f∥_{2}^{2})^{1/2}  (12)

(why is this stronger than (6)?). To see this, square both sides and take Plancherel to obtain

∫ |f̂(ξ)|^{2} dξ ∼ ∑_{k} ∫ |ψ(ξ/2^{k})|^{2}|f̂(ξ)|^{2} dξ.

Observe that for each ξ ≠ 0 there are only two values of ψ(ξ/2^{k}) which does not vanish, and that these two add up to 1. The claim follows.

Another way to rewrite (12) is

∥f∥_{2} ∼ ∥(∑_{k} |P_{k}f|_{2}^{2})^{1/2}∥_{2}.  (13)

The quantity (∑_{k} |P_{k}f|_{2}^{2})^{1/2} is known as the Littlewood-Paley square function and is sometimes denoted |Sf|. I’ll reserve the notation Sf for the vector-valued function

Sf := (P_{k}f)_{k∈Z};

thus |Sf| is the magnitude of Sf in the l^{2} norm.

We can now give the true Littlewood-Paley inequality:

Proposition 1.4 (Littlewood-Paley inequality). For any 1 < p < ∞, we have

∥Sf∥_{p} ∼ ∥f∥_{p}

with the implicit constant depending on p.

In order to prove Proposition 1.4 we will need to quickly review some Calderón-Zygmund theory.

Definition 1.5. A Calderón-Zygmund operator T is a linear operator^{7} on R^{n} of the form

T f(x) = ∫ K(x, y)f(y) dy

for some (possibly distributional, possibly matrix valued) kernel K which obeys the bounds

|K(x, y)| ≲ \frac{1}{|x − y|^{n}}  (14)

and

|∇K(x, y)| ≲ \frac{1}{|x − y|^{n+1}}  (15)

for all x ≠ y. We also require that T be bounded on L^{2}.

The function 1/|x|^{n} is just barely non-integrable on R^{n}, so that T just barely fails to automatically be bounded on L^{2} (or in fact on L^{p}, 1 ≤ p ≤ ∞). Indeed, if 1/|x|^{n} were integrable, then we would have

∥T f∥_{p} ≲ ∥ ∫ |f(x − y)|/|y|^{n} dy∥_{p} ≲ ∫ ∥f∥_{p}/|y|^{n} dy ≲ ∥f∥_{p}.

Because 1/|x|^{n} is just barely singular, an operator such as T is often referred to as a singular integral.

The classic example of a Calderón-Zygmund kernel is the Hilbert transform

Hf(x) := p.v. ∫ \frac{1}{x − y} f(y) dy

on R. This operator is a Fourier multiplier with symbol −πisgn(ξ) (exercise!), so is bounded on L^{2} by Plancherel.

A basic result of Calderón and Zygmund is that

Lemma 1.6. If T is a Calderón-Zygmund operator, then T is bounded on all L^{p}, 1 < p < ∞.

We defer the proof of this lemma to an Appendix, and return to the proof of Proposition 1.4. We observe that S is a vector-valued Calderón-Zygmund operator with vector-valued kernel

K(x, y) := (2^{nk}ψ̂(2^{k}(x − y)))_{k∈Z}.

7There are more general notions of a Calderón-Zygmund kernel, but we shall not need them in this course.

Indeed, the decay properties (14), (15) are easily verified since ψ̂ is a Schwartz function. The L^{2} boundedness properties come from (13). From Lemma 1.6 we thus have one half of Proposition 1.4, namely that

∥Sf∥_{p} ≲ ∥f∥_{p}

for all 1 < p < ∞. By duality we also have

∥S^{∗}f∥_{p} ≲ ∥f∥_{p}.

By untangling this, we see that

∥ ∑_{k} P_{k}f_{k}∥_{p} ≲ ∥(∑_{k} |f_{k}|^{2})^{1/2}∥_{p}.

Similarly we have

∥ ∑_{k} P̃_{k}f_{k}∥_{p} ≲ ∥(∑_{k} |f_{k}|^{2})^{1/2}∥_{p}

where P̃_{k} := P_{≤k+2} − P_{≤k−2}.

We apply this with f_{k} := P_{k}f. Since P̃_{k}P_{k} = P_{k}, we obtain

∥f∥_{p} ≲ ∥(∑_{k} |P_{k}f|^{2})^{1/2}∥_{p}

which is the other half of Proposition 1.4.

The Littlewood-Paley inequality can be viewed as an assertion of the form

| ∑_{k} P_{k}f| ≈ (∑_{k} |P_{k}f|^{2})^{1/2}.

Note that such a claim would be false if all the P_{k}f were scalar multiples of each other. The point is that the P_{k}f behave very differently (each P_{k}f oscillates at a frequency ∼ 2^{k}). There is a general principle which asserts that if a collection of functions f_{1}, …, f_{k} oscillate at very different rates (so that the phase of one function f_{i} is essentially independent of the phase of any other function), then | ∑_{i} f_{i}| ≈ (∑_{i} |f_{i}|^{2})^{1/2}. A typical example of this principle appears in Khinchin’s inequality, which we will not discuss in this course. The principle is also related to the law of large numbers or central limit theorem from probability theory.

Some easy consequences of the above theory are as follows. Firstly, if for each k, the operator P̃_{k} was some smooth frequency localization operator^{8}, then

∥(∑_{k} |P̃_{k}f|^{2})^{1/2}∥_{p} ≲ ∥f∥_{p}  (16)

for all 1 < p < ∞. A dual statement is that

∥ ∑_{k} P̃_{k}f_{k}∥_{p} ≲ ∥(∑_{k} |f_{k}|^{2})^{1/2}∥_{p}  (17)

for all 1 < p < ∞ and arbitrary functions f_{k}. (In other words, when adding functions of widely different frequencies, they add as if they square-summed). We leave these statements as exercises. One quick consequence is Proposition 1.7 (Hörmander-Mikhlin multiplier theorem). Let m(ξ) be a multiplier such that

8In other words, P̃_{k} is given by a bump function adapted to the annulus {|ξ| ∼ 2^{k}}. More informally, P̃_{k} looks like P_{k}.

|∇^{j}m(ξ)| ≲ |ξ|^{j}  (18)

for all j ≥ 0 (with the implicit constant depending on j). Let T_{m} be the Fourier multiplier with symbol m:

T̂_{m}f(ξ) = m(ξ)f(ξ)

then T_{m} is bounded on L^{p} for all 1 < p < ∞.

Proof We have

T_{m} = ∑_{k,k′} P_{k}T_{m}P_{k′} = ∑_{k} P̃_{k} ˜P̃_{k}

where P̃_{k} := P_{k}T_{m} and ˜ P̃_{k} := ∑_{k−2<k′<k+2} P_{k}. From (18) we see that P̃_{k} and ˜ P̃_{k} are smooth frequency localization operators to the annulus {|ξ| ∼ 2^{k}}. The claim then follows by composing (17) (for P̃_{k}) with (16) (for ˜ P̃_{k}).

A typical consequence of the Hörmander-Mikhlin multiplier theorem is the estimate

∥∂_{x_{i}}∂_{x_{j}}f∥_{p} ≲ ∥Δf∥_{p}

for all 1 < p < ∞, where Δ = ∑_{i=1}^{n} ∂_{x_{i}}^{2} is the Laplacian on R^{n}. This follows because

{ \atop ∂_{x_{i}}∂_{x_{j}}f}(ξ) = \frac{ξ_{i}ξ_{j}}{|ξ|^{2}} Δ̂f(ξ)

and the symbol m(ξ) := (ξ_{i}ξ_{j})/|ξ|^{2} satisfies the conditions (18). This particular estimate is an example of the principle known as elliptic regularity: once one controls an elliptic operator^{9} such as Δ, one automatically controls all other derivative operators of the same strength.

2. Sobolev spaces

Littlewood-Paley theory is especially good for dealing with spaces which combine L^{p} type norms with derivatives; this is basically because of the heuristic decomposition (5). The most commonly used family of such spaces are the Sobolev spaces^{10} W^{s,p}. (There are also Besov spaces, Hölder spaces, and Triebel-Lizorkin spaces, and then there are closely related spaces such as Hardy spaces and BMO (although strictly speaking, these spaces do not contain derivatives). We’ll discuss them briefly at the end of these notes).

9This begs the question as to what it means for an operator to be elliptic. A circular (but not entirely facetious) answer is that an operator is elliptic if it has some sort of elliptic regularity estimate. A slightly better answer would be that the symbol of the operator (when thought of as a Fourier multiplier) is positive definite (so that its level sets are ellipsoids, at least in the constant co-efficient second order case).

10These spaces are also called L_{s}^{p}, H^{s,p}, or W^{p,s} in the literature. When p = 2 these spaces are often just called H^{s}. Unfortunately no one individual has had enough clout to standardize the notation.

The spaces W^{s,p} are defined for s ∈ R and 1 < p < ∞. (One can also define them for p = ∞ or 0 < p ≤ 1 but some technical details come up that I’d rather not discuss). When s ≥ 0 is a non-negative integer, the spaces can be easily defined by

∥f∥_{W^{s,p}} := ∑_{j=0}^{s} ∥∇^{j}f∥_{p}.

In other words, W^{s,p} are those functions such that f and the first s derivatives of f are in L^{p}. The exponent p thus measures integrability while s measures regularity. When s = 0 we of course have W^{0,p} = L^{p}.

These spaces are ubiquitous in the study of PDE, both linear and non-linear. For instance, many elliptic PDE arise from minimizing functionals such as

∫ |f(x)|^{2} + |∇f(x)|^{2} dx,

which is essentially the W^{1,2} norm.

Before we define these spaces for non-integer s, we first try to rewrite the above definition in terms of Littlewood-Paley operators P_{k} rather than derivatives ∇. We begin by observing that

Lemma 2.1. For any j ≥ 0 and 1 < p < ∞, we have

∥∇^{j}f∥_{p} ∼ ∥ ∑_{k} |2^{jk}P_{k}f|^{2})^{1/2}∥_{p}.

Note that this lemma is consistent with Proposition 1.4 and (3).

Proof By a little Fourier analysis we may write

2^{jk}P_{k}f = P̃_{k}∇^{j}f

where P̃_{k} is an operator similar to P_{k} (i.e. a Fourier multiplier whose symbol is a bump function adapted to |ξ| ∼ 2^{k}). By (16) and Proposition 1.4 we thus have

∥ ∑_{k} |2^{jk}P_{k}f|^{2})^{1/2}∥_{p} = ∥ ∑_{k} |P̃_{k}∇^{j}f|^{2})^{1/2}∥_{p} ≲ ∥∇^{j}f∥_{p}

which is one direction of the implication. To go the other way we use a variant identity

P_{k}∇^{j}f = 2^{jk} ˜P̃_{k}P_{k}f

where ˜ P̃_{k} is another operator similar to P_{k}. By (17) we have

∥∇^{j}f∥_{p} = ∥ ∑_{k} 2^{jk} ˜P̃_{k}P_{k}f∥_{p} ≲ ∥(∑_{k} |2^{jk}P_{k}f|^{2})^{1/2}∥_{p}

as desired.

From this lemma we easily see that

∥f∥_{W^{s,p}} ∼ ∥ ∑_{k} |(1 + 2^{k})^{s}P_{k}f|^{2})^{1/2}∥_{p}.

Note that the right-hand side makes sense for all s, not just integer s. Because of this, it is natural to use the right-hand side to define the Sobolev space W^{s,p} for all s. (Of course, because we have a ∼ rather than an equality, this norm is only defined up to a constant. This, however, is good enough in practice; in most applications one is prepared to lose constant factors all over the place.)

Fractional Sobolev spaces are related to the notion of fractional derivatives. Recall that

ˆ∇f(ξ) = 2πiξf̂(ξ).

Because of this, it is natural to define the operator |∇| by

ˆ|∇|f(ξ) := 2π|ξ|f̂(ξ)

and more generally |∇|^{s} for arbitrary s ∈ R by

{ \atop |∇|^{s}f}(ξ) := (2π|ξ|)^{s}f̂(ξ).

These operators are not immediately intuitive. When s = 2 we have |∇|^{2} = −Δ. More generally we can use the functional calculus of the Laplacian Δ as |∇|^{s} = \sqrt{−Δ}^{s} (exercise!). For −n < s < 0, |∇|^{s} is known as an fractional integration operator, and has the convolution kernel representation

|∇|^{−s}f(x) = C_{n,s} ∫ \frac{f(y)dy}{|x − y|^{n−s}}

for some constant C_{n,s} depending on n, s (exercise! Note that this representation is consistent with dilations and rotations). This kernel is sometimes known as a Riesz potential, although this is confusing as Riesz potentials are distinct from Riesz transforms.

In analogy to (3) we have the heuristic

|∇|^{s}P_{k}f ≈ 2^{ks}P_{k}f;

in other words, |∇|^{s} accentuates the k^{th} frequency piece of f by a factor of 2^{ks}.

One can make this heuristic rigorous in several ways. For instance, by repeating the proof of Lemma 2.1 we have

∥|∇|^{s}f∥_{p} ∼ ∥ ∑_{k} |2^{sk}P_{k}f|^{2})^{1/2}∥_{p}.

As a consequence we see that

∥f∥_{W^{s,p}} ∼ ∥f∥_{p} + ∥|∇|^{s}f∥_{p}

for all s ≥ 0. (When s < 0 the formula is a little different; in fact we have

∥f∥_{W^{s,p}} ∼ inf_{f=g+h} ∥g∥_{p} + ∥|∇|^{s}h∥_{p}

(exercise!). Also, for any s ∈ R and σ > 0: we see that the operator |∇|^{σ} removes σ degrees of regularity:

∥|∇|^{σ}f∥_{W^{s−σ,p}} ≲ ∥f∥_{W^{s,p}}

(exercise!). Indeed, if D^{k} is any constant-coefficient operator of order k > 0, then

∥D^{k}f∥_{W^{s−k,p}} ≲ ∥f∥_{W^{s,p}}.

Because of these facts, Sobolev spaces are very good at tracking the regularity of various components of a PDE.

One can connect the Sobolev spaces of a given integrability exponent p more tightly by introducing modified fractional differentiation operators ⟨∇⟩^{s} by

{ \atop ⟨∇⟩^{s}f}(ξ) := ⟨2πξ⟩^{s}f̂(ξ)

where the Japanese bracket ⟨x⟩ is defined by ⟨x⟩ := (1 + |x|^{2})^{1/2}. In terms of the Laplacian, we have ⟨∇⟩^{s} = \sqrt{1 − Δ}^{s}. The operator ⟨∇⟩^{s} is essentially the same as |∇|^{s} for high frequencies |ξ| ≳ 1, but behaves like the identity for low frequencies |ξ| ≲ 1 (unlike |∇|^{s}, which damps low frequencies when s > 0 and accentuates them for s < 0). When s is negative, the operators ⟨∇⟩^{s} are known as smoothing operators, and the kernels are known as Bessel potentials. (Despite the name, these potentials are only tenuously connected with Bessel functions).

The reader may verify using Littlewood-Paley theory that one has the estimates

∥f∥_{W^{s,p}} ∼ ∥⟨∇⟩^{s}f∥_{p}.

In other words, the Sobolev space W^{s,p} is just the images of L^{p} under the operator ⟨∇⟩^{−s}. As a consequence, any two Sobolev spaces W^{s,p} and W^{s′,p} are isomorphic via the map ⟨∇⟩^{s′−s}. (One can now use this definition to define Sobolev spaces for p = 1 or p = ∞). As another corollary we see that the spaces W^{s,p} and W^{−s,p′} are dual for 1 < p < ∞, where p′ is defined by 1/p + 1/p′ = 1.

When p = 2 we can thus use Plancherel’s theorem to obtain a very convenient Fourier representation of a Sobolev space:

∥f∥_{W^{s,2}} ∼ (∫ (⟨ξ⟩^{s}|f(ξ)|)^{2} dξ)^{1/2}.

Because of this, the p = 2 Sobolev spaces can be manipulated directly without recourse to Littlewood-Paley theory. However, once one has the Littlewood-Paley toolbox in one’s possession, then the p ≠ 2 spaces are not much more complicated than the p = 2 spaces.

The Sobolev embedding theorems in Lemma 1.2 and Lemma 1.3 can be generalized to fractional spaces; indeed, these embedding theorems are one of the major reasons why we care about fractional Sobolev spaces in the first place. We shall simply state the theorems here, as they are proven in an almost identical fashion to their integer analogues.

Lemma 2.2 (Non-endpoint Sobolev embedding). Let 1 ≤ p < q < ∞ and s′ < s be such that

\frac{1}{p} − \frac{s − s′}{n} > \frac{1}{q}.

Then we have

∥f∥_{W^{q,s}(R^{n})} ≲ ∥f∥_{L^{p,s′} (R^{n})}  (19)

for all functions f in R^{n} for which the right-hand side is finite. (The implicit constant depends on p, q, n, s, s′).

Lemma 2.3 (Endpoint Sobolev embedding). Let 1 < p < q < ∞ and s′ < s be such that

\frac{1}{p} − \frac{s − s′}{n} = \frac{1}{q}.

Then we have

∥f∥_{W^{q,s}(R^{n})} ≲ ∥∇f∥_{L^{p}(R^{n})}  (20)

for all functions f in R^{n} for which the right-hand side is finite. (The implicit constant depends on p, q, n, s, s′).

One should think of these Sobolev embedding theorems as tools which allow one to trade in regularity^{11} for a gain in integrability. (One cannot reverse this and trade in integrability for regularity; even if a function is in the best possible integrability class, namely L^{∞}, there is no reason why it should have any differentiability properties whatsoever.)

In next week’s notes we shall start to study the multilinear and non-linear properties of Sobolev spaces. For instance, we shall see how the Sobolev class of f and g control the Sobolev class of fg.

3. Other spaces

The Sobolev spaces W^{s,p} are good examples of spaces with regularity s and integrability p, however, there are a couple other spaces with similar exponents s, p which are useful in applications. We will not use them here, but just give a very quick tour of them.

The Sobolev spaces W^{s,p} can be generalized to the Triebel-Lizorkin spaces^{12} F_{q}^{s,p}, defined for s ∈ R, 0 < p < ∞, and 1 ≤ q ≤ ∞ by

∥f∥_{F_{q}^{s,p}} := ∥(∑_{q} |(1 + 2^{k})^{s}P_{k}f|^{q})^{1/q}∥_{p}.

Thus F_{2}^{s,p} = W^{s,p} when 1 < p < ∞. When 0 < p ≤ 1 the space F_{2}^{0,p} is equal to the Hardy space H^{p}, which plays an important role in complex analysis, and is in many ways a preferable space to L^{p} in this range of exponents. The exponent q only influences things by a logarithm or so (see the footnote to (6)) and is of minor importance compared to s and p. Thus if logarithms are unimportant then we may pretend that F_{q}^{s,p} ≈ W^{s,p}.

11Remember that, ultimately, all these Sobolev embedding theorems are based on Bernstein’s inequality, which allows one to exploit frequency localization to gain integrability. Regularity can be considered as a form of frequency localization (the smoother a function is, the faster its Fourier transform decays in frequency). The physical space analogue of this is Hölder’s inequality: if a function is compactly supported in physical space, then higher L^{p} norms control lower L^{q} norms.

More generally, one use Hölder to trade in decay for a lowering of integrability.

12As with the Sobolev spaces, these spaces unfortunately do not have a standardized notation.

In light of (6) it is natural to also consider the Besov spaces B_{q}^{s,p} defined by

∥f∥_{B_{q}^{s,p}} := (∑_{q} |(1 + 2^{k})^{s}∥P_{k}f∥_{p}|^{q})^{1/q}.

Thus, for instance, (6) can be rewritten as

∥f∥_{B_{∞}^{0,p}} ≲ ∥f∥_{p} ≲ ∥f∥_{B_{1}^{0,p}}.

These spaces are handy for a number of applications, for instance one can often recover endpoint Sobolev inequalities for Besov spaces at the endpoints p = 1 or q = ∞, which are not available for ordinary Sobolev spaces. However, the differences are again only logarithmic, and if logarithms are unimportant then B_{q}^{s,p} ≈ W^{s,p}.

When s is not an integer, the spaces B_{q}^{s,p} are equivalent to the more classical Hölder spaces Λ_{q}^{s,p}. An example of this is given in the exercises.

One space which is important, but not listed in these families, is the space BMO of functions of bounded mean oscillation. In other words, BMO consists of those functions f for which the norm

∥f∥_{BMO(R^{n})} := sup_{B} \frac{1}{B} ∫_{|B|} |f − f_{B}|

is finite, where B ranges over all balls in R^{n} and f_{B} := \frac{1}{|B|} ∫_{B} f is the mean of f on B. This space often turns up as a substitute for L^{∞} (we don’t ask that the function be bounded everywhere, but only that its deviation from its mean stays bounded on the average). Clearly ∥f∥_{BMO} ≲ ∥f∥_{∞}, but the reverse inequality fails in general (e.g. f(x) := log |x| is in BMO but not L^{∞}. However, we have

Lemma 3.1. If f has frequency support on a single annulus {|ξ| ∼ 2^{k}} then we have the relationship

∥f∥_{BMO} ∼ ∥f∥_{∞}.

Proof By scaling we may take k = 0. It suffices to show that |f(x)| ≲ ∥f∥_{BMO} for all x ∈ R^{n}.

Fix x. By repeating the proof of (2) we have the reproducing formula

f(x) = ∫ f(x − y)2^{nC} ϕ̂(2^{−C}y) dy

for sufficiently large C. We also have

0 = ∫ f(x − y)2^{−nC} ϕ̂(2^{C}y) dy,

so that

f(x) = ∫ f(x − y)(2^{nC} ϕ̂(2^{−C}y) − 2^{−nC} ϕ̂(2^{C}y)) dy.

Let B be the ball B := {y : |x − y| ≤ 2^{C′}}, where C′ is a constant to be chosen later. Clearly we have

0 = ∫ f_{B}(2^{nC} ϕ̂(2^{−C}y) − 2^{−nC} ϕ̂(2^{C}y)) dy.

Thus

f(x) = ∫ (f(x − y) − f_{B})(2^{nC} ϕ̂(2^{−C}y) − 2^{−nC} ϕ̂(2^{C}y)) dy.

We split the integral into the regions y ∈ B and y ∉ B. On the region y ∈ B the integral is O(∥f∥_{BMO}), where the implicit constant depends on C and C′. On the region y ∉ B, we can estimate the integral by

2∥f∥_{∞} ∫_{|y|>C′}(2^{nC} ϕ̂(2^{−C}y) − 2^{−nC} ϕ̂(2^{C}y)) dy.

If C′ is sufficiently large depending on C, this integral is < 1/4. Thus we have

|f(x)| ≤ O(∥f∥_{BMO}) + \frac{1}{2}∥f∥_{∞}.

If we take suprema over x, the claim follows.

Unfortunately BMO does not have as nice a Littlewood-Paley representation as the other spaces listed here (even though it does turn out to be the dual of H^{1} = F_{2}^{0,1}).

Before we finish this quick tour of function spaces, we should mention that these “inhomogeneous” spaces have “homogeneous” or “pure” counterparts, in which only one order of derivative is used. For instance, the homogeneous Sobolev space Ẇ^{s,p} is defined by

∥f∥Ẇ^{s,p} ∼ ∥ ∑_{k} |(2^{k})^{s}P_{k}f|^{2})^{1/2}∥_{p}

instead of

∥f∥_{W^{s,p}} ∼ ∥ ∑_{k} |(1 + 2^{k})^{s}P_{k}f|^{2})^{1/2}∥_{p}.

Similarly one can define Ḟ_{q}^{s,p} and Ḃ_{q}^{s,p}. These homogenous spaces can be a little harder to work with (for instance, one has far fewer Sobolev embeddings for these spaces) but they scale much more nicely, and there are many situations in PDE in which the homogeneous norm is the only norm which one actually has control on. For high frequencies |ξ| ≳ 1 there is essentially no distinction between the homogeneous and inhomogeneous spaces, but the low frequencies |ξ| ≲ 1 are measured quite differently (one or the other space is stronger, depending on the sign of s).

4. Appendix: Proof of Lemma 1.6

To prove Lemma 1.6 it suffices by duality to check the case 1 < p < 2. (Individual Calderón-Zygmund operators need not be self-adjoint, but the class of Calderón-Zygmund operators is self-adjoint). By the Marcinkiewicz interpolation theorem it suffices to show that T is weak-type (1,1), i.e. that

|\{|T f| > λ\}| ≲ \frac{∥f∥_{1}}{λ} .  (21)

We now return to the proof of (21). The idea is to split f into a “good” part g, which is in L^{2} and thus amenable to the L^{2} boundedness assumption of T, and a “bad” part ∑_{I} b_{I}, which is not in L^{2} but is well localized to a collection I of intervals, and also has a mean zero condition which can be exploited via the smoothness condition (15). This decomposition is known as the Calderón-Zygmund decomposition.

Fix f (which we can take to be Schwartz) and λ. We need some notation. Define a dyadic cube I to be a cube I ⊂ R^{n} whose side-length is an integer power 2^{k} of two, and whose corners lie on the lattice 2^{k}Z^{n}. These cubes multiply cover all of Euclidean space, but are nested in the sense that if I, J are two dyadic cubes which intersect, then either I ⊆ J or J ⊆ I.

Call a dyadic cube I heavy if ∫_{I} |f| ≥ λ|I|, and call a heavy dyadic cube maximal if it is not contained in any other heavy dyadic cube. Note that the maximal heavy dyadic cubes are all disjoint and obey the bounds

λ|I| ≤ ∫_{I} |f| ≤ 2^{n}λ|I|.  (22)

so we have

λ ∑_{I} |I| ≲ ∥f∥_{1}.  (23)

We split f = g + ∑_{I} b_{I}, where

g := fχ_{R^{n}−⋃_{I} I} + ∑_{I} f_{I}χ_{I}

and

b_{I} := (f − f_{I})χ_{I},

and f_{I} := \frac{1}{I} ∫_{I} f is the mean of f on I. Thus b_{I} is the mean-zero part of f on I, and g is everything else.

We shall estimate the left-hand side of (21) by

|{|T g| ≥ λ/2}| + |{| ∑_{I} T b_{I}| ≥ λ/2}|.

The first term we can estimate using Chebyshev’s inequality by

≲ λ^{−2}∥T g∥_{2}^{2} ≲ λ^{−2}∥g∥_{2}^{2} ≲ λ^{−2}∥g∥_{1}∥g∥_{∞}.

From (22) we see that ∥b_{I}∥_{1} ≲ λ|I|. Since f = g+∑_{I} b_{I}, we thus see from (23) that ∥g∥_{1} ≲ ∥f∥_{1}. Also, from (22) and the construction of the I we see that ∥g∥_{∞} ≲ λ. Thus this term is acceptable.

Now to control the second term |{| ∑_{I} T b_{I}| ≥ λ/2}|. We expect^{13} the function ∑_{I} T b_{I} to be pretty nasty in the exceptional set ⋃_{I} 2I, but fortunately this set has small size by (23) and we can discard it. The function b_{I} is supported in I and has mean zero, so by (15) and the mean-value theorem we have the estimates

13This argument can be viewed using phase plane heuristics. A Calderón-Zygmund operator such as T, when applied to a function of frequency |ξ| ∼ 2^{k}, only spreads out the function by about 2^{−k} or so (why?). Since the function b_{I} is supported on I and has mean zero, it mostly consists of frequencies |ξ| ≳ |I|^{−1} (why?). Thus T only spreads b_{I} by about O(|I|) or so.

|T b_{I}(x)| = | ∫_{I} b_{I}(y)K(x, y) dy|
= | ∫_{I} b_{I}(y)(K(x, y) − K(x, y_{I})) dy|
≲ ∫_{I} |b_{I}||y − y_{I}|/|x − y_{I}|^{n+1} dy
≲ diam(I)∥b_{I}∥_{1}/dist(x, I)^{n+1}
≲ λdiam(I)|I|/dist(x, I)^{n+1}

whenever x ∉ 2I. Thus, outside of the exceptional set we have

∥T b_{I}∥_{1} ≲ λ|I|

so by summing using (23) we have

∥ ∑_{I} T b_{I}∥_{1} ≲ ∥f∥_{1}

and so this part is also OK by Chebyshev’s inequality. This proves (21), and so Calderón-Zygmund operators are bounded in L^{p}, 1 < p < ∞.

5. Homework for week 2 - due Friday, Feb 9

In all these problems, and n ≥ 1 is an integer, and P_{k} denotes some family of Littlewood-Paley operators on R^{n}. k ranges over the integers Z unless otherwise indicated. Implicit constants are allowed to depend on the parameters p, n, α, q, θ.

• Let n > 1, and let R^{n−1} be a hyperplane in R^{n}. Show that if 1 ≤ p ≤ ∞ and g is an L^{p} function on R^{n} with Fourier support in the ball |ξ| ≲ 2^{k} for some integer k, then

∥g∥_{L^{p}(R^{n−1})} ≲ 2^{k/p}∥g∥_{L^{p}(R^{n})}.

(Note: it may be easiest to prove this first for p = 1 and p = ∞. For general p, you may find Fubini’s theorem, Minkowski’s inequality, and Hölder’s inequality useful.) From this, conclude the Sobolev trace lemma: whenever 1 < p < ∞, s ∈ R, and σ > 1/p, we have

∥f∥_{W^{p,s−σ}(R^{n−1})} ≲ ∥f∥_{W^{p,s}(R^{n})}.

• Prove the Gagliardo-Nirenberg inequality: If 1 < p < q < ∞ is such that 1/q = 1/p + θ/n for some 0 < θ < 1, then we have

∥f∥_{q} ≲ ∥f∥_{p}^{1−θ}∥∇f∥_{p}^{θ}

for all Schwartz functions f on R^{n}. (Note: it is tempting to apply Lemma 1.3, but this only works when p < n (why?).)

• Prove the Poincare inequality: if 1 ≤ p ≤ ∞, and f is a Schwartz function supported on the unit ball {|x| ≤ 1}, then^{14}

∥f∥_{p} ≲ ∥∇f∥_{p}.

(Hint: From Lemma 1.1 one can control ∥P_{>−C}f∥_{p} well for any large constant C. To get from this partial result to control of all of ∥f∥_{p}, try to prove an inequality of the form

∥f∥_{p} ≤ C∥P_{>−C}f∥_{p} + \frac{1}{2}∥f∥_{p}.

One way to do this is to use the identity f = χ̃P_{>−C}f + ∑_{k≤−C} χ̃P_{k}f, where χ̃ is a bump function adapted to {|x| ≤ 2} which equals 1 on {|x| ≤ 1}.)

Given an informal explanation using the uncertainty principle as to why P_{>−C}f should be the “dominant” component of f.

• Let 0 < α < 1. The Hölder class Λ_{α}(R^{n}) is the space of all functions f on R^{n} for which the norm

∥f∥_{Λ_{α}} := ∥f∥_{∞} + sup_{x,y∈R^{n}:0<|x−y|<1} \frac{|f(x) − f(y)|}{|x − y|^{α}}

is finite. (Functions in Λ_{α} are called Hölder continuous of order α). These spaces are especially useful in elliptic PDE.

(a) Prove that

∥f∥_{Λ_{α}} ∼ ∥P_{≤0}f∥_{∞} + sup_{k>0} 2^{kα}∥P_{k}f∥_{∞}.

with the implicit constant depending on n, α. (Thus we have the heuristic Λ_{α} ≈ W^{α,∞}.

(b) Prove the following variant of the Sobolev embedding theorem: if n < p < ∞ and 0 < α < 1 are such that n/p < 1 − α, then

∥f∥_{Λ_{α}} ≲ ∥∇f∥_{p}.

(c) The Zygmund class Λ_{∗} is the space of all functions f whose norm

∥f∥_{Λ_{∗}} := ∥f∥_{∞} + sup_{x,h∈R^{n}:0<|h|<1} \frac{|f(x + h) − 2f(x) + f(x − h)|}{|h|}

is finite. Show that

∥f∥_{Λ_{∗}} ∼ ∥P_{≤0}f∥_{∞} + sup_{k} 2^{k}∥P_{k}f∥_{∞}.

(Thus Λ_{∗} ≈ W^{1,∞}).

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

to much more general situations.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

14This particular inequality can be proven by more elementary means, such as the fundamental theorem of calculus. However, the method of proof sketched here is far more flexible, and extends