SUMMABILITY OF FUNCTIONS

TERENCE TAO

Abstract. We discuss the summability of functions, or more precisely the behaviour of ∥ ∑_{i} f_{i}∥ on ∥f_{i}∥, and show how one can use knowledge of summability to reduce estimates.

1. Introduction

Let f_{1}, f_{2}, …, f_{N} be a collection of functions, and let ∥∥_{X} be some norm or quasi-norm. We are interested in how the quantity ∥ ∑_{i=1}^{N} f_{i}∥_{X} can be controlled by ∥f_{i}∥_{X}.

If we have

∥ ∑_{i=1}^{N} f_{i}∥_{X} ≲ (∑_{i=1}^{N} ∥f_{i}∥_{X}^{p})^{1/p}

then we say that the f_{i} are l^{p}-summable or better in X. If the reverse inequality

∥ ∑_{i=1}^{N} f_{i}∥_{X} ≳ (∑_{i=1}^{N} ∥f_{i}∥_{X}^{p})^{1/p}

holds instead, we say the f_{i} are l^{p}-summable or worse in X. If both hold then we say that the f_{i} are exactly l^{p}-summable in X.

If X is a Banach space norm, then of course any collection f_{i} of functions are at least l^{1}-summable in X, that’s just the triangle inequality. However, if there is some “disjointness” or “orthogonality” then one can often get better summability. For instance, the f_{i} have disjoint supports, then the f_{i} are exactly l^{p}-summable in L^{p}. If the f_{i} are orthogonal, then the f_{i} are exactly l^{2}-summable in L^{2}. Orthogonality, by itself, is not enough to guarantee any summability in L^{p}, p ≠ 2 beyond l^{1}-summability (see [5]), however if one assumes a bit extra, such as the f_{i} having Fourier support in rectangles whose doubles have finite overlap, then we have summability between l^{min(p,p′)} and l^{max(p,p′)} in L^{p} for all 1 ≤ p ≤ ∞ (see e.g. [4]). If the f_{i} have support in lacunary intervals (or some other Littlewood-Paley set) then we can improve this to between l^{min(p,2)} and l^{max(p,2)} summability in L^{p} for all 1 ≤ p < ∞. The same result obtains on the average if the f_{i} have randomized signs. In particular, randomized signs are exactly l^{2}-summable in L^{2}.

Non-Banach spaces are not usually l^{1}-summable, but they usually have some sort of summability. If 0 < p < 1 then any f_{i} are l^{p}-summable at least in L^{p} and in L^{p,∞} (for the latter, use the fact that min(|f_{i}|, λ) has an L^{1} norm of O(λ^{1−p}∥f_{i}∥_{p}^{p}) and use the triangle inequality followed by Chebyshev). They are also at least l log l-summable in L^{1,∞}, that’s the Stein-Weiss summability result. However, for p > 1 the L^{p,∞} spaces are actually l^{1}-summable at least, even though they don’t look like Banach spaces. That’s because they can be given the Banach space norm

1991 Mathematics Subject Classification. 42B.

∥f∥_{p,∞} ∼ sup_{E} |E|^{−1/p′} ∫_{E} |f|

where E ranges over all sets of positive measure.

Clearly, getting upper bounds on the summability can be difficult if the f_{i} can cancel each other. However if the f_{i} are non-negative then there are limits to the summability of the f_{i}. For instance, they cannot sum any better than l^{p} in L^{p} for p ≥ 1 (exercise). In particular, they are exactly l^{1}-summable in L^{1}.

It is often useful to know the summability type of sequence of functions f_{i} because of the following

Summability principle. If T is a linear operator, and the functions T f_{i} have a higher or equal summability in Y than the f_{i} have in X, then to prove the estimate

∥T ∑_{i} f_{i}∥_{Y} ≲ ∥ ∑_{i} f_{i}∥_{X}

it suffices to prove the estimates

∥T f_{i}∥_{Y} ≲ ∥f_{i}∥_{X}

individually for each i.

This principle is quite trivial, and boils down to the basic estimate

(∑_{i} a_{i}^{p})^{1/p} ≤ (∑_{i} a_{i}^{q})^{1/q} when p ≥ q

(which can be obtained by first verifying this for p = q and p = ∞, and interpolating or using Hölder. Alternatively, use Young’s inequality, noting that a = a ∗ δ, where δ is the Kronecker delta).

The point of this principle is that it allows one to reduce the problem of establishing boundedness of T for a complicated function f = ∑_{i} f_{i} to that of boundedness of T for each of its components separately, which should be an easier task (at least, if you decomposed f = ∑_{i} f_{i} intelligently).

There is a converse summability principle: if ∥T f_{i}∥_{Y}, ∥f_{i}∥_{X} are both roughly constant with respect to i, and the T f_{i} have lower summability in Y than the f_{i} have in X, then operator T cannot be bounded on ∑_{i} f_{i} uniformly in N, because the left-hand side grows faster than the right in N.

2. Examples

One basic application of the summability principle: if the f_{i} are orthogonal, and the T f_{i} are also orthogonal, to prove that T is bounded on L^{2} for ∑_{i} f_{i} it suffices to do so for each f_{i} separately. For instance, to prove the L^{2} boundedness of a Fourier multiplier it suffices to do so on plane waves, i.e. we need to show that the multiplier is bounded. For a slightly less trivial application, to show that a rotation-invariant operator is bounded in L^{2} it suffices to do so for functions which factor into a radial function multiplied by a single spherical harmonic in the angular variables. (In fact, the same principle lets you reduce mapping properties from L_{r}^{p}(L_{θ}^{2}) to L_{r}^{q}(L_{θ}^{2}) to functions which are a single harmonic in the angular variable whenever p ≤ 2 ≤ q).

Suppose that T is a convolution operator with compactly supported kernel. To show that T maps L^{p}(R^{n}) to L^{q}(R^{n}) for q ≥ p it suffices to do so for functions supported on a compact set, say the unit ball; this comes from decomposing a general function into functions supported on unit balls and using the summability principle. Conversely, if q < p then one cannot have boundedness, by the converse summability principle. This is Littlewood’s principle (“the higher exponents are always on the left”).

Another application: if p ≤ 2 ≤ q and one wishes to show that a Fourier multiplier maps L^{p} to L^{q}, it suffices to do so for functions f which are supported on a single dyadic annulus (or any other collection of sets which form Littlewood-Paley partitions). This is a basic tool in the derivation of Strichartz estimates for the wave equation, see e.g. [1].

While on the subject of Strichartz estimates, let us ask why the standard Strichartz estimate

∥e^{it\sqrt{−Δ}}f∥_{L_{t}^{q}L_{x}^{r}} ≲ ∥f∥Ḣ^{s}

needs the condition q ≥ 2 in addition to the more natural scaling conditions^{1}

\frac{n}{q} + \frac{1}{r} = \frac{n}{2} − s

and Knapp condition

\frac{(n − 1)/2}{q} + \frac{1}{r} = \frac{(n − 1)/2}{2},

together with the technical condition r < ∞. This can be explained by the converse summability principle. A basic example to test the Strichartz estimate is the focussing example, in which

f_{R} = R^{−(n−1)/2}ψ(|x| − R)

where ψ is a compactly supported bump function and R ≫ 1. This function has a Ḣ^{s} norm comparable to 1 (if an appropriate moment condition is imposed on ψ), and |e^{it\sqrt{−Δ}}f_{R}(x)| ∼ 1 when |x| ≪ 1 and |t − R| ≪ 1. So both sides of the Strichartz estimate are comparable to 1. (Think of this example as a spherical wave front of radius R and thickness 1 at time zero, and moving inward at the speed of light, so that it all focusses at time R + O(1) to a ball of radius O(1)).

Now consider f_{R} as R varies along the integers from R_{0} to 2R_{0}. These functions are l^{2}-summable in Ḣ^{s} (because the f_{R} are so smooth, the derivatives are irrelevant, and Ḣ^{s} behaves pretty much like L^{2}), whereas the portions of e^{it\sqrt{−Δ}}f_{R} discussed above have disjoint t support as R varies, and so they are l^{q}-summable in L_{t}^{q}(L_{x}^{r}). The necessity of q ≥ 2 follows. (The concrete counter-example is now a “strobe light” example, in which several concentric shells of wave fronts converge at the origin at different times).

Now let’s turn to the topic of bilinear restriction estimates. Suppose we want to prove an adjoint restriction estimate like

∥f̂dσ∥_{q} ≲ ∥f∥_{p}

1Littlewood’s principle gives r ≥ 2, but this turns out to be redundant when compared to the superior Knapp condition.

for all f on S^{n−1}, where dσ is Lebesgue measure on the sphere and 2 ≤ q ≤ 4 (which is the most interesting case). We can square and de-polarize this as

∥f̂dσĝdσ∥_{q/2} ≲ ∥f∥_{p}∥g∥_{p}.

Expand the left-hand side as

∥ ∫ ∫ f(ω)g(ω′)e^{2πi(ω+ω′)·x} dωdω′∥_{L_{x}^{q/2}} ≲ ∥f∥_{p}∥g∥_{p},

where the integrals are over the sphere. By a dyadic decomposition and giving up an epsilon somewhere, one can reduce to showing that

∥ ∫ ∫_{|ω−ω′|∼2^{−j}} f(ω)g(ω′)e^{2πi(ω+ω′)·x} dωdω′∥_{L_{x}^{q/2}} ≲ ∥f∥_{p}∥g∥_{p}

for all j. One can then essentially split this as

∥ ∑_{Q,Q′} { \atop f_{Q}dσ}{ \atop g_{Q′}dσ}∥_{q/2} ≲ ∥ ∑_{Q} f_{Q}∥_{p}∥ ∑_{Q′} g_{Q′}∥_{p}

where Q, Q′ range over a finitely overlapping collection of spherical caps of width 2^{−j}, f_{Q} and g_{Q′} are restrictions of f, g to these caps, and the sum is only over those pairs of caps whose separation is ∼ 2^{−j}.

Now use Fefferman’s observation that the Fourier transforms of { \atop f_{Q}dσ}{ \atop g_{Q′}dσ} are supported in essentially disjoint rectangles. Thus these functions are l^{q/2}-summable in L^{q/2}. Also, because of disjointness, the f_{Q} are l^{p}-summable in L^{p} and the g_{Q′} are l^{p}-summable in L^{p}. So by a bilinear formulation of the summability principle, one can reduce to a single pair of f_{Q} and g_{Q′}, i.e. we need only show

∥{ \atop f_{Q}dσ}{ \atop g_{Q′}dσ}∥_{q/2} ≲ ∥f_{Q}∥_{p}∥g_{Q′}∥_{p}

provided that \frac{2}{q} ≤ \frac{1}{p} + \frac{1}{p}, i.e. that q ≥ p. This, together with a parabolic rescaling argument to reduce to the case j = 0, is the essence of the equivalence between linear and bilinear restriction estimates in [4] (see also [6], [3], [2] for further examples of this, the latter two being concerned with null form analogues of the above ideas).

References

[1] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, International Press, 1995.

[2] T. Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, submitted, Math Z.

[3] T. Tao, A. Vargas, A bilinear approach to cone multipliers II. Applications, to appear, GAFA.

[4] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), pp. 967–1000.

[5] T. Tao, A hasty interpolation, expository note.

[6] T. H. Wolff, A sharp bilinear cone restriction estimate, to appear, Annals of Math.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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