COUNTEREXAMPLES TO THE n = 3 ENDPOINT STRICHARTZ ESTIMATE FOR THE WAVE EQUATION

TERENCE TAO

Abstract. In this note we collect various negative results concerning the false to the false L_{t}^{2}L_{x}^{∞} Strichartz estimate for the wave equation in 3 dimensions, together with its variants.

1. Introduction

For all test functions f, g in R^{3} and F in R^{3} × R, let u denote the solution to the Cauchy problem for the wave equation

\{ \begin{matrix}(−\frac{∂^{2}}{∂t^{2}} + Δ)u(t, x) = F (t, x), & (t, x) ∈ R × R^{3} \\ u(0, ·) = f, ∂_{t}u(0, ·) = g; \end{matrix}

In general dimensions one has the explicit formula

u(t) = cos(t\sqrt{−Δ})f + \frac{sin(t\sqrt{−Δ})}{\sqrt{−Δ}}g;  (1)

in three dimensions this can be rewritten as

u(x, t) = \frac{∂}{∂t}t ∫_{S^{2}} f(x+tω) dω +t ∫_{S^{2}} g(x+tω) dω +∫_{0}^{t} τ ∫_{S^{2}} F (t−τ, x+τω) dωdτ.

See for instance [10].

The (false) homogeneous endpoint Strichartz estimate asserts that

∥u∥_{L_{t}^{2}L_{x}^{∞}} ≲ ∥f∥Ḣ^{1} + ∥g∥_{L^{2}}  (2)

when F = 0, where Ḣ^{s} is the homogeneous Sobolev space \sqrt{−Δ}^{−s}L^{2}.

This estimate is known to be false (see e.g. [6]), although it is heuristically true for interpolation purposes (see e.g. [5]). In fact, an adaptation of Stein’s counterexample for the spherical maximal function (see [12], p. 472) shows that this estimate fails quite concretely, and the solution can concentrate on the lightray (t, te_{3}).

Proposition 1.1. Suppose that f = F = 0 and g is the function

g(x) = \frac{χ_{A}(x)}{|x|^{2}(1 + |log |x||)^{α}}

where \frac{1}{2} < α ≤ 1 and A is the difference of two balls:

A = B(2e_{3}, 2)\B(e_{3}, 1).

Then g is in L^{2}, but u(t, te_{3}) is infinite for all 1 < t < 2. In particular, (2) fails.

1991 Mathematics Subject Classification. 35J05.

Thus, g is supported on a set which has a cusp at the origin, and grows to infinity at a carefully controlled rate there.

We leave the proof of this proposition as an easy exercise. The the portion of g in the dyadic shell {|x| ∼ 2^{−k}} lives in a set of volume 2^{−4k} and has a height of 2^{2k}/k^{α}, and the intersection of this set with any sphere of radius t centered at te_{3} has surface area roughly 2^{−2k}. Thus the L^{2} norm of this piece is roughly 1/k^{α}, which is in l^{2}, but the contribution to u on the light ray is also roughly 1/k^{α}, which is not summable.

Note that the problem is not a global one, as one has blowup in a bounded time interval. So localizing time will not salvage this estimate.

One might think that the failure of this estimate can be circumvented by placing more derivatives on the data:

∥u∥_{L_{t}^{2}L_{x}^{∞}} ≲ ∥g∥_{H^{s}}, s large.

However, the estimate still fails, basically because the L^{2} estimate is scale invariant. By approximating the above counterexample one can find a Schwarz function g which has bounded L^{2} norm, but the L^{2}L^{∞} norm of u is arbitrarily large. Now replace g by

g_{λ}(x) = λ^{−3/2}g(x/λ)

and let λ → ∞. The L^{2} norm of g and the L^{2}L^{∞} norm of u remain constant, while the H^{s} norm of g tends to the L^{2} norm of g. Thus the above estimate also fails.

Of course, if one places more derivatives on the data and also localizes time, then one can get a finite estimate. Indeed, we have

∥u∥_{L_{T}^{2} L_{x}^{∞}} ≲ C(ε, T)∥g∥_{H^{ε}}

for any ε > 0, where

C(ε, T) = \{ \begin{matrix}C_{ε}(log T)^{1/2} & if T ≫ 1 \\ C_{ε}T^{ε} & if T ≲ 1. \end{matrix}

The T ≫ 1 estimates can be obtained by localizing in frequency and applying the T T^{∗} method as per the usual Strichartz method. The T ≲ 1 estimates can then be obtained by scaling. The Stein example shows that the former estimate is sharp, and scaling considerations show the latter estimate is sharp.

The inhomogeneous endpoint Strichartz estimate

∥u(T)∥_{L^{2}} ≲ ∥F ∥_{L_{t}^{2}L_{x}^{1}}

for f = g = 0 and arbitrary T, is essentially the adjoint of (2) and thus fails by duality. By the T T^{∗} method and symmetry, we thus see that the retarded estimate

∥u∥_{L_{t}^{2}L_{x}^{∞}} ≲ ∥F ∥_{L_{t}^{2}L_{x}^{1}}

for f = g = 0 also fails.

2. The frequency-localized or BMO estimate

One may hope to salvage (2) by replacing the L_{x}^{∞} norm with a BMO norm, or by restricting (2) to functions f, g whose frequency is localized to the unit annulus {|ξ| ≲ 1}. It turns out that both of these estimates are also false.

We first show that the BMO estimate, if true, would imply the frequencylocalized estimate. Suppose the BMO estimate held, i.e. one had

∥u∥_{L_{t}^{2}BMO_{x}} ≲ ∥f∥Ḣ^{1} + ∥g∥_{L^{2}}.  (3)

Now suppose f and g had localized frequency. Then u must also have localized spatial frequencies. But this implies that

∥u(t, ·)∥_{BMO} ∼ ∥u(t, ·)∥_{∞}

(see e.g. [1]; the point is that every Littlewood-Paley projection of a BMO function is in L^{∞}). Thus the BMO estimate implies the frequency-localized estimate.

In light of this implication, we see that to disprove both of these estimates it suffices to disprove the frequency-localized estimate. To do this we adapt the argument^{1} in [8]. By duality, it suffices to disprove the estimate

∥ ∫ e^{it\sqrt{−Δ}}F (t) dt∥_{L_{x}^{2}} ≲ ∥F ∥_{L_{t}^{2}L_{x}^{1}}  (4)

for functions F which are spatially localized in frequency.

An obvious guess for a counterexample F would be a travelling wave, possibly with a time cutoff:

F (x, t) = ψ(x − te_{3})η(t/T).

However it turns out that there is enough residual cancellation in the expression ∫ e^{it\sqrt{−Δ}}F (t) dt to ensure that this cannot be a counterexample. (Indeed, for this case (4) becomes a one-dimensional statement, equivalent to the L^{2} boundedness of the Hilbert transform). To eradicate this cancellation we have to break the translation invariance. We will use an idea of Montgomery-Smith [8], and introduce a one-dimensional Brownian motion β_{t}, with the property that for any s ≠ t the quantity β_{t} − β_{s} is a Gaussian random variable with generating function

E(e^{2πi(β_{t}−β_{s})ξ}) = e^{−2π|t−s|ξ^{2}}  (5)

for arbitrary ξ ∈ R, where E denotes the expectation.

Our modified counterexample is now

F (x, t) = ψ(x − te_{3} − β_{t}e_{1})η(t/T)  (6)

where T ≫ 1, η is a standard non-negative even cutoff function to the interval [−1, 1], and ψ is any Schwarz function whose Fourier transform is a standard cutoff function for the unit annulus \{|ξ| ∼ 1\}. We thus have ∥F ∥_{L_{t}^{2}L_{x}^{1}} ≲ \sqrt{T}. On the Fourier side, we have

F̂(ξ, t) = e^{−2πitξ_{3}}e^{−2πiβ_{t}ξ_{1}} ψ̂(ξ)η(t/T).

Suppose that (4) held for all functions of the form (6). From this and Plancherel’s theorem, we thus see that

∥ ∫ e^{2πit|ξ|}e^{−2πitξ_{3}}e^{−2πiβ_{t}ξ_{1}}η(t/T) dtψ̂(ξ)∥_{L_{ξ}^{2}} ≲ \sqrt{T} .

We square both sides. Since ψ̂ is an arbitrary cutoff function, we see that

∫_{|ξ|∼1} |∫ e^{2πit|ξ|}e^{−2πitξ_{3}}e^{−2πiβ_{t}ξ_{1}}η(t/T) dt|^{2} dξ ≲ T.

We may obviously restrict the annulus |ξ| ∼ 1 still further, to the region

A = {ξ : ξ_{3} ∼ 1, |ξ_{2}| ≲ |ξ_{1}| ≲ 1}.

1This argument was directed at the BMO estimate, but easily extends to the frequencylocalized case.

The significance of this region is that one has the “Taylor approximation”

|ξ| − ξ_{3} ∼ ξ_{1}^{2}  (7)

for all ξ ∈ A.

Expanding out the square, we obtain

∫_{A} ∫ ∫ e^{2πi(t−s)(|ξ|−ξ_{3})}e^{−2πi(β_{t}−β_{s})ξ_{1}}η(t/T)η(s/T) dtdsdξ ≲ T.

Taking expectations of both sides and using (5) we obtain

∫_{A} ∫ ∫ e^{2πi(t−s)(|ξ|−ξ_{3})}e^{−2π|t−s|ξ_{1}^{2}}η(t/T)η(s/T) dtdsdξ ≲ T.

By the change of variables u = t − s, the assumptions on η, and some calculus, we may rewrite this as

∫_{A} ∫ e^{2πiu(|ξ|−ξ_{3})}e^{−2π|u|ξ_{1}^{2}}η ∗ η(u/T) dudξ ≲ 1,

where ∗ denotes convolution. Letting T → ∞, this becomes^{2}

∫_{A} ∫ e^{2πiu(|ξ|−ξ_{3})}e^{−2π|u|ξ_{1}^{2}} dudξ ≲ 1.  (8)

However, from the identity (see e.g. [11] Ch. III, Proposition 5)

∫ e^{2πiux}e^{−2π|u|y} du = \frac{y}{π(x^{2} + y^{2})}

and (7), this becomes

∫_{A} \frac{1}{ξ_{1}^{2}} dξ ≲ 1,

which is easily seen to be false. Hence (4) must fail for some functions F of the form (6), with non-zero probability.

Notice what happens if the Brownian motion term is not present. In (8) this would mean that the factor e^{−2π|u|ξ_{1}^{2}} is absent, and (8) would become (formally at least)

∫_{A} δ(ξ_{1}^{2}) dξ ≲ 1,

which does not give a contradiction.

References

[1] M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79 (1991).

[2] D. Foschi, Lecture Notes for S. Klainerman’s Graduate Course In Nonlinear Wave Equations: Fall 1996, Princeton University, Private Communication.

[3] J. Ginibre, G. Velo, Generalized Strichartz Inequalities for the Wave Equation, Jour. Func. Anal., 133 (1995), 50–68.

[4] L. Kapitanski, Some Generalizations of the Strichartz-Brenner Inequality, Leningrad Math. J., 1 (1990), 693–676.

[5] M. Keel, T. Tao, Endpoint Strichartz Estimates, To appear, Amer. Math. J.

[6] S. Klainerman, M. Machedon, Space-time Estimates for Null Forms and the Local Existence Theorem, Comm. Pure Appl. Math., 46 (1993), 1221–1268.

[7] H. Lindblad, C. D. Sogge, On Existence and Scattering with Minimal Regularity for Semilinear Wave Equations, Jour. Func. Anal., 130 (1995 ), 357–426.

2This limit is justified because of the exponential decay factor e^{−2π|u|ξ_{1}^{2}}.

[8] S. J. Montgomery-Smith, Time Decay for the Bounded Mean Oscillation of Solutions of the Schrödinger and Wave Equation, Preprint, 1996.

[9] G. Mockenhaupt, A. Seeger, C. D. Sogge, Local Smoothing of Fourier Integrals and Carleson-Sjölin Estimates, J. Amer. Math. Soc., 6 (1993), 65–130.

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

[11] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[12] E. M. Stein, Harmonic Analysis, Princeton University Press, 1993.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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