SHARPNESS OF THE CARLESON-SJÖLIN THEOREM

TERENCE TAO

Abstract. The Carleson-Sjölin theorem gives an L^{p} bound of log(λ)^{1/2−1/p} for Carleson-Sjölin operators when 2 ≤ p ≤ 4. In the case of the parabolic restriction operator it is known from Fefferman’s disk multiplier argument (combined with Keich’s Besicovitch set construction) that this bound is sharp. We use the diffeomorphism and modulation invariance of the Carleson-Sjölin family of operators to show that in fact this bound is sharp for arbitrary Carleson-Sjölin operators. (This question was suggested by Andreas Seeger).

1. Introduction

Let ϕ(x_{1}, x_{2}, y) be a phase function satisfying the Carleson-Sjölin condition

ϕ_{x_{1}y}ϕ_{x_{2}yy} − ϕ_{x_{2}y}ϕ_{x_{1}yy} ≠ 0.  (1)

and consider the normalized operator

T_{λ}f(x_{1}, x_{2}) = λ^{1/2} ∫ e^{iλϕ(x_{1},x_{2},y)}a(x_{1}, x_{2}, y)f(y) dy

where a is a smooth cutoff function and λ ≫ 1.

The Carleson-Sjölin theorem asserts that this operator is bounded on L^{p} with norm O(log(λ)^{1/2−1/p}) when 2 ≤ p ≤ 4.

In the parabolic case

ϕ(x_{1}, x_{2}, y) = x_{1}y^{2} + x_{2}y

this bound is sharp; this is a routine application of Fefferman’s disk multiplier argument, using the Besicovitch set construction of Keich to obtain the compression of log(λ) (as opposed to log(λ)/ log log(λ) coming from the standard constructions).

Observe that the Carleson-Sjölin phases are invariant under diffeomorphisms of the x and y variables separately, or if one replaces ϕ(x, y) by ϕ(x, y) + f(x) + g(y) for arbitrary f, g. Observe that the operators T_{λ} are also essentially unaffected by such transformations, at least as far as L^{p} boundedness is concerned.

The purpose of this note is to show that an arbitrary Carleson-Sjölin phase function can be transformed by the above invariances and a certain scaling to become the parabolic phase function. As a consequence, the L^{p} bounds of the Carleson-Sjölin theorem are sharp for every phase function.

Fix ϕ. For any real y and c, the set

γ_{y,c} := {(x_{1}, x_{2}) : ϕ_{y}(x_{1}, x_{2}, y) = c}

is locally a curve (because ∇_{x_{1},x_{2}}ϕ_{y} ≠ 0). By applying a diffeomorphism we may assume that γ_{0,0} is locally equal to the x_{1} axis near the origin, i.e. that

ϕ_{y}(x_{1}, 0, 0) = 0.  (2)

1991 Mathematics Subject Classification. 42B15, 35L05.

for x_{1} near 0.

From (1) we thus see that

ϕ_{yx_{2}}(x_{1}, 0, 0) ≠ 0 and ϕ_{yyx_{1}}(x_{1}, 0, 0) ≠ 0.

By applying a diffeomorphism of the x_{1} variable only we may assume that

ϕ_{yyx_{1}}(x_{1}, 0, 0) = 1

near the origin. Subtracting off a quadratic function of y from the phase ϕ if necessary, we thus have

ϕ_{yy}(x_{1}, 0, 0) = x_{1}.  (3)

Note that the above transformations do not affect (2).

For each fixed x_{1}, we can apply a diffeomorphism of the x_{2} variable so that

ϕ_{yx_{2}}(x_{1}, 0, 0) = 1  (4)

near the origin. Note that this diffeomorphism does not affect (2), (3).

Finally, by subtracting off a harmless phase ϕ(x_{1}, x_{2}, 0) we may assume that

ϕ(x_{1}, x_{2}, 0) = 0;  (5)

note that this does not affect (2), (3), (4).

Suppose that T_{λ} is bounded on L^{p} with an operator norm of M, so that

∥ ∫ e^{iλϕ(x_{1},x_{2},y)}a(x_{1}, x_{2}, y)f(y) dy∥_{L_{x_{1},x_{2}}^{p}} ≤ Mλ^{−1/2}∥f∥_{L_{y}^{p}}.

Let μ := λ^{1/3}. We apply the change of variables

y′ := yλ^{1/2}/μ^{1/2}; x′_{2} = x_{2}λ^{1/2}/μ^{1/2}

and transform this to

∥ ∫ e^{iλϕ(x_{1},μ^{1/2}λ^{−1/2}x_{2},μ^{1/2}λ^{−1/2}y)}a(x_{1}, μ^{1/2}λ^{−1/2}x_{2}, μ^{1/2}λ^{−1/2}y′)f(y′) dy′∥_{L_{x_{1},x′_{2}}^{p}} ≤ Mμ^{−1/2}∥f∥_{L_{y′}^{p}} .

By strengthening the cutoff we obtain

∥ ∫ e^{iλϕ(x_{1},μ^{1/2}λ^{−1/2}x′_{2},μ^{1/2}λ^{−1/2}y′)}a(x_{1}, x_{2}, y′)f(y′) dy′∥_{L_{x_{1},x′_{2}}^{p}} ≲ Mμ^{−1/2}∥f∥_{L_{y′}^{p}} .

We use (2), (3), (4), (5) and Taylor expand

λϕ(x_{1}, μ^{1/2}λ^{−1/2}x′_{2}, μ^{1/2}λ^{−1/2}y′) = μ(x′_{2}y′ + x_{1}y′^{2}) + O(1)

where the O(1) error is smooth and can be absorbed into the amplitude function. Thus we reduce to showing

∥ ∫ e^{iμ(x_{2}y+x_{1}y^{2})}a(x_{1}, x_{2}, y)f(y) dy∥ ≲ Mμ^{−1/2}∥f∥_{L_{y}^{p}}.

This shows that general Carleson-Sjölin operators are at least as badly behaved as the parabolic operators, in particular

M ≳ (log μ)^{1/2−1/p} ∼ (log λ)^{1/2−1/p}.

References

[1] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972): 287–299.

[2] U. Keich, On L^{p} bounds for Kakeya maximal functions and the Minkowski dimension in R^{2}, Bull. Lond. Math. Soc. 31 (1999), 213–221.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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