Rough convolution operators on homogeneous groups

Terence Tao (UCLA)

www.math.ucla.edu/∼tao/preprints (in

preparation)

The purpose of this talk is to:

• Describe the known weak-type (1, 1) results for rough convolution operators in R^{n}

• Extend these results to general homogeneous groups Convolution operators on R^{n}

• Let K(x) be a function on R^{n} which is homogeneous of degree −n:

K(λx) = λ^{−n}K(x).

We are interested in the singular integral operator T(f) = f ∗ K.

• In order for T to be bounded even on L^{2}, it is necessary that K is L^{1} and has mean zero on the unit annulus.

• Conversely, if K is L^{1} and odd on the unit annulus, or if K is L log L, even, and mean zero on the unit annulus then T(f) is bounded on L^{p} for every 1 < p < ∞. (Calderón-Zygmund). Thus if K ∈ L log L and has mean zero, then T is bounded on every L^{p}.

• The L log L on K was weakened to H^{1} by Weiss, and recently weakened in a different manner by Grafakos and Stefanov. (The L^{2} question is completely settled).

• We consider the problem of weak-type (1, 1) bounds.

Known results

• K will always be assumed to have mean zero.

• If K has some smoothness (e.g. Hölder continuity) then the weak-type (1, 1) bound is a result of Calderön-Zygmund theory.

• Work by Hofmann, Christ, Rubio de Francia, and Seeger generalized this to obtain weak-type (1, 1) results whenever K is merely in L log L. There is still an open gap between L log L and L^{1}.

• The work of the first three authors is based on geometric T T^{∗} arguments, and was only shown for low dimensions. The method of Seeger relies on Fourier analysis, and covered all dimensions.

• We show that Christ and Rubio de Francia’s arguments actually can be extended to all dimensions. Furthermore, they apply to homogeneous groups other than R^{n}, whereas the Fourier methods of Seeger do not seem to extend easily.

L^{2} theory

• Of course, to get weak(1,1) boundedness one must first obtain L^{2} boundedness.

• When the underlying group is R^{n} we get L^{2} boundedness for L log L kernels from the work of Calderón and Zygmund.

• If K is odd then we can extend this to homogeneous groups (and for L^{1} kernels) by a variant of the method of rotations. (Of course, instead of rotating the Hilbert transform one must use something more curved, but these objects have been well studied, e.g. by Phong and Stein).

• But if K is even and in L log L we must use a different approach. The idea is to introduce pseudo-Littlewood-Paley operators f ↦ f ∗ ϕ_{j}, where ϕ_{j} is a rescaling of ϕ at scale 2^{j}, and ϕ is Schwarz and satisfies a lot of moment conditions. To prove L^{2} boundedness of f ↦ f ∗ K it suffices to do prove the boundedness of the individual Littlewood-Paley projections f ↦ f ∗ K ∗ ϕ_{j}. By dilation invariance we may take j = 0.

• The idea is to break up the kernel K into dyadic pieces K = ∑_{K} K_{k} and obtain bounds for the pieces f ↦ f ∗K_{k} ∗ϕ_{0} which have a decay factor 2^{−ε|k|}. For k < 0 this follows from the mean zero condition on K, while for k > 0 the idea is to use the radial regularity of K_{k} to eventually interact with the moment conditions in ϕ_{0}. To exploit this we have to use the T T^{∗} method several times, and look at the iterated operator

f ↦ f ∗ K_{k} ∗ ϕ_{0} ∗ …∗ K_{k} ∗ ϕ_{0}

to obtain the cancellation. One needs to convolve n copies of K_{k} because each K_{k} is smooth in only one direction.

• There is a bad contribution in which the n directions of smoothing have a small Jacobian (which means that they do not combine to form something with enough isotropic regularity to interact with the moment conditions of ϕ_{0}), but this term is ”small” and can be dealt with by crude, non-cancellatory methods.

Weak (1,1): Preliminary reductions

• We’ll work in R^{n} for simplicity, but the arguments extend, usually without difficulty, to more general groups such as the Heisenberg group.

• Fix f ∈ L^{1}. By linearity, it suffices to show that

{x : |f ∗ K(x)| ≳ 1} ≲ ∥f∥_{1}.

• Apply a Calderón-Zygmund decomposition at height 1, decomposing f into a good function g (which we can throw away by L^{2} theory) and a sum of bad functions ∑_{I} b_{I}, where the I are an essentially disjoint collection of balls, b_{I} is mean zero and has L^{1} norm ≲ |I| on I, and ∑_{I} |I| ≲ ∥f∥_{1}. So we only need show that

{x : | _{I}^{∑} b_{I} ∗ K(x)| ≳ 1} ≲ _{I}^{∑} |I|.

• Now we break up K. Let K_{0} be the restriction of K to the unit annulus. For each dyadic scale j, let K̃_{j} be the smoothed-out dilate of K_{0} at scale 2^{j}:

K̃_{j}(x) = 2^{−nj} ∫_{t∼1}K_{0}(2^{−j}tx)ϕ(2^{−j}tx)ϕ(t) \frac{dt}{t}

where ϕ is an appropriate cutoff function. Then we can write K = ∑_{j} K̃_{j}.

• Let 2^{i} denote the radius of I. We can rewrite our expression as

_{I}^{∑} b_{I} ∗ K(x) = _{s}^{∑} _{I}^{∑} b_{I} ∗ K̃_{i+s}(x).

• If s ≤ 0 then this set is supported in an exceptional set of measure O(∑_{I} |I|), so it suffices to estimate the contribution for s > 0.

• The idea is to control this contribution in L^{2}, under the stronger assumption that K_{0} is in L^{∞} (not just L log L). In fact, we will be able to get an exponential gain in s:

∥ _{I}^{∑} b_{I} ∗ K̃_{i+s}(x)∥_{2} ≲ 2^{−εs}∥K_{0}∥_{∞} _{I}^{∑} |I|.

A standard argument allows one to convert this exponential decay into a weak-type estimate for the L log L kernels:

{| _{s}^{∑} _{I}^{∑} b_{I} ∗ K̃_{i+s}(x)| ≳ 1} ≲ ∥K_{0}∥_{L log L} _{I}^{∑} |I|.

The Christ-Rubio de Francia trick

• Fix s ≫ 1. By dilation invariance we can make ∑_{I} |I| ≲ 1. We wish to prove

∥ _{I}^{∑} b_{I} ∗ K̃_{i+s}∥_{2} ≲ 2^{−εs}∥K_{0}∥_{∞}.

• The functions b_{I} ∗ K̃_{i+s} are supported on the annuli 2^{s}I_{Δ} with radii ∼ 2^{i+s} centered at I. We emphasize this by rewriting the desired estimate as

∥ _{I}^{∑} ψ_{I}(b_{I} ∗ K̃_{i+s})∥_{2} ≲ 2^{−εs}∥K_{0}∥_{∞}

where ψ_{I} is a smooth cutoff to 2^{s}I_{Δ}.

• Since the I are disjoint, we expect the 2^{s}I to have multiplicity at most O(2^{ns}), i.e. that

∥ _{I}^{∑} χ_{2^{s}I}∥_{∞} ≲ 2^{ns}.

It turns out that this estimate is not strictly speaking true (the dilates can pile up logarithmically), but we can refine the collection I by a small amount to make it true. The idea is to first prove the bound

∥ _{I}^{∑} χ̃_{2^{s}I}∥_{BMO} ≲ 2^{ns},

which is valid for any collection of disjoint balls I.

• Having made this reduction, it turns out that we can relax the L^{∞} control on K_{0} to just L^{2} control. I.e. we’ll show need to show

∥ _{I}^{∑} ψ_{I}(b_{I} ∗ K̃_{i+s})∥_{2} ≲ 2^{−εs}∥K_{0}∥_{2}.

• Now we change our viewpoint. Usually, we keep K fixed and think of these types of estimates as an operator estimate on f (or on related functions such as b_{I}). But as Christ and Rubio de Francia showed, it is often profitable to reverse this perspective, keep the b_{I} fixed, and think of this estimate as an operator estimate on K_{0}! Note that we have converted a weak (1, 1) result to a (2, 2) result.

Next step: T T^{∗}

• The Christ-Rubio de Francia trick has the non-trivial consequence of allowing one to apply the T T^{∗} method in the K variable.

• Using T T^{∗}, the estimate

∥ _{I}^{∑} ψ_{I}(b_{I} ∗ K̃_{i+s})∥_{2} ≲ 2^{−εs}∥K_{0}∥_{2}

gets transformed to

∥2^{−ns} _{I}^{∑} ψ_{I}τ_{I}Dτ_{I}^{∗}ψ_{I}F ∥_{2} ≲ 2^{−2εs}∥F ∥_{2}

where ψ_{I} is a cutoff to the annulus 2^{s}I_{Δ}, τ_{I} is convolution with the signed measure |I|^{−1}b_{I}, and D is the averaged dilation operator

DF(x) = ∫_{t∼1}f(tx)ϕ(t) dt.

• There is a probabilistic interpretation of this operator, if one is willing to be very liberal with the notion of ”probability”, namely allow total probability to become less than 1, and allow negative probability.

• For each x, randomly pick an I such that x ∈ 2^{s}I_{Δ}, each with probability 2^{−ns}. (Note that the total probability here can be less than 1). Randomly pick a point in the support of b_{I}, with (signed!) probability distribution b_{I}/|I|. Dilate F around this point by a random t ∼ 1. The expected value of this operation is the expression above.

• It’s easy to prove this bound without the 2^{−2εs}, even without exploiting any cancellation in the b_{I} or any smoothing in the D operator. Basically one uses the fact that the ψ_{I} overlap with multiplicity at most 2^{ns}. So the hope is to use the cancellation in the b_{I} and the regularity in the D to obtain some additional gain.

• Unfortunately D only smooths in one direction (the direction oriented toward the randomly chosen point in b_{I}). To get around this, we need to iterate the above operator n times to get smoothing in every direction, which will then interact with the mean zero condition of the b_{I} to get a gain of 2^{−εs}.

A model case

• Iterating the desired operator n times, we find ourselves having to estimate quantities like

n ; ∏ ; i=1 ψ_{I_{i}}τ_{I_{i}}Dτ_{I_{i}}^{∗}ψ_{I_{i}}F.

To win the 2^{−εs} gain, we need to play the cancellation in the τ_{I_{i}} against the n smoothing operators D.

• Let us consider a model case when all the I_{i} are the same size. Then the cutoff functions ψ_{I_{i}} are fairly harmless. Also, for randomly placed I_{i}, the operators D smooth in very different directions. So in most cases one gets smoothing in every direction, and so one can obtain cancellation using any one of the τ_{I_{i}}.

• There is an exceptional set of n-tuples of I_{i} which are arranged so that the smoothing effects are in a set of directions with small Jacobian. Fortunately, these ntuples are rare, and their contribution can be easily controlled.

• Now let’s consider a slightly less special case in which the I_{i} are non-decreasing in size. In the Euclidean setting this case can be handled in the same manner as previously. For general homogeneous groups there is a possible difficulty due to the non-isotropic dilation structure, in that the smoothing effects corresponding to the large I_{i} may be in very parallel directions. However, these directions will also be very long, and so there is still more than enough isotropic smoothing to interact with τ_{I_{1}}, which is the convolution operator with the best cancellation properties (since I_{1} is smaller than all the other balls).

• By self-adjointness one also can handle the case when the I_{i} are non-increasing in size.

• However, we encounter a difficulty when the balls I_{i} alternate between being very large and very small. The effect of the large balls is to create smoothing over very long, very parallel arcs, but this is then truncated by the cutoff functions associated with the small balls so that one only obtains smoothing over short, parallel arcs. This is not enough regularity to obtain the desired 2^{−εs} gain even if one uses the τ_{I_{i}} associated with the smallest ball in the I_{i}.

• To get around this problem we will take a much larger iteration of our operator - instead of iterating it n times, we shall iterate it 2^{2n−3} times! The reason for this is the following combinatorial lemma, which allows us to extract from any sequence I_{1}, …, I_{2^{2n−3}} of balls, a subsequence of I_{i_{1}}, …, I_{i_{n}} which is either non-decreasing or non-increasing in size, and such that the smoothing effects of the associated operators remain untruncated by any intervening operators.

• More precisely: Let r_{1}, …, r_{2^{2n−3}} be any collection of positive reals. Then there exists a subsequence r_{i_{1}}, …, r_{i_{n}} which is monotone, and such that r_{j} ≥ min(r_{i_{k}}, r_{i_{l}}) whenever i_{k} < j < i_{l}. (In other words, the balls between I_{i_{k}} and I_{i_{l}} do not cause any truncation problems.)