ARE L^{2}-BOUNDED HOMOGENEOUS SINGULAR INTEGRALS NECESSARILY L^{p}-BOUNDED?

LOUKAS GRAFAKOS, PETR HONZÍK, AND DMITRY RYABOGIN

Abstract. We present a dyadic one-dimensional version of the construction of even integrable functions Ω on the unit sphere S^{n−1} with mean value zero satisfying

es sup_{ξ∈S^{n−1}} ∫_{S^{n−1}} |Ω(θ)| log \frac{1}{|θ · ξ|} dθ < +∞ ,

such that the singular integral operator T_{Ω} given by convolution with the distribution p.v. Ω(x/|x|)|x|^{−n} is bounded on L^{p}(R^{n}) if and only if p = 2.

1. Introduction and statements of results

Let Ω be an even complex-valued integrable function on the sphere S^{n−1}, with mean value zero with respect to the surface measure. The classical theory of singular integral operators says that the Calderón and Zygmund principal-value singular integral initially defined for functions f in the Schwartz class S(R^{n})

(1)  T_{Ω}(f)(x) = lim_{ε→0} ∫_{|y|>ε} \frac{Ω(y/|y|)}{|y|^{n}}f(x − y) dy,

is given by a convolution with the distribution p.v. Ω(x/|x|)|x|^{−n}, whose Fourier transform is the homogeneous of degree zero function

(2)  m(Ω)(ξ) := (p.v. Ω(x/|x|)|x|^{−n}) (ξ) = ∫_{S^{n−1}} Ω(θ) log \frac{1}{|ξ · θ|} dθ .

Thus, the L^{2} boundedness of T_{Ω} is equivalent to the condition that m(Ω) is an essentially bounded function, i.e. m(Ω) ∈ L^{∞}(R^{n}). The theory of singular integrals of the form (1) was developed by Calderón and Zygmund [1], [2] who established their L^{p} boundedness in the range 1 < p < ∞ for Ω in L log L(S^{n−1}). It was proved by Weiss and Zygmund [8] that T_{Ω} may be unbounded even on L^{2} for Ω in L(log L)^{1−ε}(S^{n−1}) when ε > 0. Thus the L log L condition on Ω is the sharpest possible, in this sense, that implies the L^{p} boundedness for in the whole range of p ∈ (1, ∞). The weak type (1, 1) boundedness of such singular integrals with Ω in L log L(S^{n−1}) was studied much later by Christ and Rubio de Francia [3] and Seeger [7].

1991 Mathematics Subject Classification. Primary 42B20. Secondary 42E30.

Key words and phrases. Homogeneous Calderón-Zygmund singular integrals, rough kernels.

Grafakos’ research was partially supported by the NSF under grant DMS0400387 and by the University of Missouri Research Council. Honzík was supported by 201/03/0931 Grant Agency of the Czech Republic. Ryabogin’s research was partially supported by the NSF under grant DMS0400789.

In [5] the following result was established:

Theorem 1. There is an integrable function Ω with mean value zero on the unit sphere S^{n−1}, satisfying

(3)  es sup_{ξ∈S^{n−1}} ∫_{S^{n−1}} |Ω(θ)| log \frac{1}{|ξ · θ|} dθ < ∞ ,

but such that T_{Ω} is L^{p} bounded exactly when p = 2.

In this note we consider the one-dimensional dyadic model D_{Ω} of T_{Ω},

(4) D̂_{Ω}f(x) = m(Ω)(x)f̂(x),  m(Ω)(x) = χ_{[0,1]}(x) ∑_{I∋x} ∫_{I} Ω(y)dy,  x ∈ R.

Here the sum is extended over all dyadic subintervals I of [0, 1], and Ω is a nonnegative function in L^{1}([0, 1]). We observe that

∑_{I∋x} ∫_{I} Ω(y)dy = ∫_{0}^{1} ∑_{I∋x,y} χ_{I}(y)Ω(y)dy ≤ ∫_{0}^{1} log \frac{1}{|x − y|}Ω(y)dy,

provided x does not belong to a set of ends of dyadic intervals. We prove the following

Theorem 2. There exists a nonnegative function Ω ∈ L^{1}([0, 1]) such that m(Ω) is bounded and is not a L^{p} Fourier multiplier for any p ≠ 2.

To show that the multiplier norm ∥m(Ω)∥_{M_{p}(R)} is infinite for p ≠ 2, we use deLeeuw [4] type result which comes from the work of Lebedev and Olevski [6]:

Theorem 3. Let b be a function on the real line and let y_{j} be a sequence of real numbers such that y_{j+1} − y_{j} is a constant for all j. Assume that the function b is regulated at the points y_{j}, i.e. the average of left and right limits of b at each y_{j} coincides with b(y_{j}). Then we have

∥b∥_{M_{p}(R)} ≥ ∥{b(y_{j})}_{j}∥_{M_{p}(Z)}.

Here ∥{b(y_{j})}_{j}∥_{M_{p}(Z)} is the norm of the operator f → ∑_{j} b(y_{j})f̂(j)e^{2πijx} acting on functions f on the circle [0, 1]. For compactly supported sequences this norm is at most the size of the support of the sequence times its L^{∞} norm.

Given a compactly supported sequence {ϵ_{j}}_{j} with a large norm ∥{ϵ_{j}}_{j}∥_{M_{p}(Z)} we will construct an integrable function Ω and take an arithmetic progression {x_{j}}_{j} such that ∥{m(Ω)(x_{j})}_{j}∥_{M_{p}(Z)} ≥ c∥{ϵ_{j}}_{j}∥_{M_{p}(Z)}.

2. Proof of Theorem 2

To pick up a sequence {ϵ_{j}}_{j} with a large multiplier norm, we use the fact that the Riesz basis of L^{p}(T), {e^{2πijx}}_{j=−∞}^{+∞} is not unconditional for p ≠ 2. That means that for any K > 0 we can find a compactly supported sequence a_{j} and a sequence ε_{j} of 0’s and 1’s such that

(5)  ∥ ∑_{j} ε_{j}a_{j}e^{2πijx}∥_{p} ≥ K∥ ∑_{j} a_{j}e^{2πijx}∥_{p}.

L^{p} BOUNDS FOR SINGULAR INTEGRALS 3

Consider a decreasing sequence p_{1} > p_{2} > p_{3} > … which converges to 2 and let a_{j}^{k} be a sequence supported in {1, …, l_{k}} and ε_{j}^{k} be a sequence of zeros and ones such that (5) holds with p = p_{k} and K = k, i.e.

(6)  ∥ ∑_{j=1}^{l_{k}} ε_{j}^{k}a_{j}^{k}e^{2πijx}∥_{p_{k}} ≥ k∥ ∑_{j=1}^{l_{k}} a_{j}^{k}e^{2πijx}∥_{p_{k}}.

To construct Ω (depending on ε_{j}^{k}), we look at m(Ω) where Ω = χ_{I_{0}} is the characteristic function of any dyadic interval I_{0} ⊂ [0, 1] of length 2^{−i_{0}}. We observe that m(χ_{I_{0}})(x) = (i_{0} +1)2^{−i_{0}} for x ∈ I_{0}, and m(χ_{I_{0}})(y) ≤ n_{0}2^{−i_{0}}, for y outside I_{0}. Here n_{0} is the number of dyadic subintervals of [0, 1] that contain both I_{0} and y. This means that given any dyadic interval I and any δ > 0, one can find a centrally located (within I) dyadic subinterval J of I of length 2^{−j} and a function Ω_{δ,I} = 2^{j}χ_{J}/(j + 1) such that m(Ω_{δ,I})(x) = 1 when x ∈ J and m(Ω_{δ,I})(x) ≤ δ when x is not in I. Note that the L^{1} norm of Ω_{δ,I} is 1/(j + 1), j = − log |J|, and it can be made small.

We set

Ω = ∑_{k=0}^{∞} Ω_{I_{k}}, Ω_{I_{k}} = ∑_{j=1}^{l_{k}} ε_{j}^{k} Ω_{δ_{k,j},I_{k,j}}.

where Ω_{I_{k}} are supported in I_{k}, the dyadic subintervals of [0, 1],

I_{1} = [0, 1/2], I_{2} = [1/2, 3/4], I_{3} = [3/4, 7/8], I_{4} = [7/8, 15/16], ....,

and ε_{j}^{k} are as in (6). To define Ω_{δ_{k,j},I_{k,j}} we pick irrational points

x_{k,1} < x_{k,2} < ⋯ < x_{k,l_{k}}

inside I_{k} so that the intervals spanned by two consecutive such points have the same length. We choose small disjoint subintervals I_{k,j} of I_{k} centered at the points x_{k,j} for all j ∈ {1, 2, …, l_{k}}. Next, we select an interval J_{k,j} ⊂ I_{k,j} such that the function

Ω_{δ_{k,j},I_{k,j}} = \frac{χ_{J_{k,j}}}{|J_{k,j}|(log(1/|J_{k,j}|) + 1)}

satisfies

(7)  m(Ω_{δ_{k,j},I_{k,j}})(x) = 1  when  x ∈ J_{k,j},

and

(8)  m(Ω_{δ_{k,j},I_{k,j}})(x) ≤ δ_{k,j} = 2^{−2−j−k}/l_{k}^{2}  when x ∉ I_{k,j}.

We can also assume that J_{k,j} satisfies

(9)  log \frac{1}{|J_{k,j}|} ≥ k^{2} l_{k}.

Observe that (9) implies

∥Ω∥_{1} ≤ ∑_{k=1}^{∞} ∑_{j=1}^{l_{k}} \frac{1}{log(1/|J_{k,j}|)} ≤ ∑_{k=1}^{∞} \frac{1}{k^{2}} < ∞.

Observe also that m(Ω) is a bounded function. Indeed, let x ∈ [0, 1]. Then x ∈ I_{n} for some n ≥ 1, and

(10)  m(Ω)(x) ≤ ∑_{k=1,k≠n}^{∞} ∑_{j=1}^{l_{k}} m(Ω_{δ_{k,j},I_{k,j}})(x) + ∑_{j=1}^{l_{n}} m(Ω_{δ_{n,j},I_{n,j}})(x).

The first term in the right hand side of (10) is bounded due to the choice of I_{k,j} and δ_{k,j}, see (8). To estimate the second one, we consider two cases, a) x ∈ I_{n,s} \ J_{n,s} for some fixed s = 1, 2, …, l_{n}, and b) x ∈ J_{n,s}, or x ∈ I_{n} \ I_{n,s}. We write

(11)  ∑_{j=1}^{l_{n}} m(Ω_{δ_{n,j},I_{n,j}})(x) = ∑_{j=1,j≠s}^{l_{n}} m(Ω_{δ_{n,j},I_{n,j}})(x) + m(Ω_{δ_{n,s},I_{n,s}})(x).

In the case a) we have

m(Ω_{δ_{n,s},I_{n,s}})(x) ≤ ∑_{I∋x,J_{n,s}} ∫_{I} Ω_{δ_{n,s},I_{n,s}}(y)dy ≤ \frac{|J_{n,s}| ∑_{I∋x,J_{n,s}} 1}{|J_{n,s}|(log(1/|J_{n,s}|) + 1)} < ∞,

and the boundedness of the right-hand side in (11) follows from (8). In the case b) we use (7) and (8). Thus, the second term in (10) is bounded and m(Ω) is bounded.

It remains to show that m(Ω) is not an L^{p} Fourier multiplier for any p ≠ 2. We fix a p > 2 and pick a k_{0} so that 2 < p_{k_{0}} < p. Then ∥m(Ω)∥_{M_{p}(R)} ≥ ∥m(Ω)∥_{M_{p_{k_{0}}} (R)} and it suffices to show that the latter can become arbitrarily large.

Observe that the function m(Ω) is regulated at the points {x_{k_{0},j}}_{j=1}^{l_{k_{0}}}, (this can be easily seen by splitting m(Ω)(x_{k_{0},j}) into the sums similar to (10), (11)), and by Theorem 3 we have

∥m(Ω)∥_{M_{p_{k_{0}}} (R)} ≥ ∥{m(Ω)(x_{k_{0},j})}_{j=1}^{l_{k_{0}}}∥_{M_{p_{k_{0}}} (Z)}.

But the last expression is at least as big as

∥{m(Ω_{I_{k_{0}}})(x_{k_{0},j})}_{j=1}^{l_{k_{0}}}∥_{M_{p_{k_{0}}} (Z)} − ∥{∑_{k≠k_{0}} m(Ω_{I_{k}})(x_{k_{0},j})}_{j=1}^{l_{k_{0}}}∥_{M_{p_{k_{0}}} (Z)}.

Note that the functions ∑_{k≠k_{0}} m(Ω_{I_{k}}) are constant on the interval I_{k_{0}} and therefore

the sequence {∑_{k≠k_{0}} m(Ω_{I_{k}})(x_{k_{0},j})}_{j=1}^{l_{k_{0}}} is constant of length l_{k_{0}}. The multiplier norm of this sequence is a constant c(p_{k_{0}}) which is bounded above by a constant c(p) = cot(π/2p) independent of k_{0}. Now

m(Ω_{I_{k_{0}}})(x_{k_{0},j}) = ϵ_{j}^{k_{0}} + E_{j}^{k_{0}},

where

E_{j}^{k_{0}} = ∑_{1≤j′≠j≤l_{k_{0}}} ϵ_{j′}^{k_{0}}m(Ω_{δ_{k_{0},j′},I_{k_{0},j′}})(x_{k_{0},j}),

and (8) implies |E_{j}^{k_{0}}| ≤ 2^{−2−j−k_{0}}/l_{k_{0}}, ∥{E_{j}^{k_{0}}}_{j=1}^{l_{k_{0}}}∥_{M_{p_{k_{0}}} (Z)} ≤ 2^{−2−j−k_{0}}, due to the compactness of the support of {E_{j}^{k_{0}}}_{j}. We conclude that ∥m(Ω)∥_{M_{p}(R)} ≥ k_{0} − 1 − c(p) and this can be made arbitrarily large. Hence ∥m(Ω)∥_{M_{p}(R)} = ∞.

References

[1] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.

[2] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309.

[3] M. Christ and J.-L. Rubio de Francia, Weak type (1, 1) bounds for rough operators II, Invent. Math. 93 (1988), 225–237.

[4] K. deLeeuw, On L^{p} multipliers, Ann. Math, 81 (1964), 364–379.

[5] L. Grafakos, P. Honzík and D. Ryabogin, On the p-independence boundedness property of Calderon-Zygmund theory, Journal fur die Reine und Angewandte Mathematik, to appear.

[6] V. Lebedev and A. Olevskii, Idempotents of Fourier multiplier algebra, Geom. Funct. Anal. 4 (1994), 539–544.

[7] A. Seeger, Singular integral operators with rough convolution kernels, Jour. Amer. Math. Soc., 9 (1996), 95–105.

[8] M. Weiss and A. Zygmund, An example in the theory of singular integrals, Studia Math. 26 (1965), 101–111.

Loukas Grafakos, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

  E-mail address: loukas@math.missouri.edu

Petr Honzík, Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

  E-mail address: petrhonz@math.msu.edu

Dmitry Ryabogin, Department of Mathematics, Kansas State University, Manhattan, KS 66503 USA

  E-mail address: ryabs@math.ksu.edu