PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 3, Pages 745–751 S 0002-9939(01)06242-6 Article electronically published on August 28, 2001

SINGULAR INTEGRALS GENERATED BY ZONAL MEASURES

DMITRY RYABOGIN AND BORIS RUBIN

(Communicated by Christopher D. Sogge)

Abstract. We study L^{p}-mapping properties of the rough singular integral operator T_{ν}f(x) = ∫_{0}^{∞} dr/r ∫_{Σ_{n−1}} f(x − rθ)dν(θ) depending on a finite Borel measure ν on the unit sphere Σ_{n−1} in ℝ^{n}. It is shown that the conditions sup_{|ξ|=1} ∫_{Σ_{n−1}} log (1/|θ · ξ|)d|ν|(θ) < ∞, ν(Σ_{n−1}) = 0 imply the L^{p}- boundedness of T_{ν} for all 1 < p < ∞ provided that n > 2 and ν is zonal.

1. Introduction

Let Σ_{n−1} be the unit sphere in ℝ^{n}, Ω ∈ L^{1}(Σ_{n−1}), ∫_{Σ_{n−1}} Ω(θ)dθ = 0. Consider the classical Calderón-Zygmund singular integral operator

(T_{Ω}f)(x) = lim_{{ϵ→0 \atop ρ→∞}}(T_{Ω}^{ε,ρ}f)(x)

= lim_{{ϵ→0 \atop ρ→∞}} ∫_{ε<|y|<ρ} f(x − y)\frac{Ω(y/|y|)}{|y|^{n}}dy,

(1.1)

arising in a variety of problems (we refer the reader to the books [3], [9]–[11] and the survey article [5] for more background information). It is well known (cf. [10], p. 40) that if

(1.2)  sup_{|ξ|=1} ∫_{Σ_{n−1}} |Ω(θ)| log \frac{1}{|θ · ξ|} dθ < ∞,

then T_{Ω} is bounded from L^{2}(ℝ^{n}) into itself. In [5] L. Grafakos and A. Stefanov asked the following

Question. Does (1.2) imply the L^{p}-boundedness of T_{Ω} for some p ≠ 2?

In this note we show that (1.2) implies the L^{p}-boundedness of T_{Ω} for all p ∈ (1, ∞) in the case n > 2, provided that Ω is zonal (i.e. invariant under all rotations about the x_{n}-axis). We also consider a generalization of T_{Ω} with Ω replaced by a

\begin{matrix}c \\ ◯ \end{matrix}2001 American Mathematical Society

Received by the editors September 10, 2000.

1991 Mathematics Subject Classification. Primary 42B20; Secondary 47G10.

Key words and phrases. Singular integrals, L^{p}-boundedness.

This research was partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).

finite Borel measure ν on Σ_{n−1} (cf. [8])

\begin{aligned}(T_{ν}f)(x) & = lim_{{ϵ→0 \atop ρ→∞}}(T_{ν}^{ε,ρ}f)(x) \\ & = lim_{{ϵ→0 \atop ρ→∞}} ∫_{ε}^{ρ} \frac{dr}{r} ∫_{Σ_{n−1}} f(x − rθ)dν(θ). \end{aligned} (1.3)

Definition. Let SO(n − 1) denote the group of rotations leaving the x_{n}-axis fixed. A finite Borel measure ν on Σ_{n−1} is called zonal if ∫_{Σ_{n−1}} f(γϑ)dν(ϑ) = ∫_{Σ_{n−1}} f(ϑ)dν(ϑ) for all γ ∈ SO(n − 1) and f ∈ L^{1}(Σ_{n−1}; d|ν|).

Our main result is

Theorem. Suppose that ν is zonal, ν(Σ_{n−1}) = 0, n > 2.

(a) If

∫_{Σ_{n−1}} log \frac{1}{|θ_{n}|\sqrt{1 − θ_{n}^{2}}} d|ν|(θ) < ∞, (1.4)

then the operator T_{ν}, initially defined by (1.3) on functions f ∈ C_{c}^{∞}(ℝ^{n}), extends to a bounded operator from L^{p} into itself for all p ∈ (1, ∞).

(b) Let f ∈ L^{p}(ℝ^{n}), 1 < p < ∞. If

∫_{Σ_{n−1}} |θ_{n}|^{−β}(1 − θ_{n}^{2})^{−β/2}d|ν|(θ) < ∞ for some β ∈ (0, 1/2),

then ∥ sup_{0<ϵ<ρ<∞} |T_{ν}^{ε,ρ}f| ∥_{p} ≤ c_{p} ∥f∥_{p}, 1 < p < ∞, and the limit in (1.3) exists in the L^{p}-norm and in the a.e. sense.

The proof of this theorem employs results of D. K. Watson [12], and of J. Duoandikoetxea and J. L. Rubio de Francia [2].

Corollary. Suppose that ν is zonal, ν(Σ_{n−1}) = 0, n > 2. Then T_{ν} extends to a bounded operator from L^{p} into itself for all p ∈ (1, ∞), provided

sup_{|ξ|=1} ∫_{Σ_{n−1}} log \frac{1}{|θ · ξ|}d|ν|(θ) < ∞. (1.5)

2. Proof of the Theorem and the Corollary

2.1. Auxiliary results. The proofs of the following two lemmas are technical and given in the Appendix.

Lemma A. Suppose that Λ is an SO(n − 1)-invariant subset of Σ_{n−1} and ν is zonal. Then for β ∈ (0, 1/2) and n > 2,

\begin{matrix}sup_{|ξ|=1} ∫_{Λ} |θ · ξ|^{−β}d|ν|(θ) \\ ≤ c ∫_{Λ} |θ_{n}|^{−β}(1 − θ_{n}^{2})^{−β/2}d|ν|(θ), c = c(n, β). \end{matrix} (2.1)

Lemma B. Let ν be zonal, n > 2. Then conditions (1.4) and (1.5) are equivalent.

The next result will be used in the proof of the Theorem. By

μ̂(ξ) = ∫_{ℝ^{n}} e^{−2πix·ξ}dμ(x)

we denote the Fourier transform of the finite Borel measure μ.

Theorem C ([12], p. 3). Let {σ_{j}}_{j∈ℤ} be a sequence of finite Borel measures on ℝ^{n}, which for integers m ≥ 0 admit a splitting σ_{j} = U_{j}^{m} + L_{j}^{m} into Borel measures U_{j}^{m} and L_{j}^{m} so that

(2.2) U_{j}^{m} and L_{j}^{m} are supported in {x : |x| < c 2^{j}},

∥L_{j}^{m}∥ ≤ c, |L̂_{j}^{m}(ξ)| ≤ \frac{c 2^{am}}{(2^{j}|ξ|)^{α}} for some α > 0, (2.3)

sup_{j} ∑_{m=0}^{∞} ∥U_{j}^{m}∥ ≤ c. (2.4)

Here c and a are nonnegative constants, independent of m and j. If the operator

T f = ∑_{j=−∞}^{∞} σ_{j} ∗ f, f ∈ C_{c}^{∞}(ℝ^{n}),

extends to a bounded operator on L^{2}(ℝ^{n}), then T extends to a bounded operator on L^{p}(ℝ^{n}), 1 < p < ∞.

2.2. Proof of the main results. We start with (a). Denote

Γ_{m} = \{θ ∈ Σ_{n−1} : |θ_{n}|\sqrt{1 − θ_{n}^{2}} < 2^{−m}\}, Γ_{m}^{c} = Σ_{n−1} \ Γ_{m}, (2.5)

and set σ_{j} = U_{j}^{m} + L_{j}^{m}, where the measures L_{j}^{m} and U_{j}^{m} are defined by

(L_{j}^{m}, g) = c_{ν} ∫_{2^{j}}^{2^{j+1}} \frac{dr}{r} ∫_{Γ_{m}^{c}} g(rθ)dν(θ), c_{ν} = \frac{1}{∥ν∥ log 2}, (2.6)

(U_{j}^{m}, g) = c_{ν} ∫_{2^{j}}^{2^{j+1}} \frac{dr}{r} ∫_{Γ_{m}} g(rθ)dν(θ), (2.7)

g ∈ C_{0}(ℝ^{n}). Suppose that f ∈ C_{c}^{∞}(ℝ^{n}). Then the series

T_{ν}f(x) = ∑_{j∈ℤ} (σ_{j} ∗ f)(x)

converges for each x ∈ ℝ^{n}. By Lemma B and by reasons which are similar to [10, p. 40], T_{ν} extends to a bounded operator on L^{2}(ℝ^{n}).

Thus, by Theorem C, it suffices to check (2.2)–(2.4). The validity of (2.2) and the first condition in (2.3) is clear. To check the second inequality in (2.3), we note that

(L_{j}^{m})^{∧}(ξ) = c_{ν} ∫_{Γ_{m}^{c}} dν(θ) \{∫_{2^{j}}^{2^{j+1}} e^{−2πirθ·ξ} \frac{dr}{r} \} .

The integral in brackets is dominated by log 2 and also by 2^{−j}|θ · ξ|^{−1}. Hence, it does not exceed c (2^{−j}|θ · ξ|)^{−α} for any α ∈ (0, 1). Let α = 1/4, ξ′ = ξ/|ξ|. By Lemma A,

|(L_{j}^{m})^{∧}(ξ)| ≤ \frac{c}{(2^{j}|ξ|)^{1/4}} ∫_{Γ_{m}^{c}} \frac{d|ν|(θ)}{|θ · ξ′|^{1/4}}

≤ \frac{c}{(2^{j}|ξ|)^{1/4}} ∫_{Γ_{m}^{c}} \frac{d|ν|(θ)}{(|θ_{n}|\sqrt{1 − θ_{n}^{2}})^{1/4}} ≤ \frac{c 2^{m/4}}{(2^{j}|ξ|)^{1/4}} .

It remains to check (2.4). By (2.7),

∥U_{j}^{m}∥ ≤ c_{ν} ∫_{2^{j}}^{2^{j+1}} \frac{dr}{r} ∫_{Γ_{m}} d|ν|(θ) ≤ c ∫_{Γ_{m}} d|ν|(θ).

Hence (see (2.5))

∑_{m=0}^{∞} ∥U_{j}^{m}∥ ≤ c ∑_{m=0}^{∞} ∫_{Γ_{m}} d|ν|(θ) = c ∫_{Σ_{n−1}} d|ν|(θ) [  ∑_{m<log_{2}(1/|θ_{n}|\sqrt{1−θ_{n}^{2}})} 1]

≤ c ∫_{Σ_{n−1}} log \frac{1}{|θ_{n}|\sqrt{1 − θ_{n}^{2}}} d|ν|(θ) < ∞,

which gives (2.4). Statement (a) is proved.

In order to prove (b), we use Theorems A, B and E of [2] and follow the scheme from [2, p. 550]. It suffices to show that

|σ̂_{0}(ξ)| ≤ c |ξ| if |ξ| ≤ 2, |σ̂_{0}(ξ)| ≤ c |ξ|^{−1/4} if |ξ| > 2.

The inequality for |ξ| ≤ 2 is clear because ν(Σ_{n−1}) = 0. For |ξ| > 2, by Lemma A we obtain

\begin{aligned}|σ̂_{0}(ξ)| & ≤ c \frac{c}{|ξ|^{1/4}} ∫_{Σ_{n−1}} \frac{d|ν|(θ)}{|θ · ξ′|^{1/4}} \\ & ≤ \frac{c}{|ξ|^{1/4}} ∫_{Σ_{n−1}} \frac{d|ν|(θ)}{(|θ_{n}|\sqrt{1 − θ_{n}^{2}})^{1/4}} ≤ c |ξ|^{−1/4}. \end{aligned}

The Corollary is a consequence of the Theorem and Lemma B.

Appendix

Proposition. Let n > 2, and let Λ and ν be the same as in Lemma A. If f ∈ L^{1}(Σ_{n−1}; d|ν|), then

∫_{Λ} f(ϑ)dν(ϑ) = \frac{1}{σ_{n−2}} ∫_{Λ} dν(ϑ) ∫_{Σ_{n−2}} f(\sqrt{1 − ϑ_{n}^{2}}σ + ϑ_{n}e_{n})dσ, (3.1)

where σ_{n−2} = |Σ_{n−2}| = 2π^{(n−1)/2}/Γ((n − 1)/2) and dσ is the usual Lebesgue measure on Σ_{n−2}.

Proof. Let ϑ = (sin θ)σ + (cos θ)e_{n}, σ ∈ Σ_{n−2}, cos θ = ϑ_{n}. Then

∫_{Λ} f(ϑ)dν(ϑ) = ∫_{SO(n−1)} dγ ∫_{Λ} f(γϑ)dν(ϑ)

= ∫_{Λ} dν(ϑ) ∫_{SO(n−1)} f((sin θ)γσ + (cos θ)e_{n})dγ

= \frac{1}{σ_{n−2}} ∫_{Λ} dν(ϑ) ∫_{Σ_{n−2}} f((sin θ)σ + (cos θ)e_{n})dσ,

which gives (3.1).

Proof of Lemma A. Let ξ = (ξ̃, ξ_{n}), ξ̃ ∈ ℝ^{n−1}. By (3.1),

\begin{matrix}∫_{Λ} |θ · ξ|^{−β}d|ν|(θ) \\ = \frac{1}{σ_{n−2}} ∫_{Λ} d|ν|(θ) ∫_{Σ_{n−2}} |\sqrt{1 − θ_{n}^{2}} σ · ξ̃+ θ_{n}ξ_{n}|^{−β}dσ \\ = \frac{σ_{n−3}}{σ_{n−2}} ∫_{Λ} A(ξ, θ)d|ν|(θ), \\ A(ξ, θ) = ∫_{−1}^{1} |t\sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})} + θ_{n}ξ_{n}|^{−β}(1 − t^{2})^{n/2−2}dt. \end{matrix} (3.2)

If |θ_{n}|≥\sqrt{1 − ξ_{n}^{2}}, then |ξ_{n}|≥\sqrt{1 − θ_{n}^{2}}, and by setting b=\sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})}/|θ_{n}ξ_{n}| ≤ 1, we have

A(ξ, θ) ≤ ∫_{−1}^{1} \frac{(1 − t^{2})^{n/2−2}dt}{(|θ_{n}ξ_{n}| − |t|\sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})})^{β}}

≤ |θ_{n}ξ_{n}|^{−β} ∫_{−1}^{1} \frac{(1 − t^{2})^{n/2−2}dt}{(1 − b|t|)^{β}} ≤

≤ 2 |θ_{n}|^{−β}(1 − θ_{n}^{2})^{−β/2} ∫_{0}^{1} \frac{(1 − t^{2})^{n/2−2}dt}{(1 − t)^{β}}

= c |θ_{n}|^{−β}(1 − θ_{n}^{2})^{−β/2}, c = const.

If |θ_{n}| < \sqrt{1 − ξ_{n}^{2}}, i.e. |ξ_{n}| < \sqrt{1 − θ_{n}^{2}}, we set a = \frac{−θ_{n}ξ_{n}}{\sqrt{(1 − ξ_{n}^{2})(1 − θ_{n}^{2})}} ∈ (−1, 1)

and get

\begin{aligned}A(ξ, θ) & ≤ [(1 − θ_{n}^{2})(1 − ξ_{n}^{2})]^{−β/2} ∫_{−1}^{1} \frac{(1 − t^{2})^{n/2−2}dt}{|t − a|^{β}} \\ & ≤ |θ_{n}|^{−β}(1 − θ_{n}^{2})^{−β/2}[I(a) + I(−a)], \\ & I(a) = ∫_{−1}^{a} \frac{(1 − t^{2})^{n/2−2}dt}{(a − t)^{β}}. \end{aligned} (3.3)

By the formulas 2.2.6.1 from [7] and 9.102.2 from [4] we obtain

\begin{aligned}I(a) & = \frac{2^{n/2−2}B(n/2 − 1, 1 − β)}{(a + 1)^{1+β−n/2}}F (\frac{n}{2} − 1, 2 − \frac{n}{2}; \frac{n}{2} − β; \frac{a + 1}{2}) \\ & ≤ c(n, β) < ∞, 0 < β < 1/2. \end{aligned}

The same estimate holds for I(−a). Proof of Lemma B. Denote

R(ξ_{n}, θ_{n}) = ∫_{−1}^{1}(1 − t^{2})^{n/2−2} log \frac{1}{|t\sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})} + θ_{n}ξ_{n}|}dt.

As in (3.2),

∫_{Σ_{n−1}} log \frac{1}{|θ · ξ|}d|ν|(θ)

= \frac{σ_{n−3}}{σ_{n−2}} ∫_{Σ_{n−1}} R(ξ_{n}, θ_{n})d|ν|(θ) =^{def} \frac{σ_{n−3}}{σ_{n−2}} K(ξ_{n}).

(3.4)

Using the same notation as in the proof of Lemma A we have for |θ_{n}| ≥ \sqrt{1 − ξ_{n}^{2}} :

R(ξ_{n}, θ_{n}) ≤ ∫_{−1}^{1}(1 − t^{2})^{n/2−2} log \frac{1}{|θ_{n}ξ_{n}|(1 − |t|b)}dt

≤ c_{1} log \frac{1}{|θ_{n}ξ_{n}|} + 2 ∫_{0}^{1}(1 − t^{2})^{n/2−2} log \frac{1}{1 − t}dt

≤ c_{1} log \frac{1}{|θ_{n}|\sqrt{1 − θ_{n}^{2}}} + c_{2};

for |θ_{n}| < \sqrt{1 − ξ_{n}^{2}} :

R(ξ_{n}, θ_{n}) ≤ log \frac{1}{\sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})}} [c_{1} + ∫_{−1}^{1}(1 − t^{2})^{n/2−2} log \frac{1}{|t − a|}dt]

< log \frac{1}{|θ_{n}|\sqrt{1 − θ_{n}^{2}}} [c_{1} + c_{2} ∫_{−1}^{1} \frac{(1 − t^{2})^{n/2−2}}{|t − a|^{1/4}} dt]

≤ c log \frac{1}{|θ_{n}|\sqrt{1 − θ_{n}^{2}}} ,

c being independent of a (see the estimate of the integral in (3.3)). Hence (1.4) implies (1.5). Conversely, if (1.5) holds, then (see (3.4)) K(0) < ∞ and K(±1) < ∞. Since

K(0) = ∫_{Σ_{n−1}} log \frac{1}{\sqrt{1 − θ_{n}^{2}}} d|ν|(θ) ∫_{−1}^{1}(1 − t^{2})^{n/2−2}dt

+ ∫_{Σ_{n−1}} d|ν|(θ) ∫_{−1}^{1}(1 − t^{2})^{n/2−2} log \frac{1}{|t|}dt

and

K(±1) = ∫_{Σ_{n−1}} log \frac{1}{|θ_{n}|}d|ν|(θ) ∫_{−1}^{1}(1 − t^{2})^{n/2−2}dt,

then

∫_{Σ_{n−1}} log \frac{1}{\sqrt{1 − θ_{n}^{2}}} d|ν|(θ) < ∞, ∫_{Σ_{n−1}} log \frac{1}{|θ_{n}|}d|ν|(θ) < ∞,

and (1.4) follows.

Acknowledgment

We would like to thank Loukas Grafakos and Atanas Stefanov for sharing their knowledge of the subject with us. We are indebted to David K. Watson for giving us his preprints.

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Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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

Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

E-mail address: boris@math.huji.ac.il