Singular integrals, generated by spherical measures

Dmitry Ryabogin and Boris Rubin

Abstract. In this paper we study the L^{p}-mapping properties of the Calderón-Zygmund type 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. In particular it is shown that the conditions ν(Σ_{n−1}) = 0, sup_{|ξ|=1} ∫_{Σ_{n−1}} log (1/|θ · ξ|)d|ν|(θ) < ∞ imply the L^{p}-boundedness of T_{ν}, 1 < p < ∞ provided n > 2, and ν is zonal.

1. Introduction.

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

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

arising in a variety of problems (we refer the reader to the books [18], [19], [8], [4], [17] and the survey article [6] for more background information).

It is well-known [1], that if Ω ∈ L^{1}(Σ_{n−1}) is odd, then the limit in (1.1) exists in the L^{p}-norm and a.e., for all f ∈ L^{p}( ^{n}), 1 < p < ∞. This is a consequence of the corresponding one-dimensional result and the method of rotations. The main difficulty is connected with the case of Ω even. The following result of W. Connett [2], F. Ricci and G. Weiss [11], is well-known (see also [21], [6], [15]). Theorem 1.1. Let f ∈ L^{p}( ^{n}), 1 < p < ∞. If Ω belongs to the Hardy space H^{1}(Σ_{n−1}), and ∫_{Σ_{n−1}} Ω(θ)dθ = 0, then

(1.2)  ∥ sup_{0<ε<ρ<∞}|T_{Ω}^{ε,ρ}f| ∥_{p} ≤ c_{p}∥f∥_{p},

and the limit (1.1) exists in the L^{p}-norm and a.e.

We also mention the following L^{2}-result (cf. [18], p. 40). Theorem 1.2. If

(1.3)  ∫_{Σ_{n−1}} Ω(θ)dθ = 0 and  sup_{|ξ|=1} ∫_{Σ_{n−1}} |Ω(θ)| log \frac{1}{|θ · ξ|} dθ < ∞,

then T_{Ω} is bounded from L^{2}( ^{n}) into itself.

L. Grafakos and A. Stefanov [7] considered the class of functions Ω(θ) satisfying the following conditions:

(1.4) ∫_{Σ_{n−1}} Ω(θ)dθ = 0,  sup_{|ξ|=1} ∫_{Σ_{n−1}} |Ω(θ)|( log \frac{1}{|θ · ξ|})^{1+α}dθ < ∞,  α > 0.

They showed that this class is different from H^{1}(Σ_{n−1}) and proved the following theorem.

1 Theorem 1.3 ([7]). If Ω satisfies (1.4), then T_{Ω} extends to a bounded operator from L^{p} into itself for 2 − α/(1 + α) < p < 2 + α. If, moreover, α > 1, then (1.2) holds for 2 − (2 + 2α)/(1 + 2α) < p < 2 + (2α − 2)/3.

The method of the proof of Theorem 1.3 is based on ideas, which were developed by J. Duoandikoetxea and J. L. Rubio de Francia [3]. The following questions were posed in [7, pp. 456, 457]:

Question 1. Are the ranges of indices in Theorem 1.3 sharp?

Question 2. Does the conditions (1.3) imply the L^{p}- boundedness of T_{Ω} for some p ≠ 2 ?

In this paper we extend the aforementioned ranges of indices and show that (1.3) 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 finite Borel measure on Σ_{n−1}. More precisely, let M(Σ_{n−1}) be a space of all such measures. Given ν ∈ M(Σ_{n−1}), consider the singular integral operator

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

If ν is absolutely continuous with respect to the Lebesgue measure dθ on Σ_{n−1} , i.e. dν(θ) = Ω(θ)dθ, Ω ∈ L^{1}(Σ_{n−1}), then (1.5) coincides with (1.1).

Let us state our main results. The following theorems are related to Question 1.

Theorem A. Let

(1.6) ν(Σ_{n−1}) = 0,  sup_{|ξ|=1} ∫_{Σ_{n−1}} ( log \frac{1}{|θ · ξ|})^{1+α} d|ν|(θ) < ∞ for some α > 0.

Then the operator T_{ν} , initially defined by (1.5) on functions f ∈ C_{c}^{∞}( ^{n}), extends to a linear bounded operator from L^{p} into itself provided

(1.7)  |\frac{1}{2} − \frac{1}{p}| < \frac{α}{2(1 + α)}.

Theorem B. Suppose that f ∈ L^{p}( ^{n}), and ν ∈ M(Σ_{n−1}) satisfies (1.6) for some α > 1. Then

(1.8)  ∥ sup_{0<ε<ρ<∞}|T_{ν}^{ε,ρ}f| ∥_{p} ≤ c_{p} ∥f∥_{p},

provided

(1.9) |\frac{1}{2} − \frac{1}{p}| < \frac{α − 1}{2α},

and the limit in (1.5) exists in the L^{p}-norm, and in the a.e. sense.

As in [7] and in many other papers, related to singular integral operators, we imploy the ideas developed by J. Duoandikoetxea and J. L. Rubio de Francia in [3]. The possibility of extending the bounds for p is based on the use of the method of rotations, instead of a “bootstrap” argument (cf. [17], p. 463), which was used in [7], p. 460.

Our next result concerns Question 2. Let M_{z}(Σ_{n−1}) be the subspace of M(Σ_{n−1}), consisting of zonal measures. Theorem C. Suppose that ν ∈ M_{z}(Σ_{n−1}), ν(Σ_{n−1}) = 0, n > 2.

(a) If

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

then T_{ν} extends to a bounded operator from L^{p} into itself for all p ∈ (1, ∞).

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

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

then (1.8) holds, and the limit in (1.5) exists in the L^{p}-norm and in the a.e. sense.

The proof of part (a) of this theorem employs recent results of D. K. Watson [20]. Corollary 1.4 (cf. Theorem 1.2). Let n > 2, and let ν ∈ M_{z}(Σ_{n−1}) satisfy (1.6) with α = 0. Then T_{ν} extends to a bounded operator from L^{p} into itself for all p ∈ (1, ∞). Corollary 1.5. Let n > 2. There is an even function Ω ∉ H^{1}(Σ_{n−1}) which satisfies (1.3) and does not satisfy (1.4) for any α > 0, but, nevertheless, the relevant operator T_{Ω} extends to bounded operators from L^{p} into itself for all p ∈ (1, ∞).

The above corollary shows that the ranges of indices in (1.7) are also not sharp.

Corollary 1.6. Let n > 2. There is an even function Ω ∉ H^{1}(Σ_{n−1}), which satisfies (1.4) for all α > 0.

This result was proved in [7] for n = 2. But the proof, given there, was fairly complicated. We show that (for n > 2) examples of functions indicated in Corollary 1.6, can be easily obtained from Theorem C and geometric properties of the Hardy spaces H^{1}(Σ_{n−1}) and H^{1}( ^{n}).

We do not know if the results of Theorem C and Corollaries 1.4, 1.5 are true in the case n = 2. Another open problem is whether Corollary 1.4 holds for non-zonal ν if p ≠ 2.

The following observation related to Theorems 1.1 and 1.2 is also of interest. Namely, the second condition in (1.3) may fail, but nevertheless, T_{Ω} is bounded on L^{p} for all 1 < p < ∞. More precisely, the following statement holds.

Proposition 1.7. There is an even function Ω ∈ H^{1}(Σ_{n−1}) such that ∫_{Σ_{n−1}} Ω(θ)dθ = 0 and

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

The paper is organized as follows. In section 2 we prove Theorem A. Sections 3 and 4 are devoted to the proof of Theorem B. The proof of Theorem C and Corollaries 1.4-1.6 is given in section 5. In section 6 we prove Proposition 1.7 and in section 7 give examples of non-zonal singular measures, satisfying (1.6) for all α > 0. Notation. Let Σ_{n−1} = {x ∈ ^{n} : |x| = 1}, σ_{n−1} = |Σ_{n−1}| = 2π^{n/2}/Γ(n/2); M(X) denotes the space of all finite Borel measures on a measure space X; |μ| designates the total variation of μ ∈ M(X). The notation L^{p}(X) is standard; C_{0}( ^{n}) denotes the space of continuous on ^{n} functions, tending to zero at infinity; C_{c}^{∞}( ^{n}) is the space of infinitely differentiable on ^{n} functions, having a compact support. We define the Fourier transform of μ ∈ M( ^{n}) by μ̂(ξ) = ∫_{^{n}} e^{−2πix·ξ}dμ(x). The group of rotations leaving the x_{n}-axis fixed will be denoted by SO(n − 1); e_{n} = (0, …, 0, 1). A measure ν ∈ M(Σ_{n−1}) is called zonal if ∫_{Σ_{n−1}} f(γϑ)dν(ϑ) = ∫_{Σ_{n−1}} f(ϑ)dν(ϑ) for each γ ∈ SO(n − 1) and each f ∈ L^{1}(Σ_{n−1}, dν). The set of all zonal measures on Σ_{n−1} is denoted by M_{z}(Σ_{n−1}). The letter c designates a constant, not necessarily the same at each occurrence.

2. Proof of Theorem A.

We begin by proving some auxiliary statements. Following [3], let {ψ_{j}}_{j∈} be a smooth partition of the unity on (0, ∞) so that

a) ψ_{j} ∈ C^{1}( _{+}), 0 ≤ ψ_{j} ≤ 1, ∑_{j∈} ψ_{j}^{2}(t) = 1,

b) supp(ψ_{0}) ⊆ {t ∈ : 1/2 ≤ t ≤ 2}, ψ_{j}(t) = ψ_{0}(2^{j}t),

c) ψ_{0}(t) ≡ 1 ∀t ∈ [1, 3/2], |ψ′_{j}(t)| ≤ c/t. Suppose also that σ_{k}(∈ M( ^{n})), k ∈ , is a sequence of measures such that

(2.1)  ∥σ_{k}∥ ≤ 1, supp σ_{k} ⊆ {x ∈ ^{n} : 2^{k} ≤ |x| ≤ 2^{k+1}},

and

(2.2)  |σ̂_{k}(ξ)| ≤ \{ \begin{matrix}c |2^{k}ξ| if |2^{k}ξ| ≤ 2, \\ c log^{−1−α} |2^{k}ξ| if |2^{k}ξ| > 2, α > 0. \end{matrix}

For f ∈ C_{c}^{∞}( ^{n}), we define

(2.3)  T f(x) = ∑_{k∈} (σ_{k} ∗ f)(x),
(2.4)  (S_{j}f)^{∧}(ξ) = f̂(ξ)ψ_{j}(|ξ|), (T_{j}f)(x) = ∑_{k∈}S_{j+k}(σ_{k} ∗ S_{j+k}f)(x), j ∈ .

Lemma 2.1. Let f ∈ C_{c}^{∞}( ^{n}), and

(2.5)  ∥ sup_{k∈} (|σ_{k}| ∗ |f|) ∥_{s} ≤ c ∥f∥_{s} ∀s ∈ (1, ∞).

Then

(2.6)  ∥T_{j}f∥_{q} ≤ c ∥f∥_{q} for all q ∈ (1, ∞).

If, moreover, σ_{k}, k ∈ , satisfy (2.2), then for all λ ∈ [0, 1],

(2.7)  ∥T_{j}f∥_{p} ≤ c (1 + |j|)^{−(1+α)λ}∥f∥_{p},

provided that λ/2 < 1/p < 1 − λ/2 if 0 ≤ λ < 1 , and p = 2 if λ = 1 . The constant c in (2.6) and (2.7) is independent of j.

The estimate (2.7) for the smaller range of p’s was proved in [7], p. 460 (it was a consequence of a “bootstrap argument” and the assumption that (2.5) holds for s = 2).

But the point is that in the studying of the operator (1.5), we can always assume that the maximal estimate, corresponding to (2.5) holds for a full range of s (see (2.13)). Proof of Lemma 2.1. The estimate (2.6) was established in [3], p. 545, and we recall its proof for convenience of the reader. We have

∥T_{j}f∥_{q} ≤^{(1)} c ∥(∑_{k∈}|S_{j+k}(σ_{k} ∗ S_{j+k}f)|^{2})^{1/2}∥_{q} ≤^{(2)} c ∥(∑_{k∈}|σ_{k} ∗ S_{j+k}f|^{2})^{1/2}∥_{q} ≤

≤^{(3)} c ∥(∑_{k∈}|S_{j+k}f|^{2})^{1/2}∥_{q} ≤^{(4)} c ∥f∥_{q}.

Here (1) and (4) follow from the Littlewood-Paley theory [17], p. 267; (2) is a special case of the more general estimate (4) from [13]; (3) holds according to the lemma on p. 544 from [3]. Furthermore, by Plancherel’s theorem,

∥T_{j}f∥_{2} ≤ ∑_{k∈}∥σ̂_{k}ψ_{j+k}^{2}f̂∥_{2} ≤ ∑_{k∈} ( ∫_{supp ψ_{j+k}} |σ̂_{k}(ξ)|^{2}|f̂(ξ)|^{2}dξ)^{1/2}.

Owing to (2.2), this gives (2.7) for λ = 1, p = 2 (cf. formula (11) from [7]). For λ = 0, (2.7) coincides with (2.6). The result for 0 < λ < 1 follows by interpolation.

Lemma 2.2. Suppose that σ_{k} ∈ M( ^{n}), k ∈ , satisfy (2.1), (2.2), and (2.5). Then (2.3) extends to a linear bounded operator on L^{p}( ^{n}) provided |1/2 − 1/p| < α (2(1 + α))^{−1}.

Proof. As in [3, p. 545], for f ∈ C_{c}^{∞}( ^{n}) we have:

(2.8)  T f = ∑_{k∈}σ_{k} ∗ f = ∑_{j∈}T_{j}f,

and (2.7) yields ∥T f∥_{p} ≤ c ∥f∥_{p} ∑_{j}(1 + |j|)^{−(1+α)λ}, λ/2 < 1/p < 1 − λ/2. Assuming

λ > (1 + α)^{−1}, we obtain the required result.

Now we pass to the operator T_{ν} from (1.5). One can write T_{ν}f = ∑_{k∈} ω_{k} ∗ f, where ω_{k} ∈ M( ^{n}) are defined by

(2.9)  ∫_{^{n}} g(y)dω_{k}(y) = c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} g(rθ)dν(θ), c_{ν} = (|ν|(Σ_{n−1}) log 2)^{−1},

g ∈ C_{0}( ^{n}), and ν satisfies (1.6). Denote by B( ^{n}) the σ-algebra of all Borel measurable sets in ^{n}. As usual [14, p. 116],

(2.10)  |ω_{k}|(E) = sup_{{A_{i}}} ∑_{i=1}^{∞} |ω_{k}(A_{i})|, E ∈ B( ^{n}),

where the supremum is taken over all partitions of E by A_{i} ∈ B( ^{n}).

Lemma 2.3. Let E ∈ B( ^{n}), 1_{E}(x) = 1 for x ∈ E and 1_{E}(x) = 0 otherwise. Then

(2.11)  |ω_{k}|(E) ≤ c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} 1_{E}(rθ)d|ν|(θ) ≤ 1.

Furthermore, for f ∈ L^{s}( ^{n}), 1 < s < ∞, the following relations hold:

(2.12)  (|ω_{k}| ∗ |f|)(x) ≤^{a.e} c_{ν} ∫_{Σ_{n−1}} d|ν|(θ) ∫_{2^{k}}^{2^{k+1}}|f(x − rθ)|\frac{dr}{r} ,

(2.13)  ∥ sup_{k∈} (|ω_{k}| ∗ |f|) ∥_{s} ≤ c ∥f∥_{s}.

Proof. The first relation follows by (2.9), (2.10):

|ω_{k}|(E) ≤ c_{ν} sup_{\{A_{i}\}} ∑_{i} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} 1_{A_{i}}(rθ)d|ν|(θ) ≤

≤ c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} (sup_{\{A_{i}\}} ∑_{i} 1_{A_{i}}(rθ))d|ν|(θ) = c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} 1_{E}(rθ)d|ν|(θ) ≤ 1.

Furthermore, if f ∈ C_{0}( ^{n}), f^{x}(y) = |f(x − y)|, then by Theorem 1.17 from [14, p. 15] there is a sequence {S_{m}^{x}(y)}_{m=1}^{∞} of simple functions, such that 0 ≤ S_{1}^{x} ≤ S_{2}^{x} ≤ … ≤ S_{m}^{x} ≤ … ≤ |f^{x}| and S_{m}^{x}(y) → |f^{x}(y)| for each x and y. Hence

(|ω_{k}| ∗ |f|)(x) = (|ω_{k}|, lim_{m→∞} S_{m}^{x}) = lim_{m→∞}(|ω_{k}|, S_{m}^{x}) ≤^{(2.11)}

≤ lim_{m→∞} c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} S_{m}^{x}(rθ)d|ν|(θ) = c_{ν} ∫_{2^{k}}^{2^{k+1}} \frac{dr}{r} ∫_{Σ_{n−1}} |f(x − rθ)|d|ν|(θ).

In the general case f ∈ L^{s}( ^{n}), (2.12) then follows by the limiting argument from its validity for any convolution (|f| ∗ g_{t})(x), g_{t}(x) = t^{−n}g(x/t), g ∈ C_{c}^{∞}( ^{n}), g ≥ 0.

To prove (2.13), we denote by

(M_{θ}f)(x) = sup_{R>0} \frac{1}{R} ∫_{0}^{R} |f(x − rθ)|dr

the one-dimensional Hardy-Littlewood maximal operator in direction θ ∈ Σ_{n−1}. By the method of rotations

(2.14)  ∥M_{θ}f∥_{s} ≤ c ∥f∥_{s}, s > 1,

c being independent of θ. Then

(|ω_{k}| ∗ |f|)(x) ≤^{(2.12)} 2c_{ν} ∫_{Σ_{n−1}} [2^{−k−1} ∫_{2^{k}}^{2^{k+1}}|f(x − rθ)|dr]d|ν|(θ) ≤ 2c_{ν} ∫_{Σ_{n−1}} (M_{θ}f)(x)d|ν|(θ),

and the result follows by (2.14). Lemma 2.4. Let ν ∈ M(Σ_{n−1}) satisfy (1.6). Then there is a constant c = c(α, ν) > 0 such that for all k ∈ ,

(2.15)  |ω̂_{k}(ξ)| ≤ \{ \begin{matrix}c |2^{k}ξ|  if |2^{k}ξ| ≤ 2, \\ c log^{−1−α} |2^{k}ξ|  if |2^{k}ξ| > 2. \end{matrix}

This statement resembles the estimate (10) from [7]; see also [3, p. 550]. For convenience of the reader we prove (2.15) by completing some details, which were omitted in [7]. Proof. Since ω̂_{k}(ξ) = ω̂_{0}(2^{k}ξ), it suffices to consider k = 0. The inequality

|ω̂_{0}(ξ)| = |c_{ν} ∫_{1}^{2} \frac{dr}{r} ∫_{Σ_{n−1}} e^{−2πirθ·ξ}dν(θ)| ≤ c |ξ|, |ξ| ≤ 2,

is clear because ν(Σ_{n−1}) = 0. The inequality |ω̂_{0}(ξ)| ≤ c (log |ξ|)^{−1−α}, |ξ| > 2, follows by (1.6) from the estimate

(2.16)  A =^{def} | ∫_{1}^{2} e^{−2πirθ·ξ} \frac{dr}{r} | ≤ c(\frac{b}{a})^{γ}, a = log |ξ|, b = log \frac{3/2}{|θ · ξ′|},

c = c(γ) = const. The latter holds for all γ ≥ 0 (in our case γ = 1+α), θ·ξ ≠ 0, ξ′ = ξ/|ξ|. Let us prove (2.16). Integration by parts yields

A = | ∫_{1}^{2} \frac{d(e^{−2πirθ·ξ})}{2πirθ · ξ}| ≤ \frac{1}{π|θ · ξ|} ≤ \frac{3/2}{|θ · ξ|} = e^{b−a}.

Note also that b > log(3/2) > 1/4, i.e. 1 < 4b. If a − b ≥ 1, then a/(a − b) ≤ 1 + b ≤ 5b, and therefore

A ≤ \frac{(a − b)^{γ}e^{−(a−b)}}{(a − b)^{γ}} ≤ \frac{c_{γ}}{(a − b)^{γ}} ≤ c_{γ} (\frac{5b}{a})^{γ}.

If a − b < 1, then a/b ≤ (b + 1)/b = 1 + 1/b < 5 and we get A ≤ ∫_{1}^{2} dr/r < 1 < (5b/a)^{γ}.

It remains to note that Theorem A is a consequence of Lemmas 2.2 - 2.4 (put σ_{k} = ω_{k}, where ω_{k} are defined by (2.9)).

3. Auxiliary statements.

Suppose that T is the operator (2.3), Φ(x) is a Schwartz function and Φ_{j}(x) = 2^{−jn}Φ(2^{−j}x).

Lemma 3.1. Let f ∈ C_{c}^{∞}( ^{n}). If ∥T f∥_{p} ≤ c ∥f∥_{p}, then

(3.1)  ∥ sup_{j∈} |Φ_{j} ∗ ∑_{k=j}^{∞} σ_{k} ∗ f| ∥_{p} ≤ c ∥f∥_{p}.

The proof of this statement is given in [3], p. 548, and employs the estimate

(3.2)  | ∑_{k=−∞}^{j−1} (σ_{k} ∗ Φ_{j})(y)| = |(Φ_{j} ∗ ∑_{k=−∞}^{j−1} σ_{k})(y)| ≤ c ψ_{j}(y),

ψ_{j}(y) = 2^{−jn}/(1 + |2^{−j}y|)^{n+1}. Since this estimate will be used below and the proof of it was skipped in [3], we complete the details. For x ∈ supp σ_{k}, y ∈ ^{n}, define h_{x,y}(t) = Φ_{j}(y − tx), t ∈ [0, 1]. By (2.2), σ̂_{k}(0) = σ_{k}( ^{n}) = 0. Hence the left hand side of (3.2) does not exceed

∑_{k=−∞}^{j−1} ∫_{^{n}} |h′_{x,y}(η)|d|σ_{k}|(x) ≤ ∑_{k=−∞}^{j−1} ∫_{^{n}} ∑_{l=1}^{n} |\frac{∂Φ_{j}}{∂ξ_{l}} (y − ηx)||x_{l}|d|σ_{k}|(x),

where η = η(x, y) ∈ [0, 1], ξ_{l} = y_{l} − η_{l}x. The above expression is estimated by

c ∑_{k=−∞}^{j−1} 2^{−j(n+1)} ∫_{2^{k}<|x|<2^{k+1}} (1 + |(y − ηx)2^{−j}|)^{−n−1}|x|d|σ_{k}|(x) ≤

≤ c ∑_{k=−∞}^{j−1} 2^{−j(n+1)} ∫_{2^{k}<|x|<2^{k+1}} (\frac{1 + |ηx2^{−j}|}{1 + |y2^{−j}|})^{n+1}|x|d|σ_{k}|(x) ≤ c 2^{n+1} ∑_{k=−∞}^{j−1} 2^{−j+k+1}ψ_{j}(y),

which gives (3.2).

We recall Cotlar’s lemma, which will be used below.

Lemma 3.2 ([17], p. 280). Suppose that {Q_{ℓ}} is a finite collection of bounded operators on L^{2}( ^{n}). Assume that we are given a sequence of positive constants {γ(ℓ)}_{ℓ∈} with

(3.3)  A = ∑_{ℓ∈}γ(ℓ) < ∞,

and

(3.4)  ∥Q_{i}^{∗}Q_{k}∥ ≤ [γ(i − k)]^{2}, ∥Q_{i}Q_{k}^{∗}∥ ≤ [γ(i − k)]^{2};

here ∥ · ∥ denotes the operator norm on L^{2}. Then the operator Q = ∑_{ℓ} Q_{ℓ} satisfies ∥Q∥ ≤ A.

For Φ_{k}(x) as in Lemma 3.1 and for ω_{k} defined by (2.9), we get

(3.5)  Q_{j}f = sup_{k∈} |f_{j,k}|, f_{j,k} = ω_{j+k} ∗ f − Φ_{k} ∗ ω_{j+k} ∗ f.

Lemma 3.3. Let j ≥ 0. There is a constant c > 0, independent of j, with the following properties.

(i) If f ∈ L^{q}( ^{n}), 1 < q < ∞, then

(3.6)  ∥Q_{j}f∥_{q} ≤ c ∥f∥_{q}.

(ii) If Φ is a radial function, such that |Φ̂(ξ)| ≤ 1, Φ̂(ξ) = 1 for |ξ| ≤ 2 and Φ̂(ξ) = 0 for |ξ| > 3, and f ∈ L^{2}( ^{n}), then

(3.7)  ∥Q_{j}f∥_{2} ≤ c (1 + j)^{−α}∥f∥_{2},

provided that ν satisfies (1.6) with α > 1.

Proof. (i) We have Q_{j}f ≤ sup_{k∈}|ω_{j+k} ∗ f| + c M[sup_{k∈}|ω_{j+k} ∗ f|], where M is the Hardy-Littlewood maximal operator. Hence (3.6) follows by Lemma 2.3.

(ii) Since (sup_{k}|f_{j,k}|)^{2} ≤ sup_{k}|f_{j,k}|^{2} ≤ ∑_{k}|f_{j,k}|^{2}, then

∥Q_{j}f∥_{2}^{2} ≤ ∥(∑_{k∈}|f_{j,k}|^{2})^{1/2}∥_{2}^{2} = lim_{N→∞} ∑_{|k|≤N} ∫_{^{n}} |f_{j,k}(x)|^{2}dx.

It suffices to show that for an arbitrary N ∈ ,

(3.8)  ∑_{|k|≤N} ∫_{^{n}} |f_{j,k}(x)|^{2}dx ≤ c (1 + j)^{−2α}∥f∥_{2}^{2},

where c is independent of j and N. To prove (3.8) we make use of Lemma 3.2. Let {r_{ℓ}(t)}_{ℓ=1}^{∞} be an orthonormal system of the Rademacher functions in L^{2}[0, 1] so that r_{ℓ}(t) = sgn sin 2^{ℓ}tπ,

∑_{|k|≤N} ∫_{^{n}} |f_{j,k}(x)|^{2}dx = ∫_{0}^{1} dt ∫_{^{n}} | ∑_{|k|≤N} r_{k+N+1}(t)f_{j,k}(x)|^{2}dx,

(cf. [22], p. 176, 180). Fix N ∈ , t ∈ [0, 1], and set Q_{k,N}^{j}f = r_{k+N+1}(t)f_{j,k}. We claim that,

(3.9)  ∥(Q_{i,N}^{j})^{∗}Q_{k,N}^{j}∥_{2→2} ≤ [γ_{j}(k − i)]^{2}, γ_{j}(k − i) = c \frac{(1 + j)^{−(1+α)/2}}{(1 + j + |k − i|)^{(1+α)/2}} ,

where c is independent of r_{k+N+1} and N (the same estimate holds for Q_{i,N}^{j}(Q_{k,N}^{j})^{∗}). Suppose for a moment, that (3.9) is true. Then

∑_{ℓ∈}γ_{j}(ℓ) = c (1 + j)^{−(1+α)/2} ∑_{ℓ∈}(1 + j + |ℓ|)^{−(1+α)/2} ≤

≤ 2c (1 + j)^{−(1+α)/2} ∫_{0}^{∞}(1 + j + t)^{−(1+α)/2}dt = c (1 + j)^{−α}, α > 1,

and Lemma 3.2 yields

∥ ∑_{|k|≤N} Q_{k,N}^{j}f∥_{2} = ∥ ∑_{|k|≤N} r_{k+N+1}(t)f_{j,k}∥_{2} ≤ c (1 + j)^{−α}∥f∥_{2},

c being independent of r_{k+N+1}(t) and N. This implies (3.8).

Let us prove (3.9). By Plancherel’s Theorem and the definition of Φ_{ℓ} (Φ̂_{ℓ}(ξ) = 1 for 2^{ℓ}|ξ| ≤ 2),

∥(Q_{i,N}^{j})^{∗}Q_{k,N}^{j}f∥_{2}^{2} ≤ ∫_{^{n}} |1 − Φ̂_{k}(ξ)|^{2}|1 − Φ̂_{i}(ξ)|^{2}|ω̂_{j+k}(ξ)\overline{ω̂_{j+i}}(ξ)|^{2}|f̂(ξ)|^{2}dξ ≤

≤ ∫_{|ξ|>2^{1−min(i,k)}} |ω̂_{j+k}(ξ)\overline{ω̂_{j+i}}(ξ)|^{2}|f̂(ξ)|^{2}dξ.

By Lemma 2.4 the last integral does not exceed

∫_{|ξ|>2^{1−min(i,k)}} |f̂(ξ)|^{2}[log(2^{j+k}|ξ|) log(2^{j+i}|ξ|)]^{−2−2α}dξ ≤ c^{2} a_{i,k}^{j} ∥f̂∥_{2}^{2},

where a_{i,k}^{j} = [(j+k+1−min(i, k))(j+i+1−min(i, k))]^{−2−2α} = [(j+1)(j+1+|k−i|)]^{−2−2α}, and (3.9) follows.

Corollary 3.4. Under the conditions of Lemma 3.3 (ii),

(3.10)  ∥Q_{j}f∥_{p} ≤ c (1 + j)^{−αλ}∥f∥_{p}, j ≥ 0, f ∈ L^{p},

where λ/2 < 1/p < 1 − λ/2 if 0 ≤ λ < 1, p = 2 if λ = 1 , and c is independent of j.

Proof: Since Q_{j} is not a linear operator we cannot interpolate between (3.6) and (3.7) directly. Therefore we proceed as in [19], p. 280–281 (see also [8, p. 60]). Redenote f_{j,k} = Q_{j,k}f so that Q_{j}f = sup_{k} |Q_{j,k}f| (cf. (3.5)). Let K be the set of all measurable integer-valued functions k(x) on ^{n}. Given k(x) ∈ K, define a linear operator

Q_{j,k(x)}f(x) = c_{ν}[ ∫_{2^{j+k(x)}}^{2^{j+k(x)+1}} \frac{dr}{r} ∫_{Σ_{n−1}} f(x − rθ)dν(θ)−

− ∫_{^{n}} f(x − y)dy ∫_{2^{j+k(x)}}^{2^{j+k(x)+1}} \frac{dr}{r} ∫_{Σ_{n−1}} 2^{−nk(x)}Φ(2^{−k(x)}(y − rθ))dν(θ)],

so that

(3.11)  sup_{k∈K} |Q_{j,k(x)}f(x)| = Q_{j}f(x).

By (3.6), for f ∈ L^{q} we have

(3.12)  ∥ sup_{k∈K}|Q_{j,k(x)}f| ∥_{q} = ∥Q_{j}f∥_{q} ≤ c ∥f∥_{q} ∀q ∈ (1, ∞).

Moreover,

(3.13)  sup_{k∈K} ∥Q_{j,k(x)}f∥_{q} = ∥Q_{j}f∥_{q}.

Indeed, by (3.11) there is a sequence {k_{ℓ}(x)} ⊂ K such that lim_{ℓ→∞} |Q_{j,k_{ℓ}(x)}f(x)| = Q_{j}f(x), and therefore lim_{ℓ→∞} ∥Q_{j,k_{ℓ}(x)}f∥_{q} = ∥Q_{j}f∥_{q} (use the Lebesgue theorem on dominated convergence together with (3.12)). The last equality implies (3.13) because ∥Q_{j,k(x)}f∥_{q} ≤ ∥Q_{j}f∥_{q} ∀k ∈ K.

Since (3.6) and (3.7) are valid for Q_{j,k(x)}, then

(3.14)  ∥Q_{j,k(x)}f∥_{p} ≤ c (1 + j)^{−αλ}∥f∥_{p},

where c is independent of j and k(x), λ and p are as required. The relations (3.14) and (3.13) imply (3.10).

4. Proof of Theorem B. Step 1. Let us prove (1.8) for f ∈ C_{c}^{∞}( ^{n}). Suppose that 2^{j−1} ≤ ε < 2^{j}, 2^{ℓ−1} ≤ ρ < 2^{ℓ} for some j, ℓ ∈ . Then

(4.1)  T_{ν}^{ε,ρ}f = T_{ν}^{ε,∞}f − T_{ν}^{ρ,∞}f,

T_{ν}^{ε,∞}f(x) = ∑_{k=j}^{∞} (ω_{k} ∗ f)(x) + ∫_{ε}^{2^{j}} \frac{dr}{r} ∫_{Σ_{n−1}} f(x − rθ)dν(θ),

and

(4.2)  sup_{0<ε<ρ<∞}|T_{ν}^{ε,ρ}f(x)| ≤ 2 sup_{j∈} | ∑_{k=j}^{∞} (ω_{k} ∗ f)(x)| + 2 sup_{j∈} (|ω_{j}| ∗ |f|)(x).

By Lemma 2.3 (with s = p),

(4.3)  ∥ sup_{j∈} (|ω_{j}| ∗ |f|) ∥_{p} ≤ c ∥f∥_{p} ∀p ∈ (1, ∞).

Let us estimate the first term in the right-hand side of (4.2). We take Φ_{j}(x) = 2^{−jn}Φ(2^{−j}x) with Φ as in Lemma 3.3 (ii). Then

(4.4)  sup_{j∈} | ∑_{k=j}^{∞} ω_{k} ∗ f| ≤ sup_{j∈} |(δ − Φ_{j}) ∗ ( ∑_{k=j}^{∞} ω_{k} ∗ f)| + sup_{j∈} |Φ_{j} ∗ ( ∑_{k=j}^{∞} ω_{k} ∗ f)|,

δ being the Dirac delta function. By (3.5) and (3.10),

(4.5)  ∥ sup_{j∈} |(δ − Φ_{j}) ∗ ( ∑_{k=j}^{∞} ω_{k} ∗ f)| ∥_{p} ≤ ∥ ∑_{j=0}^{∞} Q_{j}f∥_{p} ≤ c ∥f∥_{p},

provided (1.9). Furthermore, by Theorem A and Lemma 3.1 (choose σ_{k} = ω_{k}),

(4.6)  ∥ sup_{j∈} |Φ_{j} ∗ ( ∑_{k=j}^{∞} ω_{k} ∗ f)| ∥_{p} ≤ c ∥f∥_{p}.

These estimates imply (1.8). Step 2. Suppose that f ∈ L^{p}, T̃_{ν} : L^{p} → L^{p} is an extension of the operator T_{ν}, the existence of which was stated in Theorem A. Let us prove that Δ = ∥T_{ν}^{ε,ρ}f −T̃_{ν}f∥_{p} → 0 as ε → 0, ρ → ∞ for p satisfying (1.9). This result is a consequence of the uniform estimate

(4.7)  sup_{0<ε<ρ<∞}∥T_{ν}^{ε,ρ}f∥_{p} ≤ A ∥f∥_{p}, A = const.

Indeed, if {f_{m}} ⊂ C_{c}^{∞}, lim_{m→∞} ∥f − f_{m}∥_{p} = 0, then

Δ ≤ ∥T_{ν}^{ε,ρ}(f − f_{m})∥_{p} + ∥T_{ν}^{ε,ρ}f_{m} − T̃_{ν}f_{m}∥_{p} + ∥T̃_{ν}(f_{m} − f)∥_{p} ≤
≤ A∥f − f_{m}∥_{p} + ∥T_{ν}^{ε,ρ}f_{m} − T̃_{ν}f_{m}∥_{p} + c ∥f_{m} − f∥_{p}.

The first and the last terms become small due to m, the second term tends to 0 as ε → 0 and ρ → ∞ by the Lebesgue theorem of dominated convergence which is applicable owing to Step 1.

In order to prove (4.7) we note that the uniform inequality ∥T_{ν}^{ε,ρ}ω∥_{p} ≤ A∥ω∥_{p} holds for ω ∈ C_{c}^{∞} due to Step 1. Hence it can be extended to all f ∈ L^{p}, and we get ∥T_{ν}^{ε,ρ}f∥_{p} ≤ A∥f∥_{p}. This gives (4.7). Step 3. Let us prove (1.8) for f ∈ L^{p}. It suffices to check (4.1)-(4.6) for such an f. By the reasons, which are similar to [1], p. 292, the integral T_{|ν|}^{ε,∞}|f|(x) is well-defined for a.e. x ∈ ^{n}, which implies the a.e. convergence of the series A_{j}f(x) = ∑_{k=j}^{∞} ω_{k} ∗ f(x) for each j ∈ . This gives (4.1)–(4.4). The validity of (4.5) follows from Corollary 3.4. Let us check (4.6). Note that the relation Δ → 0 in Step 2 implies T̃_{ν}f = lim_{m→∞}_{ℓ→−∞}^{(L^{p})} ∑_{k=ℓ}^{m} ω_{k} ∗ f. Suppose for

a moment that the series A_{j}f and B_{j}f = ∑_{k=−∞}^{j−1} ω_{k} ∗ f converge in the L^{p}-norm. Then A_{j}f =^{a.e.} T̃_{ν}f − B_{j}f, and the left-hand side of (4.6) does not exceed

(4.8)  ∥ sup_{j∈} |Φ_{j} ∗ T̃_{ν}f| ∥_{p} + ∥ sup_{j∈} |Φ_{j} ∗ B_{j}f| ∥_{p},

(the series A_{j}f(x) being also convergent in the a. e. sense). By [17, p. 27] and Step 2, we can estimate the first term in (4.8):

(4.9)  ∥ sup_{j∈} |Φ_{j} ∗ T̃_{ν}f| ∥_{p} ≤ c ∥M(T̃_{ν}f)∥_{p} ≤ c ∥T̃_{ν}f∥_{p} ≤ c ∥f∥_{p},

where M is the Hardy-Littlewood maximal operator.

Let us estimate the second term in (4.8). By the reasons, which are similar to [19], p. 162-163, the series B_{j}Φ_{j}(x) converges for each x. Furthermore, by (3.2) (with σ_{k} = ω_{k}),

|B_{j}Φ_{j}(x)| ≤ c ψ_{j}(x), ψ_{j}(x) = 2^{−jn}/(1 + |2^{−j}x|)^{n+1}. Hence Φ_{j} ∗ B_{j}f =^{a.e.} B_{j}Φ_{j} ∗ f (both

functions belong to L^{p} and coincide in the weak sense), and we obtain

(4.10)  ∥ sup_{j∈} |Φ_{j} ∗ B_{j}f| ∥_{p} ≤ c ∥ sup_{j∈} (|ψ_{j}| ∗ |f|) ∥_{p} ≤ c ∥Mf∥_{p} ≤ c ∥f∥_{p}.

By (4.8)–(4.10) we get (4.6).

It remains to check the L^{p}-convergence of the series A_{j}f and B_{j}f. The operators A_{j} and B_{j} extend as L^{p}-bounded operators with the norms, independent of j. To see this, one should use Lemmas 2.2 and 2.3 by putting σ_{k} = \{ \begin{matrix}ω_{k} if k ≥ j \\ 0 if k < j \end{matrix} for A_{j}, and

σ_{k} = \{ \begin{matrix}0 if k ≥ j \\ ω_{k} if k < j \end{matrix} for B_{j}. By Step 1, for f ∈ C_{c}^{∞} we have

∥ sup_{m∈} | ∑_{k=j}^{m} ω_{k} ∗ f| ∥_{p} ≤ ∥ sup_{ℓ,m} | ∑_{k=ℓ}^{m} ω_{k} ∗ f| ∥_{p} ≤ ∥ sup_{0<ε<ρ<∞}|T_{ν}^{ε,ρ}f| ∥_{p} ≤ c ∥f∥_{p}.

Hence, by the reasons, which are similar to those in Step 2, we obtain an L^{p}-convergence of A_{j}f and B_{j}f for f ∈ L^{p}.

Thus, the maximal estimate (1.8) is proved. The a.e. convergence of T_{ν}^{ε,ρ}f then follows in a standard way (use, e.g., Theorem 3.12 from [19, Chapter II]).

5. Singular integrals, generated by zonal spherical measures. 5.1 Auxiliary results. Lemma 5.1. Suppose that n > 2, Λ is an SO(n − 1)-invariant subset of Σ_{n−1}, ν ∈ M_{z}(Σ_{n−1}) (see Notation). If f ∈ L^{1}(Σ_{n−1}, dν), then

(5.1)  ∫_{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,

where σ_{n−2} = |Σ_{n−1}| 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 (5.1). Lemma 5.2. Let ν and Λ be the same as in Lemma 5.1. Then for β ∈ (0, 1/2) and n > 2,

(5.2)  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).

Proof. Let ξ = (ξ̃, ξ_{n}), ξ̃ ∈ ^{n−1}. By (5.1),

(5.3)  ∫_{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.

If |θ_{n}| ≥ \sqrt{1 − ξ_{n}^{2}}, then |ξ_{n}| ≥ \sqrt{1 − θ_{n}^{2}}, b =^{def} \sqrt{(1 − θ_{n}^{2})(1 − ξ_{n}^{2})}/|θ_{n}ξ_{n}| ≤ 1, and 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}}, then a =^{def} −θ_{n}ξ_{n}/\sqrt{(1 − ξ_{n}^{2})(1 − θ_{n}^{2})} ∈ (−1, 1),

and we get

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}}. (5.4)

By the formulas 2.2.6.1 from [10], and 9.102.2 from [5] we obtain

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. The same estimate holds for I(−a).

Lemma 5.3. Let ν ∈ M_{z}(Σ_{n−1}), n > 2. Then

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

if and only if

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

Proof. 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 (5.3) we have

(5.7)  ∫_{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}).

Using the same notation as in the proof of Lemma 5.2 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 (5.4)). Hence (5.6) implies (5.5). Conversely, if (5.5) holds, then (see (5.7)) 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 (5.6) follows.

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

Theorem 5.4 (cf. [20], p. 3). Let {σ_{j}}_{j∈} be a sequence of finite Borel measures on ^{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

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

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

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

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

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

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

5.2. Proof of Theorem C and Corollary 1.4.

We denote

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

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

(5.13)  (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};

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

g ∈ C_{0}( ^{n}). Suppose that f ∈ C_{c}^{∞}( ^{n}). Then the series T_{ν}f(x) = ∑_{j∈} (σ_{j} ∗ f)(x) , converges for each x ∈ ^{n}. By Lemma 5.3, and by the reasons, which are similar to [18, p. 40], T_{ν} extends to a bounded operator on L^{2}( ^{n}).

Thus, by Theorem 5.4, it suffices to check (5.8)–(5.10). The validity of (5.8) and the first condition in (5.9) is clear. To check the second inequality in (5.9), 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|. Then by Lemma 5.2,

|(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 (5.10). By (5.14),

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

(see the proof of (2.11)). Hence (see (5.12))

∑_{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 (5.10). The statement (a) is proved. The statement (b) follows from Theorem B by taking into account that (1.6) holds for all α > 0 owing to (1.11) and Lemma 5.2.

Corollary 1.4 is a consequence of part (a) of Theorem C and Lemma 5.3.

5.3. Proof of Corollary 1.5.

The required function can be constructed as follows. Denote

Λ_{1} = {θ ∈ Σ_{n−1} : 1/4 < θ_{n} < 3/4}, Λ_{2} = {θ ∈ Σ_{n−1} : 1/3 ≤ θ_{n} ≤ 1/2}(⊂ Λ_{1}),

and let ψ : Σ_{n−1} → be an integrable even zonal function such that ψ(θ) > 0 on Λ_{1} and |ψ| log(1 + |ψ|) ∉ L^{1}(Λ_{2}). We define

Ω(θ) = \{ \begin{matrix}λ |θ_{n}|^{−1}(log |θ_{n}|^{−1})^{−2}(log log |θ_{n}|^{−1})^{−2} if |θ_{n}| < 1/100, \\ ψ(θ) if 1/4 < |θ_{n}| < 3/4, \\ 0 otherwise, \end{matrix}

where λ(< 0) is choosen so that ∫_{Σ_{n−1}} Ω(θ)dθ = 0. For all α > 0,

sup_{|ξ|=1} ∫_{Σ_{n−1}} ( log \frac{1}{|ξ · θ|})^{1+α}|Ω(θ)|dθ ≥ ∫_{\{θ:|θ_{n}|<1/100\}} ( log \frac{1}{|θ_{n}|})^{1+α}|Ω(θ)|dθ = ∞,

i.e. (1.4) fails. Let us prove that Ω ∉ H^{1}(Σ_{n−1}). Assuming the contrary and setting g(x) = |x|^{1−n}Ω(x′), x′ = x/|x|, if |x| ≤ 2, and g(x) ≡ 0 if |x| > 2, we obtain g(x) ∈ H^{1}( ^{n}) (see Lemma 2.5 from [15]). Since g is positive on the open set

Λ̃_{1} = {x = rθ ∈ ^{n} : 1/4 < r < 2, θ ∈ Λ_{1}},

then g belongs to the class L log L on any compact K ⊂ Λ̃_{1} ([17], p. 128). By choosing K = {x = rθ ∈ ^{n} : 1/2 ≤ r ≤ 1, θ ∈ Λ_{2}} we get

∞ > ∫_{K} g(x) log(1 + g(x))dx = ∫_{1/2}^{1} dr ∫_{Λ_{2}} Ω(x′) log (1 + \frac{Ω(x′)}{r^{n−1}})dx′ ≥

≥ \frac{1}{2} ∫_{Λ_{2}} ψ(x′) log(1 + ψ(x′))dx′ = ∞

due to the choice of ψ. This contradiction shows that Ω ∉ H^{1}(Σ_{n−1}).

It remains to note that for dν(θ) = Ω(θ)dθ the operator T_{ν} extends to a bounded operator on L^{p}( ^{n}) ∀p ∈ (1, ∞) according to Corollary 1.4 and Lemma 5.3. 5.4. Proof of Corollary 1.6.

Let Λ_{1}, Λ_{2} and ψ be the same as in the previous subsection. Consider the function

(5.15)  Ω(θ) = \{ \begin{matrix}λ & if 1/5 < |θ_{n}| ≤ 1/4, \\ ψ(θ) & if 1/4 < |θ_{n}| < 3/4, \\ 0, & otherwise, \end{matrix}

where λ < 0 is such that ∫_{Σ_{n−1}} Ω(θ)dθ = 0. By Lemma 5.2 the function (5.15) satisfies (1.4) for all α > 0. On the other hand, Ω ∉ H^{1}(Σ_{n−1}) (see the proof of Corollary 1.5). Proof of Proposition 1.7.

Let x = (x_{1}, x̃) ∈ ^{n}, x̃ = (x_{2}, …, x_{n}) ∈ ^{n−1}. We set Ω(x′) = ∫_{0}^{∞} r^{n−1}g(rx′)dr, x′ = x/|x|, where g(x) = u(x_{1})v(x̃),

u(x_{1}) = \{ \begin{matrix}x_{1}^{−1}(log |x_{1}|^{−1})^{−1−ε} & if 0 < |x_{1}| < 1/100, \\ 0 & if |x_{1}| > 1/100, \end{matrix} 0 < ε < 1,

v(x̃) = \{ \begin{matrix}sgn x_{2} & if 1/100 < |x̃| < 2/100, \\ 0 & otherwise. \end{matrix}

It is clear that ∫_{Σ_{n−1}} Ω(x′)dx′ = ∫_{^{n}} g(x)dx = 0. Let us check that Ω ∈ H^{1}(Σ_{n−1}). By Lemma 2.4 from [15], it suffices to show that g ∈ H^{1}( ^{n}).

Since u ∈ H^{1}( ) (see Section 6.2 of [17], p. 178), and v ∈ H^{1}( ^{n−1}) (see Section 1.2.4. of [17], p. 92), there are Schwartz functions Φ_{1}(x_{1}) and Φ_{2}(x̃) with nonvanishing integrals such that

(6.1)  sup_{t>0}|((Φ_{1})_{t} ∗ u)(x_{1})| ∈ L^{1}( ),  sup_{t>0}|((Φ_{2})_{t} ∗ v)(x̃)| ∈ L^{1}( ^{n−1})

(cf. Theorem 1 from [17], p. 91). In view of (6.1), sup_{t>0}|(Φ_{t} ∗ g)(x)| ∈ L^{1}( ^{n}), where Φ(x) = Φ_{1}(x_{1})Φ_{2}(x̃) is a Schwartz function. This gives g ∈ H^{1}( ^{n}).

Let us check (1.12). We set a(ξ) = ∫_{Σ_{n−1}} |Ω(x′)| log(1/|ξ · x′|) dx′. Since g(x) = |g(x)| sgn x_{1} sgn x_{2} , then |Ω(x′)| = ∫_{0}^{∞} r^{n−1}|g(rx′)|dr. Hence

sup_{|ξ|=1} a(ξ) ≥ a(e_{1}) = ∫_{^{n}} |g(x)| log \frac{1}{|x′_{1}|}dx = ∫_{^{n}} |g(x)| log \frac{1}{|x_{1}|}dx − ∫_{^{n}} |g(x)| log \frac{1}{|x|}dx ≥ ∞

because the first integral in (6.2) is infinite and the second one does not exceed

∫_{}|u(x_{1})|dx_{1} ∫_{^{n−1}} |v(x̃)| log \frac{1}{|x̃|}dx̃ < ∞.

Thus we are done.

7. Examples.

Below we give examples of singular non-zonal measures, which satisfy (1.6) for all α > 0. For these measures all statements of Theorem B hold in the maximal range 1 < p < ∞. Example 7.1 (n = 2). Consider the distribution function C(x) of the middle third Cantor set on [0,1] (see [14], p. 145). Let C_{2π}(x) = 2πC(x/2π), so that C_{2π}(0) = 0 and C_{2π}(2π) = 2π. By setting

g(x) = \{ \begin{matrix}x, & x ∈ [0, 2π/6], \\ 2π/3 − x, & x ∈ [2π/6, 2π/3], \\ 0, & x ∈ [2π/3, 4π/3], \\ 4π/3 − x, & x ∈ [4π/3, 10π/6], \\ x − 2π, & x ∈ [10π/6, 2π], \end{matrix}

we define an auxiliary measure σ on [0, 2π] by

σ(E) = ∫_{E} dψ(x), ψ(x) = (g ∘ C_{2π})(x),

E being a Borel subset of [0, 2π] (since ψ is a function of bounded variation, this definition is correct). Let h : [0, 2π] → Σ_{1} be a canonical map so that h(θ) = (cos θ, sin θ) ∈ Σ_{1}. We define the required measure ν on Σ_{1} as an image of σ under the mapping h. It is clear that ν(Σ_{1}) = σ([0, 2π]) = 0. Moreover, one can readily check that the total variation V_{0}^{x} ψ of ψ on [0, x] coincides with C(x). Hence for any interval [a, b] ⊂ [0, 2π], we have

|σ|([a, b]) =^{def} V_{a}^{b} ψ = V_{0}^{b} ψ − V_{0}^{a} ψ = C(b) − C(a),

and therefore [14, p. 157]

(7.1)  |σ|([a, b]) ≤ c (b − a)^{log_{3} 2}.

Let us show that

(7.2)  sup_{|ξ|=1} ∫_{Σ_{1}} |ϑ · ξ|^{−β}d|ν|(ϑ) < ∞

for all β < log_{3} 2 (this implies (1.6) for all α > 0). Fix ξ = (cos φ, sin φ) ∈ Σ_{1}, and ε ∈ (0, 2^{−10}). Suppose for a moment that

(7.3)  ∫_{Σ_{1}} |ϑ · ξ|^{−β}d|ν|(ϑ) ≤ ∫_{0}^{2π}| cos(φ − θ)|^{−β}d|σ|(θ).

The right-hand side of (7.3) is equal to

(7.4)  ( ∫_{A_{ε}^{1}(φ)} + ∫_{A_{ε}^{2}(φ)} + ∫_{A_{ε}^{3}(φ)} ) | cos(φ − θ)|^{−β}d|σ|(θ) = I_{1} + I_{2} + I_{3},

where

A_{ε}^{1}(φ) = {θ ∈ [0, 2π] : π/2 − ε < |θ − φ| < π/2 + ε},
A_{ε}^{2}(φ) = {θ ∈ [0, 2π] : 3π/2 − ε < |θ − φ| < 3π/2 + ε},

A_{ε}^{3}(φ) = [0, 2π] \ (A_{ε}^{1}(φ) ∪ A_{ε}^{2}(φ)).

The third integral in (7.4) is dominated by (sin ε)^{−β} |σ|([0, 2π]). Furthermore, by Theorem 1.15 from [9], p. 15,

I_{1} ≤ c ∫_{A_{ε}^{1}(φ)} |θ − φ − \frac{π}{2} |^{−β}d|σ|(θ) = β(∫_{0}^{ε} + ∫_{ε}^{∞})|σ|(\{θ ∈ A_{ε}^{1}(φ) : |θ − φ − \frac{π}{2} | ≤ t\})\frac{dt}{t^{β+1}} .

Both integrals are dominated by a constant which is independent of φ. For the second integral this is obvious. For the first one the statement holds due to estimate

|σ|({θ ∈ A_{ε}^{1}(φ) : |θ − φ − π/2| ≤ t}) ≤ |σ|({θ ∈ [0, 2π] : |θ − φ − π/2| ≤ t}) ≤ c t^{log_{3} 2}

which follows from (7.1). For I_{2} the argument is similar.

It remains to prove (7.3). For N ∈ denote D_{ξ}(N) = {ϑ ∈ Σ_{1} : |ϑ · ξ|^{−β} ≤ N}, and let {S_{m}^{ξ}(ϑ)}_{m=1}^{∞} be a sequence of simple functions, such that for each ϑ ∈ D_{ξ}(N), 0 ≤ S_{1}^{ξ} ≤ … ≤ S_{m}^{ξ} ≤ … ≤ |ϑ · ξ|^{−β} and S_{m}^{ξ}(ϑ) → |ϑ · ξ|^{−β} as m → ∞ . By the reasons, which are similar to those in the proof of (2.11), we have |ν|(E) ≤ |σ|(h^{−1}(E)) ∀E ∈ B(Σ_{1}). Hence,

∫_{D_{ξ}(N)} |ϑ · ξ|^{−β} d|ν|(ϑ) = lim_{m→∞} ∫_{D_{ξ}(N)} S_{m}^{ξ}(ϑ) d|ν|(ϑ) ≤

≤ lim_{m→∞} ∫_{h^{−1}(D_{ξ}(N))} (S_{m}^{ξ} ∘ h)(θ) d|σ|(θ) = ∫_{h^{−1}(D_{ξ}(N))} | cos(φ − θ)|^{−β}d|σ|(θ),

where h^{−1}(D_{ξ}(N)) = {θ ∈ [0, 2π] : | cos(φ − θ)|^{−β} ≤ N}. Tending N to infinity, we obtain (7.3).

The next example is motivated by Corollary 4.3 from [3], p. 553. Example 7.2 (n > 2). Define a measure ν on Σ_{n−1} by

∫_{Σ_{n−1}} g(ϑ)dν(ϑ) = ∫_{Γ} g(y)Ω(y)d_{Γ}y, ∫_{Γ} Ω(y)d_{Γ}y = 0,

where Γ = {ϑ ∈ Σ_{n−1} : ϑ_{n} = 1/2}, Ω ∈ L^{q}(Γ) for some q > 1, g ∈ C(Σ_{n−1}); d_{Γ}y is the induced Lebesgue measure on Γ. By the reasons, which are similar to those in the proof of (7.3), and by Hölder’s inequality we have

∫_{Σ_{n−1}} |ξ · ϑ|^{−β}d|ν|(ϑ) ≤ ∫_{Γ} |ξ · y|^{−β}|Ω(y)|d_{Γ}y ≤ K^{1/p} ∥ Ω ∥_{L^{q}(Γ)},

where 1/p+1/q = 1 and K = ∫_{Γ} |ξ ·y|^{−pβ}d_{Γ}y is bounded uniformly in ξ for 0 < β < 1/2 p (cf. Lemma 5.2). By Theorem A the relevant singular integral operator T_{ν} is bounded on L^{p} for all 1 < p < ∞. References.

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

[2] Connett, W. C., Singular integrals near L^{1}, Proc. Sympos. Pure Math., Amer. Math. Soc. (S. Wainger and G. Weiss, eds.), Vol 35, I (1979), 163–165.

[3] Duoandikoetxea, J. and Rubio de Francia, J.L., Maximal and singular integral operators via Fourier transform estimates, Invent. math. 84 (1986), 541–561.

[4] Garcia-Cuerva, J., Rubio de Francia, J.L., Weighted norm inequalities and related topics, Notas de Matem. 116, North-Holland, Amsterdam, 1985.

[5] Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series, and products, Academic Press., 1980.

[6] Grafakos, L. and Stefanov, A., Convolution Calderón-Zygmund singular integral operators with rough kernels, in Analysis of Divergence, Control and Management of Divergent processes, (W. O. Bray, C. V. Stanojević eds.), p. 119–143 (1999).

[7] Grafakos, L. and Stefanov, A., L^{p} bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J., 47(2), p. 455–469 (1998).

[8] de-Guzman, M., Real Variable Methods in Fourier Analysis, Notas de Matem. 46, North-Holland, Amsterdam, 1981.

[9] Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge University Press, Cambridge, 1995.

[10] Prudnikov, A.P., Brychkov, Yu. A. and Marichev O. I., Integrals and series, Moscow, Nauka, 1981.

[11] Ricci, F. and Weiss, G., A characterization of H^{1}(Σ_{n−1}), Proc. Sympos. Pure Math., Amer. Math. Soc. (S. Wainger and G. Weiss, eds.), Vol 35, I (1979), 289–294.

[12] Rubin, B., Fractional integrals and potentials, Addison Wesley Longman, Essex, U.K., 1996.

[13] Rubio de Francia, J.L.,Vector valued inequalities for operators in L^{p} spaces, Bull. London Math. Soc. 12 (1980), p. 211–215.

[14] Rudin, W., Real and Complex Analysis, McGraw-Hill International editions, third edition, 1987.

[15] Ryabogin, D. and Rubin, B., Singular integral operators generated by wavelet transform, The Integral Equations and Operator Theory (to appear).

[16] Stefanov, A., Characterizations of H^{1} and applications to Singular Integrals, (1998), submitted.

[17] Stein, E.M., Harmonic analysis, real variable methods, orthogonality, and oscillation integrals, Princeton Univ. Press, Princeton, N.J., 1993.

[18] Stein, E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.

[19] Stein, E.M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971.

[20] Watson, D.K., Norm inequalities for rough Calderón-Zygmund operators, having no Fourer transform decay, preprint.

[21] Watson, D.K., The Hardy space kernel conditions for rough singular integrals, (1994), preprint.

[22] Wojtaszczyk, P., A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge (1997).

Dmitrii Ryabogin and Boris Rubin, Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

E-mail addresses: ryabs@math.huji.ac.il boris@math.huji.ac.il