SINGULAR INTEGRAL OPERATORS GENERATED BY WAVELET TRANSFORMS

Dmitri RYABOGIN and Boris RUBIN^{1}

Let g ∈ L^{1}( ^{n}), g_{t}(x) = t^{−n}g(x/t). If ∫ g = 0, then g is called the wavelet function, and the convolution operator f → f ∗ g_{t} is called the wavelet transform of f generated by the wavelet g. For a large class of functions g and f ∈ L^{p}( ^{n}), 1 < p < ∞, it is shown that ∫_{ε}^{ρ}(f ∗g_{t})(x)dt/t converges as ε → 0 and ρ → ∞ in the L^{p}-norm and in the a.e. sense to a limit I(f, g) = cf +T f, where c ≡ c(g) = const, and T ≡ T (g) is the Calderón-Zygmund singular integral operator. The particular case T ≡ 0 corresponds to Calderón’s reproducing formula. Each “rough” singular integral operator T_{θ}f = p.v. |x|^{−n}θ(x/|x|) ∗ f, with θ ∈ H^{1}(Σ_{n−1}) (the Hardy space on the unit sphere) can be represented as I(f, g) with a suitable wavelet g. A new proof of the L^{p}-boundedness of T_{θ}, θ ∈ H^{1}(Σ_{n−1}), is given.

1 INTRODUCTION.

The present article is motivated by the following problems, which are closely

related to each other.

1. If f ∈ L^{p}( ^{n}), 1 < p < ∞, and g is an integrable radial function, then ([12],

[13])

lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} ∫_{ε}^{ρ} \frac{f ∗ g_{t}}{t} dt = c_{g}f,  c_{g} = ∫_{^{n}} g(x) log \frac{1}{|x|}dx,  (1.1)

1Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). provided

∫_{^{n}} g(x)dx = 0 and ∫_{^{n}} |g(x) log |x||dx < ∞.  (1.2)

The first equality in (1.1) is a variation of the Calderón reproducing formula ([5], p. 8). What can one say about the limit in (1.1) if g is not radial? For which nonradial functions g does the reproducing formula (1.1) still hold?

2. If g belongs to the Schwartz space S( ^{n}) and ∫ g = 0, then ([15], p. 45)

∫_{ε}^{ρ} g_{t}dt/t tends (as ε → 0 and ρ → ∞) to the distribution of the form cδ + p.v.K, where c = c(g) = const, δ is the Dirac delta function, and K is a certain Calderón-Zygmund kernel, having the form Ω(y/|y|)/|y|^{n} with the characteristic Ω depending on g. The expected L^{p}-analogue of this statement reads :

lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} ∫_{ε}^{ρ} \frac{f ∗ g_{t}}{t} dt = cf + T f, f ∈ L^{p},  (1.3)

T ≡ T (g) being a singular integral operator defined by T f = p.v. K ∗ f . The problem is to prove (1.3) and to find a possibly large class of integrable functions g for which (1.3) holds.

In the case n = 1 these problems were studied in ([11]; Sec. 12). One should

mention the earlier papers by M. J. Fisher [6, 7], in which the L^{p}-convergence of integrals similar to that in (1.1) was studied in the more general setting for functions f defined on a Hilbert space H. The results from [6, 7], reformulated for the special case H = n, are less general than those presented below. For example, we interpret the limit of ∫_{ε}^{ρ}(f ∗ g_{t})dt/t not only in the L^{p}-norm but also in the “almost everywhere” sense. Furthermore, the integral c_{g} (see (1.1)) in our consideration can be regarded in the sense of duality between BMO( ^{n}) (we recall that log(1/|x|) ∈ BMO( ^{n})) and the Hardy space H^{1}( ^{n}). The “standard” case, when c_{g} is represented as an absolutely convergent integral, is also considered. Our proofs seem to be simpler than those in [6, 7], and employ different ideas.

In order to state main results we denote: Σ_{n−1} = {x ∈ ^{n} : |x| = 1}. The

Hilbert transform H_{y′}f of f in the direction y′ = y/|y| ∈ Σ_{n−1}, and the Riesz transforms R_{j}f are defined by

(H_{y′}f)(x) = \frac{1}{πi} p.v. ∫_{−∞}^{∞} \frac{f(x − ty′)}{t} dt,  (1.4)

(R_{j}f)(x) = \frac{Γ ((n + 1)/2)}{iπ^{(n+1)/2}} p.v. ∫_{^{n}} \frac{x_{j} − y_{j}}{|x − y|^{n+1}} f(y)dy , j = 1, …, n.  (1.5)

The notation H^{1}( ^{n}) and H^{1}(Σ_{n−1}) is used for the coresponding Hardy spaces [3, 4, 10]. The letter c designates a constant which can be different at each occurence. We will use the notation I_{ε,ρ}(f, g)(x) = ∫_{ε}^{ρ}(f ∗ g_{t})(x)dt/t, 0 < ε < ρ < ∞.

Theorem 1.1. Let f ∈ L^{p}( ^{n}), 1 < p < ∞, g ∈ H^{1}( ^{n}). Then
∥ sup_{0<ε<ρ<∞} I_{ε,ρ}(f, g)| ∥_{p} ≤ c∥g∥_{H^{1}}∥f∥_{p},  (1.6)

and the limit lim_{{ε→0 \atop ρ→∞}} I_{ε,ρ}(f, g) exists in the L^{p}-norm and in the a.e. sense.

In order to state our next results, we need the Calderón-Zygmund singular in-

tegral operators defined by

(T_{θ}f)(x) = lim_{{ε→0 \atop ρ→∞}}(T_{θ}^{ε,ρ}f)(x), (T_{θ}^{ε,ρ}f)(x) = ∫_{ε<|y|<ρ} f(x − y)\frac{θ(y′)}{|y|^{n}}dy.  (1.7)

Theorem 1.2. Suppose that θ ∈ H^{1}(Σ_{n−1}), ∫_{Σ_{n−1}} θ(x′)dx′ = 0, f ∈ L^{p}( ^{n}),

1 < p < ∞. Then ∥ sup_{0<ε<ρ<∞} |T_{θ}^{ε,ρ}f|∥_{p} ≤ c ∥θ∥_{H^{1}(Σ_{n−1})}∥f∥_{p}, and the limit T_{θ}f =

lim_{{ε→0 \atop ρ→∞}} T_{θ}^{ε,ρ}f exists in the L^{p}-norm and in the a.e. sense.

This theorem was first proved by Calderón and Zygmund [1] for θ belonging to

L log^{+} L(Σ_{n−1}) . The more general case θ ∈ H^{1}(Σ_{n−1}), was studied in [2, 3, 8-10, 17]. In Section 4 we give an alternative proof of Theorem 1.2 based on Theorem 1.1.

Theorem 1.3. Assume that g(x) is a measurable function satisfying (1.2) and

such that Ω(x′) = ∫_{0}^{∞} r^{n−1}g(rx′)dr ∈ H^{1}(Σ_{n−1}). For f ∈ L^{p}( ^{n}), 1 < p < ∞, the following statements hold:

(i)  I(f, g) ≡ lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} I_{ε,ρ}(f, g) = c_{g}f + T_{Ω}f, c_{g} = ∫_{^{n}} g(x) log \frac{1}{|x|}dx,  (1.8)

where T_{Ω}f = lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} T_{Ω}^{ε,ρ}f (cf. (1.7)).

(ii) If, moreover,

∫_{|x|<1/2} |g(x)| | log |x| |^{1+δ}dx < ∞ for some δ > 0,  (1.9)

then ∥ sup_{0<ε<ρ<∞} |I_{ε,ρ}(f, g)| ∥_{p} ≤ c ∥f∥_{p}, c = c(g), and (i) holds with convergence inter-

preted in the a.e. sense.

This theorem can be extended to g ∈ H^{1}( ^{n}) as follows.

Theorem 1.4. Let g ∈ H^{1}( ^{n}) and Ω(x′) = ∫_{0}^{∞} r^{n−1}g(rx′)dr. Then (1.8)

holds in the L^{p}-norm and in the a.e. sense with

c_{g} = \frac{iπ^{(n+1)/2}}{σ_{n−1} Γ((n + 1)/2)} ∑_{j=1}^{n} ∫_{^{n}} (R_{j}g)(y)\frac{y_{j}}{|y|}dy, σ_{n−1} = |Σ_{n−1}|.  (1.10)

Theorems 1.3 and 1.4 are multidimensional generalizations of the similar result

for n = 1 from [11, Sec. 12]. They give a partial answer to problems 1 and 2 mentioned above, and link together Calderón’s reproducing formula with singular integrals.

Example 1.5. Let g(x) = ψ(|x|)θ(x′) be such that

θ ∈ L^{1}(Σ_{n−1}), ∫_{0}^{∞}r^{n−1}ψ(r)dr = 0, ∫_{0}^{∞}r^{n−1}|ψ(r) log r|dr < ∞.  (1.11)

Then the conditions (1.2) are satisfied and Ω(x′) ≡ ∫_{0}^{∞}r^{n−1}g(rx′)dx′ ≡ 0. By Theorem 1.3, for f ∈ L^{p}( ^{n}), 1 < p < ∞, we have

lim_{{ε→0 \atop ρ→∞}} ∫_{ε}^{ρ} \frac{f ∗ g_{t}}{t} dt = c_{g}f, c_{g} = ∫_{0}^{∞}r^{n−1}ψ(r) log \frac{1}{r}dr ∫_{Σ_{n−1}} θ(x′)dx′ ,

in the L^{p}-norm and in the a.e. sense ( the latter holds if ∫_{0}^{1/2} r^{n−1}|ψ(r)|| log r|^{1+δ}dr < ∞ for some δ > 0 ).

This example generalizes the corresponding results (for radial wavelets) from

[12, 13] and shows that the reproducing formula (1.1) also holds for nonradial degenerate wavelet functions g(x) = ψ(|x|)θ(x′), satisfying (1.11) (more generally, one can take any function g satisfying Theorem 1.3 or Theorem 1.4 with Ω ≡ 0).

Example 1.6. Consdier the Calderón-Zygmund singular integral operator (1.7)

with θ ∈ H^{1}(Σ_{n−1}), ∫_{Σ_{n−1}} θ(x′)dx′ = 0. Let ψ(r) be a positive integrable decreasing function on (0, ∞), such that ∫_{0}^{∞} ψ(r)dr = 1. Then g(x) = |x|^{1−n}ψ(|x|)θ(x′) ∈ H^{1}( ^{n}) (see Lemma 2.6 below), and Theorem 1.4 yields c_{g} = 0, Ω(x′) = θ(x′),

T_{θ}f = lim_{{ε→0 \atop ρ→∞}} ∫_{ε}^{ρ} \frac{f ∗ g_{t}}{t} dt, f ∈ L^{p}( ^{n}), 1 < p < ∞,  (1.12)

in the L^{p}-norm and a.e. Thus, each Calderón-Zygmund operator T_{θ} can be represented in “wavelet form” (1.12).

The paper is organized as follows. Section 2 contains auxiliary results, mainly

related to the Hardy spaces H^{1}. In Section 3 we prove Theorem 1.1. A new proof of Theorem 1.2 and a proof of Theorems 1.3 and 1.4 are given in Section 4.

Regarding some perspectives, it is our hope that close results can be obtained in

different settings for bilinear operators I(f, g)(x) = ∫ (W_{g}f)(x, t)dν(t) generated by suitable continuous wavelet transforms (W_{g}f)(x, t), which are integrated with respect to the relevant measure dν(t). The case, when g is replaced by a wavelet measure or distribution, also seems to be interesting. One may expect that a suitable choice of g in I(f, g) will lead to certain new singular integrals corresponding to the initial setting.

Acknowledgments. We are very grateful to Prof. L. Grafakos and Dr. A.

Stefanov for sharing with us their knowledge of the subject. The second author expresses his gratitude to Prof. E.M. Stein and Prof. R.S. Strichartz for useful discussions and the hospitality during his visit in summer 1997. Special thanks go to the referee for valuable remarks.

2 AUXILIARY RESULTS.

Recall some basic facts related to the Hardy spaces H^{1}( ^{n}) and H^{1}(Σ_{n−1}) (for

more details see [15], p. 106; [3], p. 591; [4], p. 233).

Definition 2.1. A function a(x) on ^{n} supported by a ball B ⊂ ^{n} is called

an atom on ^{n} if

∫_{^{n}} a(x)dx = 0 and |a(x)| ≤ |B|^{−1}.  (2.1)

Definition 2.2. A regular atom on Σ_{n−1} is a function α(x′) on Σ_{n−1}

supported by a spherical cap B ⊂ Σ_{n−1} and satisfying the relations

∫_{Σ_{n−1}} α(x′)dx′ = 0, |α(x′)| ≤ |B|^{−1}.  (2.2)

An exceptional atom on Σ_{n−1} is the constant function α(x′), having the value |Σ_{n−1}|^{−1}.

Theorem 2.3. An integrable on ^{n} function g belongs to H^{1}( ^{n}) if and only

if

g = ∑_{k=1}^{∞} λ_{k}a_{k}  (2.3)

where a_{k} is an atom on ^{n} and Σ_{k=1}^{∞}|λ_{k}| < ∞. If g ∈ H^{1}( ^{n}), then ∥g∥_{H^{1}} ∼ inf ∑_{k=1}^{∞} |λ_{k}| where the infimum is taken over all decompositions (2.3).

A similar statement holds for functions on Σ_{n−1} provided that atoms in the

equality of the (2.3)-type are regular or exceptional (see [3], pp. 591, 592).

We prove now some auxiliary lemmas which are of independent interest.

Lemma 2.4. If g ∈ H^{1}( ^{n}), then Ω(x′) = ∫_{0}^{∞} t^{n−1}g(tx′)dt ∈ H^{1}(Σ_{n−1}), and

∥Ω∥_{H^{1}(Σ_{n−1})} ≤ c ∥g∥_{H^{1}( ^{n})} .  (2.4)

Proof. Let g be an atom, supported by a ball B of radius δ, r = dist(0, B̄).

It is clear that ∫_{Σ_{n−1}} Ω(x′)dx = ∫_{^{n}} g(x)dx = 0. Let us show that there is a geodesic ball B ⊂ Σ_{n−1} such that

supp Ω ⊂ B and |Ω| ≤ \frac{A}{|B|}, A = A(n).

Put r = kδ for some k ≥ 0. Then

|Ω(x′)| = | ∫_{r}^{r+2δ}t^{n−1}g(tx′)dt| ≤ c_{n} \frac{(r + 2δ)^{n} − r^{n}}{δ^{n}} = c_{n}P_{n−1}(k),  (2.5)

where P_{n−1}(k) = c_{n}((k + 2)^{n} − k^{n}) if k > 0 and P_{n−1}(k) = c_{n} if k = 0. Let k_{0} be a real number such that P_{n−1}(k)/(k + 1)^{n−1} ≤ A(n) for k ≥ k_{0}. Consider two cases: 1) k ≥ k_{0}, and 2) k < k_{0}. In the first case, we choose B to be the projection of B on Σ_{n−1}. If ρ is the radius of B, then |B| ∼ (sin ρ)^{n−1} = (δ/(r + δ))^{n−1} = (κ + 1)^{1−n}. Hence by (2.5),

|Ω(x′)| ≤ \frac{c_{n}P_{n−1}(k)}{|B|(k + 1)^{n−1}} ≤ \frac{A(n)}{|B|}.

Let k < k_{0}. Then (2.5) shows that one can take B = Σ_{n−1} with |Ω(x′)| ≤

A/|B|, A = c_{n}P_{n−1}(κ_{0}). If g = ∑_{j=1}^{∞} λ_{j}a_{j} is an atomic decomposition of g, then Ω(g) = ∑_{j=1}^{∞} λ_{j}Ω(a_{j}) = A ∑_{j=1}^{∞} λ_{j}Ω̃(a_{j}), where Ω̃(a_{j}) = Ω(a_{j})/A are regular atoms on Σ_{n−1}, and we are done.

The next statement can be regarded as an inverse of the previous one in a certain

sense.

Lemma 2.5. If θ(x′) ∈ H^{1}(Σ_{n−1}), ∫_{Σ_{n−1}} θ(x′)dx′ = 0, and φ(r) is an inte-

grable positive decreasing function on _{+}, then

g(x) {def \atop =} θ(x′)\frac{φ(|x|)}{|x|^{n−1}} ∈ H^{1}( ^{n}) and ∥g∥_{H^{1}( ^{n})} ≤ c ∥θ∥_{H^{1}(Σ_{n−1})}.  (2.6)

Remark. This statement was inspired by the argument from [15], p. 178,

where θ(x′) was a bounded function.

Proof. Let first θ(x′) be a regular atom, supported by the geodesic ball B

with the center at σ ∈ Σ_{n−1} of radius ρ. We claim that G(x) = θ(x′)φ(|x|)|x|^{1−n} admits a decomposition G = ∑_{k=−∞}^{∞} c_{κ}G_{κ}, where G_{κ} are atoms in H^{1}( ^{n}) and ∑_{k=−∞}^{∞} |c_{k}| ≤ c < ∞ with c independent of θ. Given a number a ∈ (1, 2], which will be specified later, we set

G_{k}(x) = \frac{a^{1−k}}{(a − 1) φ(a^{k})} \frac{θ(x′)φ(|x|)}{|x|^{n−1}}  if  a^{k} ≤ |x| < a^{k+1},

and G_{k}(x) ≡ 0 elsewhere. Then G = ∑_{k=−∞}^{∞} c_{k}G_{k}, where c_{k} = a^{k−1}(a − 1)φ(a^{k}), and

∑_{k=−∞}^{∞} c_{k} = ∑_{k=−∞}^{∞} φ(a^{k}) a^{k−1}(a − 1) < ∑_{k=−∞}^{∞} ∫_{a^{k−1}}^{a^{k}} φ(r)dr = ∫_{0}^{∞}φ(r)dr < ∞.  (2.7)

Let us show that G_{k} is an atom for a suitable a = a(ρ). Consider two cases: 1) ρ ≥ ρ_{0} = π/100, and 2) ρ < ρ_{0}. In the first case we put a = 2 and obtain

|G_{k}| ≤ \frac{2^{−k}}{(2^{k})^{n−1}|B|} ≤ \frac{c}{|B_{k}|}, c = c(n, ρ_{0}), B_{k} = \{x: |x| < 2^{k+1}\}.

Since supp G_{k} ⊂ B_{k} and ∫ G_{k} = 0, then c^{−1}G_{k} is an atom. If ρ < ρ_{0} we proceed as follows. Put a = cos ρ/(1 − 2 tan ρ). Then there is a cube Q_{k} around D_{k} ≡ {x: a^{k} < |x| < a^{k+1}, x/|x| ∈ B} with the axis of symmetry along the vector σ, centered at the point q_{k} = 2^{−1}(a^{k+1} +a^{k} cos ρ)σ, and having two parallel faces on the hyperplanes ⟨x, σ⟩ = a^{k+1} and ⟨x, σ⟩ = a^{k} cos ρ. A simple calculation shows that the side of Q_{k} equals a^{k}(a − cos ρ). Let B_{k} = {x: |x − q_{k}| < r_{k}}, r_{k} = 2^{−1/2}a^{k}(a − cos ρ), be the smallest ball containing Q_{k}. Then supp G_{k} ⊂ B_{k} and

|G_{k}| ≤ \frac{a^{1−k}}{(a − 1) a^{k(n−1)}|B|} ≤ \frac{c_{n}}{|B_{k}|}ω(ρ), ω(ρ) = \frac{(a − cos ρ)^{n}}{(a − 1)|B|} .

Since for small ρ, |B| ∼ (sin ρ)^{n−1}, a − cos ρ ∼ sin ρ and a − 1 ∼ sin ρ, then sup_{0<ρ<ρ_{0}} ω(ρ) ≤ c(n, ρ_{0}) and G_{k} is an atom (up to a constant multiple independent of k

and ρ).

In the general case we have θ(x′) = ∑_{j=1}^{∞} λ_{j}α_{j}(x′) (see Definition 2.2), where

α_{j}(x′) is a regular atom such that supp α_{j} ⊂ B_{j}, |α_{j}| ≤ |B_{j}|^{−1}, B_{j} being a geodesic ball of radius ρ_{j}. We set

g(x) = ∑_{ρ_{j}≥ρ_{0}} λ_{j} ∑_{k=−∞}^{∞} c_{k,j}G_{k,j}(x) + ∑_{ρ_{j}<ρ_{0}} λ_{j} ∑_{k=−∞}^{∞} c_{k,j}G_{k,j}(x), ρ_{0} = π/100,

with c_{k,j} and G_{k,j}(x) constructed above according to the size of ρ_{j}. Due to (2.7),

∑_{j} ∑_{k} c_{k,j}|λ_{j}| ≤ c ∑_{j} |λ_{j}| ≤ c ∥θ∥_{H^{1}(Σ_{n−1})},

and the proof is complete.

Lemma 2.6. Let g be an integrable function, such that (a) g satisfies (1.2)

or (b) g ∈ H^{1}( ^{n}). Then (Kg)(x) = |x|^{−n} ∫_{|y|<|x|} g(y)dy ∈ L^{1}( ^{n}) and the constant

c_{g} {def \atop =} ∫_{^{n}} (Kg)(x)dx can be evaluated as follows:

c_{g} = ∫_{^{n}} g(x) log \frac{1}{|x|} dx  (2.8)

in the case (a) and

c_{g} = \frac{iπ^{(n+1)/2}}{σ_{n−1} Γ((n + 1)/2)} ∑_{j=1}^{n} ∫_{^{n}} (R_{j}g)(y)\frac{y_{j}}{|y|}dy  (2.9)

in the case (b).

The proof of this lemma can be found in [12, Theorem 3].

Lemma 2.7. Given a function g ∈ L^{1}( ^{n}), let

k^{0}(x) = \{ \begin{matrix}|x|^{−n} ∫_{0}^{|x|} r^{n−1}g(rx′)dr & if |x| < 1, \\ −|x|^{−n} ∫_{|x|}^{∞} r^{n−1}g(rx′)dr & if |x| > 1. \end{matrix}  (2.10)

If g satisfies (1.2), then k^{0} ∈ L^{1}( ^{n}) and

∫_{^{n}} k^{0}(x)dx = ∫_{^{n}} g(x) log \frac{1}{|x|} dx.  (2.11)

Moreover, if g satisfies (1.9) and

∥g∥_{δ,log} = ∫_{|x|<1/2} |g(x)| | log |x| |^{1+δ}dx + ∫_{|x|>1} |g(x)| log |x| dx + ∥g∥_{1},  (2.12)

then for any f ∈ L^{p}( ^{n}), 1 < p < ∞,

∥ sup_{ε>0} |k_{ε}^{0} ∗ f| ∥_{p} ≤ c ∥g∥_{δ,log}∥f∥_{p},  k_{ε}^{0}(x) = ε^{−n}k^{0}(x/ε).  (2.13)

Proof. We have

∫_{|x|<1} |k^{0}(x)|dx ≤ ∫_{Σ_{n−1}} dx′ ∫_{0}^{1} \frac{dr}{r} ∫_{0}^{r} t^{n−1}|g(tx′)|dt = ∫_{|x|<1} |g(x)| log \frac{1}{|x|}dx,

and

∫_{|x|>1} |k^{0}(x)|dx ≤ ∫_{Σ_{n−1}} dx′ ∫_{1}^{∞} \frac{dr}{r} ∫_{r}^{∞}t^{n−1}|g(tx′)|dt = ∫_{|x|>1} |g(x)| log |x|dx.

The proof of (2.11) is similar. Furthermore, since

|k^{0}(x)| ≤ \frac{1}{|x|^{n}} \{ \begin{matrix}| log |x||^{−1−δ} ∫_{0}^{1/2} r^{n−1}|g(rx′)|| log r|^{1+δ}dr & if |x| ≤ 1/2, \\ ∫_{0}^{1} r^{n−1}|g(rx′)|dr & if 1/2 < |x| ≤ 1, \\ ∫_{|x|}^{∞} r^{n−1}|g(rx′)|dr & if |x| > 1, \end{matrix}

and ∫_{^{n}} |k^{0}(x)|dx ≤ ∥g∥_{δ,log}, then (2.13) holds by Proposition 1 from [15], p. 71.

3 PROOF OF THEOREM 1.1.

We keep on the notation (1.4), (1.5), and remind that g ∈ L^{1}( ^{n}), f ∈ L^{p}( ^{n}),

1 < p < ∞, I_{ε,ρ}(f, g)(x) = ∫_{ε}^{ρ}(f ∗ g_{t})(x)dt/t, 0 < ε < ρ < ∞. The required result will be derived from the following two lemmas.

Lemma 3.1. If g is odd, then

∥ sup_{0<ε<ρ<∞} |I_{ε,ρ}(f, g)|∥_{p} ≤ c ∥g∥_{1}∥f∥_{p},  (3.1)

and

lim_{{ε→0 \atop ρ→∞}} I_{ε,ρ}(f, g)(x) = \frac{πi}{2} ∫_{^{n}} g(y)(H_{y′}f)(x)dy  (3.2)

in the L^{p}-norm and almost everywhere.

Proof. By changing the order of integration and taking into account that g is

odd, we have

I_{ε,ρ}(f, g)(x) = \frac{1}{2} ∫_{^{n}} g(y)dy ∫_{ε|y|<|t|<ρ|y|} \frac{f(x − ty′)}{t} dt.

It remains to apply an argument which is similar to that in [16], Ch. VI, Sec. 2.

Lemma 3.2. If g ∈ H^{1}( ^{n}) is even, then (3.1) holds with ∥g∥_{1} replaced by

∥g∥_{H^{1}} and

lim_{{ε→0 \atop ρ→∞}} I_{ε,ρ}(f, g)(x) = \frac{πi}{2} ∑_{j=1}^{n} ∫_{^{n}} (R_{j}g)(y)(H_{y′}R_{j}f)(x)dy  (3.3)

in the L^{p}-norm and almost everywhere.

Proof. Following the Calderón-Zygmund idea, we pass from the “even case” to

the “odd one” by employing the Riesz transforms R_{j}. Since ∥R_{j}f∥_{p} ≤ c∥f∥_{p}, ∥R_{j}g∥_{H^{1}} ≤ c∥g∥_{H^{1}}, and R_{j}g is odd, then (3.1) (with ∥g∥_{H^{1}}) and (3.3) are consequences of the equality I_{ε,ρ}(f, g) = ∑_{j=1}^{n} I_{ε,ρ}(R_{j}f, R_{j}g) (use Lemma 3.1). The latter can be easily checked for Schwartz functions f and g orthogonal to all polynomials, by applying the Fourier transform to both sides and using the equalities

(R_{j}f)^{∧}(ξ) = f̂(ξ) \frac{ξ_{j}}{|ξ|}, j = 1, ..., n,  ∑_{j=1}^{n} \frac{ξ_{j}^{2}}{|ξ|^{2}} = 1.

We recall that the space Φ( ^{n}) of all such functions is dense in L^{p}( ^{n}) and in H^{1}( ^{n}) ([15], p. 128).

In order to prove Theorem 1.1 we write g in the form g = g_{+} + g_{−}, g_{±}(x) =

(g(x) + g(−x)) /2. The result then follows by Lemmas 3.1 and 3.2.

4 PROOF OF THEOREMS 1.2, 1.3 AND 1.4.

The following lemma will be used repeatedly as a bridge between Theorems

1.1 - 1.3. We assume that f ∈ L^{p}( ^{n}), 1 < p < ∞, g ∈ L^{1}( ^{n}) satisfies (1.2), and keep on the notation I_{ε,ρ}(f, g) and T_{Ω}^{ε,ρ}f from Theorem 1.3.

Lemma 4.1. (i) For ε → 0 and ρ → ∞, the integrals I_{ε,ρ}(f, g) and T_{Ω}^{ε,ρ}f

converge in the L^{p}-norm simultaneously. If the limit T_{Ω}f = lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} T_{Ω}^{ε,ρ}f exists, then

lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} I_{ε,ρ}(f, g) = c_{g}f + T_{Ω}f, c_{g} = ∫_{^{n}} g(x) log \frac{1}{|x|}dx.  (4.1)

(ii) Under the additional assumption (1.9) the following assertions hold :

(a)  ∥ sup_{0<ε<ρ<∞} |T_{Ω}^{ε,ρ}f|∥_{p} ≤ c_{1} (∥g∥_{δ,log} + ∥g∥_{H^{1}})∥f∥_{p}  (4.2)

if and only if

∥ sup_{0<ε<ρ<∞} |I_{ε,ρ}(f, g)|∥_{p} ≤ c_{2} (∥g∥_{δ,log} + ∥g∥_{H^{1}})∥f∥_{p}  (4.3)

where c_{1}, c_{2} are some constants independent of f and g, and ∥g∥_{δ,log} is defined by (2.12).

(b) The statements in (i) are valid with the L^{p}-convergence replaced by that in

the a.e. sense.

Proof. (i) A simple calculation yields

I_{ε,ρ}(f, g) = f ∗ k_{ε,ρ}, where k_{ε,ρ}(x) = |x|^{−n} ∫_{|x|/ρ}^{|x|/ε}r^{n−1}g(rx′)dr (∈ L^{1}( ^{n})).  (4.4)

Let k(x) = |x|^{−n} ∫_{0}^{|x|} r^{n−1}g(rx′)dr. One can write k(x) = k^{0}(x) + q^{0}(x), where k^{0}(x) is the function (2.10), q^{0}(x) ≡ 0 for |x| < 1 and q^{0}(x) = |x|^{−n}Ω(x′) for |x| > 1. Then (4.4) reads:

I_{ε,ρ}(f, g) = k_{ε}^{0} ∗ f − k_{ρ}^{0} ∗ f + T_{Ω}^{ε,ρ}f  (4.5)

(the subscripts indicate the relevant dilations, e.g., k_{ε}^{0}(x) = ε^{−n}k^{0}(x/ε)), and the results follow by Lemma 2.7.

Proof of Theorem 1.2. Let us prove the maximal estimate

∥ sup_{0<ε<ρ<∞} |T_{θ}^{ε,ρ}f| ∥_{p} ≤ c ∥θ∥_{H^{1}(Σ_{n−1})}∥f∥_{p}.  (4.6)

We set g(x) = e^{−|x|}|x|^{1−n}θ(x′). By Lemma 2.5, g ∈ H^{1}( ^{n}) and ∥g∥_{H^{1}( ^{n})} ≤ c∥θ∥_{H^{1}(Σ_{n−1})}. Further, Theorem 1.1 yields ∥ sup_{0<ε<ρ<∞} |I_{ε,ρ}(f, g)| ∥_{p} ≤ c∥g∥_{H^{1}( ^{n})} ∥f∥_{p}. By taking all of

these into account, we make use of Lemma 4.1 (ii a). In our case Ω(x′) = θ(x′), and

∥g∥_{δ,log} = ∥θ∥_{1}( ∫_{0}^{1/2} | log r|^{1+δ}e^{−r}dr + ∫_{1}^{∞} | log r| e^{−r}dr + 1) ≤ c ∥θ∥_{H^{1}(Σ_{n−1})}.

Hence (4.6) follows. The estimate (4.6) enables us to obtain an a.e. convergence of T_{θ}^{ε,ρ}f by taking into account that for compactly supported smooth f,

(T_{θ}^{ε,ρ}f)(x) ≡ ∫_{ε<|y|<1} (f(x − y) − f(x))\frac{θ(y′)}{|y|^{n}}dy + ∫_{1<|y|<ρ} f(x − y)\frac{θ(y′)}{|y|^{n}}dy

converges pointwise as ε → 0 and ρ → ∞. The L^{p}-convergence of T_{θ}^{ε,ρ}f, f ∈ L^{p}( ^{n}), then follows by the Lebesgue dominated convergence theorem.

Remark 4.2. Variants of Theorem 1.2 are formulated sometimes in terms

of the truncated integral (T_{θ}^{ε}f)(x) = ∫_{|y|>ε} f(x − y)θ(y′)dy/|y|^{n} (without truncation at infinity). For f ∈ L^{p}( ^{n}), 1 ≤ p < ∞, and each θ ∈ L^{1}(Σ_{n−1}) such an integral is well-defined a.e. for ε > 0 [1, p. 292]. Since

∥ sup_{ε>0} |T_{θ}^{ε}f|∥_{p} = ∥ sup_{ε>0} | lim_{ρ→∞} T_{θ}^{ε,ρ}f|∥_{p} ≤ ∥ sup_{0<ε<ρ<∞} |T_{θ}^{ε,ρ}f|∥_{p},

then Theorem 1.2 can be reformulated in terms of T_{θ}^{ε}f.

Proof of Theorem 1.3. By Theorem 1.2, T_{Ω}f = lim_{{ε→0 \atop ρ→∞}} T_{Ω}^{ε,ρ}f exists in the L^{p}-

norm and a.e., and ∥ sup_{0<ε<ρ<∞} |T_{Ω}^{ε,ρ}f| ∥_{p} ≤ c ∥Ω∥_{H^{1}(Σ_{n−1})}∥f∥_{p}. Hence the first statement

of the theorem holds by Lemma 4.1 (i). According to (4.5),

sup_{0<ε<ρ<∞} |I_{ε,ρ}(f, g)| ≤ 2 sup_{ε>0} |k_{ε}^{0} ∗ f| + sup_{0<ε<ρ<∞} |T_{Ω}^{ε,ρ}f|,

and the second statement follows by Lemma 2.7.

Proof of Theorem 1.4. We first prove the part of the theorem, related to

the L^{p}-convergence. Let g ∈ Φ( ^{n}) (the space of Schwartz functions orthogonal to all polynomials). We recall that Φ( ^{n}) is dense in L^{p}( ^{n}) and in H^{1}( ^{n}) ([15], p. 128). Then (1.8) gives

I(f, g) = c_{g}f + T_{Ω}f, f ∈ L^{p}( ^{n}), g ∈ Φ( ^{n}),  (4.7)

where, by Lemma 2.6,

c_{g} = ∫_{^{n}} (Kg)(x)dx = \frac{iπ^{(n+1)/2}}{σ_{n−1} Γ((n + 1)/2)} ∑_{j=1}^{n} ∫_{^{n}} (R_{j}g)(y)\frac{y_{j}}{|y|}dy.

Fix f ∈ L^{p}( ^{n}) and consider the mappings g → I(f, g), g → c_{g}f, g → T_{Ω}f as operators from H^{1}( ^{n}) to L^{p}( ^{n}) (the first and the third operators are defined as L^{p}-limits on g ∈ Φ( ^{n}) for fixed f ∈ L^{p}( ^{n})). By Theorem 1.1, ||I(f, g)||_{p} ≤ ||g||_{H^{1}( ^{n})} ||f||_{p}. Note also that ||c_{g}f||_{p} ≤ ||g||_{H^{1}( ^{n})} ||f||_{p} (due to the boundedness of R_{j} : H^{1}( ^{n}) → L^{1}( ^{n})). Furthermore, by Theorem 1.2 and Lemma 2.4,

||T_{Ω}f||_{p} ≤ c ||Ω||_{H^{1}(Σ_{n−1})}||f||_{p} ≤ c ||g||_{H^{1}( ^{n})} ||f||_{p}.

Owing to these relations, one can extend (4.7) to all g ∈ H^{1}( ^{n}). It remains to show that our extensions, say, Ĩ(f, g) and T̃_{Ω}f, coincide with

I(f, g) = lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} I_{ε,ρ}(f, g) and T_{Ω}f = lim_{{ε→0 \atop ρ→∞}}^{(L^{p})} T_{Ω}^{ε,ρ}f

respectively for an arbitrary g ∈ H^{1}( ^{n}). Given g ∈ H^{1}( ^{n}), let {g_{k}}_{k=1}^{∞} ⊂ Φ( ^{n}) be a sequence such that ||g − g_{k}||_{H^{1}( ^{n})} → 0 as k → ∞. Then

||Ĩ(f, g) − I(f, g)||_{p} ≤ ∑_{j=1}^{4} Δ_{j} where Δ_{1} = ||Ĩ(f, g) − I(f, g_{k})||_{p},

Δ_{2} = ||I(f, g_{k}) − I_{ε,ρ}(f, g_{k})||_{p}, Δ_{3} = ||I_{ε,ρ}(f, g_{k} − g)||_{p},

Δ_{4} = ||I_{ε,ρ}(f, g) − I(f, g)||_{p}.

For sufficiently large k, Δ_{1} becomes arbitrary small by definition of Ĩ(f, g). The same holds for Δ_{3} due to (1.6). The quantities Δ_{2} and Δ_{4} tend to 0 as ε → 0 and ρ → ∞. Hence Ĩ(f, g) = I(f, g). Similarly, T̃_{Ω}f = T_{Ω}f. Thus, we have (4.7) with I(f, g) and T_{Ω}f defined as the L^{p}-limits. The existence of these limits in the a.e. sense follows by Theorems 1.1 and 1.2.

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Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel E-mail address: ryabs@math.huji.ac.il, boris@math.huji.ac.il

MSC 1991: Primary: 42B20. Secondary: 42B30, 47G10.