GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 1995, 445-468

NECESSARY AND SUFFICIENT CONDITIONS FOR WEIGHTED ORLICZ CLASS INEQUALITIES FOR

MAXIMAL FUNCTIONS AND SINGULAR INTEGRALS. II

A. GOGATISHVILI AND V. KOKILASHVILI

Abstract. This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [1].

§ 1. Weak Type Weighted Inequalities for Singular Integrals

Our further discussion will involve singular integrals with kernels which in homogeneous type spaces are analogues of Calderon–Zygmund kernels.

It will be assumed that k : X ×X\{(x, x) : x ∈ X} → ℝ^{1} is a measurable function satisfying the conditions

|k(x, y)| ≤ \frac{c_{1}}{μB(x, d(x, y))}  (1.1)

and there exists positive constants c_{2} and b_{0} such that

|k(x′, y) − k(x, y)| + |k(y, x′) − k(y, x)| ≤
≤ c_{2}ω(\frac{d(x, x′)}{d(x, y)}) \frac{1}{μB(x, d(x, y))}  (1.2)

for arbitrary x, y and x′ with the condition d(x, y) > b_{0}d(x′, x). Here ω : (0, 1) → ℝ^{1} is a nondecreasing function with the condition ω(0) = 0, ω(2t) ≤ cω(t) and

∫_{0}^{1} \frac{ω(t)}{t} dt < ∞.

445

1072-947X/95/0900-0445$7.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1991 Mathematics Subject Classification. 42B25, 26D15, 46E30.

Key words and phrases. Singular integrals, maximal functions, homogeneous type spaces, Orlicz class, weak type inequality, strong type inequality.

Definition 1.1. A kernel k will be said to belong to the class CZ, (k ∈ CZ), if conditions (1.1), (1.2) are fulfilled and the singular integral

Kf(x) = lim_{ε→0} K_{ε}f(x) = lim_{ε→0} ∫_{d(x,y)>ε} k(x, y)f(y) dμ

generates a continuous operator in L_{p_{0}}(X, μ) for some p_{0}, 1 < p_{0} < ∞.

Singular integrals with such kernels were treated in [2–6]. We set

K^{∗}f(x) = sup_{ε>0} |K_{ε}f(x)|.

The following theorem is well known.

Theorem B [5]. If k ∈ CZ, then for an arbitrary w ∈ A_{p} (1 < p < ∞) we have the inequality

∫_{X} (K^{∗}f(x))^{p}w(x) dμ ≤ c_{3} ∫_{X} |f(x)|^{p}w(x) dμ,  (1.3)

where the constant c_{3} is independent of f.

We shall begin our investigation of weighted problems for singular integrals of the above-mentioned kind with weak type inequalities.

Theorem 1.1. Let φ ∈ Φ ∩ Δ_{2} and k ∈ CZ. If there exists a constant c_{4} > 0 such that

∫_{B} φ̃ (\frac{∫_{B} φ(λw_{1}(y))w_{2}(y)dμ}{λμBw_{1}(x)w_{2}(x)}) w_{2}(x) dμ ≤

≤ c_{4} ∫_{B} φ(λw_{1}(x))w_{2}(x) dμ  (1.4)

for an arbitrary λ > 0 and any ball B, then we have the inequality

∫_{{x:|Kf(x)|>λ}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c_{5} ∫_{X} φ(f(x)w_{1}(x))w_{2}(x) dμ, (1.5)

where the constant c_{5} is independent of λ and f.

Proof. Let λ > 0 and f : X → ℝ^{1} be a μ-measurable function with a compact support. We set

M̃f(x) = sup_{r>0} \frac{1}{μB(x, r)} ∫_{B(x,r)} |f(y)| dμ

to be a centered maximal function.

Let further Ω = {x : M̃f(x) > λ}. One can easily verify that the set Ω is open and bounded. If X = Ω, then

∫_{{x:|Kf(x)|>λ}} φ(λw_{1}(x))w_{2}(x) dμ ≤ ∫_{{x:_{M}f(x)>λ}} φ(λw_{1}(x))w_{2}(x) dμ

and the validity of Theorem 1.1 follows from Theorem 3.1 from [1].

Assume that Ω ≠ X. By virtue of Lemma 4.1 from Part I for the set Ω and the constant C = a_{1}(1 + b_{0}a_{0}) there exists a sequence of balls B_{j} = B(x_{j}, r_{j}) such that

Ω = ⋃_{j=1}^{∞} CB_{j}, ∑_{j=1}^{∞} χ_{CB_{j}} (x) ≤ η,

B̃_{j} = B(x_{j}, 3Ca_{1}r_{j}) ∩ (X\Ω) ≠ ∅, j = 1, 2, …,

where the constant b_{0} is from the definition of k while the numbers a_{0} and a_{1} are from the definition of X.

Set F = X\Ω. Since B̃_{j} ∩ F ≠ ∅, we have

|f|_{B_{j}} ≤ c|f|_{B_{j}} ≤ cλ,  (1.6)

where the constant c is independent of λ and j.

Let

g(x) = f(x)χ_{F} (x) + ∑_{j} (f)_{B_{j}}χ_{B_{j}} (x),

ψ(x) = f(x) − g(x) = ∑_{j} (f(x) − (f)_{B_{j}})χ_{B_{j}} (x) = ∑_{j} ψ_{j}(x).

By virtue of

|Kf(x)| ≤ |Kg(x)| + |Kψ(x)|,

we have

∫_{\{x:|Kf(x)|>λ\}} φ(λw_{1}(x))w_{2}(x) dμ ≤ ∫_{\{x:|Kg(x)|> \frac{λ}{2} \}} φ(λw_{1}(x))w_{2}(x) dμ +

+  ∫_{\{x:|Kψ(x)|> \frac{λ}{2} \}} φ(λw_{1}(x))w_{2}(x) dμ.  (1.7)

Since by Proposition 3.5 of Part I the function φ(λw_{1})w_{2} ∈ A_{∞} uniformly with respect to λ, there exists p such that φ(λw_{1})w_{2} ∈ A_{p} uniformly with respect to λ. Therefore by Theorem B and inequality (1.6) we have

∫_{\{x:|Kg(x)|> \frac{λ}{2} \}}φ(λw_{1}(x))w_{2}(x)dμ≤cλ^{−p} ∫_{X}|Kg(x)|^{p}φ(λw_{1}(x))w_{2}(x)dμ≤

≤cλ^{−p}∫_{X}|g(x)|^{p}φ(λw_{1}(x))w_{2}(x)dμ≤ cλ^{−p}∫_{F}|f(x)|^{p}φ(λw_{1}(x))w_{2}(x)dμ+

+c ∫_{X} φ(λw_{1}(x))w_{2}(x) dμ.  (1.8)

Next, due to the fact that |f(x)| < λ almost everywhere on F, in the sense of the μ-measure, using (2.2) from [1] and (1.8) we conclude that

λ^{−p}|f(x)|^{p}φ(λw_{1}(x)) ≤ cφ(f(x)w_{1}(x))  (1.9)

holds for almost all x ∈ F and sufficiently large p.

Thus (1.8) and (1.9) give the estimate

∫_{\{x:|Kg(x)|> \frac{λ}{2} \}}φ(λw_{1}(x))w_{2}(x)dμ≤c ∫_{X}φ(f(x)w_{1}(x))w_{2}(x)dμ. (1.10)

Now we shall estimate |Kψ(x)| on the set F. We have

Kψ(x) = ∑_{j} ∫_{B_{j}} k(x, y)ψ_{j}(y) dμ.

From the definition of ψ_{j} we obtain ∫_{B_{j}} ψ_{j}(x) dμ = 0. Therefore

Kψ_{j}(x) = ∫_{B_{j}} (k(x, y) − k(x, x_{j}))ψ_{j}(y) dμ.  (1.11)

Choosing balls B_{j} appropriately we have x ∉ B(x_{j}, a_{1}(1+b_{0}a_{0})r_{j}) if x ∈ F. Hence for an arbitrary x ∈ F we have d(x_{j}, x) > a_{1}(1 + b_{0}a_{0})r_{j}.

By the last inequality we have

a_{0}(1+b_{0}a_{0})r_{j} <d(x_{j}, x)≤a_{1}(d(x_{j}, y)+d(y, x)) ≤
≤ a_{1}r_{j} +a_{1}a_{0}d(x, y)  (1.12)

for an arbitrary y ∈ B_{j}.

From (1.12) we conclude that d(x, y) > b_{0}r_{j}, i.e., b_{0}d(x_{j}, y) ≤ d(x, y) for x ∈ F and y ∈ B(x_{j}, r_{j}).

Using the above reasoning for x ∈ F and condition (1.2), we write the kernel in the form

|Kψ_{j}(x)| ≤ c ∑_{j} |ψ_{j}(y)|ω(\frac{d(x_{j}, y)}{d(x_{j}, x)}) \frac{1}{μB(x_{j}, d(x_{j}, x))} dμ ≤

≤ cω(\frac{r_{j}}{d(x_{j}, x)}) \frac{1}{μB(x_{j}, d(x_{j}, x))} ∫_{B_{j}} |ψ_{j}(y)| dμ.

On the other hand, by virtue of (1.6)

|ψ_{j}|_{B_{j}} ≤ 2|f|_{B_{j}} ≤ 2cλ.

Hence from the preceding inequality for x ∈ F we obtain

|Kψ_{j}(x)| ≤ cλω(\frac{r_{j}}{d(x_{j}, x)}) \frac{μB(x_{j}, r_{j})}{μB(x_{j}, d(x_{j}, x))} (j = 1, 2, …).

On summing these inequalities, we find

|Kψ(x)| ≤ cλI_{ω}(x)  (1.13)

for x ∈ F.

Now, applying Theorem 4.1 of [1], from (1.13) we derive

∫_{\{x∈X:|Kψ(x)|> \frac{λ}{2} \}}φ(λw_{1}(x))w_{2}(x)dμ≤ ∫_{\{x∈F:|Kψ(x)|> \frac{λ}{2} \}}φ(λw_{1}(x))w_{2}(x)dμ+

+ ∫_{Ω} φ(λw_{1}(x))w_{2}(x) dμ.  (1.14)

By virtue of Proposition 3.5 (see Part I) the condition of the theorem implies that φ(λw_{1})w_{2} ∈ A_{∞} uniformly with respect to λ. Therefore, as said above, there exists p > 1 such that φ(λw_{1})w_{2} ∈ A_{p} uniformly with respect to λ. Thus using (1.13) and Corollary 4.2 from [1] we obtain

∫_{\{x∈F:|Kψ(x)|> \frac{λ}{2} \}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c ∫_{F} (I_{ω}(x))^{p}φ(λw_{1}(x))w_{2}(x) dμ ≤

≤ c ∫_{Ω} φ(λw_{1}(x))w_{2}(x) dμ.

Therefore (1.14) gives the estimate

∫_{\{x∈F:|Kψ(x)|> \frac{λ}{2} \}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c ∫_{Ω} φ(λw_{1}(x))w_{2}(x) dμ.

Taking into account the definition of the set Ω and Theorem 3.1 of [1], we obtain the estimate

∫_{\{x∈F:|Kψ(x)|> \frac{λ}{2} \}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c ∫_{X} φ(λw_{1}(x))w_{2}(x) dμ.

Finally, the last inequality together with (1.7) and (1.10) imply that the theorem is valid.

Theorem 1.2. Let φ ∈ Φ ∩ Δ_{2}, k ∈ CZ. Then from condition (1.4) it follows that there exists a constant c_{1} > 0 such that the inequality

∫_{{x∈X:K^{∗}f(x)>λ}}φ(λw_{1}(x))w_{2}(x) dμ≤c_{1} ∫_{X}φ(f(x)w_{1}(x))w_{2}(x) dμ(1.15)

is fulfilled for any λ > 0 and μ-measurable function f : X → ℝ^{1}.

Proof. This theorem, which is more general than Theorem 1.1, is proved quite similarly to the latter provided that we show that the inequality

K^{∗}ψ(x) ≤ cλI_{ω}(x) + cMf(x)  (1.16)

holds for x ∈ F.

We have

K_{ε}ψ(x)=∑_{j}  ∫_{{y∈B_{j}:d(x,y)>ε}} ψ_{j}(y)k(x, y)dμ=  ∑_{{j:dist(x,B_{j})>ε}}∫_{B_{j}}ψ(y)k(x, y)dμ+

+ ∑_{{j:dist(x,B_{j})≥ε}}  ∫_{{y∈B_{j}:d(x,y)>ε}} ψ_{j}(y)k(x, y) dμ = A_{ε}(x) + B_{ε}(x).

In proving Theorem 1.1, it was shown that

sup_{ε>0} |A_{ε}(x)| ≤ cλI_{ω}(x)

for x ∈ F.

Further for x ∈ F, y ∈ B_{j} and z ∈ B_{j} we have

d(x, y) ≤ a_{1}(d(x, z) + d(z, y)) ≤ a_{1}(d(x, z) + a_{1}(d(z, x_{j}) + d(x_{j}, y))) ≤
≤ a_{1}(d(x, z) + a_{1}(a_{0} + 1)r_{j}).

Since z is an arbitrary point from B_{j}, for dist(x, B_{j}) ≤ ε the last inequality implies

d(x, y) ≤ a_{1} dist(x, B_{j}) + a_{1}^{2}(a_{0} + 1)r_{j} ≤ c_{0}ε,

where c_{0} = a_{1} + 2a_{0}a_{1}^{2}b_{0}^{−1}.

Therefore, due to (1.1), for x ∈ F we have

|B_{ε}(x)| ≤ ∫_{ε<d(x,y)<c_{0}ε} \frac{∑_{j=1}^{∞} |ψ_{j}(y)|}{μB(x, d(x, y))} dμ ≤

≤ \frac{c}{μB(x, ε)} ∫_{ε<d(x,y)≤c_{0}ε} |ψ(y)| dμ ≤ cMψ(x).

Now repeating the arguments from the proof of Theorem 1.1 and using inequalities (1.16), (1.17) and Theorem 3.1 from [1], we arrive at (1.14).

§ 2. Criteria of Weak Type Weighted Inequalities for Singular Integrals

In this section, from the class of Calderon–Zygmund kernels we single out a subclass of kernels such that for the corresponding singular integrals we succeed in obtaining necessary and sufficient conditions ensuring the validity of weak type inequalities.

Definition 2.1. A kernel k : X × X\{(x, x), x ∈ X} → ℝ^{1} belongs to the class S_{1} if for an arbitrary ball B = B(z, r) there is a ball B′ = B(z′, r) such that B ∩ B′ = ∅, d(z, z′) ≤ c_{1}r and for any x ∈ B and y ∈ B′ we have the inequality

k(x, y) ≥ \frac{c_{2}}{μB(x, d(x, y))},  (2.1)

where the constant c_{2} is independent of the ball B, x and y.

It is easy to see that in the above condition the ball B′ can be chosen so that dist(B, B′) > r.

Indeed, in addition to the ball B(z, r) let us consider the ball B(z, mr), where m = a_{1} + a_{1}^{2}(1 + a_{0}). By assumption, there exists a ball B(z_{0}, mr) such that B(z, mr) ∩ B(z_{0}, mr) = ∅ and (2.1) is fulfilled for arbitrary x ∈ B(z, mr) and y ∈ B(z_{0}, mr). Now for a given ball B(z, r) we shall take B(z_{0}, r) as B′. Then already dist(B, B′) > r. Indeed, for arbitrary x ∈ B and y ∈ B′ we have

a_{1}(1 + a_{1}(1 + a_{0}))r ≤ d(z, z_{0}) ≤ a_{1}(d(z, x) + d(x, z_{0})) ≤

≤ a_{1}(r + a_{1}(d(x, y) + d(y, z_{0}))) ≤ a_{1}r + a_{1}^{2}d(x, y) + a_{1}^{2}a_{0}d(z_{0}, y) ≤

≤ (a_{1} + a_{1}^{2}a_{0})r + a_{1}^{2}d(x, y).

Hence it follows that d(x, y) ≥ r and therefore r < dist(B, B′).

Definition 2.2. We shall say that k ∈ S_{2} if for an arbitrary ball B(z, r) there exists a ball B′ = B(z′, r) such that B ∩ B′ = ∅, d(z, z′) < c_{2}r and for arbitrary y ∈ B and x ∈ B′ we have the inequality

k(x, y) ≥ \frac{c_{4}}{μB(x, d(x, y))},  (2.1′)

where the constant c_{4} is independent of x and y.

It is easy to verify that if k ∈ S_{1}, then k̃ ∈ S_{2}, where k̃(x, y) = k(y, x). When X is compact, condition (2.1) is to be fulfilled for balls with a

sufficiently small radius.

Further assume

K̃f(x) = ∫_{X} k(y, x)f(y) dμ.

We have

Theorem 2.1. Let φ ∈ Φ and the kernel k ∈ S_{1} ∪ S_{2}. If the inequalities

∫_{{x:|Kf(x)|>λ}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c ∫_{X} φ(cf(x)w_{1}(x))w_{2}(x) dμ, (2.2)

∫_{{x:|_{K}f(x)|>λ}} φ(λw_{1}(x))w_{2}(x) dμ ≤ c ∫_{X} φ(cf(x)w_{1}(x))w_{2}(x) dμ,(2.2′)

where the constant c is independent of λ > 0 and f, are fulfilled, then φ ∈ Δ_{2}, φ is quasiconvex, and the following condition holds: there exists a constant c_{0} such that

∫_{B} φ̃ (\frac{∫_{B} φ(λw_{1}(y)w_{2}(y)dμ}{λμBw_{1}(x)w_{2}(x)}) w_{2}(x) dμ ≤ c_{0} ∫_{B} φ(λw_{1}(x))w_{2}(x) dμ

for any λ > 0 and ball B.

In §1 the last inequality figured as formula (1.4). In what follows by referring to this condition we shall mean (1.4).

The proof of the theorem is based on

Proposition 2.1. Let E be a set of positive μ-measure not containing atoms. Assume that k ∈ S_{1} ∪ S_{2}.

If there exists a constant c > 0 such that the inequality

φ(λ)μ{x ∈ E\E_{1} : |Kf(x)| > λ} ≤ c ∫_{E_{1}} φ(cf(x)) dμ  (2.3)

holds for any λ > 0, μ-measurable E_{1} ⊆ E and μ-measurable function f, supp f ⊂ E_{1}, then φ is quasiconvex and satisfies the condition Δ_{2}.

Proof. Let first k ∈ S_{1}. Assume that z_{0} is a density point of the set E. From the property of the kernel it follows that for each B(z_{0}, r) there exists a ball B(z′_{0}, r) such that B(z_{0}, r) ∩ B(z′_{0}, r) = ∅, d(z_{0}, z′_{0}) < c_{1}r, and condition (2.1) is fulfilled for x ∈ B(z_{0}, r) and y ∈ B(z′_{0}, r).

We shall show that the number r can be chosen so small that

μB(z′_{0}, r) ∩ E ≥ \frac{1}{2} μB(z′_{0}, r).

First note that the condition d(z_{0}, z′_{0}) < c_{1}r implies

B(z′_{0}, r) ⊂ B(z_{0}, a_{1}(c_{1} + 1)r).

Indeed, if x ∈ B(z′_{0}, r), then d(z_{0}, x) ≤ a_{1}(d(z_{0}, z′_{0}) + d(z′_{0}, x)) ≤ a_{1}(c_{1} + 1)r. On the other hand,

B(z_{0}, a_{1}(c_{1} + 1)r) ⊂ B(z′_{0}, c_{3}r),

where c_{3} = a_{1}(a_{0}c_{1} + a_{1}(c_{1} + 1)).

Therefore

B(z′_{0}, r) ⊂ B(z_{0}, a_{1}(c_{1} + 1)r) ⊂ B(z′_{0}, c_{3}r).

Since z_{0} is a density point of E, for ε > 0 there exists δ > 0 such that if r < δ then

μB(z_{0}, a_{1}(c_{1} + 1)r)\E ≤ εμB(z_{0}, a_{1}(c_{1} + 1)r).

Therefore

μB(z′_{0}, r)\E ≤ μB(z_{0}, a_{1}(c_{1} + 1)r)\E ≤ εμB(z_{0}, a_{1}(c_{1} + 1)r) ≤
≤ εμB(z_{0}, c_{3}r) ≤ εbμB(z′_{0}, r),

where the constant b is from the doubling condition.

If εb < \frac{1}{2}, the last inequality implies

μB(z′_{0}, r) ∩ E > (1 − εb)μB(z′_{0}, r) ≥ \frac{1}{2}μB(z′_{0}, r).

Fix some ball B(z_{0}, r) with the condition 0 < r < δ. Let B(z′_{0}, r) be the ball existing by virtue of the condition k ∈ S_{1}. Now if f ≥ 0, supp f ⊂ B(z′_{0}, r) ∩ E, for any x ∈ B(z_{0}, r) we obtain

T f(x) = ∫_{X} k(x, y)f(y) dy ≥ c_{2} ∫_{B(z′_{0},r)} \frac{f(y)dμ}{μB(x, d(x, y))}.  (2.4)

Moreover, if x ∈ B(z_{0}, r), y ∈ B(z′_{0}, r) and z ∈ B(x, d(x, y)), we have

d(z′_{0}, z) ≤ a_{1}(d(z′_{0}, x) + d(x, z)) ≤ a_{1}^{2}(d(z′_{0}, z_{0}) + d(z_{0}, x) + a_{1}d(x, y)) ≤
≤ a_{1}^{2}c_{1}r + a_{1}^{2}r + a_{1}^{2}(d(x, z_{0}) + d(z_{0}, y)) ≤
≤ a_{1}^{2}(c_{1} + 1)r + a_{1}^{2}a_{0}r + a_{1}^{3}(d(z_{0}, z′_{0}) + d(z′_{0}, y)) ≤ c_{4}r.

Thus B(x, d(x, y)) ⊂ B(z′_{0}, c_{4}r). Hence by the doubling condition we find that μB(x, d(x, y)) ≤ c_{5}μB(z′_{0}, r). Therefore (2.4) implies that if r < δ, then for B(z_{0}, r) there exists a ball B(z′_{0}, r) such that for x ∈ B(z_{0}, r) we obtain the estimate

T f(x) ≥ \frac{c_{2}}{c_{5}} \frac{1}{μB(z′_{0}, r)} ∫_{B(z′_{0},r)} f(y) dμ.  (2.5)

Moreover, μB(z′_{0}, r) ∩ E > \frac{1}{2}μB(z_{0}, r), d(z_{0}, z′_{0}) < c_{1}r and dist(B(z′_{0}, r), B(z_{0}, r)) > r.

Let now r_{k} = \frac{δ}{a_{1}^{k}(c_{1}+1)^{k}} and B(z_{0}^{k}, r_{k}) be a ball corresponding to the ball B(z_{0}, r_{k}) in condition (2.5).

For x ∈ B(z_{0}^{k}, r_{k}) we have

d(z_{0}, x) ≤ a_{1}(d(z_{0}, z_{0}^{k}) + d(z_{0}^{k}, x)) ≤ a_{1}(c_{1} + 1)r_{k} = r_{k} − 1,

which implies B(z_{0}^{k}, r_{k}) ⊂ B(z_{0}, r_{k−1}). Also, B(z_{0}, r_{k−1}) ∩ B(z_{0}^{k−1}, r_{k−1}) = ∅. Therefore B(z_{0}^{k}, r_{k})∩B(z_{0}^{k−1}, r_{k−1})=∅. Similarly, B(z_{0}^{j}, r_{j})∩B(z_{0}^{i}, r_{i})= ∅, i ≠ j. Further B(z_{0}, r_{k}) ⊆ B(z_{0}, r_{i}), i = 1, 2, …, k, and B(z_{0}, r_{i}) ∩ B(z_{0}^{i}, r_{i}) = ∅. Thus B(z_{0}, r_{k}) ∩ B(z_{0}^{i}, r_{i}) = ∅ (i = 1, 2, …, k).

We set

f(x) = \frac{λ}{c}χ_{k_{∪} ; i=1B(z^{i}_{0},r_{i})∩E},

where the constant is from (2.3) and k is chosen so that \frac{kc_{2}}{2cc_{5}} > 2. Then by (2.5) for x ∈ B(z_{0}, r_{k}) we obtain

T f(x) = ∫_{X} k(x, y)f(y) dμ = ∑_{i=1}^{k} ∫_{B(z_{0}^{i},r_{i})} k(x, y)f(y) dμ ≥

≥ \frac{λc_{2}}{cc_{5}} ∑_{i=1}^{k} \frac{μB(z_{0}^{i}, r_{i}) ∩ E}{μB(z_{0}^{i}, r_{i})} ≥ \frac{λc_{2}}{2cc_{5}} > 2λ.  (2.6)

Now substituting E_{1} = ∪^{k}_{i=1}B(z_{0}^{i}, r_{i}) ∩ E in (2.3) and taking into account

that by virtue of the above reasoning B(z_{0}, r_{k}) ∩ E ⊂ E\E_{1}, by (2.6) we have

φ(2λ)μB(z_{0}, r_{k}) ∩ E ≤ cφ(λ) ∑_{i=1}^{k} μB(z_{0}^{i}, r_{i}) ∩ E

which implies the fulfillment of the condition Δ_{2}.

Next we want to prove that the function φ is quasiconvex. Let z_{0} ∈ E be a density point of this set. As shown above, there exists a ball B(z′_{0}, r_{0}) such that dist(B(z_{0}, r_{0}), B(z′_{0}, r_{0})) > r_{0},

μB(z_{0}, r_{0}) ∩ E ≥ \frac{1}{2} μB(z_{0}, r_{0}),

μB(z′_{0}, \frac{r_{0}}{2a_{1}} ) ∩ E > \frac{1}{2} μB(z′_{0}, \frac{r_{0}}{2a_{1}} ),

and for arbitrary x ∈ B(z_{0}, r_{0}), f ≥ 0, supp f ⊂ B(z′_{0}, r_{0}) we have

T f(x) ≥ \frac{c_{2}}{c_{5}} \frac{1}{μB(z′_{0}, r_{0})} ∫_{B(z′_{0},r_{0})} f(y) dμ.  (2.7)

Let z_{1} ∈ B(z′_{0}, \frac{r}{2a_{1}} ) ∩ E be a density point of the set E.

One can readily verify that

B(z_{1}, \frac{r_{0}}{2a_{1}} ) ⊂ B(z′_{0}, r_{0}) ⊂ B(z_{1}, (\frac{a_{0}}{2} + a_{1})r_{0}).

Let r_{1} be a positive number so small that r_{1} < \frac{r_{0}}{2a_{1}} and for any r ≤ r_{1} we have

μB(z_{1}, r) ∩ E ≥ \frac{1}{2} μB(z_{0}, r).

Let further 0 ≤ t_{1} < t_{2} < ∞ and

r_{2} = sup \{r : μB(z_{1}, r) ≤ \frac{t_{1}}{t_{2}} μB(z_{1}, r_{1})\}.

Then

μB(z_{1}, r_{2}) ≤ bμB(z_{1}, \frac{r_{2}}{2} ) ≤ b \frac{t_{1}}{t_{2}} μB(z_{1}, r_{1}) ≤

≤ bμB(z_{1}, 2r_{2}) ≤ b^{2}μB(z_{1}, r_{2}).  (2.8)

We write

f(x) = \frac{2c_{5}bμB (z_{1}, (\frac{a_{0}}{2} + a_{1}) r_{0})}{c_{2}μB(z_{1}, r_{1})} · t_{2}χ_{B(z_{1},r_{2})∩E}.

For x ∈ B(z_{0}, r_{0}) inequalities (2.7) and (2.8) give

T f(x) ≥ \frac{c_{2}}{c_{5}μB (z_{1}, (\frac{a_{0}}{2} + a_{1}) r_{0})} ∫_{B(x,r_{2})} f(x) dμ =
= \frac{2bt_{2}μB(z_{1}, r_{2}) ∩ E}{μB(z_{1}, r_{1})} > t_{1}.  (2.9)

Set

c_{6} = \frac{2c_{5}bμB (z_{1}, (\frac{a_{0}}{2} + a_{1}) r_{0})}{c_{2}μB(z_{1}, r_{1})}.

Taking into account that B(z_{1}, r_{1}) ⊂ B(z_{0}, r_{0}), B(z′_{0}, r_{0})∩B(z_{0}, r_{0}) = ∅, we obtain B(z_{1}, r_{1}) ∩ B(z_{0}, r_{0}) = ∅. On the other hand, by the definition of the number r_{2} we have r_{2} < r_{1} and therefore B(z_{1}, r_{2}) ∩ B(z_{0}, r_{0}) = ∅. Now putting E_{1} = B(z_{1}, r_{2}) ∩ E in (2.3), by (2.9) we obtain

φ(t_{1})μB(z_{0}, r_{0}) ∩ E ≤ cφ(cc_{6}t_{2})μB(z_{1}, r_{2}) ∩ E ≤ cφ(cc_{6}t_{2})b \frac{t_{1}}{t_{2}} μB(z_{1}, r_{1})

which implies that the function \frac{φ(t)}{t} quasiincreases and thus by Lemma 2.1 from [1] φ is quasiconvex.

The case with k ∈ S_{2} is proved similarly and hence omitted.

Proof of Theorem 2.1. By Proposition 2.1 either of conditions (2.2) and (2.2′) guarantees the quasiconvexity of φ and the fulfillment of the condition φ ∈ Δ_{2}. Indeed, choose a number such that E = {x : k^{−1} ≤ w_{1}(x), k^{−1} ≤ w_{2}(x)} has a positive measure. Then for any set E_{1} ⊂ E, μE_{1} > 0, and function f, supp f ⊂ E_{1}, say from (2.2), we find that (2.3) is fulfilled and therefore by Proposition 2.1 φ is quasiconvex and φ ∈ Δ_{2}.

It remains to show that condition (1.4) holds. Under the condition of the theorem for an arbitrary ball B = B(z_{0}, r) there exists a ball B′ = B(z′_{0}, r) such that d(z_{0}, z′_{0}) < c_{1}r and for x ∈ B(z_{0}, r) and y ∈ B(z′_{0}, r) (2.1) holds. Therefore if g ≥ 0 and supp g ⊂ B(z′_{0}, r), then for x ∈ B(z_{0}, r) we have

Kg(x) ≥ \frac{c}{μB′} ∫_{B′} g(y) dμ.

For such functions (2.2) gives the estimate

∫_{B} φ(g_{B′} w_{1}(x))w_{2}(x) dμ ≤ c ∫_{B′} φ(cg(x)w_{1}(x))w_{2}(x) dμ. (2.10)

On the other hand, for g̃ ≥ 0, supp g̃ ⊂ B(z_{0}, r), and x ∈ B(z′_{0}, r) we have

K̃g̃(x) ≥ \frac{c}{μB} ∫_{B} g̃(y) dμ.

Therefore (2.2′) implies

∫_{B} φ(g̃_{B}w_{1}(x))w_{2}(x) dμ ≤ c ∫_{B′} φ(cg̃(x)w_{1}(x))w_{2}(x) dμ. (2.11)

Let now f ≥ 0 be an arbitrary locally summable function. Substituting g = χ_{B′} f_{B} in (2.10), we obtain

∫_{B} φ(f_{B}w_{1}(x))w_{2}(x) dμ ≤ c ∫_{B′} φ(f_{B}w_{1}(x))w_{2}(x) dμ.  (2.12)

Substituting the function fχ_{B} in (2.11) gives

∫_{B′} φ(f_{B}w_{1}(x))w_{2}(x) dμ ≤ c ∫_{B} φ(f(x)w_{1}(x))w_{2}(x) dμ. (2.13)

From (2.12) and (2.13) we conclude that the inequality

∫_{B} φ(f_{B}w_{1}(x))w_{2}(x) dμ ≤ c ∫_{B} φ(f(x)w_{1}(x))w_{2}(x) dμ

holds for any locally summable function f, supp f ⊂ B.

By virtue of Theorem 3.1 from [1] the last inequality implies that (1.4) is valid.

In the concrete cases the necessary and sufficient conditions for weak type weighted inequalities for singular integrals acquire a rather transparent form. Namely, by virtue of Theorem 3.2 of Part I, from Theorems 1.1 and 2.1 we immediately conclude that the statements below are valid.

Theorem 2.2. Let φ ∈ Φ and the kernel k ∈ CZ ∩S_{1} ∪ S_{2}. Then the following statements are equivalent:

(i) φ is quasiconvex, φ ∈ Δ_{2}, and w ∈ A_{p(φ)};

(ii) there exists a positive constant c_{1} > 0 such that for any λ > 0 and μ-measurable function we have

φ(λ)w{x : |Kf(x)| > λ} ≤ c_{1} ∫_{X} φ(cf(x))w(x) dμ,
φ(λ)w{x : |K̃f(x)| > λ} ≤ c_{1} ∫_{X} φ(cf(x))w(x) dμ.

Theorem 2.3. Let φ ∈ Φ, k ∈ CZ ∩S_{1} ∪ S_{2}. Then the following statements are equivalent:

(i) φ is quasiconvex, φ ∈ Δ_{2}, w^{p(φ)} ∈ A_{p(φ)}, w^{−p(_{φ})} ∈ A_{p(_{φ})};

(ii) we have the inequalities

∫_{{x:|Kf(x)|>λ}} φ(λw(x)) dμ ≤ c ∫_{X} φ(cf(x)w(x)) dμ,

∫_{{x:|_{K}f(x)|>λ}} φ(λw(x)) dμ ≤ c ∫_{X} φ(cf(x)w(x)) dμ,

where the constant c is independent of f and λ > 0.

Theorem 2.4. Let φ ∈ Φ and k ∈ CZ ∩S_{1} ∪ S_{2}. Then the following conditions are equivalent:

(i) φ is quasiconvex, φ ∈ Δ_{2}, and w ∈ A_{p(_{φ})};

(ii) there exists a constant c > 0 such that for any λ > 0 and μ-measurable function f : X → ℝ^{1} we have the inequalities

∫_{\{x:|Kf(x)|>λ\}} φ(\frac{λ}{w(x)})w(x) dμ ≤ c ∫_{X} φ(c \frac{f(x)}{w(x)})w(x) dμ,

∫_{\{x:|_{K}f(x)|>λ\}} φ(\frac{λ}{w(x)})w(x) dμ ≤ c ∫_{X} φ(c \frac{f(x)}{w(x)})w(x) dμ.

§ 3. Criteria for Strong Type Weighted Inequalities for Maximal Functions and Singular Integrals

Using the previous results as well as the general interpolation theorem to be given below, in this section we are able to obtain a solution of the problem, to give a full description of classes of the function φ and weights w ensuring the validity of strong type weighted inequalities for maximal functions and singular integrals defined on homogeneous type spaces.

For maximal functions we have

Theorem 3.1. Let φ ∈ Φ. Then the following statements are equivalent:

(i) φ^{α} is quasiconvex for some α, 0 < α < 1, w^{p(φ)} ∈ A_{p(φ)}, and w^{−p(_{φ})} ∈

A_{p(_{φ})};

(ii) there exists c_{1} > 0 such that the inequality

∫_{X} φ(Mf(x)w(x)) dμ ≤ c_{1} ∫_{X} φ(c_{1}f(x)w(x)) dμ  (3.1)

is fulfilled for any locally μ-summable function f : X → ℝ^{1}.

For singular integrals the solution of one-weighted problems is given by the statements as follows.

Theorem 3.2. Let φ ∈ Φ and k ∈ CZ. If φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, and w ∈ A_{p(φ)}, then there is c_{2} > 0 such that the following inequalities hold:

∫_{X} φ(Kf(x))w(x) dμ ≤ c_{2} ∫_{X} φ(f(x))w(x) dμ,  (3.2)
∫_{X} φ̃(\frac{Kf(x)}{w(x)})w(x) dμ ≤ c_{3} ∫_{X} φ̃(\frac{f(x)}{w(x)})w(x) dμ.  (3.3)

Similar statements hold for the operator K̃ as well.

Theorem 3.3. Let φ ∈ Φ and k ∈ CZ ∩S_{1} ∪ S_{2}. Then the following conditions are equivalent:

(i) the inequality (3.2) is fulfilled for K and K̃; (ii) the inequality (3.3) is fulfilled for K and K̃; (iii) φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, and w ∈ A_{p(φ)}.

Theorem 3.4. If φ ∈ Φ∩Δ_{2} and k ∈ CZ, φ^{α} is quasiconvex for some α, 0 < α < 1, w^{p(φ)} ∈ A_{p(φ)}, and w^{−p(_{φ})} ∈ A_{p(_{φ})}, then there exists a constant c_{4} > 0 such that the following inequalities are fulfilled:

∫_{X} φ(Kf(x)w(x)) dμ ≤ c_{4} ∫_{X} φ(f(x)w(x)) dμ,  (3.4)
∫_{X} φ(K̃f(x)w(x)) dμ ≤ c_{4} ∫_{X} φ(f(x)w(x)) dμ.  (3.5)

Theorem 3.5. Let φ ∈ Φ and k ∈ CZ ∩S_{1} ∪ S_{2}. Then the following conditions are equivalent:

(i) φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, w^{p(φ)} ∈ A_{p(φ)} and

w^{−p(_{φ})} ∈ A_{p(_{φ})};

ii) inequalities (3.4) and (3.5) hold.

To prove the above-formulated theorems we next give the interpolation theorem.

Let (M, S, ν) be a space with measure. Let further the subadditive operator T be the one mapping the space D of functions measurable on (M, S, ν) into the space of functions measurable and defined on another space (M_{1}, S_{1}, ν_{1}) with measure.

Theorem 3.6. Let φ ∈ Φ and be quasiconvex. Let further 1 ≤ r < p(φ) ≤ p′(φ̃) < s < ∞ and for the case s = ∞ assume that p′(φ̃) ≤ ∞.

Let there exist constants c_{1} and c_{2} such that for any λ > 0 and f ∈ L^{r}+^{·}L^{s} we have the inequalities

∫_{{x∈M_{1}:|T f(x)|>λ}} dν_{1} ≤ c_{1}λ^{−r} ∫_{M} |f(x)|^{r} dν,  (3.6)

∫_{{x∈M_{1}:|T f(x)|>λ}} dν_{1} ≤ c_{2}λ^{−s} ∫_{M} |f(x)|^{s} dν  (3.7)

and for s = ∞ we have

∥T f∥_{L^{∞}} ≤ c_{2}∥f∥_{L^{∞}}.

Then there exists a constant c_{3} > 0 such that the following inequality holds:

∫_{M_{1}} φ(T f(x)) dν_{1} ≤ c_{3} ∫_{M} φ(f(x)) dν.  (3.8)

Proof. The theorem is proved by the standard arguments. Let s < ∞. By virtue of the definition of p(φ) there exists p_{1}, r < p_{1} < p(φ), such that t^{−p_{1}}φ(t) almost increases. Similarly, there exists ε > 0 such that t^{−(p(_{φ})−ve)}φ̃(t) almost increases, which by virtue of Lemma 2.5 from [1] means that t^{−(p(_{φ})−ε)′}φ(t) almost decreases. Therefore there exists p_{2} such that p̃′(φ̃) < p_{2} < s so that the function t^{−p_{2}}φ(t) will almost decrease. Based on these facts, we readily conclude that

∫_{0}^{t} \frac{dφ(τ)}{τ^{r}} ≤ c \frac{φ(t)}{t^{r}} and ∫_{t}^{∞} \frac{dφ(τ)}{τ^{s}} ≤ c \frac{φ(t)}{t^{s}}.  (3.9)

For each λ > 0 we put

f_{1}(x) = \begin{cases}f(x) & if |f(x)| ≥ \frac{λ}{2c_{2}} , \\ 0 & if |f(x)| < \frac{λ}{2c_{2}} , \end{cases} f_{2}(x) = \begin{cases}f(x) & if |f(x)| ≤ \frac{λ}{2c_{2}} , \\ 0 & if |f(x)| > \frac{λ}{2c_{2}} . \end{cases}

Further we have

∫_{M_{1}} φ(T f(x)) dν_{1} = ∫_{0}^{∞} ν_{1}{x ∈ M_{1} : |T f(x)| > λ} dφ(λ) ≤

≤ c (∫_{0}^{∞} ν_{1}\{x ∈ M : |T f_{1}(x)| > \frac{λ}{2c_{2}} \} dφ(λ)+

+ ∫_{0}^{∞} ν_{1}\{x ∈ M_{1} : |T f_{2}(x)| > \frac{λ}{2c_{2}} \} dφ(λ)) = I_{1} + I_{2}.

On account of (3.6) and the first inequality of (3.9) we have

I_{1} ≤ c ∫_{0}^{∞} \frac{2^{r}c_{1}}{λ^{r}} ( ∫_{M} |f_{1}(x)|^{r}dν)dφ(λ) =

= c ∫_{0}^{∞} \frac{2^{r}c_{1}}{λ^{r}} ( ∫_{\{x:2c_{2}|f(x)|>λ\}} |f(x)|^{r}dν)dφ(λ) =

= cc_{1} ∫_{M} |f(x)|^{r}( ∫_{0}^{2c_{2}|f(x)|} \frac{dφ(λ}{λ^{r}}) dν ≤ c_{3} ∫_{M} φ(f(x)) dν.

Similarly, (3.7) and the second inequality of (3.9) imply

I_{2} ≤ c_{3} ∫_{M} φ(f(x)) dν

and as a result we have (3.8).

If s = ∞, then ∥f∥_{∞} < \frac{λ}{2c_{2}} and therefore I_{2} = 0.

Proof of Theorem 3.1. Let us show that the implication (i)⇒(ii) holds. From the condition w^{p(φ)} ∈ A_{p(φ)} it follows that w^{p(φ)−ε} ∈ A_{p(φ)−ε}. If p(φ̃) > 1,

then the condition w^{−p(_{φ})} ∈ A_{p(_{φ})} implies w^{−(p(_{φ})−ε)} ∈ A_{p(_{φ})−ε} for ε > 0.

Therefore w^{(p(_{φ})−ε)′} ∈ A_{(p(_{φ})−ε)′}.

Consider the operator

T f = wM(\frac{f}{w} ).

Then due to Proposition 3.2 from [1] we obtain the inequalities

∫_{X} |T f(x)|^{p(φ)−ε}dμ ≤ c_{1} ∫_{X} |f(x)|^{p(φ)−ε}dμ,

∫_{X} |T f(x)|^{(p(_{φ})−ε)′}dμ ≤ c_{2} ∫_{X} |f(x)|^{(p(_{φ})−ε)′}dμ.

For p(φ̃) = 1 the function w^{−1} belongs to the class A_{1} and it is clear that

∥T f∥_{∞} ≤ c_{2}∥f∥_{∞}.

Since p(φ) − ε < p(φ) < p′(φ) < (p(φ̃) − ε)′, by Theorem 3.6 the above inequalities imply that (3.1) is valid.

As to the implication (ii)⇒(i), note that by virtue of the second part of Theorem 3.2 from [1] we find from the condition of the theorem that φ is

quasiconvex, w^{p(φ)} ∈ A_{p(φ)}, and w^{−p(_{φ})} ∈ A_{p(_{φ})}.

To show that condition (ii) implies the quasiconvexity of φ^{α} for some α, 0 < α < 1, we have to prove the following propositions.

Proposition 3.1. Let k ∈ S_{1} ∪ S_{2}, φ ∈ Φ, and μE > 0. If we have the inequality

∫_{E} φ(Kf(x)) dμ ≤ c ∫_{E} φ(cf(x)) dμ, supp f ⊂ E,  (3.10)

with constant c independent of f, then φ^{α} is qusiconvex for some α, 0 < α < 1, i.e., φ̃ ∈ Δ_{2}.

Proof. (3.10) implies (2.3) and hence φ is quasiconvex and satisfies the condition Δ_{2}. Therefore by (2.10) from [1] there exists c_{1} > 0 such that

φ(c \frac{φ̃(t)}{t}) ≤ c_{1}φ̃(t), t > 0.  (3.11)

Assume now that E_{1} is any μ-measurable subset of E and supp f ⊂ E_{1}. Applying the Young inequality, (3.11) and (3.10), we obtain the chain of inequalities

∫_{E\E_{1}} φ̃(K̃f(x))dμ = ∫_{E\E_{1}} \frac{φ̃(K̃f(x))}{K̃f(x)} K̃f(x) dμ =

= ∫_{E\E_{1}} K(\frac{φ̃(K̃f(x))}{K̃f(x)})(x)f(x) dμ ≤

≤ \frac{1}{2c_{1}c} ∫_{E\E_{1}}φ (K(\frac{φ̃(K̃f(x))}{K̃f(x)}χ_{E\E_{1}} )(x)) dμ+ \frac{1}{2cc_{1}} ∫_{E_{1}}φ̃(2cc_{1}f(x))≤

≤ \frac{1}{2c_{1}} ∫_{E\E_{1}} φ (c\frac{φ̃(K̃f(x))}{K̃f(x)}) dμ + \frac{1}{2cc_{1}} ∫_{E_{1}} φ̃(2cc_{1}f(x)) dμ ≤

≤ \frac{1}{2} ∫_{E\E_{1}} φ̃(K̃f(x)) dμ + \frac{1}{2cc_{1}} ∫_{E_{1}} φ̃(2cc_{1}f(x)) dμ,

which allow us to conclude that

∫_{E\E_{1}} φ̃(K̃f(x)) dμ ≤ \frac{1}{cc_{1}} ∫_{E_{1}} φ̃(2cc_{1}f(x)) dμ.

Since E_{1} is an arbitrary measurable subset of E, from the last inequality we find by Proposition 2.1 that φ̃ ∈ Δ_{2}, i.e., φ^{α} is quasiconvex for some α, 0 < α < 1.

Proposition 3.2. Let k satisfy condition (2.1) or (2.1′), φ ∈ Φ. If for some weight functions w_{i} (i = 1, 2, 3, 4) we have the inequality

∫_{X} φ(Kf(x)w_{1}(x))w_{2}(x) dμ ≤ c ∫_{X} φ(cf(x)w_{3}(x))w_{4}(x) dμ

with the constant c independent of f, then φ̃ ∈ Δ_{2}.

Proof. It is sufficient to choose the number m such that the set E = {x ∈ X : m^{−1} ≤ w_{1}(x), m^{−1} ≤ w_{2}(x), w_{3}(x) ≤ m, w_{4}(x) ≤ m} has a positive measure and to apply Proposition 3.1 for the set E.

Proof of Theorem 3.2. From the condition w ∈ A_{p(φ)} it follows that w ∈ A_{p(φ)−ε} for some ε, and since p(φ) ≤ p′(φ̃) we have w ∈ A_{p′(_{φ})+ε}. Next, applying interpolation Theorem 3.6 for the operator T = K, we obtain (3.2).

Now we shall prove (3.3). Note that the condition w ∈ A_{p(φ)−ε} implies w^{1−(p(φ)−ε)′} ∈ A_{(p(φ)−ε)′}. Since p(φ) < p′(φ̃) ≤ (p(φ̃) − ε)′, we have w ∈ A_{(p(_{φ})−ε)′}. Hence it follows that w^{1−(p(_{φ})−ε)} ∈ A_{p(_{φ})−ε}. Consider the operator

T f = \frac{1}{w} K̃(fw).

By Theorem B we have the inequalities

∫_{X} |T f(x)|^{p(_{φ})−ε}w(x) dμ ≤ c_{1} ∫_{X} |f(x)|^{p(_{φ})−ε}w(x) dμ,

∫_{X} |T f(x)|^{(p(φ)−ε)′}w(x) dμ ≤ c_{2} ∫_{X} |f(x)|^{(p(φ)−ε)′}w(x) dμ.

On the other hand, since p(φ̃) − ε < p(φ̃) < (p(φ) − ε)′ we conclude by Theorem 3.2 that inequality (3.3) holds.

Proof of Theorem 3.3. The validity of this theorem follows from the preceding theorem, Theorems 2.2, 2.3 and Proposition 3.2.

Proof of Theorem 3.4. This theorem is proved similarly to the proof of Theorem 3.1 using Theorem B.

Proof of Theorem 3.5. The validity of this theorem follows from the preceding theorem, Theorem 2.3, and Proposition 3.2.

Remark 1. In establishing the criteria for weak and strong type inequalities, we virtually used the following property of the operator: for any ball B there exists a ball B′ of the same volume such that the distance between these balls does not exceed c_{1} rad B. For any f ≥ 0, supp f ⊆ B′, the inequality

Kf(x) ≥ \frac{c}{νB′} ∫_{B′} f(y) dμ

holds for any x ∈ B′.

§ 4. Applications to Classical Singular Integrals

First we shall consider the Hilbert transform

Hf(x) = ∫_{−∞}^{∞} \frac{f(y)}{x − y} dy.

Given X = ℝ^{1}, μ is the Lebesgue measure and d is the Euclidean metric, we conclude from Theorem 3.3 that the statements below are valid.

Theorem 4.1. Let φ ∈ Φ. In order that there exist a constant c_{1} > 0 such that either of the inequalities

∫_{−∞}^{∞} φ(Hf(x))w(x) dx ≤ c_{1} ∫_{−∞}^{∞} φ(f(x))w(x) dx,  (4.1)
∫_{−∞}^{∞} φ̃(\frac{Hf(x)}{w(x)})w(x) dx ≤ c_{1} ∫_{−∞}^{∞} φ̃(\frac{f(x)}{w(x)})w(x) dx  (4.2)

is fulfilled, it is necessary and sufficient that φ ∈ Δ_{2}, φ^{α} be quasiconvex for some α, 0 < α < 1, and w ∈ A_{p(φ)}.

The validity of our next statement follows from Theorem 3.1.

Theorem 4.2. Let φ ∈ Φ. In order that there exist a constant c_{2} > 0 such that the inequality

∫_{−∞}^{∞} φ(Hf(x)w(x)) dx ≤ c_{2} ∫_{−∞}^{∞} φ(f(x)w(x)) dx  (4.3)

is fulfilled, it is necessary and sufficient that φ ∈ Δ_{2}, φ^{α} be quasiconvex for

some α, 0 < α < 1, w^{p(φ)} ∈ A_{p(φ)}, and w^{−p(_{φ})} ∈ A_{p(_{φ})}.

Let now X = [0, ∞), d(x, y) = |x^{2} − y^{2}|, the measure dμ = x dx. It can be easily seen that intervals will be balls in X and μB(x, r) ∼ r.

The kernel

k(x, y) = \frac{1}{x^{2} − y^{2}}

satisfies conditions (1.1), (1.2), and (2.1).

For odd and even functions the Hilbert transforms are respectively written as

H_{o}f(x) = \frac{2}{π} ∫_{0}^{∞} \frac{yf(y)}{x^{2} − y^{2}} dy,

H_{e}f(x) = \frac{2}{π} ∫_{0}^{∞} \frac{xf(y)}{x^{2} − y^{2}} dy.

Obviously, H_{o}f = Kf and H_{e}f(x) = xK(\frac{f}{·} )(x). Also note that the function w(x) = |x| ∈ A_{2} and therefore the operator K is bounded in L^{2}(X, x dx). Now we are able to apply Theorems 3.3 and 3.5 and to conclude that the statements below are valid.

Theorem 4.3. Let φ ∈ Φ. Then the following conditions are equivalent:

(i) φ ∈ Δ_{2}, φ^{α} is qusiconvex for some α, 0 < α < 1, and

sup (\frac{1}{b^{2} − a^{2}} ∫_{a}^{b}w(x) dx)(\frac{1}{b^{2} − a^{2}} ∫_{a}^{b}w^{1−p′(φ)}x^{p′(φ)}dx)^{p(φ)−1}, (4.4)

where the exact upper bound is taken with respect to all intervals [a, b] ⊂ [0, ∞);

(ii) there exists c > 0 such that

∫_{0}^{∞} φ(H_{o}f(x))w(x) dx ≤ c ∫_{0}^{∞} φ(f(x))w(x) dx;  (4.5)

(iii) there exists c > 0 such that

∫_{0}^{∞} φ̃(\frac{xH_{o}f(x)}{w(x)})w(x) dx ≤ c ∫_{0}^{∞} φ̃(\frac{xf(x)}{w(x)})w(x) dx.  (4.6)

On the other hand, since

H_{e}f(x) = xH_{o}(\frac{f}{y} )(x),

by Theorem 4.2 we conclude that the following theorem is valid.

Theorem 4.4. Let φ ∈ Φ. Then the following two conditions are equivalent:

(i) the inequality

∫_{0}^{∞} φ(\frac{H_{e}f(x)}{w(x)})w(x) dx ≤ c ∫_{0}^{∞} φ(\frac{f(x)}{w(x)})w(x) dx  (4.7)

holds; (ii) φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, and we have

sup(\frac{1}{b^{2}−a^{2}} ∫_{a}^{b}w(x)dx)(\frac{1}{b^{2}−a^{2}} ∫_{a}^{b}w^{1−p′(_{φ})}x^{p′(_{φ})}dx)^{p(_{φ})−1}<∞, (4.8)

where the exact upper bound is taken with respect to all intervals [a, b] ⊂ [0, ∞).

For Calderon–Zygmund singular integrals in ℝ^{n} we conclude by Theorem 3.1 that the following statement is true.

Theorem 4.5. Let

Kf(x) = ∫_{ℝ^{n}} k(x − y)f(y) dy,

where the kernel k satisfies the conditions:

∥k̂∥_{∞} < c, |k(x)| ≤ c|x|^{−n}

|k(x) − k(x − y) ≤ \frac{cω (\frac{|y|}{|x|})}{|x|^{n}} for |y| < \frac{|x|}{2},

where ω is an increasing function satisfying the condition

∫_{0}^{1} \frac{ω(s)}{s} ds < ∞.

If φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, w ∈ A_{p(φ)}, then we have the inequalities

∫_{ℝ^{n}} φ(Kf(x))w(x) dx ≤ c ∫_{ℝ^{n}} φ(f(x))w(x) dx,  (4.9)
∫_{ℝ^{n}} φ̃(\frac{Kf(x)}{w(x)})w(x) dx ≤ c ∫_{ℝ^{n}} φ̃(\frac{f(x)}{w(x)})w(x) dx  (4.10)

with the constant c independent of f.

Further, if φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, w^{p(φ)} ∈ A_{p(φ)}, w^{−p(_{φ})} ∈ A_{p(_{φ})}, then we have the inequality

∫_{−∞}^{∞} φ(Kf(x)w(x)) dx ≤ c ∫_{−∞}^{∞} φ(f(x)w(x)) dx.  (4.11)

Theorem 4.6. If

k(x) = \frac{Ω(x)}{|x|^{n}},

where Ω : ℝ^{n} → ℝ^{1} is a homogeneous function of degree zero not vanishing identically on the sphere S^{n−1} and satisfying the Lipschitz condition of order α, 0 < α ≤ 1, on the sphere S^{n−1},

∫_{S^{n−1}} Ω(x′) dx′ = 0.

Then the simultaneous fulfillment of inequality (4.9) for K and K̃ is equivalent to that of the combination of conditions: φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, and w ∈ A_{p(φ)}.

An analogous statement holds for (4.10).

Theorem 4.7. Under the conditions of Theorem 4.6 for the kernel k the simultaneous fulfillment of (4.11) for K and K̃ is equivalent to that of the combination of conditions: φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α,

0 < α < 1, w^{p(φ)} ∈ A_{p(φ)} and w^{−p(_{φ})} ∈ A_{p(_{φ})}.

In particular, the above-formulated theorems are valid for Riesz transforms

R_{j}f(x) = ∫_{ℝ^{n}} \frac{x_{j} − y_{j}}{|x − y|^{n+1}} f(y) dy,

j = 1, …, n, where x = (x_{1}, …, x_{n}).

Let now Γ ⊂ ℂ be a connected rectifiable curve and ν be an arc-length measure on Γ. By definition, Γ is regular if

ν(Γ ∩ B(z, r)) ≤ cr

for every z ∈ ℂ and r > 0.

For r less than half the diameter of Γ we have the reverse inequality

ν(Γ ∩ B(z, r)) ≥ r

for all z ∈ Γ. When equipped with ν and the Euclidean metric, a regular curve becomes a homogeneous type space. Then k(z_{1}, w) = (z −w)^{−1} is the standard kernel and the Cauchy integral

S_{Γ}f(t) = ∫_{Γ} \frac{f(t)}{t − τ} dν

is defined as the Calderon–Zygmund singular operator on a homogeneous type space.

G. David [7] has shown that the operator S_{Γ} is bounded in L^{p}(Γ, ν) iff Γ is regular.

Theorem 4.8. Let Γ be a regular curve. The following conditions are equivalent:

(i) φ ∈ Δ_{2}, φ^{α} is quasiconvex for some α, 0 < α < 1, and

sup (\frac{1}{r} ∫_{B(z,r)∩Γ}w(x) dx)(\frac{1}{r} ∫_{B(z,r)∩Γ}w^{1−p′(φ)}dν)^{p(φ)−1} < ∞, (4.12)

where the supremum is taken over all balls; (ii) there exists a positive constant c > 0 such that

∫_{Γ} φ(S_{Γ}f(t))w(t) dν ≤ c ∫_{Γ} φ(f(t))w(t) dν  (4.13)

with the constant independent of f.

An analogue of Theorem 4.2 also holds for the operator S_{Γ}.

Proof of Theorem 4.8. The implication (4.12)⇒(4.13) follows from David’s result, Theorem 3 from [8], and Theorem 3.2 above.

On the other hand, for an arbitrary point z ∈ Γ fix a ball B(z_{0}, r) of sufficiently small radius r. Let a point z′_{0} be chosen such that |z′_{0} −z_{0}| = 3r. For any z ∈ B(z_{0}, r) and z′ ∈ B(z′_{0}, r) we have |z − z′| > r and, as one can easily see, Re (z − z′) or Im (z − z′) preserve their sign and are greater than r. Therefore condition (2.1) is fulfilled for the kernel Re \frac{1}{z−z′} or Im \frac{1}{z−z′}. Since the inequality

|S_{Γ}f(t)| ≥ |Re (S_{Γ}f)(t)| + |Im (S_{Γ}f)(t)| ≥ \frac{c}{r} ∫_{B(z′,r)∩Γ}f(τ) dν

is fulfilled for f ≥ 0, supp f ⊂ B(z′_{0}, r) ∩ Γ, and t ∈ B(z_{0}, r) ∩ Γ, the validity of the implication (ii)⇒(i) follows from Theorem 3.4 and Remark 3.1.

§ 5. Appendix

The above-described methods of investigation make it possible to extend the results of §§2, 3, and 4 to vector-valued functions as well.

Let X be a homogeneous type space. Consider the kernels k(x, y) defined on X × X with values in L(A, B) which is a space of all bounded operators from the Banach space A to the Banach space B. Introduce the operator norm | · | = | · |_{L(A,B)}. It is assumed that |k(x, ·)| is locally integrable apart from x and the standard conditions (1.1) and (1.2) with the norm | · | for Calderon–Zygmund kernels are satisfied. Moreover, the operator

Kf(x) = ∫_{X} k(x, y)f(y) dμ

will be assumed to be bounded from L_{A}^{p_{0}}(X, μ) into L_{B}^{p_{0}}(X, μ) for some p_{0}, 1 < p_{0} < ∞.

To illustrate the above we give

Theorem 5.1. Let φ ∈ Φ∩Δ_{2}, φ^{α} be quasiconvex for some α, 0<α< 1. Then the inequality

∫_{X} φ(|Kf(x)|_{B})w(x) dμ ≤ c ∫_{X} φ(|f(x)|_{A})w(x) dμ

is fulfilled.

An analogue of Theorem 3.4 also holds.

Let us further consider the case where sufficient conditions convert to criteria. Assume that A = B is a Banach space with an unconditional base (b_{j})_{j∈ℕ}. Then a B-valued measurable function f(x) is the same as a sequence of measurable functions (f_{j}(x))_{j∈ℕ} such that ∑_{j} f_{j}(x)b_{j} ∈ B for every x ∈ X. If B is reflexive, then the dual base (b_{j}^{∗})_{j∈ℕ} is an unconditional base of B^{∗}.

Define

T f(x) = ∑_{j} kf_{j}(x)b_{j}

If the kernel k : X × X → ℝ^{1}, then by condition (2.1) (or (2.1′)) we have criteria of weighted inequalities similar to Theorems 3.3 and 3.1. It is obvious that conversions of theorems of type 5.1 are obtained from the corresponding theorems for scalar-valued functions.

References

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2. R. A. Macías and C. Segovia, Singular integrals on generalized Lipschitz and Hardy spaces. Studia Math. 65(1979), No. 1, 55–71.

3. R. A. Macías, C. Segovia, and J. L. Torrea, Singular integral operators with not necessarily bounded kernels on spaces of homogeneous type. Advances in Math. 93(1992), 25–60.

4. R. A. Macías and C. Segovia, A well-defined quasi-distance for spaces of homogeneous type. Trab. Mat. Inst. Argentina Mat. 32(1981), 1–14.

5. S. Hoffman, Weighted norm inequalities and vector-valued inequalities for certain rough operators. Indiana University Math. J. 42(1993), No. 1, 1–14.

6. F. J. Ruiz-Torrea and J. L. Torrea. Vector-valued Calderon–Zygmund theory and Calderon measures on spaces of homogeneous nature. Studia Math. 88(1988), 221–243.

7. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm. Sup. (4)17 (1984), No. 1, 157– 189.

8. Qian Tao, Weighted inequalities concerning the Radon measure of the arc length of curves on the complex plane. J. Sys. Sci. Math. Sci. 6(1986), No. 2, 146–152.

(Received 23.02.1994)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Z. Rukhadze St., Tbilisi 380093

Republic of Georgia