GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 65-82

TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS IN LORENTZ SPACES DEFINED ON HOMOGENEOUS GROUPS

V. KOKILASHVILI AND A. MESKHI

Abstract. The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.

Introduction

In this paper, the optimal sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In [1–7], analogous problems were studied in the Lebesgue spaces for the Hilbert transform and singular integrals, while in [8] the sufficient conditions are found for pairs of weights, which guarantee the fulfilment of two-weighted inequalities for singular integrals in the Lorentz spaces defined on Euclidean spaces. In this paper, the necessary and sufficient conditions are also found for pairs of weights, which provide the boundedness of the generalized Hardy operators in the weighted Lorentz spaces. Analogous problems were considered in the Lebesgue spaces in [9–14] and in the Lorentz spaces in [8], [15] and [16] (see also [17]).

Finally, we would like to note that most of the results obtained in this paper are new for the classical singular integrals as well.

65 Homogeneous Groups and the Spaces of Functions Defined on

1991 Mathematics Subject Classification. 42B20, 43A80, 46E40, 47G10.

Key words and phrases. Hilbert transform, singular integrals, Hardy operator, Lorentz spaces, two-weighted inequality, homogeneous groups.

1072-947X/99/0100-0065$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

Them. Some Known Results

In this section, we give the notion of homogeneous groups and define the Lorentz spaces. Some known results on the Lorentz spaces and singular integrals are also presented.

Definition 1 (see [18], p. 5). A homogeneous group is a connected, simply connected nilpotent Lie group G on whose Lie algebra g a oneparametric group of extensions δ_{t} = exp(A ln t), t > 0, is given, where A is the diagonizable operator on g whose eigenvalues are positive.

For the homogeneous group G the mappings exp ∘δ_{t} ∘ exp^{−1}, t > 0, are the automorphisms on G which will again be denoted by δ_{t}.

The number Q = trA is called a homogeneous dimension of the group G. Homogeneous groups can be exemplified by an n-dimensional Euclidean

space, Heisenberg groups and so on.

Onto the homogeneous group G we introduce a homogeneous norm, i.e., a continuous function r : G → [0, ∞) which is smooth on G\{e} and satisfies the following conditions:

1. r(x) = r(x^{−1}) for any x ∈ G;

2. r(δ_{t}x) = tr(x) for arbitrary x ∈ G and t > 0;

3. r(x) = 0 ⇔ x = e;

4. there exists a constant c_{0} > 0 such that r(xy) ≤ c_{0}(r(x) + r(y)) for arbitrary x and y from G.

For x ∈ G and ρ > 0 we set B(x, ρ) = {y ∈ G : r(xy^{−1}) < ρ}. Further, let S(x, ρ) = {y ∈ G : r(xy^{−1}) = ρ}. Note that δ_{ρ}B(e, 1) = B(e, ρ).

Onto the group G we fixed the normed Haar measure in such a manner that the measure of the unit ball B(e, 1) be equal to 1. The Haar measure of any measurable set E ⊂ G will be denoted by |E|, and the integral on E with respect to this measure by ∫_{E} f(x)dx.

It readily follows that

|δ_{t}E| = t^{Q}|E|, d(δ_{t}x) = t^{Q}dx.

In particular, for arbitrary x ∈ G and ρ > 0 we have |B(e, ρ)| = ρ^{Q}.

Definition 2. An almost everywhere positive, locally integrable function w : G → ℝ^{1} will be called a weight.

In what follows, we shall denote by L_{w}^{pq}(G) the Lorentz space with weight w which is a class of all measurable functions f : G → ℝ^{1} for which

∥f∥_{L_{w}^{pq}(G)} = (q ∫_{0}^{∞}( ∫_{\{x∈G\}:|f(x)|>λ\}} w(x) dx)^{\frac{q}{p}} λ^{q−1}dλ)^{\frac{1}{q}} < ∞

when 1 ≤ p ≤ ∞, 1 ≤ q < ∞, and

∥f∥_{L_{w}^{p∞}(G)} = sup_{λ>0} λ( ∫_{\{x∈G:|f(x)|>λ\}} w(x) dx)^{\frac{1}{p}} < ∞

when 1 ≤ p < ∞.

For p = q, L_{w}^{pq}(G) is the Lebesgue space which it is commonly accepted to denote by L_{w}^{p}(G).

Theorem A (see [18], p. 14). Let G be a homogeneous group, S_{G} = {x ∈ G : r(x) = 1}. There exists a unique countably additive measure σ defined on S_{G} such that the equality

∫_{G} f(x) dx = ∫_{0}^{∞}t^{Q−1}( ∫_{S_{G}} f(δ_{t}ξ) dσ(ξ))dt

holds for any f ∈ L^{1}(G).

Let k : G → ℝ^{1} be a measurable function such that:

(1) |k(x)| ≤ \frac{c}{r(x)^{Q}} for an arbitrary element x ≠ e;

(2) there exists a positive constant c_{1} such that for arbitrary x, y ∈ G with the condition r(xy^{−1}) < \frac{1}{2}r(x) we have the inequality

|k(x) − k(y)| ≤ c_{1}ω(\frac{r(xy^{−1})}{r(x)}) \frac{1}{r(x)^{Q}} ,

where ω : [0, 1] → ℝ^{1} is a nondecreasing function such that ω(0) = 0,

ω(2t) ≤ c_{2}ω(t) for any t > 0 and ∫_{0}^{1} \frac{ω(t)}{t} dt < ∞.

It will be assumed that the kernel k together with the above-given conditions satisfy the condition: the singular integral

T f(x) = p.v. ∫_{G} k(xy^{−1})f(y) dy =^{def} lim_{ε→0} ∫_{G\B(x,ε)} k(xy^{−1})f(y) dy

defines the bounded operator in the space L^{2}(G).

The lemmas below are valid.

Lemma A ([19]). Let E ⊂ G be an arbitrary measurable set, w a weight function on G and f, f_{1}, f_{2} measurable functions on G. Then we have:

(1) ∥χ_{E}(·)∥_{L_{w}^{pq}(G)} = ( ∫_{E} w(x) dx)^{\frac{1}{p}};

(2) ∥f∥_{L_{w}^{pq_{1}} (G)} ≤ ∥f∥_{L_{w}^{pq_{2}} (G)},

for fixed p and q_{2} ≤ q_{1};

(3) ∥f_{1}f_{2}∥_{L_{w}^{pq}(G)} ≤ c∥f_{1}∥_{L_{w}^{p_{1}q_{1}} (G)}∥f_{2}∥_{L_{w}^{p_{2}q_{2}} (G)},

where \frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} , \frac{1}{q} = \frac{1}{q_{1}} + \frac{1}{q_{2}} .

Lemma B. Let E_{k} be measurable sets of the homogeneous group G such that the inequality ∑_{k} χ_{E_{k}} ≤ cχ_{{∪ \atop k}E_{k}} with constant c is fulfilled. Then:

(1) for any f we have the inequality

∑_{k} ∥f(·)χ_{E_{k}} (·)∥_{L_{w}^{rs}(G)}^{λ} ≤ c_{1}∥f(·)χ_{{∪ \atop k}E_{k}} (·)∥_{L_{w}^{rs}(G)}^{λ},

where the constant c_{1} does not depend on f and max(r, s) ≤ λ; (2) for any f we have the inequality

∥ ∑_{k} f(·)χ_{E_{k}} (·)∥_{L_{w}^{pq}(G)}^{γ} ≤ c_{2} ∑_{k} ∥f(·)χ_{E_{k}} (·)∥_{L_{w}^{pq}(G)}^{γ},

where the constant c_{2} does not depend on f and 0 < γ ≤ min(p, q).

Lemma B is proved as Lemma 2 from [8]. For the case c = 1 analogous result was obtained in [20], [15] (see also [16]).

Definition 3. A function v : G → ℝ_{+}^{1} is called radial if there exists a function β : ℝ_{+}^{1} → ℝ_{+}^{1} such that the equality v(x) = β(r(x)) holds for any x ∈ G. In what follows, instead of β we shall use the notation v.

Definition 4. Let 1 < p < ∞. A weight function w belongs to A_{p}(G) if

sup (\frac{1}{|B|} ∫_{B} w(x) dx) (\frac{1}{|B|} ∫_{B} w^{1−p′}(x) dx)^{p−1} < ∞,

where the least upper bound is taken with respect to all balls B, B ⊂ G.

Theorem B ([21]). Let 1 < p < ∞, w ∈ A_{p}(G). Then the operator T is bounded in L_{w}^{p}(G).

Theorem C (see [22], p. 207). Let 1 < p, q < ∞. If w ∈ A_{p}(G), then the operator T is bounded in L_{w}^{pq}(G). If the Hilbert transform acts continuously in L_{w}^{pq}(ℝ), then w ∈ A_{p}(ℝ).

We shall need

Lemma C ([4], [7]). Let 1 < p < ∞, ρ ∈ A_{p}(G), 0 ≤ c_{1} < c_{2} ≤ c_{3} < c_{4} < ∞. Then there exists a positive constant c such that the inequality

∫_{{x∈G:c_{3}t<r(x)<c_{4}t}} ρ(x) dx ≤ c ∫_{{x∈G:c_{1}t<r(x)<c_{2}t}} ρ(x) dx

holds for any t, t > 0.

1. Two-Weighted Inequalities for the Hardy Operator in the Lorentz Spaces Defined on Homogeneous Groups

In this paragraph, the necessary and sufficient conditions are found for the boundedness of the operators

Hf(x) = a(x) ∫_{r(y)≤r(x)} b(y)f(y)w(y) dy,

H^{∗}f(x) = b(x) ∫_{r(y)≥r(x)} a(y)f(y)w(y) dy

from L_{w}^{rs}(G) into L_{v}^{pq}(G), where a, b, w and v are measurable non-negative functions on G.

Theorem 1.1. Let r = s = 1 or r = s = ∞ or r ∈ (1, ∞) and s ∈ (1, ∞), p = q = 1 or p = q = ∞ or p ∈ (1, ∞) and q ∈ (1, ∞); max(r, s) ≤ min(p, q). For the inequality

∥Hf(·)∥_{L_{v}^{pq}(G)} ≤ c∥f(·)∥_{L_{w}^{rs}(G)},  (1.1)

where the constant c does not depend on f, to be valid it is necessary and sufficient that the condition

sup_{t>0} ∥a(·)χ_{{r(y)>t}}(·)∥_{L_{v}^{pq}(G)} ∥b(·)χ_{{r(y)<t}}(·)∥_{L_{w}^{r′s′} (G)} < ∞  (1.2)

(r′ = \frac{r}{r−1}, s′ = \frac{s}{s−1}) be fulfilled.

Proof. Sufficiency. Assume that condition (1.2) is fulfilled. We take f ≥0. If ∫_{G} f(x)b(x)w(x) dx < ∞, then it belongs to the interval (2^{m}, 2^{m+1}] for some

integer m. By virtue of Theorem A we can choose a sequence {x_{k}}_{k=−∞}^{m} such that

2^{k} = ∫_{r(y)≤x_{k}}b(y)f(y)w(y) dy =  ∫_{x_{k}<r(y)≤x_{k+1}} b(y)f(y)w(y) dy for k ≤ m − 1 (1.3)

and

2^{m} = ∫_{r(y)≤x_{m}}b(y)f(y)w(y) dy.  (1.4)

Let G_{k} = {y : x_{k} < r(y) ≤ x_{k+1}}, k ≤ m and x_{m+1} = ∞. Then the sets G_{k} do not intersect pairwise and, since we can assume that lim_{k→−∞} x_{k} = 0,

we obtain

⋃_{k≤m}G_{k} = G \ {e}.  (1.5)

If ∫_{G} b(y)f(y)w(y) dy = ∞, then we choose a sequence {x_{k}}_{k=−∞}^{+∞} such that

(1.3) is fulfilled for any integer k (in this case m = +∞). By (1.3) and (1.4) we have

Hf(x) ≤ a(x) · 2^{k+1} for x ∈ G_{k}, k ≤ m.  (1.6)

Choose a number σ such that max(r, s) ≤ σ ≤ min(p, q). Using (1.5), Lemma B, (1.6), Hölder’s inequality, we obtain

∥Hf(·)∥_{L_{v}^{pq}(G)}^{σ} = ∥ ∑_{k≤m}(Hf)(·)χ_{G_{k}} (·)∥_{L_{v}^{pq}(G)}^{σ} ≤

≤ ∑_{k≤m} ∥(Hf)(·)χ_{G_{k}} (·)∥_{L_{v}^{pq}(G)}^{σ} ≤

≤ ∑_{k≤m} 2^{σ(k+1)}∥a(·)χ_{G_{k}} (·)∥_{L_{v}^{pq}(G)}^{σ} = 2^{2σ} ∑_{k≤m} 2^{σ(k−1)}∥a(·)χ_{G_{k}} (·)∥_{L_{v}^{pq}(G)}^{σ} ≤

≤ 4^{σ} ∑_{k≤m} ( ∫_{x_{k−1}<r(y)≤x_{k}} b(y)f(y)w(y) dy)^{σ}∥a(·)χ_{{r(y)>x_{k}}}(·)∥_{L_{v}^{pq}(G)}^{σ} ≤

≤ 4^{σ} ∑_{k≤m} ∥f(·)χ_{G_{k−1}} (·)∥_{L_{w}^{rs}(G)}^{σ}∥b(·)χ_{{r(y)<x_{k}}}(·)∥_{L_{w}^{r′s′} (G)}^{σ} ×

×∥a(·)χ_{{r(y)>x_{k}}}(·)∥_{L_{v}^{pq}(G)}^{σ} ≤ c∥f(·)∥_{L_{w}^{rs}(G)}^{σ}

where the constant c does not depend on f.

Necessity. Let inequality (1.1) be fulfilled. If ∥f∥_{L_{w}^{rs}(G)} ≤ 1 and t ∈ (0, ∞), then we have

c ≥ c∥f∥_{L_{w}^{rs}(G)} ≥ ∥Hf∥_{L_{v}^{pq}(G)} ≥ ∥(Hf)(·)χ_{{r(y)>t}}(·)∥_{L_{v}^{pq}(G)} ≥

≥ ∫_{r(y)<t} b(y)f(y)w(y) dy ∥a(·)χ_{{r(y)>t}}(·)∥_{L_{v}^{pq}(G)}.

Taking the least upper bound with respect to all such f and t, we obtain condition (1.2).

If we apply the dual arguments (see also [8]), then we easily obtain

Theorem 1.2. Let the numbers r, s, p, q satisfy the conditions of Theorem 1.1. For the inequality

∥H^{∗}f(·)∥_{L_{v}^{pq}(G)} ≤ c∥f(·)∥_{L_{w}^{rs}(G)},

where the constant c does not depend on f, to be valid it is necessary and sufficient that the condition

sup_{t>0} ∥b(·)χ_{{r(y)<t}}(·)∥_{L_{v}^{pq}(G)} ∥a(·)χ_{{r(y)>t}}(·)∥_{L_{w}^{r′s′} (G)} < ∞

be fulfilled.

Analogous results for the operator Hf(x) = a(x) ∫_{0}^{x} b(t)f(t)w(t) dt are

given in [16]. A certain analog in ℝ^{n} was obtained in [8].

Corollary 1.1. Let 1 < r, s, p, q < ∞, max(r, s) ≤ min(p, q), γ < −\frac{Q}{p}, β = γr + Qr(\frac{1}{r′} + \frac{1}{p}). Then the inequality

∥r(·)^{γ} ∫_{r(y)≤r(·)} f(y) dy∥_{L^{pq}(G)} ≤ c∥f(·)∥_{L_{r(·)^{β}}^{rs} (G)}

holds, where the constant c does not depend on f.

2. Weighted Inequalities for Singular Integrals in the Lorentz Spaces Defined on Homogeneous Groups

In this paragraph, the sufficient conditions are found for pairs of weights, which provide the boundedness of a singular integral operator in the weighted Lorentz spaces defined on homogeneous groups.

First we investigate the existence of T f(x).

Lemma 1. Let 1 < s ≤ p < ∞, ρ ∈ A_{p}(G). If the weight functions w and w_{1} satisfy the conditions:

(1) there exists a positive increasing on (0, ∞) function σ such that for almost all x ∈ G we have the inequality

σ(r(x))ρ(x) ≤ bw(x)w_{1}^{p}(x),

where the positive constant b does not depend on x;

(2) ∥\frac{1}{w(·)w_{1}(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′s′} (G)} < ∞ for any t > 0,

then for arbitrary φ with the condition ∥φ(·)w_{1}(·)∥_{L_{w}^{ps}(G)} < ∞, T φ(x) exists almost everywhere on G.

Proof. Fix the number α, α > 0, and let

S_{α} = \{x ∈ G : r(x) > \frac{α}{2}\}.

Take a function φ with the condition ∥φ(·)w_{1}(·)∥_{L_{w}^{ps}(G)} < ∞. We write ϕ as

φ(x) = φ_{1}(x) + φ_{2}(x),

where φ_{1}(x) = φ(x) · χ_{Sα} (x), φ_{2}(x) = φ(x) − φ_{1}(x).

For φ_{1} we obtain

∫_{G} |φ_{1}(x)|^{p}ρ(x) dx = \frac{σ(\frac{α}{2})}{σ(\frac{α}{2})} ∫_{S_{α}} |φ(x)|^{p}ρ(x) dx ≤

≤ \frac{1}{σ(\frac{α}{2})} ∫_{S_{α}} |φ(x)|^{p}ρ(x)σ(r(x)) dx ≤ \frac{b}{σ(\frac{α}{2})} ∫_{S_{α}} |φ(x)|^{p}w_{1}^{p}(x)w(x) dx ≤

≤ \frac{b}{σ(\frac{α}{2})}∥φ(·)w_{1}(·)∥_{L_{w}^{p}(G)}^{p} ≤ \frac{b}{σ(\frac{α}{2})}∥φ(·)w_{1}(·)∥_{L_{w}^{ps}(G)}^{p}.

We find by Theorem B that T φ_{1} ∈ L_{ρ}^{p}(G) and T φ_{1}(x) exists almost everywhere on G.

Now we shall show that T φ_{2}(x) converges absolutely for r(x) > αc_{0}. Note that when r(x) > αc_{0} and r(y) < \frac{α}{2}, we have r(x) ≤ c_{0}(r(xy^{−1}) + r(y)) ≤ c_{0}(r(xy^{−1}) + \frac{α}{2}) ≤ c_{0}(r(xy^{−1}) + \frac{r(x)}{2c_{0}}) and \frac{α}{2} < \frac{r(x)}{2c_{0}} ≤ r(xy^{−1}).

We obtain

|T φ_{2}(x)| ≤ c_{1} ∫_{\{y:r(y)< \frac{α}{2} \}} \frac{φ(y)}{r(xy^{−1})^{Q}} dy ≤ \frac{c_{2}}{α^{Q}} ∫_{\{y:r(y)< \frac{α}{2} \}} |φ(y)| dy =

= \frac{c_{2}}{α^{Q}} ∫_{\{y:r(y)< \frac{α}{2} \}} φ(y) \frac{1}{w_{1}(y)w(y)} w_{1}(y)w(y)dy ≤

≤ \frac{c_{2}}{α^{Q}} ∥φ(·)w_{1}(·)∥_{L_{w}^{ps}(G)} ∥χ_{\{y:r(y)< \frac{α}{2} \}}(·) \frac{1}{w_{1}(·)w(·)}∥_{L_{w}^{p′s′} (G)} < ∞.

Since we can take α arbitrarily small, T φ(x) exists almost everywhere on G.

The next lemma is proved in a similar manner.

Lemma 2. Let 1 < p, s < ∞, s ≤ p, ρ ∈ A_{p}(G). If the weight functions w and w_{1} satisfy the conditions:

(1) there exists a positive decreasing on (0, ∞) function σ such that for almost all x ∈ G we have the inequality

σ(r(x))ρ(x) ≤ bw(x)w_{1}^{p}(x),

where the positive constant b does not depend on x;

(2) ∥\frac{r(·)^{−Q}}{w(·)w_{1}(·)} χ_{\{r(y)>t\}}(·)∥_{L_{w}^{p′s′} (G)} < ∞ for any t > 0,

then for arbitrary φ with the condition ∥φ(·)w_{1}(·)∥_{L_{w}^{ps}(G)} < ∞, T φ(x) exists almost everywhere on G.

Lemma 1 readily implies

TWO-WEIGHTED INEQUALITIES FOR INTEGRAL OPERATORS 73

Lemma 3. Let 1 < p, s, < ∞, s ≤ p. If u and u_{1} are positive increasing functions on (0, ∞) and

∥\frac{1}{u_{1}(r(·))u(r(·))} χ_{\{r(y)<t\}}(·)∥_{L_{u(r(·))}^{p′s′}(G)} < ∞

for arbitrary t > 0, then for any φ with the condition ∥φ(·)u_{1}(r(·))∥_{L_{u(r(·))}^{ps}(G)} < ∞, T φ(x) exists almost everywhere on G.

Analogously, Lemma 2 implies

Lemma 4. Let 1 < p, s < ∞, s ≤ p. If u and u_{1} are positive decreasing functions on (0, ∞) and

∥\frac{r(·)^{−Q}}{u(r(·))u_{1}(r(·))} χ_{\{r(y)>t\}}(·)∥_{L_{u(r(·))}^{p′s′}(G)} < ∞

for arbitrary t > 0, then for φ with the condition φ(·)u_{1}(r(·)) ∈ L_{u(r(·))}^{ps}(G) < ∞, T φ(x) exists almost everywhere on G.

Now we shall formulate and prove our basic theorems.

Theorem 2.1. Let 1 < s ≤ p ≤ q < ∞, σ be a positive increasing function on (0, ∞), the function ρ ∈ A_{p}(G), w a weight function on G, v(x) = σ(r(x))ρ(x). Let the following conditions be fullfilled:

(1) there exists a positive constant b such that almost for all x ∈ G σ(2c_{0}r(x))ρ(x) ≤ bw(x);

(2) ∥r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v}^{pq}(G)} ∥\frac{1}{w(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′s′} (G)} ≤ c < ∞.

Then there exists a positive constant c such that the inequality

∥T f(·)∥_{L_{v}^{pq}(G)} ≤ c∥f(·)∥_{L_{w}^{ps}(G)}  (2.1)

holds for any f ∈ L_{w}^{ps}(G). Proof. Without loss of generality we can write the function σ as

σ(t) = σ(0) + ∫_{0}^{t} φ(τ) dτ,

where σ(0) = lim_{t→0} σ(t) and φ(τ) ≥ 0 on (0, ∞). We have

∥T f(·)∥_{L_{v}^{pq}(G)} ≤ c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{\{x:|T f(x)|>λ\}} ρ(x)σ(0) dx)^{\frac{q}{p}} dλ)^{\frac{1}{q}} +

+c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{\{x:|T f(x)|>λ\}} ρ(x)( ∫_{0}^{r(x)}φ(t) dt)dx)^{\frac{q}{p}} dλ)^{\frac{1}{q}} =

= I_{1} + I_{2}.

If σ(0) = 0, then I_{1} = 0, and if σ(0) ≠ 0, then by Theorem C and Lemma A we obtain

I_{1} = c_{1}σ^{\frac{1}{p}} (0)∥T f(·)∥_{L_{ρ}^{pq}(G)} ≤ c_{2}σ^{\frac{1}{p}} (0)∥f(·)∥_{L_{ρ}^{pq}(G)} ≤
≤ c_{2}σ^{\frac{1}{p}} (0)∥f(·)∥_{L_{ρ}^{ps}(G)} ≤ c_{3}∥f(·)∥_{L_{w}^{ps}(G)}.

Now we shall estimate I_{2}. Let f_{1t}(x) = f(x) · χ_{\{r(x)> \frac{t}{2c_{0}} \}}(x), f_{2t}(x) =

f(x) − f_{1t}(x). We have

I_{2} = c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{∞}φ(t)( ∫_{\{x:r(x)>t, |T f(x)|>λ\}} ρ(x) dx)dt)^{\frac{q}{p}} dλ)^{\frac{1}{q}} ≤

≤c_{4}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{∞}φ(t)( ∫_{\{x:r(x)>t\}} χ\{x : |T f_{1t}(x)| > \frac{λ}{2}\}ρ(x)dx)dt)^{\frac{q}{p}} dλ)^{\frac{1}{q}} +

+c_{4}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{∞}φ(t)( ∫_{\{x:r(x)>t\}} χ\{x : |T f_{2t}(x)| > \frac{λ}{2}\}ρ(x)dx)dt)^{\frac{q}{p}} dλ)^{\frac{1}{q}} =

= I_{21} + I_{22}.

Applying Minkowski’s inequality twice (\frac{q}{p} ≥ 1, \frac{p}{s} ≥ 1) and Theorem C, we obtain

I_{21} ≤ c_{5}( ∫_{0}^{∞}φ(t)( ∫_{0}^{∞}λ^{q−1}( ∫_{\{x:|T f_{1t}(x)|>λ\}} ρ(x) dx)^{\frac{q}{p}} dλ)^{\frac{p}{q}} dt)^{\frac{1}{p}} ≤

≤ c_{6}( ∫_{0}^{∞}φ(t)∥f_{1t}(·)∥_{L_{ρ}^{pq}(G)}^{p}dt)^{\frac{1}{p}} ≤ c_{6}( ∫_{0}^{∞}φ(t)∥f_{1t}(·)∥_{L_{ρ}^{ps}(G)}^{p}dt)^{\frac{1}{p}} ≤

≤ c_{6}( ∫_{0}^{∞}λ^{s−1}( ∫_{0}^{∞}φ(t)( ∫_{\{x:|f(x)|>λ\}} ρ(x)χ_{\{r(y)> \frac{t}{2c_{0}} \}}(x) dx)dt)^{\frac{s}{p}} dλ)^{\frac{1}{s}} =

= c_{6}( ∫_{0}^{∞}λ^{s−1}( ∫_{\{x:|f(x)|>λ\}} ρ(x)( ∫_{0}^{2c_{0}r(x)}φ(t) dt)dx)^{\frac{s}{p}} dλ)^{\frac{1}{s}} ≤ c_{7}∥f∥_{L_{w}^{ps}(G)}.

Next, we shall estimate I_{22}. Note that if r(x) > t and r(y) < \frac{t}{2c_{0}} , then r(x) ≤ 2c_{0}r(xy^{−1}). By Theorem 1.1 we obtain

I_{22} = c_{4}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{∞}φ(t)( ∫_{\{x: \frac{(2c_{0})^{Q}}{r(x)^{Q}} ∫_{\{y:r(y)<r(x)\}} f(y)dy> \frac{λ}{2} \}} ρ(x)χ_{\{r(y)>t\}}(x) dx)dt)^{\frac{q}{p}} dλ)^{\frac{1}{q}} ≤

= c_{8}∥\frac{1}{r(·)^{Q}} ∫_{\{r(y)<r(·)\}} |f(y)| dy∥_{L_{v}^{pq}(G)} ≤ c_{9}∥f(·)∥_{L_{w}^{ps}(G)}.

If we write the function σ as

σ(t) = σ(+∞) + ∫_{t}^{∞}ψ(τ)dτ, where σ(+∞) = lim_{t→+∞} σ(t), ψ(τ) ≥ 0,

on (0, ∞), again apply Theorem C, Lemma A and Theorem 1.2, then we shall have

Theorem 2.2. Let 1 < s ≤ p ≤ q < ∞, σ be a positive decreasing function on (0, ∞), w a weight function on G, ρ ∈ A_{p}(G), v(x) = σ(r(x))ρ(x). Let the following conditions be fulfilled:

(1) there exists a positive constant b such that the inequality

ρ(x)σ(\frac{r(x)}{2c_{0}}) ≤ bw(x)

holds for almost all x ∈ G;

(2) sup_{t>0} ∥χ_{\{r(y)<t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{r(·)^{−Q}}{w(·)} χ_{\{r(y)>t\}}(·)∥_{L_{w}^{p′s′} (G)} < ∞.

Then inequality (2.1) is valid.

It is proved in [8] that if inequality (2.1) is fulfilled for the Hilbert transform, then the inequality v(x) ≤ b_{1}w(x) holds almost everywhere on ℝ^{1}.

Corollary 2.1. Let 1 < s ≤ p ≤ q < ∞, σ_{1} and σ_{2} be positive increasing functions on (0, ∞), ρ ∈ A_{p}(G), v(x) = σ_{2}(r(x))ρ(x), w(x) = σ_{1}(r(x))ρ(x). If the conditions

(1) there exists a positive constant b such that the inequality

σ_{2}(2c_{0}t) ≤ bσ_{1}(t)

holds for any t > 0;

(2) sup_{t>0} ∥r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{1}{w(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′s′} (G)} < ∞

are fulfilled, then inequality (2.1) is valid.

Corollary 2.2. Let 1 < s ≤ p ≤ q < ∞, σ_{1} and σ_{2} be positive decreasing functions on (0, ∞), ρ ∈ A_{p}(G), v(x) = σ_{2}(r(x))ρ(x), w(x) = σ_{1}(r(x))ρ(x). If the conditions

(1) σ_{2}(\frac{t}{2c_{0}} ) ≤ bσ_{1}(t) for any t > 0;

(2) sup_{t>0} ∥χ_{\{r(y)<t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{r(·)^{−Q}}{w(·)} χ_{\{r(y)>t\}}(·)∥_{L_{w}^{p′s′} (G)} < ∞

are fulfilled, then inequality (2.1) is valid.

Theorem 2.3. Let 1 < p ≤ q < ∞, σ_{1} and σ_{2} be positive increasing functions on (0, ∞), ρ ∈ A_{p}(G), v(x) = σ_{2}(r(x))ρ(x), w(x) = σ_{1}(r(x))ρ(x). If v and w satisfy the condition

sup_{t>0} ∥r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{1}{w(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′} (G)} < ∞

then the inequality

∥T f(·)∥_{L_{v}^{pq}(G)} ≤ c∥f(·)∥_{L_{w}^{p}(G)}

holds, where the positive constant c does not depend on f.

Proof. By Corollary 2.1. it is sufficient to show that the inequality

σ_{2}(2c_{0}t) ≤ bσ_{1}(t)

holds for any t > 0.

Using Hölder’s inequality and Lemma C (ρ^{1−p′} ∈ A_{p′}(G)), we obtain

\frac{σ_{2}(2c_{0}t)}{σ_{1}(t)} ≤

≤ c_{1} \frac{σ_{2}(2c_{0}t)}{σ_{1}(t)} t^{−Qp}(  ∫_{\{x:2c_{0}t<r(x)<4c_{0}t\}} ρ(x) dx)(  ∫_{\{x:2c_{0}t<r(x)<4c_{0}t\}} ρ^{1−p′}(x) dx)^{p−1} ≤

≤ c_{2} \frac{σ_{2}(2c_{0}t)}{σ_{1}(t)} t^{−Qp}( ∫_{\{x:2c_{0}t<r(x)<4c_{0}t\}} ρ(x) dx)( ∫_{\{x:r(x)<t\}}ρ^{1−p′}(x) dx)^{p−1} ≤

≤ c_{2}t^{−Qp}( ∫_{{x:2c_{0}t<r(x)<4c_{0}t}} v(x) dx)( ∫_{{x:r(x)<t}}w^{1−p′}(x) dx)^{p−1} =

= c_{2}t^{−Qp}∥χ_{\{x:2c_{0}t<r(x)<4c_{0}t\}}(·)∥_{L_{v}^{pq}(G)}^{p}∥\frac{1}{w(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′} (G)}^{p} ≤

≤ c_{3}∥r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v}^{pq}(G)}^{p}∥\frac{1}{w(·)} χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′} (G)}^{p} < ∞.

Theorem 2.4. Let 1 < s ≤ p ≤ q < ∞; σ_{1}, σ_{2}, u_{1} and u_{2} be weight functions on G, ρ ∈ A_{p}(G), v = σ_{2}ρ, w = σ_{1}ρ. Let the following three conditions be fulfilled:

(1) there exists a positive constant b such that for any t > 0 we have the inequality

sup_{F_{t}} σ2^{\frac{1}{p}} (x) sup_{F_{t}} u_{2}(x) ≤ b inf_{F_{t}} σ1^{\frac{1}{p}} (x) inf_{F_{t}} u_{2}(x),

where F_{t} = \{x ∈ G : \frac{t}{c_{0}} < r(x) < 8c_{0}t\};

(2) sup_{t>0} ∥u_{2}(·)r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{1}{u_{1}(·)w(·)}χ_{\{r(y)<t\}}(·)∥_{L_{w}^{p′s′} (G)} <∞;

(3) sup_{t>0} ∥u_{2}(·)χ_{\{r(y)<t\}}(·)∥_{L_{v}^{pq}(G)}∥\frac{1}{u_{1}(·)w(·)}r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{w}^{p′s′} (G)} <∞.

Then the inequality

∥u_{2}(·)T f(·)∥_{L_{v}^{pq}(G)} ≤ c∥u_{1}(·)f(·)∥_{L_{w}^{ps}(G)}  (2.2)

is valid, where the positive constant c does not depend on f.

Proof. Let E_{k} = \{x ∈ G : 2^{k} < r(x) ≤ 2^{k+1}\}, G_{k1} = \{x ∈ G : r(x) ≤ \frac{2^{k−1}}{c_{0}} \},

G_{k2} = \{x ∈ G : \frac{2^{k−1}}{c_{0}} < r(x) ≤ c_{0}2^{k+2}\}, G_{k3} = \{x ∈ G : r(x) > c_{0}2^{k+2}\}. We estimate the left-hand side of inequality (2.2) as follows:

∥u_{2}(·)T f(·)∥_{L_{v}^{pq}(G)}^{p} = ∥ ∑_{k∈ℤ} T f(·)u_{2}(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(G)}^{p} ≤

≤ c_{1}∥ ∑_{k∈ℤ} u_{2}(·)T (f · χ_{G_{k1}} )(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(G)}^{p} +

+c_{1}∥ ∑_{k∈ℤ} u_{2}(·)T (f · χ_{G_{k2}} )(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(G)}^{p} +

+c_{1}∥ ∑_{k∈ℤ} u_{2}(·)T (f · χ_{G_{k3}} )(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(G)}^{p} = c_{1}(S_{1}^{p} + S_{2}^{p} + S_{3}^{p}).

Next, we shall estimate S_{1}^{p}. Note that for x ∈ E_{k} and y ∈ G_{k1} we have r(y) ≤ \frac{2^{k−1}}{c_{0}} = \frac{2^{k}}{2c_{0}} ≤ \frac{r(x)}{2c_{0}} < r(x). Moreover, r(xy^{−1}) ≥ \frac{r(x)}{2c_{0}} and we obtain

|T (f · χ_{G_{k1}} )(x)| = | ∫_{G} k(xy^{−1})f(y)χ_{G_{k1}} (y) dy| ≤

≤ c_{2} ∫_{G} \frac{|f(y)|χ_{G_{k1}} (y)}{r(xy^{−1})^{Q}} dy ≤ c_{3} \frac{1}{r(x)^{Q}} ∫_{r(y)<r(x)} |f(y)| dy

for any x ∈ E_{k}. By Theorem 1.1 we find that

S_{1}^{p} ≤ c_{3}^{p}∥u_{2}(·)r(·)^{−Q}( ∫_{{r(y)≤r(x)}} f(y) dy)∥_{L_{v}^{pq}(G)}^{p} ≤ c_{4}∥u_{1}(·)f(·)∥_{L_{w}^{ps}(G)}^{p}.

Let us estimate S_{3}^{p}. As is easy to verify, for x ∈ E_{k} and y ∈ G_{k3} we have r(y) ≥ r(x) and r(xy^{−1}) ≥ \frac{r(y)}{2c_{0}}. For x ∈ E_{k} we obtain

|T (f · χ_{G_{k3}} )(x)| ≤ c_{5} ∫_{\{r(y)≥r(x)\}} \frac{|f(y)|}{r(y)^{Q}} dy.

By Theorem 1.2 we find that

S_{3}^{p} ≤ c_{5}^{p}∥u_{2}(·) ∫_{\{r(y)≥r(x)\}} \frac{f(y)}{r(y)^{Q}} dy∥_{L_{v}^{pq}(G)}^{p} ≤ c_{6}∥u_{1}(·)f(·)∥_{L_{w}^{ps}(G)}^{p}.

Now we shall estimate S_{2}^{p}. By Lemma B (second part) we obtain

S_{2}^{p} ≤ ∑_{k∈ℤ} ∥u_{2}(·)T (f · χ_{G_{k2}} )(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(G)}^{p} = ∑_{k} S_{k2}^{p}.

Introducing the notation u_{2k} = sup_{x∈E_{k}} u_{2}(x), σ_{2k} = sup_{x∈E_{k}} σ_{2}(x), u_{1k} =

inf_{x∈E_{k}} u_{1}(x), σ_{1k} = inf_{x∈E_{k}} σ_{1}(x), by Lemma A and Theorem C we obtain

S_{k2} ≤ u_{2k}σ2k^{\frac{1}{p}}∥T (f · χ_{G_{k2}} )(·)∥_{L_{ρ}^{pq}(G)} ≤ c_{7}u_{2k}σ2k^{\frac{1}{p}}∥f(·)χ_{G_{k2}} (·)∥_{L_{ρ}^{pq}(G)} ≤

≤ c_{7}u_{2k}σ2k^{\frac{1}{p}}∥f(·)χ_{G_{k2}} (·)∥_{L_{ρ}^{ps}(G)} ≤ c_{8}u_{1k}σ1k^{\frac{1}{p}}∥f(·)χ_{G_{k2}} (·)∥_{L_{ρ}^{ps}(G)} ≤

≤ c_{9}[ ∑_{j} 2^{js}( ∫_{G_{k2}∩\{|f|>2^{j}\}} u_{1k}^{p}σ_{1k}ρ(x) dx)^{\frac{s}{p}} ]^{\frac{1}{s}} ≤

≤c_{9}[ ∑_{j}(u_{1k}2^{j})^{s}( ∫_{G_{k2}∩\{u_{1}f>u_{1k}2^{j}\}} w(x) dx)^{\frac{s}{p}} ]^{\frac{1}{s}} ≤c_{10}∥u_{1}(·)f(·)χ_{G_{k2}} (·)∥_{L_{w}^{ps}(G)}.

By Lemma B (first part) we have

S_{2}^{p} ≤ c_{11}∥u_{1}(·)f(·)∥_{L_{w}^{ps}(G)}^{p}.

In [8] (see also [1], [4]) it is shown that for the Hilbert transform conditions (2) and (3) of Theorem 2.4 are necessary for inequality (2.2) to be fulfilled.

Remark 1. One can easily verify that Theorem 2.4 remains in force when condition (1) is replaced by the condition

(1′) sup_{\frac{r(x)}{4c_{0}} <r(y)≤4c_{0}r(x)} (σ_{2}(y)u_{2}^{p}(y)) ≤ b_{1}σ_{1}(x)u_{1}^{p}(x) for a.a. x ∈ G.

Now let us consider the case with radial weight functions.

Corollary 2.3. Let 1 < s ≤ p ≤ q < ∞; u_{1}, u_{2} and v be positive increasing functions on (0, ∞) which satisfy the condition

sup_{t>0} ∥u_{2}(·)r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L_{v(r(·))}^{pq}(G)}∥\frac{1}{u_{1}(r(·))} χ_{\{r(y)<t\}}(·)∥_{L^{p′s′} (G)} =

= sup_{t>0} B(t) < ∞.

Then the inequality

∥T f(·)u_{2}(r(·))∥_{L_{v(r(·))}^{pq}(G)} ≤ c∥f(·)u_{1}(r(·))∥_{L^{ps}(G)}  (2.3)

holds, where the positive constant c does not depend on f.

Proof. First we shall show that the inequality

u_{2}(8c_{0}t)v^{\frac{1}{p}} (8c_{0}t) ≤ bu_{1}(\frac{t}{c_{0}} )

holds, where the positive constant b does not depend on t. Indeed, using Lemma A and Theorem A, we obtain

B(t) ≥ ∥u_{2}(r(·))r(·)^{−Q}χ_{{t<r(y)<2t}}(·)∥_{L_{v(r(·))}^{pq}(G)} ×

×∥\frac{1}{u_{1}(r(·))} χ_{\{r(y)< \frac{t}{8c^{2}_{0}} \}} ∥_{L^{p′s′} (G)} ≥ c_{1}t^{−Q}u_{2}(t)( ∫_{t}^{2t}v(τ)τ^{Q−1}dτ)^{\frac{1}{p}} ×

×\frac{1}{u_{1}(\frac{t}{8c_{0}^{2}} )} |\{y : r(y) < \frac{t}{8c_{0}^{2}} \}|^{\frac{1}{p′}} ≥ c_{2}u_{2}(t)v^{\frac{1}{p}} (t) \frac{1}{u_{1}(\frac{t}{8c_{0}^{2}} )}.

Moreover,

B_{1}(t) = ∥u_{2}(r(·))χ_{{r(y)<t}}(·)∥_{L_{v(r(·))}^{pq}(G)} ×

×∥\frac{1}{u_{1}(r(·))} r(·)^{−Q}χ_{\{r(y)>t\}} ∥_{L^{p′s′} (G)} ≤

≤ c_{3}u_{2}(t)v^{\frac{1}{p}} (t)t^{\frac{Q}{p}} \frac{1}{u_{1}(\frac{t}{8c_{0}^{2}} )} ∥r(·)^{−Q}χ_{\{r(y)> \frac{t}{8c^{2}_{0}} \}}(·)∥_{L^{p′s′} (G)}.

It is easy to verify that

∥r(·)^{−Q}χ_{\{r(y)> \frac{t}{8c^{2}_{0}} \}}(·)∥_{L^{p′s′} (G)} ≤ c_{4}t^{− \frac{Q}{p}} ,

where c_{4} does not depend on t > 0. We have sup_{t>0} B_{1}(t) ≤ c_{5}. Using

Theorem 2.4, we obtain inequality (2.3).

An analogous reasoning is used to prove

Corollary 2.4. Let 1 < s ≤ p ≤ q < ∞; u_{1}, u_{2} and v be positive decreasing functions on (0, ∞) and

sup_{t>0} ∥u_{2}(r(·))χ_{\{r(y)<t\}}(·)∥_{L_{v(r(·))}^{pq}(G)}∥\frac{1}{u_{1}(r(·))} r(·)^{−Q}χ_{\{r(y)>t\}}(·)∥_{L^{p′s′} (G)} < ∞.

Then inequality (2.3) holds.

Now we shall give examples illustrating pairs of weights.

Example 1. Let 1 < p < q < ∞, v(t) = t^{p−1}, w(t) = t^{p−1} ln^{γ} \frac{2π}{t}, where p − 1 < γ < p and γ = \frac{p}{q} + p − 1. Then the pair (v, w) satisfies the condition

sup_{0<t<π} ∥| · |^{−1}χ_{\{t<|y|<π\}}(·)∥_{L_{v(|·|)}^{pq}} ∥\frac{1}{w^{\frac{1}{p}} (| · |)} χ_{\{|y|<t\}}(·)∥_{L^{p′}} < ∞

and therefore the inequality

∥f̃(·)∥_{L_{v(|·|)}^{pq}} ≤ c∥f(·)∥_{L_{w(|·|)}^{p}}

holds, where f̃is the conjugate function and the constant c does not depend on f.

Example 2. Let 1 < s ≤ p < ∞, \frac{p}{s′} ≤ γ < p, s = \frac{p}{p+1−γ} (γ = \frac{p}{s′} + 1). Then the inequality

∥f̃(·)| · |^{\frac{p−1}{p}} ∥_{L^{p}} ≤ c∥f(·)| · |^{\frac{p−1}{p}} ln^{\frac{γ}{p}} \frac{2π}{| · |}∥_{L^{ps}}

holds, where the positive constant c does not depend on f.

Example 3. Let 1 < s ≤ p ≤ q < ∞, γ = p(\frac{1}{q} + 1 − \frac{1}{s}). Then the inequality

∥f̃(·)| · |^{\frac{p−1}{p}} ∥_{L^{pq}} ≤ c∥f(·)| · |^{\frac{p−1}{p}} ln^{\frac{γ}{p}} \frac{2π}{| · |}∥_{L^{ps}}  (2.4)

holds, where the constant c > 0 does not depend on f. Inequality (2.4) remains in force for γ > p(\frac{1}{q} + 1 − \frac{1}{s}) too.

Acnowledgement

The first author was partially supported by the Georgian Academy of Sciences under Grant No. 1.7 and the second by the grant of the Soros International Educational Program.

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(Received 21.11.1996)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia