Georgian Mathematical Journal 1(94), No. 4, 367-376

TWO-WEIGHTED L_{p}-INEQUALITIES FOR SINGULAR INTEGRAL OPERATORS ON HEISENBERG GROUPS

V. S. GULIEV

Abstract. Some sufficient conditions are found for a pair of weight functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.

Estimates for singular integrals of the Calderon–Zygmund type in various spaces (including weighted spaces and the anisotropic case) have attracted a great deal of attention on the part of researchers. In this paper we will deal with singular integral operators T on the Heisenberg group H^{n} which have an essentially different character as compared with operators of the Calderon–Zygmund type. We have obtained the two-weighted L_{p}- inequality with monotone weights for singular integral operators T on H^{n}. Applications are given.

Let H^{n} be the Heisenberg group (see [1], [2]) realized as a set of points x = (x_{0}, x_{1}, …, x_{2n}) = (x_{0}, x′) ∈ ℝ^{2n+1} with the multiplication

xy = (x_{0} + y_{0} + \frac{1}{2} ∑_{i=1}^{n}(x_{i}y_{n+i} − x_{n+i}y_{i}), x′ + y′).

The corresponding Lie algebra is generated by the left-invariant vector fields

X_{0} = \frac{∂}{∂x_{0}} , X_{i} = \frac{∂}{∂x_{i}} + \frac{1}{2}x_{n+i} \frac{∂}{∂x_{0}} ,

X_{n+i} = \frac{∂}{∂x_{n+i}} − \frac{1}{2}x_{i} \frac{∂}{∂x_{0}} , i = 1, …, n,

which satisfy the commutation relation

[X_{i}, X_{n+i}] = \frac{1}{4}X_{0},

[X_{0}, X_{i}] = [X_{0}, X_{n+i}] = [X_{i}, X_{j}] = [X_{n+i}, X_{n+j}] = [X_{i}, X_{n+j}] = 0, 367

1991 Mathematics Subject Classification. 42B20, 42B25, 42B30.

1072-947X/94/0700-0367$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

i, j = 1, …, n i ≠ j.

The dilation δ_{t} : δ_{t}x = (t^{2}x_{0}, tx′), t > 0, is defined on H^{n}. The Haar measure on this group coincides with the Lebesgue measure dx = dx_{0}dx_{1} ⋯ dx_{2n}. The identity element in H^{n} is e = 0 ∈ ℝ^{2n+1}, while the element x^{−1} inverse to x is (−x).

The function f defined in H^{n} is said to be H-homogeneous of degree m, on H^{n}, if f(δ_{t}x) = t^{m}f(x), t > 0. We also define the norm on H^{n}

|x|_{H} = [x_{0}^{2} + ( ∑_{i=1}^{2n} x_{i}^{2})^{2}]^{1/4}

which is H-homogeneous of degree one. This also yields the distance function, namely, the distance

d(x, y) = d(y^{−1}x, e) = |y^{−1}x|_{H},

|y^{−1}x|_{H} = [(x_{0} − y_{0} − \frac{1}{2} ∑_{i=1}^{n}(x_{i}y_{n+i} − x_{n+i}y_{i}))^{2} +

+( ∑_{i=1}^{2n}(x_{i} − y_{i})^{2})^{2}]^{1/4}.

d is left-invariant in the sense that d(x, y) remains unchanged when x and y are both left-translated by some fixed vector in H^{n}. Furthermore, d satisfies the triangle inequality d(x, z) ≤ d(x, y) + d(y, z), x, y, z ∈ H^{n}. For r > 0 and x ∈ H^{n} let

B(x, r)={y ∈H^{n}; |y^{−1}x|_{H} <r} (S(x, r)={y ∈H^{n}; |y^{−1}x|_{H} =r})

be the H-ball (H-sphere) with center x and radius r.

The number Q = 2n + 2 is called the homogeneous dimension of H^{n}. Clearly, d(δ_{t}x) = t^{Q}dx.

Given functions f(x) and g(x) defined in H^{n}, the Heisenberg convolution (H-convolution) is obtained by

(f ∗ g)(x) = ∫_{H^{n}} f(y)g(y^{−1}x)dy = ∫_{H^{n}} f(xy^{−1})g(y)dy,

where dy is the Haar measure on H^{n}.

The kernel K(x) admitting the estimate |K(x)| ≤ C|x|_{H}^{α−Q} is summable in the neighborhood of e for α > 0 and in that case K ∗ g is defined for the function g with bounded support. If however the kernel K(x) has a singularity of order Q at zero, i.e., |K(x)| ∼ |x|_{H}^{−Q} near e, then there arises a singular integral on H^{n}.

TWO-WEIGHTED L_{p}-INEQUALITIES 369

Let ω(x) be a positive measurable function on H^{n}. Denote by L_{p}(H^{n}, ω) a set of measurable functions f(x), x ∈ H^{n}, with the finite norm

∥f∥_{L_{P} (H^{n},ω)} = ( ∫_{H^{n}} |f(x)|^{p}ω(x)dx)^{1/p}, 1 ≤ p < ∞.

We say that a locally integrable function ω : H^{n} → (0, ∞) satisfies Muckenhoupt’s condition A_{p} = A_{p}(H^{n}) (briefly, ω ∈ A_{p}), 1 < p < ∞, if there is a constant C = C(ω, p) such that for any H-ball B ⊂ H^{n}

(|B|^{−1} ∫_{B} ω(x)dx)(|B|^{−1} ∫_{B} ω^{1−p′}(x)dx) ≤ C, \frac{1}{p} + \frac{1}{p′} = 1,

where the second factor on the left is replaced by ess sup{ω^{−1}(x) : x ∈ B} if p = 1.

Let K(x) be a singular kernel defined on H^{n}\{e} and satisfying the conditions: K(x) is an H-homogeneous function of degree −Q, i.e., K(δ_{t}x) = t^{−Q}K(x) for any t > 0 and ∫_{S_{H}} K(x)dσ(x) = 0, where dσ(x) is a measure element on S_{H} = S(e, 1).

Denote by ω_{K}(δ) the modulus of continuity of the kernel on S_{H}:

ω_{K}(δ) = sup{|K(x) − K(y)| : x, y ∈ S_{H}, |y^{−1}x|_{H} ≤ δ}.

It is assumed that

∫_{0}^{1} ω_{K}(t)\frac{dt}{t} < ∞.

We consider the singular integral operator T:

T f(x) = ∫_{H^{n}} K(xy^{−1})f(y)dy =: lim_{ε→0+} ∫_{|xy^{−1}|_{H}>ε} K(xy^{−1})f(y)dy.

As is known, T acts boundedly in L_{p}(H^{n}), 1 < p < ∞ (see [3], [4]). For singular integrals with Cauchy–Szegö kernels the weighted estimates were established in the norms of L_{p}(H^{n}, ω) with weights ω satisfying the condition A_{p} [5]. These results extend to the more general kernels considered above [4].

Theorem 1 [4]. Let 1 < p < ∞ and ω ∈ A_{p}; then T is bounded in L_{p}(H^{n}, ω).

In the sequel we will use

Theorem 2. Let 1 ≤ p ≤ q < ∞ and U(t), V (t) be positive functions on (0, ∞).

1) The inequality

( ∫_{0}^{∞} U(t)| ∫_{0}^{t} φ(τ)dτ|^{q}dt)^{1/q} ≤ K_{1}( ∫_{0}^{∞} |φ(t)|^{p}v(t)dt)^{1/p}

with the constant K_{1} not depending on φ holds iff the condition

sup_{t>0} ( ∫_{t}^{∞} U(τ)dτ)^{p/q}( ∫_{0}^{t} V (τ)^{1−p′}dτ)^{p−1} < ∞

is fulfilled; 2) The inequality

( ∫_{0}^{∞} U(t)| ∫_{t}^{∞} φ(τ)dτ|^{q}dt)^{1/q} ≤ K_{2}( ∫_{0}^{∞} |φ(t)|^{p}V (t)dt)^{1/p}

with the constant K_{2} not depending on φ holds iff the condition

sup_{t>0} ( ∫_{0}^{t} U(τ)dτ)^{p/q}( ∫_{t}^{∞} V (τ)^{1−p′}dτ)^{p−1} < ∞

is fulfilled.

Note that Theorem 2 was proved by G.Talenti, G.Tomaselli, B.Muckenhoupt [7] for 1 ≤ p = q < ∞, and by J.S.Bradley [8], V.M.Kokilashvili [9], V.G.Maz’ya [10] for p < q.

We say that the weight pair (ω, ω_{1}) belongs to the class Ã_{pq}(γ), γ > 0, if either of the following conditions is fulfilled: a) ω(t) and ω_{1}(t) are increasing functions on (0, ∞) and

sup_{t>0} ( ∫_{t}^{∞} ω(τ)τ^{−1−γq/p′}dτ)^{p/q}( ∫_{0}^{t/2} ω(τ)^{1−p′}τ^{γ−1}dτ)^{p−1} < ∞;

b) ω(t) and ω_{1}(t) are decreasing functions on (0, ∞) and

sup_{t>0} ( ∫_{0}^{t/2} ω_{1}(τ)τ^{γ−1}dτ)^{p/q}( ∫_{t}^{∞} ω(τ)^{1−p′}τ^{−1−γp′/q}dτ)^{p−1} < ∞.

Theorem 3. Let 1 < p < ∞ and the weight pair (ω, ω_{1}) ∈ Ã_{p}(Q) ≡ Ã_{pp}(Q). Then for f ∈ L_{p}(H^{n}, ω(|x|_{H})) there exists T f(x) for almost all x ∈ H^{n} and

∫_{H^{n}} |T f(x)|^{p}ω_{1}(|x|_{H})dx ≤ C ∫_{H^{n}} |f(x)|^{p}ω(|x|)_{H})dx,  (1)

where the constant C does not depend on f.

Corollary. If ω(t), t > 0 is increasing (decreasing) and the function ω(t)t^{−β} is decreasing (increasing) for some β ∈ (0, Q(p − 1)) (β ∈ (−Q, 0)), then T is bounded on L_{p}(H^{n}, ω(|x|_{H})).

Proof of Theorem 3. Let f ∈ L_{p}(H^{n}, ω(|x|_{H})) and ω, ω_{1} be positive increasing functions on (0, ∞). We will prove that T f(x) exists for almost all x ∈ H^{n}. We take any fixed τ > 0 and represent the function f in the norm of the sum f_{1} + f_{2}, where

f_{1}(x) = \begin{cases}f(x), & if |x|_{H} > τ/2 \\ 0, & if |x|_{H} ≤ τ/2 \end{cases} , f_{2}(x) = f(x) − f_{1}(x).

Let ω(t) be a positive increasing function on (0, ∞) and f ∈ L_{p}(H^{n}, ω(|x|_{H})). Then f_{1} ∈ L_{p}(H^{n}) and therefore T f_{1}(x) exists for almost all x ∈ H^{n}. Now we will show that T f_{2} converges absolutely for all x : |x|_{H} ≥ τ. Note that C(K) = sup_{x∈S_{H}} |K(x)| < ∞. Hence

\begin{matrix}|T f_{2}(x)| ≤ C(K) ∫_{|y|_{H}≤τ/2} \frac{|f(y)|}{|xy_{−1}|_{H}^{Q}} dy ≤ \\ ≤ (\frac{2}{τ} )^{\frac{Q}{p}} ∫_{|y|_{H}≤τ/2} \frac{|f(y)|ω(|y|_{H})^{\frac{1}{p}}}{ω(|y|_{H})^{\frac{1}{p}}} dy, \end{matrix}  (2)

since |xy^{−1}|_{H} ≥ |x|_{H} − |y|_{H} ≥ τ/2. Thus, by the Hölder inequality we can estimate (2) as

|T f_{2}(x)| ≤ Cτ^{−Q/p}∥f∥_{L_{p}(H^{n},ω(|x|_{H}))}( ∫_{0}^{τ/2} ω(t)^{1−p′}t^{Q−1}dt)^{1/p′}.

Therefore T f_{2}(x) converges absolutely for all x : |x|_{H} ≥ τ and thus T f(x) exists for almost all x ∈ H^{n}. Assume ω̄_{1}(t) to be an arbitrary continuous increasing function on (0, ∞) such that ω̄_{1}(t) ≤ ω_{1}(t), ω̄_{1}(0) = ω_{1}(0+) and ω̄_{1}(t) = ∫_{0}^{t} φ(τ)dτ + ω̄_{1}(0), t ∈ (0, ∞) (it is obvious that such ω̄_{1}(t) exists; for example, ω̄_{1}(t) = ∫_{0}^{t} ω′_{1}(τ)dτ + ω_{1}(t)).

We observe that the condition a) implies

∃C_{1} > 0, ∀t > 0, ω_{1}(t) ≤ C_{1}ω(t/2).  (3)

Indeed, from

\begin{matrix}∃C_{2} > 0, ∀t > 0, \\ ( ∫_{t}^{∞}φ(τ)τ^{−Q(p−1)}dτ)( ∫_{0}^{t/2}ω(τ)^{1−p′}τ^{Q−1}dτ)^{p−1}≤C_{2} \end{matrix}  (4)

we obtain (3), since

∫_{t}^{∞} ω_{1}(τ)τ^{−1−Q(p−1)}dτ ≥ Cω_{1}(t)t^{−Q(p−1)},
( ∫_{0}^{t/2} ω(τ)^{1−p′}τ^{Q−1}dτ)^{p−1} ≤ Cω(t/2)^{−1}t^{Q(p−1)}

and, besides,

\frac{1}{Q(p − 1)} ∫_{t}^{∞} φ(τ)τ^{−Q(p−1)}dτ = ∫_{t}^{∞} φ(τ)dτ ∫_{τ}^{∞} λ^{−1−Q(p−1)}dλ =

= ∫_{t}^{∞} λ^{−1−Q(p−1)}dλ ∫_{t}^{λ} φ(τ)dτ ≤ ∫_{t}^{∞} ω_{1}(τ)τ^{−1−Q(p−1)}dτ.

We have

∥T f∥_{L_{p},ω̄_{1}(H^{n})} ≤ ( ∫_{H^{n}} |T f(x)|^{p}dx ∫_{0}^{|x|_{H}} φ(t)dt)^{1/p} +

+(ω̄_{1}(0) ∫_{H^{n}} |T f(x)|^{p}dx)^{1/p} = A_{1} + A_{2}.

If ω(0+) > 0, then L_{p}(H^{n}, ω(|x|_{H})) ⊂ L_{p}(H^{n}), and if ω(0+) = 0, then ω̄(t) ≤ ω_{1}(t) ≤ Cω(t/2) implies ω̄_{1}(0) = 0. Therefore in the case ω(0+) = 0 we have A_{2} = 0.

If ω(0) > 0, then f ∈_{L_{p}(ℝ^{n})} and we have

A_{2} ≤ C(ω̄_{1}(0) ∫_{H^{n}} |f(x)|^{p}dx)^{1/p} ≤ C( ∫_{H^{n}} |f(x)|^{p}ω_{1}(|x|_{H})dx)^{1/p} ≤

≤ C∥f∥_{L_{p}(H^{n},ω(|x|_{H}))}.

Now we can write

A_{1} ≤ ( ∫_{0}^{∞} φ(t)dt ∫_{|x|_{H}>t} |T f(x)^{p}dx)^{1/p} ≤ A_{11} + A_{12}.

where

A_{11}^{p} = ∫_{0}^{∞} φ(t)dt ∫_{|x|_{H}>t} | ∫_{|y|_{H}>t/2} K(x, y^{−1})f(y)dy|^{p}dx,

A_{12}^{p} = ∫_{0}^{∞} φ(t)dt ∫_{|x|_{H}>t} | ∫_{|y|_{H}<t/2} K(x, y^{−1})f(y)dy|^{p}dx.

TWO-WEIGHTED L_{p}-INEQUALITIES 373

The relation

∫_{|y|_{H}>t/2} |f(y)|^{p}dy ≤ \frac{1}{ω(t/2)} ∫_{|y|_{H}>t/2} |f(y)|^{p}ω(|y|_{H})dy

implies f ∈ L_{p}({y ∈ H^{n} : |y|_{H} > t}) for any t > 0.

Hence, on account of (3), we have

A_{11} ≤ C( ∫_{0}^{∞} φ(t)dt ∫_{|x|_{H}>t/2} |f(x)|^{p}dx)^{1/p} =

= C( ∫_{H^{n}} |f(x)|^{p}dx ∫_{0}^{2|x|_{H}} φ(t)dt)^{1/p} ≤

≤ C( ∫_{H^{n}} |f(x)|^{p}ω_{1}(2|x|_{H})dx)^{1/p} ≤ C∥f∥_{L_{p,ω(|x|_{H})}}(H^{n}).

Obviously, if |x|_{H} > t, |y|_{H} < t/2, then \frac{1}{2}|x|_{H} ≤ |y^{−1}x|_{H} ≤ \frac{3}{2}|x|_{H}. Therefore

∫_{|x|_{H}>t} | ∫_{|y|_{H}<t/2} K(xy^{−1})f(y)dy|^{p}dx ≤

≤ C(K) ∫_{|x|_{H}>t} ( ∫_{|y|_{H}<t/2} |xy^{−1}|_{H}^{−Q}|f(y)|dy)^{p}dx ≤

≤ 2^{Qp}C(K) ∫_{|x|_{H}>t} |x|_{H}^{−Qp}dx( ∫_{|y|_{H}<t/2} |f(y)|dy)^{p}.

Taking the H-polar coordinates x = δ_{ϱ}x̄, ϱ = |x|_{H}, x̄ ∈ S_{H} we can write

∫_{|x|_{H}>t} |x|_{H}^{−Qp}dx = ∫_{S_{H}} dσ(x̄) ∫_{0}^{∞} ϱ^{Q−1−Qp}dϱ = Ct^{Q−Qp}.

For α > Q(1 + \frac{1}{p′}), by virtue of the Hölder inequality, we have

∫_{|y|_{H}<t/2} |f(y)|dy = α ∫_{S_{H}} dσ(ȳ) ∫_{0}^{t/2} ϱ^{Q−α−1}|f(δ_{ϱ}ȳ)|dϱ ∫_{0}^{ϱ} s^{α−1}ds =

= α ∫_{0}^{t/2} s^{α−1}ds ∫_{s<|y|_{H}<t/2} |f(y)| |y|_{H}^{−α}dy ≤

≤ ∫_{0}^{t/2} s^{α−1}ds( ∫_{s<|y|_{H}<t/2} |f(y)|^{p}|y|_{H}^{−Qp}dy)^{1/p} ×

×( ∫_{s<|y|_{H}<t/2} |y|_{H}^{(Q−α)p′}dy)^{1/p′} ≤

≤ C ∫_{0}^{t/2} s^{Q+ \frac{Q}{p′}} ( ∫_{s<|y|_{H}<t/2} |f(y)|^{p}|y|_{H}^{−Qp}dy)^{1/p}ds.

Consequently

A_{12} ≤ C{ ∫_{0}^{∞} φ(2t)t^{−Q(p−1)} ×

×[ ∫_{0}^{t} s^{Q(1+ \frac{1}{p′})}( ∫_{|y|_{H}≥s} |f(y)|^{p}|y|_{H}^{−Qp}dy)^{1/p}ds]^{p}dt\}^{1/p}.

By (4) and Theorem 2

A_{12} ≤ C[ ∫_{0}^{∞} s^{Qp(1+ \frac{1}{p′})}( ∫_{|y|_{H}>s} |f(y)|^{p} ×

×|y|^{−Qp}dy)ω(s)s^{−(Q−1)(p−1)}ds]^{1/p} =

= C( ∫_{0}^{∞} s^{−1+Qp}ω(s)ds ∫_{|y|_{H}>s} |f(y)|^{p}|y|_{H}^{−Qp}dy)^{1/p} =

= C( ∫_{H^{n}} |f(y)|^{p}|y|_{H}^{−Qp} ∫_{0}^{|y|_{H}} ω(s)s^{−1+Qp}ds)^{1/p} ≤

≤ C( ∫_{H^{n}} |f(y)|^{p}ω(|y|_{H})dy)^{1/p}.

Hence we obtain (1) for ω_{1}(t) = ω̄_{1}(t). Now, by the Fatou theorem, the inequality (1) is fulfilled.

Theorem 3 was earlier announced in [11]. A similar reasoning can be used to prove the analogue of Theorem 3 for

the operator T_{α} : f → T_{α}f where

T_{α}f(x) = ∫_{H^{n}} |xy^{−1}|_{H}^{α−Q}f(y)dy, 0 < α < Q.

Namely, we have

Theorem 4. Let 0 < α < Q, 1 < p < \frac{Q}{α}, \frac{1}{p} − \frac{1}{q} = \frac{α}{Q} and the weights (ω, ω_{1}) be monotone positive functions on (0, ∞). Then the inequality

( ∫_{H^{n}} |T_{α}f(x)|^{q}ω_{1}(|x|_{H})dx)^{1/q} ≤ C( ∫_{H^{n}} |f(x)|^{p}ω(|x|_{H})dx)^{1/p}

holds if and only if (ω, ω_{1}) ∈ Ã_{p,q}(Q).

Remark. In the case of a homogeneous group the analogue of Theorem 4 is also valid (see [12]).

For monotone weights one can find the weighted L_{p}-estimates for a Calderon–Zygmund operator in [13] and [14], and for the anisotropic case in [15].

As known [16], if f ∈ C_{0}^{∞}(H^{n}), then the function

g(x) = C_{n} ∫_{H^{n}} |xy^{1}|_{H}^{−2n}f(y)dy

is a solution of the equation L_{0}g = f, where L_{0} = − ∑_{i=1}^{2n} X_{j}^{2}. In particular, our results lead to

Theorem 5. Let 1 < p < ∞, (ω, ω_{1}) ∈ Ã(Q), f ∈ L_{p}(H^{n}, ω(|x|_{H})), and L_{0}(g) = f. Then

∥X_{0}g∥_{L_{p}(H^{n},ω_{1}(|x|_{H}))} ≤ c∥f∥_{L_{p}(H^{n},ω(|x|_{H}))},
∥X_{i}X_{j}g∥_{L_{p}(H^{n},ω_{1}(|x|)_{H}))} ≤ C∥f∥_{L_{p}(H^{n},ω(|x|_{H}))},
i, j = 1, 2, …, 2n.

Theorem 6. Let 1 < p < q < ∞, \frac{1}{p} − \frac{1}{q} = \frac{1}{Q}, (ω, ω_{1}) ∈ Ã_{pq}(Q), f ∈ L_{p}(H^{n}, ω(|x|_{H})), and L_{0}g = f. Then

∥X_{i}g∥_{L_{q}(H^{n},ω_{1}(|x|_{H}))} ≤ C∥f∥_{L_{p}(H^{n},ω(|x|_{H}))}, i = 1, 2, …, 2n.

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(Received 15.02.1993)

Author’s address:

Faculty of Mechanics and Mathematics

Baku State University

23 Lumumba St., Baku 370073

Azerbaijan