Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 557–568

A Characterization of a Two-Weight Inequality for Discrete Two-Dimensional Hardy Operators

Y. Rakotondratsimba

Abstract. We obtain a characterization of non-negative double sequences V = (V(n_{1}, n_{2}))_{n_{1},n_{2}} and U = (U(n_{1}, n_{2}))_{n_{1},n_{2}} for which the two-dimensional discrete Hardy operator H is bounded from ℓ^{p}(V) into ℓ^{q}(U) whenever 1 < p ≤ q < ∞.

Keywords: Inequalities, weights, discrete Hardy operators AMS subject classification: 26 D 15

1. Introduction

The discrete two-dimensional Hardy operator is defined by

(HF)(n_{1}, n_{2}) = ∑_{i_{1}=0}^{n_{1}} ∑_{i_{2}=0}^{n_{2}} F_{i_{1}i_{2}} (n_{1}, n_{2} ∈ ℕ_{0}).

For the sake of simplicity and as in the continuous case, the non-negative double sequence F_{i_{1}i_{2}} is merely denoted by F(i_{1}, i_{2}).

Our purpose in this work is to derive necessary and sufficient condition on nonnegative double sequences U(n_{1}, n_{2}) and V(n_{1}, n_{2}) for which there is a constant C > 0 such that

( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}(HF)^{q}(n_{1}, n_{2})U(n_{1}, n_{2}))^{\frac{1}{q}} ≤ C( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{1}{p}} (1.1)

for all double sequences F(·, ·) ≥ 0 and with 1 < p ≤ q < ∞. As in the continuous setting, U(·, ·) and V(·, ·) will be called weights. As a rule, the boundedness (1.1) will be also denoted by H : ℓ_{V}^{p} → ℓ_{U}^{q}.

Inequality (1.1) is involved in many parts of Analysis as in questions of convergence and summability of double series, and in analysis of random walks on infinite graphs [2, 5, 7].

Y. Rakotondratsimba: Institut Polytechnique St.-Louis, E.P.M.I., 13 bd de l’Hautil, 95 092 Cergy-Pontoise cedex, France ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

According to B. Pachpatte [7], the boundedness H : ℓ_{V}^{p} → ℓ_{U}^{p} is true for V(n_{1}, n_{2}) = 1 and U(n_{1}, n_{2}) = n_{1}^{−p}n_{2}^{−p} with 1 < p < ∞. Later D. Y. Hwang [2] proved that H : ℓ_{V}^{p} → ℓ_{U}^{p} whenever U(n_{1}, n_{2}) = λ_{1}(n_{1})λ_{2}(n_{2})[λ_{1}(1) + … + λ_{1}(n_{1})]^{−p}[λ_{2}(1) + … + λ_{2}(n_{2})]^{−p} and V(n_{1}, n_{2}) = [λ_{1}(n_{1})λ_{2}(n_{2})]^{1−p} with λ_{1}(n_{1}) and λ_{2}(n_{2}) > 0. So for this case the inequality in question is a two-dimensional version of the well-known Copson inequality

∑_{k=1}^{∞} [ ∑_{l=1}^{k} λ_{l}φ(l)]^{p} \frac{λ_{k}}{[λ_{1} + … + λ_{k}]^{p}} ≤ (\frac{p}{p − 1})^{p} ∑_{k=1}^{∞} φ^{p}(k)λ_{k} (φ(k) ≥ 0).

Recently Z. Németh [6] proved that H : ℓ_{V}^{p} → ℓ_{V}^{p} whenever V(n_{1}, n_{2}) = v_{1}(n_{1})v_{2}(n_{2}) and v_{1}(n_{1}), v_{2}(n_{2}) are quasi-geometrically decreasing.

For all of these quoted results in [2, 6, 7] the weights are of product types in the sense that V(n_{1}, n_{2}) = v_{1}(n_{1})v_{2}(n_{2}) and U(n_{1}, n_{2}) = u_{1}(n_{1})u_{2}(n_{2}). These twodimensional results with weights of product types are actually consequences of the one-dimensional ones [4]. And the crux of the matter is about weights not of product types like W(n_{1}, n_{2}) = (n_{1} + n_{2})3^{−(n_{1}+n_{2})} and W(n_{1}, n_{2}) = (n_{1} + n_{2})^{−(n_{1}+n_{2})}. A first investigation in this direction was done by A. Kamaly and the author in [4].

In this work we find a necessary and sufficient condition on weights U(·, ·) and V(·, ·) for which H : ℓ_{V}^{p} → ℓ_{U}^{q}. Indeed, until this paper, a characterization result seems not available in the literature. However, the analogue integral inequality

\begin{matrix}(∫_{0}^{∞} ∫_{0}^{∞} [ ∫_{0}^{x_{1}} ∫_{0}^{x_{2}} f(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ C(∫_{0}^{∞} ∫_{0}^{∞} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p}} \end{matrix} (∀ f(·, ·) ≥ 0) (1.2)

was already solved fifteen years ago by E. Sawyer [8]. Consequently, the present work aims to fill this gap between the discrete inequality (1.1) and its continuous version (1.2). To derive the characterization for H : ℓ_{V}^{p} → ℓ_{U}^{q}, as in [3] (see Section 3), our idea will make use of the fact that there is some equivalence between (1.1) and (1.2) for some judicious choice of the weights u(·, ·) and v(·, ·).

Our results are stated in the next Section 2, and their proofs will be performed in Section 3.

2. The results

Throughout this paper it is always assumed that

1 < p ≤ q < ∞,  p′ = \frac{p}{p − 1},  q′ = \frac{q}{q − 1}

and

U(n_{1}, n_{2}), V(n_{1}, n_{2}) are weight functions defined on ℕ_{0}^{2}.

To simplify and avoid some inconsistencies, it will be supposed that V(n_{1}, n_{2}) ≠ 0.

Recall that our purpose in this work is to derive a characterization of weights U(n_{1}, n_{2}) and V(n_{1}, n_{2}) for which H : ℓ_{V}^{p} → ℓ_{U}^{q}. This last means that for some constant C > 0

( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}(HF)^{q}(n_{1}, n_{2})U(n_{1}, n_{2}))^{\frac{1}{q}} ≤ C( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{1}{p}} (2.1)

for all double sequences F(·, ·) ≥ 0.

Our main result reads as follows.

Theorem 2.1. Suppose that H : ℓ_{V}^{p} → ℓ_{U}^{q}. Then for some constant A > 0 the three conditions

( ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=N_{2}}^{∞} U(n_{1}, n_{2}))^{\frac{1}{q}} ( ∑_{m_{1}=0}^{N_{1}} ∑_{m_{2}=0}^{N_{2}} V^{1−p′}(m_{1}, m_{2}))^{\frac{1}{p′}}

≤ A for all integers N_{1}, N_{2} ≥ 0

(2.2)

\begin{matrix}(∫_{0}^{R_{1}} ∫_{0}^{R_{2}} [ ∫_{0}^{x_{1}} ∫_{0}^{x_{2}} v^{1−p′}(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ A(∫_{0}^{R_{1}} ∫_{0}^{R_{2}} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p}} for all R_{1}, R_{2} > 0 \end{matrix} (2.3)

\begin{matrix}(∫_{R_{1}}^{∞} ∫_{R_{2}}^{∞} [ ∫_{x_{1}}^{∞} ∫_{x_{2}}^{∞} u(y_{1}, y_{2}) dy_{1}dy_{2}]^{p′}v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p′}} \\ ≤ A(∫_{R_{1}}^{∞} ∫_{R_{2}}^{∞} u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q′}} for all R_{1}, R_{2} > 0. \end{matrix}  (2.4)

are satisfied with weights u(x_{1}, x_{2}) and v(x_{1}, x_{2}) defined by

u(x_{1}, x_{2}) = ∑_{m_{1}=0}^{∞} ∑_{m_{2}=0}^{∞} U(m_{1}, m_{2})1I_{[m_{1},m_{1}+1)×[m_{2},m_{2}+1)}(x_{1}, x_{2})  (2.5)

v(x_{1}, x_{2}) = ∑_{m_{1}=0}^{∞} ∑_{m_{2}=0}^{∞} V(m_{1}, m_{2})1I_{[m_{1},m_{1}+1)×[m_{2},m_{2}+1)}(x_{1}, x_{2}).  (2.6)

Here 1I_{[m_{1},m_{1}+1)×[m_{2},m_{2}+1)} denotes the characteristic function of the rectangle [m_{1}, m_{1}+ 1)×[m_{2}, m_{2}+1). And actually in these conditions, A = C provided that C is the constant involved in (2.1).

Conversely, the boundedness H : ℓ_{V}^{p} → ℓ_{U}^{q} does hold whenever all conditions (2.2) − (2.4) are satisfied. Precisely, (2.1) remains true with C = c(p, q)A where c(p, q) > 0 depends only on the indexes p and q.

However, it is still an open problem whether conditions (2.3) and (2.4) respectively can be replaced by

( ∑_{n_{1}=0}^{N_{1}} ∑_{n_{2}=0}^{N_{2}} [ ∑_{m_{1}=0}^{n_{1}} ∑_{m_{2}=0}^{n_{2}} V^{1−p′}(m_{1}, m_{2})]^{q}U(n_{1}, n_{2}))^{\frac{1}{q}}

≤ A( ∑_{m_{1}=0}^{N_{1}} ∑_{m_{2}=0}^{N_{2}} V^{1−p′}(m_{1}, m_{2}))^{\frac{1}{p}} for all integers N_{1}, N_{2} ≥ 0

(2.7)

and

( ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=N_{2}}^{∞} [ ∑_{m_{1}=n_{1}}^{∞} ∑_{m_{2}=n_{2}}^{∞} U(m_{1}, m_{2})]^{p′}V^{1−p′}(n_{1}, n_{2}))^{\frac{1}{p′}}

≤ A( ∑_{m_{1}=N_{1}}^{∞} ∑_{m_{2}=N_{2}}^{∞} U(m_{1}, m_{2}))^{\frac{1}{q′}} for all integers N_{1}, N_{2} ≥ 0.

(2.8)

Although a characterization result is theoretically of great interest, it would be noted that a necessary and sufficient condition as the one obtained in Theorem 2.1 is not in general easy to handle in explicit computations. It means that other investigations on sufficient conditions (easily computable) for H : ℓ_{V}^{p} → ℓ_{U}^{q} deserve to be done. This will be achieved by the author in a forthcoming paper.

For many problems in analysis in product spaces, it is useful to consider variants of the operator H like

\begin{aligned}(H_{∗∗}F)(n_{1}, n_{2}) & = ∑_{i_{1}=n_{1}}^{∞} ∑_{i_{2}=n_{2}}^{∞} F(i_{1}, i_{2}) \\ (H_{1∗}F)(n_{1}, n_{2}) & = ∑_{i_{1}=0}^{n_{1}} ∑_{i_{2}=n_{2}}^{∞} F(i_{1}, i_{2}) \\ (H_{∗1}F)(n_{1}, n_{2}) & = ∑_{i_{1}=n_{1}} ∑_{i_{2}=0}^{n_{2}} F(i_{1}, i_{2}). \end{aligned}

A characterization for H_{∗∗} : ℓ_{V}^{p} → ℓ_{U}^{q} can be immediately obtained by using the above Theorem 2.1. Indeed, by duality arguments, the former boundedness is equivalent to H : ℓ_{V_{1}}^{p_{1}} → ℓ_{U_{1}}^{q_{1}} where p_{1} = q′, q_{1} = p′ (so p_{1} ≤ q_{1}) and V_{1}(n_{1}, n_{2}) = U^{1−q′}(n_{1}, n_{2}), U_{1}(n_{1}, n_{2}) = V^{1−p′}(n_{1}, n_{2}).

Our next result is related to the boundedness H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q}.

Theorem 2.2. Suppose that H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q}. Then for some constant A > 0 the three conditions

( ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=0}^{N_{2}} U(n_{1}, n_{2}))^{\frac{1}{q}} ( ∑_{m_{1}=0}^{N_{1}} ∑_{m_{2}=N_{2}}^{∞} V^{1−p′}(m_{1}, m_{2}))^{\frac{1}{p′}}

≤ A for all integers N_{1}, N_{2} ≥ 0

(2.9)

\begin{matrix}(∫_{0}^{R_{1}} ∫_{R_{2}}^{∞} [ ∫_{0}^{x_{1}} ∫_{x_{2}}^{∞} v^{1−p′}(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ A(∫_{0}^{R_{1}} ∫_{R_{2}}^{∞} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p}} for all R_{1}, R_{2} > 0 \end{matrix} (2.10)

\begin{matrix}(∫_{R_{1}}^{∞} ∫_{0}^{R_{2}} [ ∫_{x_{1}}^{∞} ∫_{0}^{x_{2}} u(y_{1}, y_{2}) dy_{1}dy_{2}]^{p′}v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p′}} \\ ≤ A(∫_{R_{1}}^{∞} ∫_{0}^{R_{2}} u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q′}} for all R_{1}, R_{2} > 0 \end{matrix} (2.11)

are satisfied with weights u(x_{1}, x_{2}) and v(x_{1}, x_{2}) defined as in (2.5) and (2.6). And actually, in these conditions, A = C provided that C is the constant involved in H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q}.

Conversely, the boundedness H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q} does hold whenever all the conditions (2.9) − (2.11) are satisfied. Precisely, the constant involved in the boundedness takes the form C = c(p, q)A where c(p, q) > 0 only depends on the indexes p and q.

A characterization for H_{∗1} : ℓ_{V}^{p} → ℓ_{U}^{q} can be deduced (by duality argument) from this last result, since this boundedness is equivalent to H_{1∗} : ℓ_{V_{1}}^{p_{1}} → ℓ_{U_{1}}^{q_{1}} where p_{1}, q_{1} and U_{1}(n_{1}, n_{2}), V_{1}(n_{1}, n_{2}) are defined as above.

3. Proofs of results

In this section we will give the details of the proof of Theorem 2.1 and sketch the arguments for Theorem 2.2.

Proof of Theorem 2.1. For convenience the inequality

\begin{matrix}(∫_{0}^{∞} ∫_{0}^{∞} [ ∫_{0}^{x_{1}} ∫_{0}^{x_{2}} f(y_{1}, y_{2}) dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ C(∫_{0}^{∞} ∫_{0}^{∞} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p}} for all f(·, ·) ≥ 0 \end{matrix}  (3.1)

will be denoted by H : L_{v}^{p} → L_{u}^{q}. Our plan is to prove Theorem 2.1 as follows.

A) First we show that the boundedness H : ℓ_{V}^{p} → ℓ_{U}^{q} implies condition (2.2) and H : L_{v}^{p} → L_{u}^{q} where the weights u(x_{1}, x_{2}) and v(x_{1}, x_{2}) are defined from U(n_{1}, n_{2}) and V(n_{1}, n_{2}) as in identities (2.5) and (2.6). Thus conditions (2.3) and (2.4) follow readily from the boundedness H : L_{v}^{p} → L_{u}^{q} as it is well-known in [8]. Consequently, the necessary part in the theorem is verified.

B) Next we check that condition (2.2) implies

(∫_{R_{1}}^{∞} ∫_{R_{2}}^{∞} u(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{q}} (∫_{0}^{R_{1}} ∫_{0}^{R_{2}} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{1}{p′}} ≤ A  (3.2)

for all R_{1}, R_{2} > 0. Then, because of [8: Theorem 1], the boundedness H : L_{v}^{p} → L_{u}^{q} does hold under conditions (2.3), (2.4) and (3.2).

C) Finally, we prove that condition (2.2) and H : L_{v}^{p} → L_{u}^{q} lead to the boundedness H : ℓ_{V}^{p} → ℓ_{U}^{q}. Therefore steps B) and C) yield the proof of the sufficient part in the theorem.

Proof of Part A. Suppose that H : ℓ_{V}^{p} → ℓ_{U}^{q}. It means that inequality (2.1) is true for some constant C > 0. Consider integers N_{1}, N_{2} ≥ 0. Applying (2.1) for any non-negative (double) sequence F(n_{1}, n_{2}) whose support is {0, …, N_{1}} × {0, …, N_{2}}, then

\begin{matrix}( ∑_{m_{1}=0}^{N_{1}} ∑_{m_{2}=0}^{N_{2}} F(m_{1}, m_{2}))( ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=N_{2}}^{∞} U(n_{1}, n_{2}))^{\frac{1}{q}} \\ ≤ C( ∑_{m_{1}=0}^{N_{1}} ∑_{m_{2}=0}^{N_{2}} F^{p}(m_{1}, m_{2})V(n_{1}, n_{2}))^{\frac{1}{p}} . \end{matrix}  (3.3)

This last inequality implies immediately condition (2.2) (with A = C) by taking F(n_{1}, n_{2}) = V^{1−p′}(n_{1}, n_{2}) in its support {0, …, N_{1}} × {0, …, N_{2}} and by using the identity (1 − p′)p + 1 = (1 − p′).

Next the boundedness H : L_{v}^{p} → L_{u}^{q} is implied by H : ℓ_{V}^{p} → ℓ_{U}^{q} as the following chain of computation shows:

∫_{0}^{∞} ∫_{0}^{∞} [ ∫_{0}^{x_{1}} ∫_{0}^{x_{2}} f(y_{1}, y_{2}) dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2}

= ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} [ ∫_{0}^{x_{1}} ∫_{0}^{x_{2}} f(y_{1}, y_{2}) dy_{1}dy_{2}]^{q}u(x_{1}, x_{2}) dx_{1}dx_{2}

≤ ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}[∫_{0}^{n_{1}+1} ∫_{0}^{n_{2}+1} f(y_{1}, y_{2}) dy_{1}dy_{2}]^{q}(∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} u(x_{1}, x_{2}) dx_{1}dx_{2})

= ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}[ ∑_{m_{1}=0}^{n_{1}} ∑_{m_{2}=0}^{n_{2}} F(m_{1}, m_{2})]^{q}U(n_{1}, n_{2})

(by using (2.5) and setting F(m_{1},m_{2})=∫_{m1}^{m1+1} ∫_{m2}^{m2+1} f(y_{1},y_{2}) dy_{1}dy_{2})

≤ C^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}}

(due to the boundedness H:ℓ_{V}^{p} →ℓ_{U}^{q} with the constant C>0)

≤ C^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}(∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2}) dx_{1}dx_{2})

× V(n_{1}, n_{2})(∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} v^{1−p′}(y_{1}, y_{2}) dy_{1}dy_{2})^{p−1})^{\frac{q}{p}}

(by the definition of F(m_{1},m_{2}) and using the Hölder inequality)

\begin{matrix}= C^{q}(∫_{0}^{∞} ∫_{0}^{∞} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2}) dx_{1}dx_{2})^{\frac{q}{p}} \\ (since V(n_{1},n_{2})(∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} v^{1−p′} (y_{1},y_{2}) dy_{1}dy_{2})^{p−1}=1). \end{matrix}

Proof of Part B. To check inequality (3.2), let us consider R_{1}, R_{2} > 0 such that N_{1} ≤ R_{1} < N_{1} +1 and N_{2} ≤ R_{2} < N_{2} +1 for some integers N_{1}, N_{2} ≥ 0. Then condition (2.2), with the constant A > 0, leads to the test inequality (3.2) with the same constant since

∫_{R_{1}}^{∞} ∫_{R_{2}}^{∞} u(x_{1}, x_{2}) dx_{1}dx_{2} ≤ ∫_{N_{1}}^{∞} ∫_{N_{2}}^{∞} u(x_{1}, x_{2}) dx_{1}dx_{2}

= ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=N_{2}}^{∞} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} u(x_{1}, x_{2}) dx_{1}dx_{2}

= ∑_{n_{1}=N_{1}}^{∞} ∑_{n_{2}=N_{2}}^{∞} U(n_{1}, n_{2})

and

∫_{0}^{R_{1}} ∫_{0}^{R_{2}} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2} ≤ ∫_{0}^{N_{1}+1} ∫_{0}^{N_{2}+1} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2}

= ∑_{n_{1}=0}^{N_{1}} ∑_{n_{2}=0}^{N_{2}} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} v^{1−p′}(x_{1}, x_{2}) dx_{1}dx_{2}

= ∑_{n_{1}=0}^{N_{1}} ∑_{n_{2}=0}^{N_{2}} V^{1−p′}(n_{1}, n_{2}).

Proof of Part C. First some remarks can be done. For instance, (2.2) implies that for some A > 0

( ∑_{n_{1}=N_{1}}^{∞} U(n_{1}, N_{2}))^{\frac{1}{q}} ( ∑_{m_{1}=0}^{N_{1}} V^{1−p′}(m_{1}, N_{2}))^{\frac{1}{p′}} ≤ A  (3.4)

( ∑_{n_{2}=N_{2}}^{∞} U(N_{1}, n_{2}))^{\frac{1}{q}} ( ∑_{m_{2}=0}^{N_{2}} V^{1−p′}(N_{1}, m_{2}))^{\frac{1}{p′}} ≤ A  (3.5)

for all N_{1}, N_{2} ∈ ℕ_{0}. It is well-known that condition (3.4) implies the one-dimensional discrete weighted Hady inequality

( ∑_{n_{1}=0}^{∞}[ ∑_{m_{1}=0}^{n_{1}} φ_{1}(m_{1})]^{q}U(n_{1}, N_{2}))^{\frac{1}{q}} ≤ c_{0}A( ∑_{n_{1}=0}^{∞} φ_{1}^{p}(n_{1})V(n_{1}, N_{2}))^{\frac{1}{p′}}  (3.6)

for all φ_{1}(·) ≥ 0, where c_{0} = c_{0}(p, q) = (1 + \frac{q}{p′} )^{\frac{1}{q}} (1 + \frac{p′}{q} )^{\frac{1}{p′}}. Similarly, from condition (3.5) then

( ∑_{n_{2}=0}^{∞}[ ∑_{m_{2}=0}^{n_{2}} φ_{2}(m_{2})]^{q}U(N_{1}, n_{2}))^{\frac{1}{q}} ≤ c_{0}A( ∑_{n_{2}=0}^{∞} φ_{2}^{p}(n_{2})V(N_{1}, n_{2}))^{\frac{1}{p′}}  (3.7)

for all φ_{2}(·) ≥ 0.

Next we are now in the position to derive the boundedness H : ℓ_{V}^{p} → ℓ_{U}^{q} by using condition (2.2) and H : L_{v}^{p} → L_{u}^{q}. To do this let us split things as

\begin{matrix}∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}(HF)^{q}(n_{1}, n_{2})U(n_{1}, n_{2}) \\ = ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞}[ ∑_{m_{1}=0}^{n_{1}} ∑_{m_{2}=0}^{n_{2}} F(m_{1}, m_{2})]^{q}U(n_{1}, n_{2}) \\ = S_{1} + S_{2} + S_{3} + S_{4} \end{matrix}  (3.8)

with

\begin{aligned}S_{1} & = F^{q}(0, 0)U(0, 0) \\ S_{2} & = ∑_{n_{2}=0}^{∞} [ ∑_{m_{2}=0}^{n_{2}} F(0, m_{2})]^{q}U(0, n_{2}) \\ S_{3} & = ∑_{n_{1}=0}^{∞} [ ∑_{m_{1}=0}^{n_{1}} F(m_{1}, 0)]^{q}U(n_{1}, 0) \\ S_{4} & = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞} [ ∑_{m_{1}=0}^{n_{1}} ∑_{m_{2}=0}^{n_{2}} F(m_{1}, m_{2})]^{q}U(n_{1}, n_{2}). \end{aligned}

Consequently, our task remains to estimate each of S_{i} (i ∈ {1, 2, 3, 4}) by a term like

(cA)^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}}

where c > 0 is a constant which only depends on the indexes p and q.

Estimate of S_{1}. The conclusion follows from condition (2.2) since

S_{1} = (F^{p}(0, 0)V(0, 0))^{\frac{q}{p}} [U^{\frac{1}{q}} (0, 0)(V^{1−p′}(0, 0))^{\frac{1}{p′}} ]^{q}

≤ A^{q}(F^{p}(0, 0)V(0, 0))^{\frac{q}{p}} (by condition (2.2) with N_{1}=N_{2}=0)

≤ A^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} .

Estimate of S_{2}. The main point is the Hardy inequality (3.7) (with N_{1} = 0). Indeed,

S_{2} = ∑_{n_{2}=0}^{∞}[ ∑_{m_{2}=0}^{n_{2}} F(0, m_{2})]^{q}U(0, n_{2})

≤ (c_{0}A)^{q}( ∑_{n_{2}=0}^{∞} F^{p}(0, n_{2})V(0, n_{2}))^{\frac{q}{p}}

≤ (c_{0}A)^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} .

Estimate of S_{3}. As for S_{2}, the proof is based on the Hardy inequality (3.6) (with N_{2} = 0). The details are left to the readers.

Estimate of S_{4}. It can be noted that for some constant c_{1} > 0, which depends only on q, then

S_{4} = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}[ ∑_{m_{1}=0}^{n_{1}} ∑_{m_{2}=0}^{n_{2}} F(m_{1}, m_{2})]^{q}U(n_{1}, n_{2}) ≤ c_{1}(S_{41} + S_{42} + S_{43} + S_{44})

where

\begin{aligned}S_{41} & = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞} F^{q}(n_{1}, n_{2})U(n_{1}, n_{2}) \\ S_{42} & = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}[ ∑_{m_{2}=0}^{n_{2}−1} F(n_{1}, m_{2})]^{q}U(n_{1}, n_{2}) \\ S_{43} & = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}[ ∑_{m_{1}=0}^{n_{1}−1} F(m_{1}, n_{2})]^{q}U(n_{1}, n_{2}) \\ S_{44} & = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}[ ∑_{m_{1}=0}^{n_{1}−1} ∑_{m_{2}=0}^{n_{2}−1} F(m_{1}, m_{2})]^{q}U(n_{1}, n_{2}) \end{aligned}

The term S_{41} can be estimated just by using condition (2.2) since

S_{41} = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}(F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} [U^{\frac{1}{q}} (n_{1}, n_{2})(V^{1−p′}(n_{1}, n_{2}))^{\frac{1}{p′}} ]^{q}

≤ A^{q} ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}(F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} (by (2.2))

≤ A^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} (since \frac{q}{p} ≥ 1).

The estimate of S_{42} is essentially based on the Hardy inequality (3.7). Indeed,

S_{42} ≤ ∑_{n_{1}=1}^{∞} ∑_{n_{2}=0}^{∞}[ ∑_{m_{2}=0}^{n_{2}} F(n_{1}, m_{2})]^{q}U(n_{1}, n_{2})

≤ (c_{0}A)^{q} ∑_{n_{1}=1}^{∞}( ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} (by (3.7) with N_{1}=n_{1})

≤ (c_{0}A)^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}} (since \frac{q}{p} ≥1).

The term S_{43} can be bounded similarly as S_{42} by using the Hardy inequality (3.6). The estimate of S_{44} makes use of the boundedness H : L_{v}^{p} → L_{u}^{q}, say with the constant c_{2}A where c_{2} > 0 depends only on the indexes p and q. Indeed,

S_{44} = ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞}[ ∑_{m_{1}=0}^{n_{1}−1} ∑_{m_{2}=0}^{n_{2}−1} F(m_{1}, m_{2})]^{q} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} u(x_{1}, x_{2})dx_{1}dx_{2}
(by the definition of u(x_{1},x_{2}) in (2.5))

= ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1}[ ∑_{m_{1}=0}^{n_{1}−1} ∑_{m_{2}=0}^{n_{2}−1}

∫_{m_{1}}^{m_{1}+1} ∫_{m_{2}}^{m_{2}+1} f(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2})dx_{1}dx_{2}

(where f(y_{1},y_{2})=∑_{k1=0}^{∞} ∑_{k2=0}^{∞} F(k_{1},k_{2})1I_{[k1,k1+1)×[k2,k2+1)}(y_{1},y_{2}))

= ∑_{n_{1}=1}^{∞} ∑_{n_{2}=1}^{∞} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1}[∫_{0}^{n_{1}} ∫_{0}^{n_{2}} f(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2})dx_{1}dx_{2}

≤ ∫_{0}^{∞} ∫_{0}^{∞}[∫_{0}^{x_{1}} ∫_{0}^{x_{2}} f(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2})dx_{1}dx_{2}

≤ (c_{2}A)^{q}(∫_{0}^{∞} ∫_{0}^{∞} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{q}{p}} (since H:L_{v}^{p}→L_{u}^{q})

= (c_{2}A)^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} ∫_{n_{1}}^{n_{1}+1} ∫_{n_{2}}^{n_{2}+1} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{q}{p}}

= (c_{2}A)^{q}( ∑_{n_{1}=0}^{∞} ∑_{n_{2}=0}^{∞} F^{p}(n_{1}, n_{2})V(n_{1}, n_{2}))^{\frac{q}{p}}

(by the definitions of f(x_{1},x_{2}) and v(x_{1},x_{2}) in (2.6)).

This way Theorem 2.1 is completely proved

Proof of Theorem 2.2. We will restrict to outline the main lines of the proof, since the arguments are the same as those used for Theorem 2.1. To do this, it can be noted that the boundedness H_{1∗} : L_{v}^{p} → L_{u}^{q}, i.e.

\begin{matrix}(∫_{0}^{∞} ∫_{0}^{∞}[∫_{0}^{x_{1}} ∫_{x_{2}}^{∞} f(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ C(∫_{0}^{∞} ∫_{0}^{∞} f^{p}(x_{1}, x_{2})v(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{p}} for all f(·, ·) ≥ 0 \end{matrix}  (3.9)

is equivalent to the three conditions

\begin{matrix}(∫_{R_{1}}^{∞} ∫_{0}^{R_{2}} u(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{q}} (∫_{0}^{R_{1}} ∫_{R_{2}}^{∞} v^{1−p′}(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{p′}} \\ ≤ A for all R_{1}, R_{2} > 0 \end{matrix}  (3.10)

Inequalities for Two-Dimensional Hardy Operators 567

\begin{matrix}(∫_{0}^{R_{1}} ∫_{R_{2}}^{∞}[∫_{0}^{x_{1}} ∫_{x_{2}}^{∞} v^{1−p′}(y_{1}, y_{2})dy_{1}dy_{2}]^{q}u(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{q}} \\ ≤ A(∫_{0}^{R_{1}} ∫_{R_{2}}^{∞} v^{1−p′}(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{p}} for all R_{1}, R_{2} > 0 \end{matrix}  (3.11)

\begin{matrix}(∫_{R_{1}}^{∞} ∫_{0}^{R_{2}}[∫_{x_{1}}^{∞} ∫_{0}^{x_{2}} u(y_{1}, y_{2})dy_{1}dy_{2}]^{p′}v^{1−p′}(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{p′}} \\ ≤ A(∫_{R_{1}}^{∞} ∫_{0}^{R_{2}} u(x_{1}, x_{2})dx_{1}dx_{2})^{\frac{1}{q′}} for all R_{1}, R_{2} > 0. \end{matrix}  (3.12)

The proof of Theorem 2.2 will be fulfilled after doing the following three steps.

A) First we have to show that the boundedness H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q} implies condition (2.9) and H_{1∗} : L_{v}^{p} → L_{u}^{q} where the weights u(x_{1}, x_{2}) and v(x_{1}, x_{2}) are defined from U(n_{1}, n_{2}) and V(n_{1}, n_{2}) as in (2.5) and (2.6). Thus conditions (2.10) and (2.11) follow readily from (3.11) and (3.12), respectively. These last conditions are implied by H_{1∗} : L_{v}^{p} → L_{u}^{q} as it is noted above. And consequently the necessary part in the Theorem is verified.

B) Next we have to check that condition (2.9) implies (3.10). Consequently, the boundedness H_{1∗} : L_{v}^{p} → L_{u}^{q} does hold because of (3.10), (2.10) and (2.11).

C) Finally, we have to show that condition (2.9) and H_{1∗} : L_{v}^{p} → L_{u}^{q} lead to the discrete boundedness H_{1∗} : ℓ_{V}^{p} → ℓ_{U}^{q}.

Therefore Steps B) and C) yield the proof of the sufficient part in Theorem 2.2.

Acknowledgement. The author would like to thank the referee for having pointed him out that the idea to solve the discrete problem via the corresponding continuous one was previously used by G. Hardy, J. Littlewood and G. Pólya in [1].

References

[1] Hardy J., Littlewood J. E. and Pólya G.: Inequalities (Cambridge Mathematical Library). Cambridge: Univ. Press 1934.

[2] Hwang, D. Y.: A many variable generalization of the discrete Hardy’s inequality. Tamkang J. Math. 27 (1996), 125 – 132.

[3] Kamaly, A. and Y. Rakotondratsimba: Weighted inequalities for discrete and integral Hardy operators. SUT J. Math 33 (1997), 183 – 188.

[4] Kamaly, A. and Y. Rakotondratsimba: Weighted inequalities for two-dimensional discrete Hardy operators. Thesis by A. Kamaly. Stockholm: KTH 1998.

[5] Leindler, L.: On the converses of inequalities of Hardy and Littlewood. Acta Sci. Math (Szeged) 58 (1993), 191 – 196.

[6] Németh, Z.: Inequalities of Hardy-Littlewood type for double series. Acta Sci. Math (Szeged) 62 (1993), 153 – 159.

[7] Pachpatte, B.: A many variable generalization of Hardy’s inequality concerning a serie of terms. Tamkang J. Math. 23 (1992), 349 – 354.

[8] Sawyer, E.: Weighted inequalities for the two-dimensional Hardy operator. Studia Math 82 (1985), 1 – 16.

Received 28.09.1998