A note on two-weight inequalities for multiple Hardy-type operators

Necessary and sufficient conditions on a pair of weights guaranteeing two-weight estimates for the multiple Riemann-Liouville transforms are established provided that the weight on the right–hand side satisfies some additional conditions.


Introduction
In 1985 E. Sawyer [15] solved the two-weight problem for the twodimensional Hardy transform Namely he proved the following statement: Theorem A. Let 1 < p ≤ q < ∞.Then for the boundedness of the operator H 2 from L p w (R 2 + ) to L q v (R 2 + ) it is necessary and sufficient that the following three independent conditions are satisfied: where σ =: for all a, b > 0 , where In her doctoral thesis A. Wedestig [20] derived a two-weight criterion for the operator H 2 when the weight on the right-hand side is a product of two functions of separate variables.In particular, she proved Theorem B. Let 1 < p ≤ q < ∞ and let s 1 , s 2 ∈ (1, p).Suppose that the weight function w on R 2  + has the form w(x, y) = w 1 (x)w 2 (y).Then for the boundedness of the operator H 2 from L p w (R 2 + ) to L q v (R 2 + ) it is necessary and sufficient that A(s 1 , s 2 ) =: sup t1,t1>0 where Earlier some sufficient conditions for the validity of the two-weight inequality for H 2 were established in [16] and [19].
Necessary and sufficient conditions on the weight function v on R 2 + governing the trace inequality for the Riemann-Liouville operator with multiple kernels where α, β > 1/p, have been obtained in [8].Analogous problem has been solved in [9] for 0 < α < 1/p and β > 1/p.
In this paper we establish boundedness criteria for the operator ) when the weight w satisfies the onedimensional doubling condition uniformly with respect to another variable.As a corollary we conclude that under this restriction the two-weight inequality for the operator H 2 holds if and only if the condition (1.1) is satisfied.When the weight function w has the form w(x, y) = w 1 (x)w 2 (y) we show that also in this case a two-weight criterion for H 2 is (1.1) .

Preliminaries
Let ρ be an almost everywhere positive function on a subset E of R n .We denote by L p ρ (E), 1 < p < ∞, the set of all measurable functions f : E → R 1 for which the norm Let us recall some well-known results for one-dimensional Hardy-type transforms.
A solution of the two-weight problem for the one-dimensional Hardy transform has been given by B. Muchenhoupt [13] for p = q ; by V. Kokilashvili [6], J. Bradley [2] and V. Maz'ya [11,Chapter 1] for p ≤ q .Namely the following statement holds.
with the positive constant c independent of f holds if and only if Moreover, if c is the best constant in (2.1), then there exists a positive constant b depending only on p and q such that the inequality A ≤ c ≤ bA holds.
Later on F. J. Martin-Reyes and E. Sawyer [10] and V. Stepanov [17] proved the next statement, which gives two-weight criteria for the Riemann-Liouville transform where α > 1.

and only if the following two conditions
hold.Moreover, there exist positive constants c 1 and c 2 depending only on α , p and q such that c Criteria for the boundedness of R α from L p (R + ) to L q v (R + ) when 1 < p ≤ q < ∞ and α > 1/p have been obtained in [12] (see also [14]), while the similar result has been derived in [7], [3, Chapter 2], for 1 < p ≤ q < ∞ and 0 < α < 1/p.When 1 < p < q < ∞ a solution of the two-weight problem for potential operators has been given in [5].
The next statement concerning the discrete Hardy operator defined on Z perhaps is already known, but we give the proof of the theorem for the completeness (see also [1], [4] for two-weight criteria for the Hardy transform on Z + ): Moreover, if c is the best constant in (2.2), then . By Minkowsky's inequality q p ≥ 1 we obtain For the intrinsic sum we have Moreover, the next easily verifiable inequality Further, the latter sum does not exceed , can also be easily verified.Therefore we obtain Finally we have In order to prove necessity we take the sequence k>n.

Then we have
On the other hand, and finally B < ∞.

Analogously it follows
Theorem F. Let 1 < p ≤ q < ∞ and let m be an integer.Suppose that {a n } m n=−∞ , {b n } m n=−∞ are positive sequences.Then the two-weight inequality Moreover, if c is the best constant in (2.3), then

The Main Results
In order to formulate the main results of this paper we need the following definition: Definition.A nonnegative function ρ : R 2 + → R 1 is said to be a weight function with doubling condition uniformly with respect to x ∈ R + if there exists a positive constant c such that for arbitrary t > 0 and almost all x > 0 the inequality Note that if the weight ρ is integrable on [0, a] 2 , a > 0, then ρ ∈ DC(y) is equivalent to the condition: there exists a constant c > 0 such that for all intervals of finite length I ⊂ R + and all t > 0 the inequality (ii) (ii) ) if and only if (1.1) holds.More general form of this corollary is the next statement: Then the boundedness of The following theorem states that if the weight function w has the form w(x, y) = w 1 (x)w 2 (y), then the boundedness of the operator ) is equivalent to the first condition in the E. Sawyer's theorem.

Proof of the Main Results
In this section we present the proofs of the results formulated in the previous section.
Proof of Theorem 3.1 .Sufficiency.First of all note that (see e.g., [18]) the condition w 1−p ∈ DC(y) implies the reverse doubling condition for w 1−p uniformly with respect to x, i.e., there exists the constants η 1 , η 2 > 1 such that for all t > 0 and a.e.x ∈ R + the inequality (4.1) In the sequel we shall use the notation: Let f ≥ 0 .Then taking into account the fact α ≥ 1 and using Theorem D we find that On the other hand, we have Indeed, (4.1) and the condition w 1−p ∈ DC(y) lead to the inequality: Consequently, by virtue of Theorem E and Hölder's inequality we find that Necessity.Let f ≥ 0 and let a, b > 0. It is easy to see that Using the latter inequality and the boundedness of R α,β on the class of functions f Hence this inequality and the condition w ∈ CD(y) give us the condition A 1 < ∞.
Taking into account the arguments used above and the fact that the operator R α,β is bounded from L p w (R 2 + ) to L q v (R 2 + ) if and only if its dual operator Proof of Theorem 3.2 .The proof is similar to that of Theorem 3.1.
Proof of Theorem 3.3.Necessity is obvious.In order to prove sufficiency, it is enough to take the sequence 2 k instead of η k 1 in the proof of Theorem 3. x k+1 x k v(x, y)dy; F k (t) =: Further, it is obvious that x k w 2 (τ )f p (x, τ )dτ dx q/p = cD q f q L p (R 2 + ) .
, y)dy holds.In this case we write ρ ∈ DC(y).Analogously we define the class DC(x).