Criteria of Two-weighted Inequalities for Multidimensional Hardy Type Operator in Weighted Musielak-Orlicz Spaces and Some Application

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type necessary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.


Criteria of Two-weighted Inequalities for Multidimensional Hardy Type Operator in Weighted Musielak-Orlicz Spaces and Some Application
The spaces L φ is called Musielak-Orlicz space.
Note that the space L φ is a Banach function spaces with respect to the norm (1.1) (more detail see [8]). In particular, the Musielak-Orlicz spaces include the classical Lebesgue spaces for φ(x, t) = t p (1 ≤ p < ∞), the Orlicz spaces for φ(x, t) = φ(t), and the Lebesgue spaces with variable exponent for φ(x, t) = t p(x) , where p : Ω → [1, ∞] is a Lebesgue measurable functions.
Let ω be a weight function on Ω, i.e., ω is a non-negative, almost everywhere positive function on Ω. In this paper we considered the weighted Musielak-Orlicz spaces. We denote It is obvious that the norm in this spaces is given by By L p, ω (R n ) (1 ≤ p < ∞) we denote the spaces of measurable functions f on R n such that . Definition 3. [27,7] Let φ ∈ Φ. Then for any y ∈ Ω we denote by φ * (y, t) the conjugate function of φ(y, t), i.e., φ * (y, u) = sup t≥0 (t u − φ(y, t)) , for all u ≥ 0 and y ∈ Ω.
In [27] was proved the following Lemma 1.
For function f on Ω we define the functional Lemma 2. [27,7] is valid.

Proof.
First we prove that from condition φ ( is a Musielak-Orlicz spaces. By the convexity of φ ( x, t 1/p ) and φ (x, 0) = 0 it follows from the is convex at t ≥ 0. Therefore using the inequality All other properties of φ-function is satisfied automatically. Therefore φ (x, t) ∈ Φ. Further we Then by Lemma 1 and Lemma 2 Hence,  [27]. In the case φ(x, t) = t q(x) and

Main Results
We consider the multidimensional Hardy type operator and its dual operator Now we prove a two-weight criteria for multidimensional Hardy type operator acting from the weighted Musielak-Orlicz spaces to weighted Lebesgue spaces.
holds, for every f ≥ 0 if and only if there exists α ∈ (0, 1) such that Proof. Sufficiency. Passing to the polar coordinates , we have where dξ the surface element on the unit sphere. Obviously, h(y) = h(|y|), i.e., h(y) is a radial function.
Applying Hölder's inequality for L p (R n ) spaces and after some standard transformations, we Applying Lemma 3, we have By switching to polar coordinates and after some calculations, we get

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Criteria of Two-weighted Inequalities for Multidimensional Hardy Type Operator in Weighted Musielak-Orlicz Spaces and Some Application Therefore by the condition (2.2), we obtain Necessity. Let f ∈ L p,v (R n ) , f ≥ 0 and the inequality (2.1) is valid. We choose the test function as where t > 0 is a fixed number and It is obvious that Again by switching to polar coordinates from the right hand side of inequality (2.1) we get that After some calculations from the left hand side of inequality (2.2), we have Hence, implies that This completes the proof of Theorem 1.
For the dual operator a theorem below is proved analogously.
weights on R n . Then the inequality holds, for every f ≥ 0 if and only if there exists γ ∈ (0, 1) such that Moreover, if C > 0 is the best possible constant in (2.3) then and s ∈ (1, p) was proved in [34]. In the case φ(x, t) = t q(x) and 1 < p ≤ q(x) ≤ ess sup x∈R n q(x) < ∞ Theorem 1 was proved in [3] (see also [2]).
Now we consider the multidimensional geometric mean operator defined as We formulate a two-weight criteria on boundedness of multidimensional geometric mean operator in weighted Musielak-Orlicz spaces.
are weights functions on R n . Then the inequality holds, if and only if there exists s ∈ (1, p) such that Moreover, if C > 0 is the best possible constant in (2.4) 2), we find that for 1 < s < p β By L'Hospital rule, we get This completes the proof of Theorem 3.