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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 130958, 12 pages doi:10.1155/2009/130958 Research Article Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel Predrag Vukovi ´ c Faculty of Teacher Education, University of Zagreb, Savska cesta 77, 10000 Zagreb, Croatia Correspondence should be addressed to Predrag Vukovi ´ c, predrag.vukovic@vus-ck.hr Received 11 July 2009; Revised 10 November 2009; Accepted 18 November 2009 Recommended by Radu Precup We consider the Hilbert-type inequalities with nonconjugate parameters. The obtaining of the best possible constants in the case of nonconjugate parameters remains still open. Our generalization will include a general homogeneous kernel. Also, we obtain the best possible constants in the case of conjugate parameters when the parameters satisfy appropriate conditions. We also compare our results with some known results. Copyright q 2009 Predrag Vukovi ´ c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let 1/p  1/q  1 p>1,f,g≥ 0, 0 <  ∞ 0 f p  x  dx < ∞, 0 <  ∞ 0 g q  x  dx < ∞. 1.1 The well-known Hardy-Hilbert’s integral inequality see 1 is given by  ∞ 0 f  x  g  y  x  y dx dy < π sin  π/p    ∞ 0 f p xdx  1/p   ∞ 0 g q xdx  1/q , 1.2 and an equivalent form is given by  ∞ 0   ∞ 0 fx x  y dx  p dy <  π sin  π/p   p  ∞ 0 f p  x  dx, 1.3 where the constant factors π/sinπ/p and π/sinπ/p p are the best possible. 2 Journal of Inequalities and Applications During the previous decades, the Hilbert-type inequalities were discussed by many authors, who either reproved them using various techniques or applied and generalized them in many different ways. For example, we refer to a paper of Yang see 2.Ifn ∈ N \{1} ,p i > 1,  n i1 1/p i 1,s>0,f i ≥ 0, satisfy 0 <  ∞ 0 x p i −s−1 f p i i  x  dx < ∞  i  1, 2, ,n  , 1.4 then   0,∞  n  n i1 f i  x i    n j1 x j  s dx 1 ···dx n < 1 Γ  s  n  i1 Γ  s p i   ∞ 0 x p i −s−1 f p i i xdx  1/p i , 1.5 where the constant factor 1/Γs  n i1 Γs/p i  is the best possible. Our generalization will include a general homogeneous kernel Kx 1 , ,x k  : R  n  k → R, where k ≥ 2, with k being nonconjugate parameters. The techniques that will be used in the proofs are mainly based on classical real analysis, especially on the well-known H ¨ older’s inequality and on Fubini’s theorem. The obtaining of the best possible constants in the case of nonconjugate parameters seems to be a very difficult problem and it remains still open. Let us recall the definition of nonconjugate exponents see 3.Letp and q be real parameters, such that p>1,q>1, 1 p  1 q ≥ 1, 1.6 and let p  and q  , respectively, be their conjugate exponents, that is, 1/p  1/p   1and 1/q  1/q   1. Further, define λ  1 p   1 q  1.7 and note that 0 <λ≤ 1 for all p and q values as in 1.6. In particular, λ  1 holds if and only if q  p  , that is, only when p and q are mutually conjugate. Otherwise, 0 <λ<1, and in such cases p and q will be referred to as nonconjugate exponents. Considering p, q,andλ as in 1.6 and 1.7, Hardy et al. 1, proved that there exists a constant C p,q , dependent only on the parameters p and q, such that the following Hilbert-type inequality holds for all nonnegative functions f ∈ L p R   and g ∈ L q R  :  ∞ 0 f  x  g  y   x  y  λ dx dy ≤ C p,q   f   L p R     g   L q R   . 1.8 Journal of Inequalities and Applications 3 Conventions Throughout this paper we suppose that all the functions are nonnegative and measurable, so that all integrals converge. We also introduce the following notations: R  n  { x   x 1 ,x 2 , ,x n  ; x 1 ,x 2 , ,x n > 0 } , | x | α   x 1 α  x 2 α  ··· x n α  1/α ,α>0, 1.9 and let |S n−1 | α  2 n Γ n 1/α/α n−1 Γn/α be an area of unit sphere in R n in view of α−norm. 2. Main Results Before presenting our idea and results, we repeat the notion of general nonconjugate exponents from 3.Letp i ,i 1, 2, ,k,be the real parameters which satisfy k  i1 1 p i ≥ 1,p i > 1,i 1, 2, ,k. 2.1 Further, the parameters p  i , i  1, 2, ,kare defined by the equations 1 p i  1 p  i  1,i 1, 2, ,k. 2.2 Since p i > 1, i  1, 2, ,k, it is obvious that p  i > 1, i  1, 2, ,k. We define λ : 1 k − 1 k  i1 1 p  i . 2.3 It is easy to deduce that 0 <λ≤ 1. Also, we introduce the parameters q i , i  1, 2, ,k, defined by the relations 1 q i  λ − 1 p  i ,i 1, 2, ,k. 2.4 In order to obtain our results we need to require q i > 0,i 1, 2, ,k. 2.5 It is easy to see that the above conditions do not automatically apply 2.5. Further, it follows λ  k  i1 1 q i , 1 q i  1 − λ  1 p i ,i 1, 2, ,k. 2.6 4 Journal of Inequalities and Applications Of course, if λ  1, then  k i1 1/p i 1; so the conditions 2.1–2.4 reduce to the case of conjugate parameters. Results in this section will be based on the following general form of Hardy-Hilbert’s inequality proven in 4. All the measures are assumed to be σ-finite on some Ω measure space. Theorem 2.1. Let k, n ∈ N,k≥ 2, and λ, p i ,p  i ,q i ,i 1, 2, ,k, be real numbers satisfying 2.1– 2.5.LetK : Ω k → R and φ ij : Ω → R, i, j  1, ,k, be nonnegative measurable functions such that  k i,j1 φ ij x j 1. Then, for any nonnegative measurable functions f i , i  1, 2, ,k,the following inequalities hold and are equivalent:  Ω k K λ  x 1 , ,x k  k  i1 f i  x i  dμ 1  x 1  ···dμ k  x k  ≤ k  i1   Ω  φ ii F i f i  p i  x i  dμ i  x i   1/p i , 2.7 ⎛ ⎝  Ω  1 φ kk F k x k   Ω k−1 K λ x 1 , ,x k  k−1  i1 f i x i dμ 1 x 1  ···dμ k−1 x k−1   p  k dμ k x k  ⎞ ⎠ 1/p  k ≤ k−1  i1   Ω  φ ii F i f i  p i x i dμ i x i   1/p i , 2.8 where F i  x i   ⎛ ⎝  Ω k−1 Kx 1 , ,x k  · n  j1,j /  i φ q i ij x j dμ 1 x 1  ···dμ i−1 x i−1 dμ i1 x i1  ···dμ k x k  ⎞ ⎠ 1/q i , i  1, ,k. 2.9 In the same paper the authors discussed the case of equality in inequalities 2.7 and 2.8. They proved that the equality holds in 2.7and analogously in 2.8 if and only if f i  x i   C i φ ii  x i  q i /1−λq i  F i  x i   1−λ  q i ,C i ≥ 0,i 1, ,k. 2.10 In the following theorem we give t he most important case where ΩR  n ,the measures μ i ,i  1, ,k, are Lebesgue measures, K α : 0, ∞ k → R is a nonnegative homogeneous function of degree −s, s > 0, and the functions φ ij represent the form φ ij x j  |x j | A ij α where A ij ∈ R, i, j  1, ,n. In order to obtain the generalizations of some known results we define k α  β 1 , ,β k−1  :   0,∞  k−1 K α  1,t 1 , ,t k−1  t β 1 1 ···t β k−1 k−1 dt 1 ···dt k−1 , 2.11 where we suppose that k α β 1 , ,β k−1  < ∞ for β 1 , ,β k−1 > −1andβ 1  ··· β k−1  k<s 1. Journal of Inequalities and Applications 5 Due to technical reasons, we introduce real parameters A ij ,i,j 1, 2, ,ksatisfying k  i1 A ij  0,j 1, 2, ,k. 2.12 We also define α i  k  j1 A ij ,i 1, 2, ,k. 2.13 Theorem 2.2. Let k, n ∈ N,k≥ 2, and λ, p i ,p  i ,q i ,i 1, 2, ,k, be real numbers satisfying 2.1–2.5.LetK α : 0, ∞ k → R be nonnegative measurable homogeneous function of degree −s, s>0, and let A ij ,i,j 1, ,k, and α i ,i 1, ,k be real parameters satisfying 2.12 and 2.13.Iff i : R  n → R, f i /  0, i  1, ,kare nonnegative measurable functions, then the following inequalities hold and are equivalent:  R  n  k K λ α  | x 1 | α , , | x k | α  k  i1 f i  x i  dx 1 ···dx k <L k  i1   R  n | x i | p i /q i k−1n−sp i α i α f p i i x i dx i  1/p i ,  R  n | x k | −p  k /q k   k−1  n−s  −p  k α k α    R  n  k−1 K λ α  | x 1 | α , , | x k | α  · k−1  i1 f i x i dx 1 ···dx k−1  p  k dx k <L p  k k−1  i1   R  n | x i | p i /q i k−1n−sp i α i α f p i i x i dx i  p  k /p i , 2.14 where L    S n−1   k−1λ α 2 k−1nλ k α  n − 1  q 1 A 12 , ,n− 1  q 1 A 1k  1/q 1 · k α  s − k − 1n − 1 − q 2 α 2 − A 22 ,n− 1  q 2 A 23 , ,n− 1  q 2 A 2k  1/q 2 ···k α  n − 1  q k A k2 , ,n− 1  q k A k,k−1 ,s− k − 1n − 1 − q k α k − A kk   1/q k , 2.15 q i A ij > −n, i /  j and q i A ii − α i  > k − 1n − s. 6 Journal of Inequalities and Applications Proof. Set Kx 1 , ,x k K α |x 1 | α , ,|x k | α  and φ ij x j |x j | A ij in Theorem 2.1, where  k i1 A ij  0 for every j  1, ,k. It is enough to calculate the functions F i x i , i  1, ,k. By using the n-dimensional spherical coordinates we find F q 1 1  x 1     R  n  k−1 K α  | x 1 | α , , | x k | α  k  j2   x j   q 1 A 1j dx 2 ···dx k    S n−1   k−1 α 2  k−1  n   0,∞  k−1 K α  | x 1 | α ,t 2 , ,t k  k  j2 t n−1q 1 A 1j j dt 2 ···dt k . 2.16 Using homogeneity of the function K α and the substitutions u i  t i /|x 1 | α ,i 2, ,k, we have F q 1 1  x 1     S n−1   k−1 α 2 k−1n   0,∞  k−1 | x 1 | −s α K α  1,u 2 , ,u k  · k  j2  | x 1 | α u j  n−1q 1 A 1j | x 1 | k−1 α du 2 ···du k    S n−1   k−1 α 2 k−1n | x 1 | k−1n−sq 1 α 1 −A 11  α k α  n − 1  q 1 A 12 , ,n− 1  q 1 A 1k  . 2.17 Similarly, by applying the n-dimensional spherical coordinates and homogeneity of the function K α we have F q 2 2  x 2     R  n  k−1 K α  | x 1 | α , , | x k | α  k  j1,j /  2   x j   q 2 A 2j dx 1 dx 3 ···dx k    S n−1   k−1 α 2  k−1  n   0,∞  k−1 t −s 1 K α  1, | x 2 | α t 1 , t 3 t 1 , , t k t 1  · k  j1,j /  2 t n−1q 2 A 2j j dt 1 dt 3 ···dt k . 2.18 Using the change of variables t 1  | x 2 | α u −1 2 ,t i  | x 2 | α u −1 2 u i ,i 3, ,k, so ∂  t 1 ,t 3 , ,t k  ∂  u 2 ,u 3 , ,u k   | x 2 | k−1 α u −k 2 , 2.19 Journal of Inequalities and Applications 7 where ∂t 1 ,t 3 , ,t k /∂u 2 ,u 3 , ,u k  denotes the Jacobian of the transformation, we have F q 2 2  x 2     S n−1   k−1 α 2 k−1n | x 2 | k−1n−sq 2 α 2 −A 22  α ·   0,∞  k−1 K α  1,u 2 , ,u k  u s−k−1n−q 2 α 2 −A 22  2 k  j3 u n−1q 2 A 2j j du 2 ···du k    S n−1   k−1 α 2 k−1n | x 2 | k−1n−s−q 2 α 2 −A 22  α · k α  s −  k − 1  n − 1 − q 2  α 2 − A 22  ,n− 1  q 2 A 23 , ,n− 1  q 2 A 2k  . 2.20 In a similar manner we obtain F q i i  x i     S n−1   k−1 α 2 k−1n | x i | k−1n−sq i α i −A ii  α · k α  n − 1  q i A i2 , ,n− 1  q i A i,i−1 ,s−  k − 1  n − 1 − q i  α i − A ii  , n−1  q i A i,i1 , ,n− 1  q i A ik  2.21 for i  3, ,k. This gives inequalities 2.14 with inequality sign ≤. Condition 2.10 immediately gives that nontrivial case of equality in 2.14 leads to the divergent integrals. This completes the proof. Remark 2.3. Note that the kernel K α |x 1 | α , ,|x k | α   k i1 |x i | β α  −s is a homogeneous function of degree −βs. In this case we have k α  β 1 , ,β k−1     0,∞  k−1  k−1 i1 t β i i  1   k−1 i1 t β i i  s dt 1 ···dt k−1  1 β k−1 Γ  s  Γ  s − k−1  i1 β i  1 β  k−1  i1 Γ  β i  1 β  , 2.22 where we used the well-known formula f or gamma function see, e.g., 5, Lemma 5.1.Now, by using Theorem 2.2 and 2.22 we obtain the result of Krni ´ cetal.see 6. 3. The Best Possible Constants in the Conjugate Case In this section we consider the inequalities in Theorem 2.2. In such a way we shall obtain the best possible constants for some general cases. It follows easily that Theorem 2.2 in the conjugate case λ  1,p i  q i  becomes as follows. 8 Journal of Inequalities and Applications Theorem 3.1. Let k, n ∈ N,k≥ 2 and let p 1 , ,p k be conjugate parameters such that p i > 1,i 1, ,k. Let K α : 0, ∞ k → R be nonnegative measurable homogeneous function of degree −s, s>0, and let A ij ,i,j 1, ,k,and α i ,i 1, ,kbe real parameters satisfying 2.12 and 2.13. If f i : R  n → R, f i /  0, i  1, ,k are nonnegative measurable functions, then the following inequalities hold and are equivalent:   R  n  k K α  | x 1 | α , , | x k | α  k  i1 f i  x i  dx 1 ···dx k <M k  i1   R  n | x i | k−1n−sp i α i α f p i i x i dx i  1/p i ,  R  n | x k | 1−p  k k−1n−s−p  k α k α    R  n  k−1 K α  | x 1 | α , , | x k | α  · k−1  i1 f i x i dx 1 ···dx k−1  p  k dx k <M p  k k−1  i1   R  n | x i | k−1n−sp i α i α f p i i x i dx i  p  k /p i , 3.1 where M    S n−1   k−1 α 2 k−1n k α  n − 1  p 1 A 12 , ,n− 1  p 1 A 1k  1/p 1 · k α  s − k − 1n − 1 − p 2 α 2 − A 22 ,n− 1  p 2 A 23 , ,n− 1  p 2 A 2k  1/p 2 ···k α  n − 1  p k A k2 , ,n− 1  p k A k,k−1 ,s− k − 1n − 1 − p k α k − A kk   1/p k , 3.2 p i A ij > −n, i /  j and p i A ii − α i  > k − 1n − s. To obtain a case of the best inequality it is natural to impose the following conditions on the parameters A ij : n  p j A ji  s −  k − 1  n − p i  α i − A ii  ,j /  i, i, j ∈ { 1, 2, ,k } . 3.3 In that case the constant M from Theorem 3.1 is simplified to the following form: M ∗    S n−1    k−1  α 2  k−1  n k α  n − 1   A 2 , ,n− 1   A k  , 3.4 where  A i  p 1 A 1i for i /  1,  A 1  p k A k1 . 3.5 Journal of Inequalities and Applications 9 Further, by using 3.4 and 3.5, the inequalities 3.1 with the parameters A ij , satisfying the relation 3.3, become   R  n  k K α  | x 1 | α , , | x k | α  k  i1 f i  x i  dx 1 ···dx k <M ∗ k  i1   R  n | x i | −n−p i  A i α f p i i x i dx i  1/p i , 3.6 ⎡ ⎣  R  n | x k | 1−p  k −n−p k  A k  α    R  n  k−1 K α  | x 1 | α , , | x k | α  · k−1  i1 f i  x i  dx 1 ···dx k−1  p  k dx k ⎤ ⎦ 1/p  k <M ∗ k−1  i1   R  n | x i | −n−p i  A i α f p i i x i dx i  1/p i . 3.7 Theorem 3.2. Suppose that the real parameters A ij ,i,j 1, ,ksatisfy conditions in Theorem 3.1 and conditions given in 3.3. If the kernel K α t 1 , ,t k  is as in Theorem 3.1 and for every i  2, ,k K α  1,t 2 , ,t i , ,t k  ≤ CK α  1,t 2 , ,0, ,t k  , 0 ≤ t i ≤ 1,t j ≥ 0,j /  i 3.8 for some C>0, then the constant M ∗ is the best possible in inequalities 3.6 and 3.7. Proof. Let us suppose that the constant factor M ∗ given by 3.4 is not the best possible in the inequality 3.6. Then, there exists a positive constant M 1 <M ∗ , such that 3.6 is still valid when we replace M ∗ by M 1 . We define the real functions  f i,ε : R n → R by the formulas  f i,ε  x i   ⎧ ⎨ ⎩ 0, | x i | α < 1, | x i | α  A i −ε/p i , | x i | α ≥ 1, i  1, ,k, 3.9 where 0 <ε<min 1≤i≤k {p i  p i  A i }. Now, we shall put these functions in inequality 3.6. By using the n-dimensional spherical coordinates, the right-hand side of the inequality 3.6 becomes M 1 k  i1   | x i | α ≥1 | x i | −n−ε α dx i  1/p i  M 1   S n−1   α 2 n  ∞ 1 t −1−ε dt  M 1   S n−1   α 2 n ε . 3.10 10 Journal of Inequalities and Applications Further, let J denotes the left-hand side of the inequality 3.6, for the above choice of the functions  f i,ε . By applying the n-dimensional spherical coordinates and the substitutions u i  t i /t 1 ,i /  2, we find J   | x 1 | α ≥1 ···  | x k | α ≥1 K α  | x 1 | α , , | x k | α  k  i1 | x i |  A i −ε/p i α dx 1 ···dx k    S n−1   k α 2 kn  ∞ 1 ···  ∞ 1 K α  t 1 , ,t k  k  i1 t n−1  A i −ε/p i i dt 1 ···dt k    S n−1   k α 2 kn  ∞ 1 t −1−ε/β 1   ∞ 1/t 1 ···  ∞ 1/t 1 K α  1,u 2 , ,u k  k  i2 u n−1  A i −ε/p i i du 2 ···du k  dt 1 . 3.11 Now, it is easy to see that the following inequality holds: J ≥   S n−1   k α 2 kn  ∞ 1 t −1−ε 1   ∞ 0 ···  ∞ 0 K α  1,u 2 , ,u k  k  i2 u n−1  A i −ε/p i i du 2 ···du k  dt 1 −   S n−1   k α 2 kn  ∞ 1 t −1−ε 1 k  j2 I j  t 1  dt 1 , 3.12 where for j  2, ,k, I j t 1  is defined by I j  t 1    D j K α  1,u 2 , ,u k  k  i2 u n−1  A i −ε/p i i du 2 ···du k , 3.13 satisfying D j  {u 2 , ,u k ;0<u j < 1/t 1 , 0 <u l < ∞,l /  j}. Without losing generality, we only estimate the integral I 2 t 1 . For k  2 w e have I 2  t 1    1/t 1 0 K α  1,u 2  u n−1  A 2 −ε/p 2 2 du 2 ≤ C  1/t 1 0 u n−1  A 2 −ε/p 2 2 du 2  C  n   A 2 − ε p 2  −1 t ε/p 2 −n−  A 2 1 , 3.14 [...]... Journal of Mathematics, vol 2, pp 135–150, 1951 4 I Brneti´ , M Krni´ , and J Peˇ ari´ , “Multiple Hilbert and Hardy-Hilbert inequalities with nonc c c c conjugate parameters,” Bulletin of the Australian Mathematical Society, vol 71, no 3, pp 447–457, 2005 5 B Yang and Th M Rassias, “On the way of weight coefficient and research for the Hilbert-type inequalities, ” Mathematical Inequalities & Applications,... , J Peˇ ari´ , and P Vukovi´ , “On some higher-dimensional Hilbert’s and Hardy-Hilbert’s c c c c integral inequalities with parameters,” Mathematical Inequalities & Applications, vol 11, no 4, pp 701– 716, 2008 7 B Sun, A multiple Hilbert-type integral inequality with the best constant factor,” Journal of Inequalities and Applications, vol 2007, Article ID 71049, 14 pages, 2007 8 I Peri´ and P Vukovi´... of Inequalities and Applications References 1 G H Hardy, J E Littlewood, and G Polya, Inequalities, Cambridge University Press, Cambridge, UK, ´ 2nd edition, 1967 2 B Yang, “On a new multiple extension of Hilbert’s integral inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol 6, no 2, article 39, pp 1–8, 2005 3 F F Bonsall, Inequalities with non-conjugate parameters,” The Quarterly...Journal of Inequalities and Applications 11 and for k > 2 we find k I 2 t1 ≤ C 0,∞ C n− k−2 Kα 1, 0, u3 , , uk ε p2 ui n−1 Ai −ε/pi 1/t1 du3 · · · duk −1 A2 u2 n−1 A2 −ε/p2 du2 0 i 3 t1 ε/p2 A2 −n kα n − 1 A3 − ε , ,n − 1 p3 Ak − ε , pk 3.15 where kα n − 1 A3 − ε/p3 , , n − 1 Ak − ε/pk is well defined since obviously A3 Ak < s − k − 2 n Hence, we have Ij t1 ≤ consequently ∞ 1 t−1−ε 1 k ε/pj −n−Aj... for ε → 0 , j ∈ {2, , k}, and Ij t1 dt1 ≤ O 1 3.16 j 2 We conclude, by using 3.10 , 3.12 , and 3.16 , that M∗ ≤ M1 which is an obvious contradiction It follows that the constant M∗ in 3.6 is the best possible Finally, the equivalence of the inequalities 3.6 and 3.7 means that the constant M∗ is also the best possible in the inequality 3.7 That completes the proof Remark 3.3 If we put k 2, Kα x,... Aij satisfy the conditions from Theorem 3.2, then the constant M∗ kα A2 , , Ak is the best possible For example, setting Kα x1 , , xk x1 · · · xk −s , s > 0, Ai s − pi /pi , i 3.17 , we obtain Yang’s result 1.5 from introduction 2, , k, in the inequality Acknowledgment This research is supported by the Croatian Ministry of Science, Education and Sports, Grant no 058-1170889-1050 12 Journal... |y|s , A1 s/q − n and A2 α α s/p − n in the inequalities 3.6 and 3.7 applying Theorem 3.2, we obtain the result of Baoju Sun see 7 Further, by putting n 1 in Theorems 3.1 and 3.2 we obtain appropriate results from 8 More precisely, the inequality 3.6 becomes k 0,∞ k Kα x1 , , xk i 1 fi xi dx1 · · · dxk < M∗ k i 1 ∞ 0 −1−pi Ai xi p fi i xi dxi 1/pi 3.17 If the kernel Kα x1 , , xk and the parameters... inequality with the best constant factor,” Journal of Inequalities and Applications, vol 2007, Article ID 71049, 14 pages, 2007 8 I Peri´ and P Vukovi´ , “Hardy-Hilbert’s inequality with general homogeneous kernel,” Mathematical c c Inequalities & Applications, vol 12, no 3, pp 525–536, 2009 . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 130958, 12 pages doi:10.1155/2009/130958 Research Article Multidimensional Hilbert-Type Inequalities. higher-dimensional Hilbert’s and Hardy-Hilbert’s integral inequalities with parameters,” Mathematical Inequalities & Applications, vol. 11, no. 4, pp. 701– 716, 2008. 7 B. Sun, A multiple Hilbert-type. C p,q   f   L p R     g   L q R   . 1.8 Journal of Inequalities and Applications 3 Conventions Throughout this paper we suppose that all the functions are nonnegative and measurable, so that all integrals converge. We also introduce