Proceedings of the Edinburgh Mathematical Society (2008) 5, 26 c DOI:0.07/S0030950500690 Printed in the United Kingdom INEQUALITIES OF THE HILBERT TYPE IN WITH NON-CONJUGATE EXPONENTS ALEKSANDRA ČIŽMEŠIJA, IVAN PERIĆ2 AND PREDRAG VUKOVIĆ3 Department of Mathematics, University of Zagreb, Bijenička cesta 30, 0000 Zagreb, Croatia (cizmesij@math.hr) 2 Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 0000 Zagreb, Croatia (iperic@pbf.hr) 3 Teacher Training College Čakovec, Ante Starčevića 55, 40000 Čakovec, Croatia (predrag.vukovic@vus-ck.hr) (Received 22 November 2005) Abstract In this paper we state and prove a new general Hilbert-type inequality in with k 2 non-conjugate exponents. Using Selberg s integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy Littlewood Sobolev inequality and some further Hilberttype inequalities for k non-negative functions and non-conjugate exponents. Keywords: inequalities; Hilbert s inequality; doubly weighted Hardy Littlewood Sobolev inequality; Selberg s integral formula; gamma function; beta function 2000 Mathematics subject classification: Primary 26D0; 26D5. Introduction Let p and q be real parameters such that p>, q >, p +, (.) q and let p and q respectively be their conjugate exponents, that is, p + p = and q + q =. Furthermore, define λ = p + q (.2) and note that 0 <λ for all p and q as in (.). In particular, λ = holds if and only if q = p, that is, only when p and q are mutually conjugate. Otherwise, we have 0 <λ<, and in such cases p and q will be referred to as non-conjugate exponents.
2 A. Čižmešija, I. Perić and P. Vuković Considering p, q and λ as in (.) and (.2), Hardy et al. [3] 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 non-negative functions f L p (R + ) and g L q (R + ): 0 0 f(x)g(y) (x + y) λ dx dy C p,q f L p (R +) g L q (R +). (.3) However, the original proof did not provide any information about the value of the best possible constant C p,q. That drawback was improved by Levin [4], who obtained an explicit upper bound for C p,q : C p,q ( π cosec π ) λ λp. (.4) This was an interesting result since the right-hand side of (.4) reduces to the previously known sharp constant π cosec(π/p ) when the exponents p and q are conjugate. A simpler proof of (.4), based on a single application of Hölder s inequality, was given later by Bonsall []. Moreover, in the same paper, with the same assumptions on p, q, λ, f and g, he proved another interesting Hilbert-type inequality, 0 0 f(x)g(y) dx dy (x + y) λ B λ ( p, q ) f p/q L p (R +) g q/p L q (R +) [ 0 0 ] λ x /p y /q (x + y) λ f p (x)g q (y)dxdy, (.5) with the best possible constant B λ (/p, /q ), where B is the usual beta function. The idea used in the proof of (.5) has to some extent guided us in the research we present here. During the following decades, the Hilbert-type inequalities were discussed by several authors, who either re-proved them using various techniques, or applied and generalized them in many different ways. Here, we emphasize one of the most important such results, the so-called doubly weighted Hardy Littlewood Sobolev inequality of Stein and Weiss [9], f(x)g(y) x α x y s y β dx dy C α,β,p,q,n f L p ( ) g L q ( ), (.6) which holds for n N, p, q > such that /p +/q >, λ as in (.2), 0 α < n/p, 0 β < n/q, s = nλ α β, and all non-negative functions f L p ( ) and g L q ( ). In [5], Lieb proved the existence of optimizers for (.6), that is, functions f and g which, when inserted into (.6), give equality with the smallest possible constant C α,β,p,q,n. Moreover, for p = q and α = β = 0, the constant and maximizing functions were explicitly computed in [5]. In particular, Lieb obtained C 0,0,p,p,n = π Γ ( n/p 2 n [ n/p ) Γ ( 2 n) ] 2/p, Γ (n/p) Γ (n)
Inequalities of the Hilbert type in with non-conjugate exponents 3 where Γ is the gamma function. Unfortunately, neither C α,β,p,q,n nor the optimizers are known for any other case of the parameters in (.6). It was shown only (see, for example, [6]) that for the classical Hardy Littlewood Sobolev inequality, that is, for (.6) with α = β = 0 and s = nλ, the estimate holds, where C 0,0,p,q,n (p ) λ +(q ) λ ( ) λ λ ( λ)pq n Sn (.7) S n = 2πn/2 Γ ( 2 n) is the area of the unit sphere S n in. For further details, see [5, 6]. On the other hand, here we also refer to [0], in which a general Hilbert-type inequality was obtained for k 2 conjugate exponents, that is, real parameters p,...,p k >, such that k =. p i Namely, let µ,...,µ k be positive σ-finite measures on Ω and let K : Ω k R and φ ij : Ω R, i, j =,...,k, be non-negative measurable functions. If k i,j= φ ij(x j )= a.e. on Ω k, then the inequality Ω k K(x,...,x k ) f i (x i )dµ (x ) dµ k (x k ) ( /pi F i (x i )(φ ii f i ) pi (x i )dµ i (x i )) (.8) Ω holds for all non-negative measurable functions f,...,f k : Ω R, where F i (x i )= K(x,...,x k ) φ pi ij (x j)dµ (x ) dµ i (x i )dµ i+ (x i+ ) dµ k (x k ), Ω k j i for i =,...,k. If there exists l {,...,k} such that p l < 0 and p i > 0 for i l, then the sign of the inequality in (.8) is reversed. In this paper we extend (.8) to a more general case with k 2 non-conjugate exponents, introduced in the following section, and point out that relations (.5) and (.8) are only special cases of our result. The technique we establish appears to be fruitful since, by choosing particular parameters in the general Hilbert-type inequality obtained and applying the well-known Selberg integral formula (see, for example, [8]), we obtain explicit upper bounds for the general case of the doubly weighted Hardy Littlewood Sobolev inequality (.6). Moreover, using a similar approach, we derive some further generalizations of (.3), that is, Hilbert-type inequalities for k non-negative functions defined on and non-conjugate exponents.
4 A. Čižmešija, I. Perić and P. Vuković.. Conventions Throughout this paper, let r be the conjugate exponent to a positive real number r, that is, r + r = or r = r r. The Euclidean norm of the vector x will be denoted by x. Furthermore, empty sums are assumed to equal zero, all measures are assumed to be positive and σ-finite and all functions are assumed to be non-negative and measurable. Finally, we shall denote by f the symmetric-decreasing rearrangement of a function f : R vanishing at infinity. 2. A general Hilbert-type inequality Before presenting our idea and results, we introduce the notion of general non-conjugate exponents. Let k N, k 2, and let real parameters p,...,p k be such that Define λ = k p,...,p k >, k p i and k p i. (2.) = λ q i p, i =,...,k. (2.2) i Obviously, we always have 0 <λ, while the equality λ = holds if and only if k /p i =. As we stated in, in this special case the parameters p,...,p k will be called conjugate exponents. Otherwise, that is, for k /p i >, they are nonconjugate. Observe that for conjugate exponents we have q i = p i, i =,...,k, while, for k =2,p = p and p 2 = q, relations (2.) and (2.2) respectively reduce to (.), (.2), q = q and q 2 = p. On the other hand, for any choice of the parameters in (2.), it follows from (2.2) that +( λ) =, i =,...,k, (2.3) q i p i and k +( λ) =. (2.4) q i Hence, in order to apply Hölder s inequality with exponents q,...,q k and /( λ), we require that > 0, i =,...,k. (2.5) q i Note that for k 3 conditions (2.) and (2.2) do not automatically imply (2.5). More precisely, since (2.) and (2.2) give only > 2 k, i =,...,k, q i k p i
Inequalities of the Hilbert type in with non-conjugate exponents 5 some of the q i can be negative. For example, for p = 2 and p 2 = p 3 = 20 9 we have /q = 5 < 0. Therefore, condition (2.5) is not redundant. We are now ready to state and prove our basic result, a general Hilbert-type inequality. Theorem 2.. Let k N, k 2, and parameters p,...,p k, λ and q,...,q k be as in (2.), (2.2) and (2.5). Let µ,...,µ k be positive σ-finite measures on Ω. IfK is a non-negative measurable function on Ω k, F,...,F k are positive measurable functions on Ω k, and φ ij, i, j =,...,k, are non-negative measurable functions on Ω, such that then the inequality Ω k K(x,...,x k ) i,j= f i (x i )dµ (x ) dµ k (x k ) φ ij (x j )= a.e. on Ω k, (2.6) [ (KF pi qi i )(x,...,x k )(φ ii f i ) pi (x i ) ] /qi φ qi ij (x j)dµ (x ) dµ k (x k ) Ω k j i [ ] λ K(x,...,x k ) F pi i (x,...,x k )(φ ii f i ) pi (x i )dµ (x ) dµ k (x k ) Ω k holds for all non-negative measurable functions f,...,f k on Ω. (2.7) Proof. Note that from (2.3) we have p i q i + p i ( λ) =, i =,...,k. Using this and (2.4), the left-hand side of (2.7) can be written as Ω k K(x,...,x k ) f i (x i )dµ (x ) dµ k (x k ) = K k (/qi)+ λ (x,...,x k ) f (pi/qi)+pi( λ) i (x i ) Ω k F (pi/qi) +pi( λ) i (x,...,x k ) j i φ ij (x j )dµ (x ) dµ k (x k ) φ (pi/qi)+pi( λ) ii (x i )
6 A. Čižmešija, I. Perić and P. Vuković = [ Ω k (KF pi qi i )(x,...,x k )(φ ii f i ) pi (x i ) φ qi ij (x j) j i [ K(x,...,x k ) ] /qi F pi i (x,...,x k )(φ ii f i ) pi (x i )] λ dµ (x ) dµ k (x k ). The inequality (2.7) now follows by using Hölder s inequality with the exponents q,...,q k and /( λ). Remark 2.2. Observe that, without loss of generality, condition (2.6) from the statement of Theorem 2. can be replaced by φ ij (x j ) = a.e. on Ω, for j =,...,k, since (2.6) implies that where c c k =. φ ij (x j )=c j = const., j =,...,k, (2.8) Remark 2.3. Obviously, (2.7) becomes an equality if at least one of the functions involved in its left-hand side is a zero function. To discuss other non-trivial cases of equality in (2.7), we can assume without loss of generality that the functions K and f i, i =,...,k, are positive. Otherwise, instead of Ω k, we consider the set S = { x =(x,...,x k ) Ω k : K(x) } f i (x i ) > 0, which has a positive measure. Now, note that the equality in (2.7) holds if and only if it holds in Hölder s inequality, that is, only if the functions KF pi qi i (φ ii f i ) pi j i φ qi k ij, i =,...,k, and K (F i φ ii f i ) pi are effectively proportional. Therefore, equality in (2.7) occurs if and only if there exist positive constants α i, β ij, i, j =,...,k, j i, such that and KF pi qi i KF pi qi i (φ ii f i ) pi l i (φ ii f i ) pi l i φ qi il = α ik φ qi (F l φ ll f l ) p l, i =,...,k, (2.9) l= il = β ijkf pj qj j (φ jj f j ) pj φ qj jl, i j. (2.0) l j
Inequalities of the Hilbert type in with non-conjugate exponents 7 Moreover, the relation (2.9) is equivalent to F qi i = α i l i φ qi il (F l φ ll f l ) p l, i =,...,k. (2.) In the special case when F i F i (x i ), i =,...,k, the functions f i and φ ij from (2.0) and (2.) can be expressed explicitly in terms of φ ii. More precisely, from (2.) we have F i const., i =,...,k, (2.2) directly, since the right-hand side of this relation depends on x l, l i, while the left-hand side in this setting is a function of x i. Considering this, (2.0) becomes (φ ii f i ) pi φ qj ji = γ ij (φ jj f j ) pj for some positive constants γ ij. Thus, l i,j φ qj jl l i φ qi il, i j, (2.3) (φ ii f i ) pi φ qj ji const., i =,...,k, j i, (2.4) where again we have exploited the fact that the left-hand side of (2.3) depends only on x i, while its right-hand side is a function of x l, l =,...,k, l i. The relation (2.4) further implies that φ qj ji φ q l li = const., i =,...,k, j, l i, which, combined with (2.8), gives φ ii φ qj l i /q l ji const., i =,...,k, j i. (2.5) Since by (2.2) and (2.4) we have (2.5) can be transformed into q j l i ) = q j (λ qi q l = q j p, i φ p i ii φqj ji const., i =,...,k, j i, (2.6) while (2.4) becomes f pi i C i φ (pi+p i ) ii, i =,...,k, (2.7) for some positive constants C i, i =,...,k. Hence, if F i F i (x i ), the conditions (2.2), (2.6) and (2.7) are necessary and sufficient for equality in (2.7). Remark 2.4. If the parameters p,...,p k in Theorem 2. are such that 0 <p i <, k p i +< k j= p j, i =,...,k, (2.8)
8 A. Čižmešija, I. Perić and P. Vuković and λ and q,...,q k are defined by (2.2), then the sign of inequality in (2.7) is reversed. To justify this assertion, observe that the first inequality in (2.8) gives /p i < 0, i =,...,k, so we have λ<0. Similarly, from the second relation in (2.8) it follows that = λ q i p = ( k k ) + < 0, i =,...,k. i k p i p j= j Therefore, q i < 0, i =,...,k, and 0 < /( λ) <, so (2.7) holds with the reversed sign of inequality as a direct consequence of Hölder s inequality (for details on the socalled reversed Hölder s inequality, see, for example, [7, Chapter V]). The same result is also achieved with the parameters p,...,p k satisfying k p i < and 0 <p l <, k p i +< k j= p j, i l, (2.9) for some l {,...,k}, since from (2.9) we obtain /( λ) < 0, q l > 0, and q i < 0, i l. Remark 2.5. Note that in the case of conjugate exponents (λ = ) the inequality (2.7) becomes K(x,...,x k ) f i (x i )dµ (x ) dµ k (x k ) Ω k [ K(x,...,x k )(φ ii f i ) pi (x i ) /qi φ pi ij (x j)dµ (x ) dµ k (x k )], Ω k j i whence, by Fubini s theorem, we obtain the right-hand side of (.8). This means that Theorem 2. may be regarded as a generalization of the mentioned result from [0]. To conclude this section, we restate Theorem 2. for the case when k = 2. This result is interesting in its own right, since it will be applied in the following section, where we consider a particular kernel K. Theorem 2.6. Let p, q and λ be as in (.) and (.2). Let µ and µ 2 be positive σ-finite measures on Ω. IfK, F and G are non-negative measurable functions on Ω 2, and ϕ and ψ are non-negative measurable functions on Ω, then the inequality K(x, y)f(x)g(y)dµ (x)dµ 2 (y) Ω 2 [ ] /q )(x, y)ψ q (y)(ϕf) p (x)dµ (x)dµ 2 (y) Ω 2 (KF p q [ (KG q p Ω 2 ] /p )(x, y)ϕ p (x)(ψg) q (y)dµ (x)dµ 2 (y) [ ] λ (KF p G q )(x, y)(ϕf) p (x)(ψg) q (y)dµ (x)dµ 2 (y) Ω 2 (2.20) holds for all non-negative measurable functions f and g on Ω.
Inequalities of the Hilbert type in with non-conjugate exponents 9 Proof. The proof follows directly from Theorem 2. using substitutions p = p, p 2 = q, q = q, q 2 = p, φ = ϕ and φ 22 = ψ. Observe that from φ φ 2 = and φ 2 φ 22 = we have φ 2 =/ϕ and φ 2 =/ψ. Remark 2.7. If we rewrite Theorem 2.6 with Ω = R +, Lebesgue measures, kernel K(x, y) =(x + y) λ and functions F (x, y) =G(x, y), ϕ(x) =x /pp and ψ(y) = y /qq, we obtain (.5). Hence, Theorem 2.6 can be seen as a generalization of the Bonsall result mentioned in []. 3. Explicit upper bounds for the doubly weighted Hardy Littlewood Sobolev inequality In this section, we consider a special case of Theorem 2.6 with Ω = and the kernel K(x, y) = x α x y s y β. More precisely, we use our general result to obtain a form of the doubly weighted Hardy Littlewood Sobolev inequality (.6) with an explicit constant factor on its right-hand side. In fact, we derive explicit upper bounds for the sharp constant C α,β,p,q,n for (.6). Our results in the following two sections will be based on Theorems 2. and 2.6 and the well-known Selberg integral formula R kn x k α k n ( k x i+ x i αi n ) x y α0 n dx dx k = Γ n(α 0 ) Γ n (α k ) y α0+ +α k n, (3.) Γ n (α 0 + + α k ) for arbitrary k, n N, y, and 0 <α 0,...,α k <nsuch that 0 < k i=0 α i <n, where Γ n (α) = πn/2 2 α Γ ( 2 α) Γ ( 2 n 2 α). In [8], Stein derived the Selberg integral formula with two parameters using the Riesz potential (see also [2]). Observe that Γ n (n α) = (2π)n, 0 <α<n. (3.2) Γ n (α) In the next lemma we give a form of (3.), more suitable for our computations. Lemma 3.. Suppose that k, n N, 0 <β,...,β k,s<nare such that k β i +s > kn and y. Then x β x k βk x + + x k + y s dx dx k R kn = Γ n(n β ) Γ n (n β k )Γ n (n s) Γ n ((k +)n β β k s) y kn β β k s. (3.3)
20 A. Čižmešija, I. Perić and P. Vuković Proof. Set α i = n β i+, i =0,...,k, and α k = n s. Substituting first t = x +y and then t 2 = t + x 2, the left-hand side of (3.3) becomes R kn x β x k βk x + + x k + y s dx dx k Rkn t y α0 n x 2 α n x k α k n = t + x 2 + + x k n α dt dx 2 dx k k Rkn t y α0 n t 2 t α n x 3 α2 n x k α k n = t 2 + x 3 + + x k n α dt dt 2 dx 3 dx k. (3.4) k After the sequence of similar substitutions t i = t i +x i, i =2,...,k, the last line of (3.4) is finally equal to R kn t k α k n ( k t i+ t i αi n ) t y α0 n dt dt k = Γ n(α 0 ) Γ n (α k ) Γ n (α 0 + + α k ) y α0+ +α k n = Γ n(n β ) Γ n (n β k )Γ n (n s) Γ n ((k +)n β β k s) y kn β β k s, where the last two equalities are obtained by Selberg s integral formula (3.) and by replacing α i by the corresponding expressions including β i. Since the case k = of Lemma 3. will be of special interest to us, we state it as a separate result. Lemma 3.2. Let n N and y.if0 <β,s<nare such that β + s>n, then Rn x β x + y s dx = Γ n(n β)γ n (n s) y n β s. Γ n (2n β s) We can now obtain the doubly weighted Hardy Littlewood Sobolev inequality (.6) mentioned above. The first step is to consider the case in which the function g L q ( ) on its left-hand side is symmetric-decreasing, that is, g(x) g(y) whenever x y. For such a function and y, y 0,wehave g q (y) B( y ) B( y ) g q (x)dx g q (x)dx B( y ) = n S n y n g q L q ( ), (3.5) where B( y ) denotes the ball of radius y in, centred at the origin, and B( y ) = y n S n /n is its volume.
Inequalities of the Hilbert type in with non-conjugate exponents 2 Theorem 3.3. Let n N, p> and q>such that /p +/q >, and set λ as in (.2). Let 0 α < n/p, 0 β < n/q and s = nλ α β. Then the inequality f(x)g(y) (2π) 2n ( S n /n) λ R x α x y s dx dy y β Γ n n (n/p + α)γ n (n/q + β)γ n (s) f L p ( ) g L q ( ) (3.6) holds for all non-negative functions f L p ( ) and symmetric-decreasing functions g L q ( ). Proof. Suppose that in Theorem 2.6 we have Ω =, K(x, y) = x α x y s y β, F (x, y) =G(x, y), ϕ(x) = x n/pp, ψ(y) = y n/qq and the Lebesgue measure dx. Then the left-hand side of (2.20) reads f(x)g(y) L = R x α x y s dx dy, (3.7) y β n while its right-hand side is a product R = I /q I /p 2 I3 λ, where R2n x n/p y n/q I = x α x y s y β f p (x)dxdy, R2n x n/p y n/q I 2 = x α x y s y β gq (y)dxdy, R2n x n/p y n/q I 3 = x α x y s y β f p (x)g q (y)dxdy. Therefore, applying Fubini s theorem, Lemma 3.2, identity (3.2), and the fact that α + β + s = nλ, respectively, we have Rn I = x (n/p ) α f p y ((n/q)+β) (x) R x y s dy dx n Rn = x (n/p ) α f p z ((n/q)+β) (x) R z + x s dzdx n = Γ n(n n/q β)γ n (n s) x (n/p ) α+n (n/q) β s f p (x)dx Γ n (2n n/q β s) (2π) 2n = Γ n (n/p + α)γ n (n/q + β)γ n (s) f p L p ( ). Analogously, (2π) 2n I 2 = Γ n (n/p + α)γ n (n/q + β)γ n (s) g q L q ( ), and, by (3.5), I 3 n Rn S n g q L q ( ) x (n/p ) α f p y ((n/q)+β) (x) R x y s dy dx n = n S n (2π) 2n Γ n (n/p + α)γ n (n/q + β)γ n (s) f p L p ( ) g q L q ( ).
22 A. Čižmešija, I. Perić and P. Vuković Finally, (3.6) follows by combining (3.7) and the expressions we have obtained for the integrals I, I 2 and I 3. To obtain an analogous result for arbitrary non-negative functions f L p ( ) and g L q ( ), we use the fact that for parameter γ > 0 the function h : R, h(x) = x γ, is symmetric-decreasing and vanishes at infinity. Hence, h = h. Theorem 3.4. Let n N, p> and q> such that /p +/q >, and set λ as in (.2). Suppose that 0 α < n/p, 0 β < n/q and s = nλ α β. Then the inequality (3.6) holds for all non-negative functions f L p ( ) and g L q ( ). Proof. Since x x α, x x s and x x β are symmetric-decreasing functions vanishing at infinity, the general rearrangement inequality (see, for example, [6]) implies that f(x)g(y) x α x y s dx dy y β f (x)g (y) x α x y s dx dy. (3.8) y β Clearly, by Theorem 3.3, the right-hand side of (3.8) is not greater than K α,β,p,q,n f Lp ( ) g Lq ( ) = K α,β,p,q,n f Lp ( ) g Lq ( ), (3.9) where K α,β,p,q,n is the constant from the right-hand side of (3.6). To obtain equality in (3.9), we have used the fact that the symmetric-decreasing rearrangement is normpreserving. Remark 3.5. Note that C α,β,p,q,n K α,β,p,q,n, where C α,β,p,q,n is the sharp constant for (.6) and K α,β,p,q,n is the constant factor involved in the right-hand side of (3.6). Hence, we obtained new explicit upper bounds for the doubly weighted Hardy Littlewood Sobolev inequality. In particular, for α = β = 0 we have K 0,0,p,q,n = while for p = q the constant (3.0) becomes K 0,0,p,p,n = (2π)2n ( S n /n) (2/p ) Γn(n/p)Γ 2 n (2n/p ) = π n/p Γ ( 2 n n/p ) Γ (n/p ) (2π)2n ( S n /n) λ Γ n (n/p)γ n (n/q)γ n (nλ), (3.0) [ Γ (n/2p ) Γ (n/2p) ] 2 [Γ ( 2 n + )] (2/p ). Although (.7) provides a better estimate for C 0,0,p,q,n than (3.0), it is important to note that our result covers all admissible choices of the parameters p, q, α and β in (.6), so our main contribution is in extending Lieb s result given in.
Inequalities of the Hilbert type in with non-conjugate exponents 23 4. Multiple Hilbert-type inequalities In the previous section, by applying Theorem 2.6 and Lemma 3.2, we obtained inequalities with two non-conjugate parameters. Our aim here is to establish some new specific Hilbert-type inequalities related to k 2 non-conjugate parameters and k non-negative functions. To derive such results, we shall use Theorem 2., Lemma 3. and the kernel K(x,...,x k )= x + + x k (k )nλ on R kn. The corresponding Hilbert-type inequality is given in the following theorem. Theorem 4.. Let n N, k N, k 2, and let parameters p,...,p k, λ and q,...,q k be as in (2.), (2.2) and (2.5). If 0 <λ</(k ), then the inequality f (x ) f k (x k ) R x kn + + x k dx (k )nλ dx k (2π)kn ( S n /n) (k )(λ ) Γ n ((k )nλ) k Γ n(n/p i ) f L p ( ) f k L pk ( ) (4.) holds for all non-negative functions f i L pi ( ), i =,...,k. Proof. First, we consider a simpler special case of the functions involved in (4.). Namely, suppose that f 2,...,f k are symmetric-decreasing functions. To prove our result, we rewrite Theorem 2. with Ω =, K(x,...,x k ) = x + + x k (k )nλ, F i (x,...,x k ) and φ ij (x j )= x j Aij, where n p i p, i = j, i A ij = n, i j, q i p j (4.2) and with Lebesgue measures dx i, for i, j =,...,k. Then the left-hand side of (2.7) becomes f (x ) f k (x k ) L = R x kn + + x k dx (k )nλ dx k, (4.3) while the right-hand side of this inequality is the product of k + factors, R = I /q I /q k k I λ k+, (4.4) where and I i = Rkn x i n/p i j i x j n/pj x + + x k I k+ = R kn (k )nλ f pi k x i n/p i x + + x k (k )nλ i (x i )dx dx k, i =,...,k, f pi i (x i )dx dx k.
24 A. Čižmešija, I. Perić and P. Vuković Before calculating these integrals, observe that, from (2.), (2.2) and (2.5), we find that 0 < n/p i <nand n +(k )nλ = n ( ) + λ = n ( ) + p j p j q j j i j i j i =(k )n + n > (k )n, q j j i for all i {,...,k}. Moreover, the conditions from the statement of Theorem 4. also imply that 0 < (k )nλ<n. Therefore, applying Fubini s theorem, Lemma 3. and (3.2), for i =,...,k, respectively, we obtain I i = x i n/p i (x i ) i f p i R (k )n = Γ n(n (k )nλ) j i Γ n(n n/p j ) Γ n (kn j i n/p j (k )nλ) x i (n/p j i x j n/pj x + + x k (k )nλ dx dx i dx i+ dx k dx i i )+(k )n j i (n/pj) (k )nλ f pi i (x i )dx i (2π) kn = Γ n ((k )nλ) k j= Γ n(n/p j ) f i pi L p i ( ). (4.5) To estimate the last integral, I k+, in (4.4) we use the assumption that the functions f 2,...,f k are symmetric-decreasing. Hence, we can use (3.5), so f pi i (x i ) n S n x i n f i pi L p i ( ) holds for all x i, x i 0. Again, according to Fubini s theorem, Lemma 3. and the identity (3.2), by a procedure similar to that used in (4.5) we obtain I k+ = = = ( ) k n k Rkn x S n f i pi n/p k i=2 x i (n/p i ) n p L p i ( ) f x i=2 + + x k (k )nλ (x )dx dx k ( ) k n k S n f i pi L p i ( ) i=2 x n/p ( n S n (x ) f p R (k )n k i=2 x i n/pi x + + x k (k )nλ dx 2 dx k dx ) k Γ n (n (k )nλ) k i=2 Γ n(n n/p i ) Γ n (kn k i=2 (n/p i) (k )nλ) ( n ) k (2π) kn S n Γ n ((k )nλ) k Γ n(n/p i ) f i pi L p i ( ) f i pi L p i ( ). (4.6)
Inequalities of the Hilbert type in with non-conjugate exponents 25 Now we arrive at inequality (4.) for this case by combining (4.3) (4.6). To complete the proof, we need to consider the general case, that is, arbitrary non-negative functions f 2,...,f k. Since the function x x (k )nλ is symmetric-decreasing and vanishes at infinity, by the general rearrangement inequality we have f (x ) f k (x k ) R x kn + + x k dx (k )nλ dx k f (x ) fk (x k) R x kn + + x k dx (k )nλ dx k (2π)kn ( S n /n) (k )(λ ) Γ n ((k )nλ) k Γ n(n/p i ) f L p ( ) f k L pk ( ) = (2π)kn ( S n /n) (k )(λ ) Γ n ((k )nλ) k Γ n(n/p i ) f L p ( ) f k L pk ( ). As in the proof of Theorem 3.4, here we used the fact that f2,...,fk decreasing functions and that the mapping f f is norm-preserving. are symmetric- Remark 4.2. Note that the proof of Theorem 4. is, in fact, based on Selberg s integral formula (3.). Some further applications of this formula can be found in [2], for example. Setting k = 2 and k = 3 in Theorem 4., we get the following corollaries. Corollary 4.3. Let n N, p> and q> be such that /p +/q >, and let λ be defined by (.2). Then f(x)g(y) x + y nλ dx dy (2π)2n ( S n /n) λ Γ n (nλ)γ n (n/p)γ n (n/q) f L p ( ) g L q ( ) (4.7) holds for all non-negative functions f L p ( ) and g L q ( ). In particular, if n =, then (4.7) becomes f(x)g(y) x + y λ dx dy 2λ π B(/2p, /2q ) B(/2p, /2q) Γ ( 2 2 λ) Γ ( 2 λ) f L p (R) g L q (R). Corollary 4.4. Let n N and parameters p, p 2, p 3, λ, q, q 2 and q 3 be as in (2.), (2.2) and (2.5). If 0 <λ< 2, then R 3n f(x)g(y)h(z) dx dy dz x + y + z 2nλ (2π) 3n ( S n /n) 2(λ ) Γ n (2nλ)Γ n (n/p )Γ n (n/p 2 )Γ n (n/p 3 ) f L p ( ) g L p 2 ( ) h L p 3 ( ) (4.8)
26 A. Čižmešija, I. Perić and P. Vuković holds for all non-negative functions f L p ( ), g L p2 ( ) and h L p3 ( ).In particular, if n =, then (4.8) reads f(x)g(y)h(z) dx dy dz x + y + z 2λ 2 2(λ ) π Γ ( 2 λ) Γ (λ) 3 Γ (/2p i ) Γ (/2p i ) f L p (R) g L p 2 (R) h L p 3 (R). Remark 4.5. Note that F i, i =,...,k, in all the presented applications of Theorem 2. that we considered, while Theorem 2.6 was applied with F = G. Obviously, according to the conditions from the statements of these theorems, we can use any other non-negative functions F i and, consequently, take the infimum of the righthand sides of the inequalities obtained over all such functions. Therefore, to conclude this paper, we mention the following open problem: can this approach give sharp Hilbert-type inequalities, that is, do there exist such functions F i that the related inequalities are obtained with the best possible constants on their right-hand sides? Acknowledgements. This research is supported by the Croatian Ministry of Science, Education and Sports, grant no. 058-70889-050. References. F. F. Bonsall, Inequalities with non-conjugate parameters, Q. J. Math. (2) 2 (95), 35 50. 2. L. Grafakos and C. Morpurgo, A Selberg integral formula and applications, Pac. J. Math. 9 (999), 85 94. 3. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd edn (Cambridge University Press, 967). 4. V. Levin, On the two-parameter extension and analogue of Hilbert s inequality, J. Lond. Math. Soc. (936), 9 24. 5. E. H. Lieb, Sharp constants in the Hardy Littlewood Sobolev and related inequalities, Ann. Math. 8 (983), 349 374. 6. E. H. Lieb and M. Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, Volume 4 (American Mathematical Society, Providence, RI, 200). 7. D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and new inequalities in analysis (Kluwer Academic, Dordrecht, 993). 8. E. M. Stein, Singular integrals and differentiability properties of functions (Princeton University Press, 970). 9. E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, J. Math. Mech. 7 (958), 503 54. 0. B. Yang, I. Brnetić, M. Krnić and J. Pečarić, Generalization of Hilbert and Hardy Hilbert integral inequalities, Math. Inequal. Appl. 8 (2005), 259 272.