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A multiple Hilbert-type integral inequality with a non-homogeneous kernel. (English) Zbl 1302.47006

For the Hardy-Hilbert integral operator \(T:L^p(0,\infty) \rightarrow L^p(0,\infty)\) defined by \[ Tf(y)=\int_0^\infty \frac{1}{x+y} f(x) \,dx, \, y\in (0,\infty), \] we have the following.
Theorem. Suppose that \(n\in \mathbb{N}\backslash \{ 1\}\), \(p_i\in \mathbb{R}\backslash \{0, 1\}\), \(i=1,\dots ,n\), \(\sum_{i=1}^n\frac{1}{p_i}=1\), \(\frac{1}{q_i}=1-\frac{1}{p_i}\). Suppose that, for any \((\lambda_1, \dots , \lambda_n)\in \mathbb{R}^n\), \(\lambda_n=\sum_{i=1}^{n-1}\lambda_i =\lambda/2\), \(k_\lambda (x_1, \dots , x_n)({\geq 0})\) is a measurable function of \(-\lambda\)-degree in \(\mathbb{R}^n_+\) such that \[ 0<k_\lambda =\int_{\mathbb{R}^{n-1}_+} k_\lambda (u_1, \dots , u_{n-1}, 1) \prod_{j=1}^{n-1} u_j^{\lambda_j-1} \,du_1 \cdots du_{n-1} < \infty. \] If \(f_i (\geq 0) \in L_{\phi_i}^{p_i}(0,\infty)\), \(\|f\|_{p_i, \phi_i} >0\) \((i=1, \dots ,n)\), then
(i)
for \(p_i>1\), \(i=1, \dots ,n\), we have \(\|T\|=k_\lambda\) and the following equivalent inequalities: \[ \|T(f_1, \dots, f_{n-1})\|_{q_n,\phi_n^{1/(1-p_n)}} < k_\lambda \prod_{j=1}^{n-1}\|f\|_{p_i, \phi_i}, \]
\[ (T(f_1, \dots, f_{n-1}), f_n) < k_\lambda \prod_{j=1}^{n}\|f\|_{p_i, \phi_i}, \] where \(\|f\|_{p, \phi}=\left( \int_0^\infty \phi(x)f^p(x)\, dx\right)^{1/p}\) and the constant factor \(k_\lambda\) is the best possible,
(ii)
for \(0<p_1<1\), \(p_i<0\), \(i=2, \dots ,n\), using the formal symbols as in the case of (i), we have the equivalent reverses of the above-mentioned inequalities with the same best constant factor.
Those results are applied in some particular cases. Let the following assumptions are valid: \(\lambda >0\), \(\lambda_i = \frac{\lambda}{r_i}\), \(i=1, \dots ,n\), \(r_n=2\), \(\sum_{i=1}^n \frac{1}{r_i}=1\).
(a)
If \(k_\lambda(x_1, \dots , x_n)=\frac{1}{(\sum_{i=1}^n x_i )^\lambda}\), then \(k_\lambda=\frac{1}{\Gamma(\lambda)} \prod_{i=1}^n\Gamma(\frac{\lambda}{r_i}).\)
(b)
If \(k_\lambda(x_1, \dots , x_n)=\frac{1}{(\max_{1\leq j\leq n} (x_i) )^\lambda}\), then \(k_\lambda=\frac{ \prod_{i=1}^n r_i}{\lambda^{n-1}}.\)
(c)
If \(k_\lambda(x_1, \dots , x_n)=(\min_{1\leq j\leq n} (x_i) )^\lambda\), then \(k_\lambda=\frac{ \prod_{i=1}^n r_i}{\lambda^{n-1}}.\)

MSC:

47A07 Forms (bilinear, sesquilinear, multilinear)
26D15 Inequalities for sums, series and integrals
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References:

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