The Improvement of Hölder’s Inequality with r -Conjugate Exponents Relative to the L p -Norm

This paper investigates Hölder’s inequality under the condition of r -conjugate exponents in the sense that ∑ k = 1 s 1 / p k = 1 / r . Successively, we have, under r -conjugate exponents relative to the L p -norm, investigated generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized s -order Hölder’s inequality which is an expansion of the known Hölder’s inequality.

GENERALIZED BESSEL AND FRAME MEASURES

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

On the multidimensional Hilbert-type inequalities involving the hardy operator

This paper deals with the multidimensional Hilbert-type inequalities involving the Hardy operator and homogeneous kernels. The main results are established in the setting with the non-conjugate exponents. After reduction to the conjugate case, the inequalities with the best possible constant factors are obtained in some general cases. As an application, some particular settings are considered in order to obtain the multidimensional extensions of some recent results, known from the literature.

INEQUALITIES OF THE HILBERT TYPE IN $\mathbb{R}^{n}$ WITH NON-CONJUGATE EXPONENTS

In this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ 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 Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.