Narrow Results

In this paper, we explore a broad extension of Minkowski-type inequalities to the left-sided fractal-fractional integral operator. Our findings serve not only to generalize previously established results but also to introduce a significant array of new inequalities applicable to fractal-fractional, fractional, and classical integrals. These results deepen the understanding of fractal-fractional integral operators and expand the scope of their applications across different mathematical frameworks. The study concludes with several practical applications.

The paper is devoted to the analysis of a drift-diffusion-Schrödinger–Poisson (DDSP) system. From the physical point of view, it describes the transport of a quasi-bidimensional electron gas confined in a nanostructure. Existence, uniqueness and long-time behavior of a weak solution were already obtained in Ref. 8 for constant scalar diffusion matrices. Here, we develop an L log L existence theory for the DDSP system for a general class of smooth diffusion matrices. Our argument relies on a Trudinger estimate for the entropy functional and a sharp bound on the Hamiltonian's spectrum.

This paper investigates certain new weighted Young- and Pólya–Szegö-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. A large quantity of usable classical inequalities in the literature are included in the main results of this paper. Meanwhile, two types of new generalized weighted fractional integral operators are introduced to establish some new weighted Young- and Pólya–Szegö-type inequalities. As applications, several estimates of Chebyshev-type weighted unified fractional integral inequalities with two unknown functions are obtained by employing the Heaviside unit step function. Finally, some relations between main results and known inequalities for different kinds of fractional integral operators are provided.

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Hölder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Hölder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Hölder inequality and classic Hölder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite-dimensional framework.

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian⁷ and Poissonian¹² cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

In this paper we give an extension of Young inequality establishing lower and upper bound.