Narrow Results

In this paper, by using the atomic decomposition theory of Hardy space H¹(ℝⁿ) and weak Hardy space WH¹(ℝⁿ), we give the boundedness properties of some operators with variable kernels such as singular integral operators, fractional integrals and parametric Marcinkiewicz integrals on these spaces, under certain logarithmic type Lipschitz conditions assumed on the variable kernel Ω(x, z).

Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a specific medium.

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Gasdynamic equations for fractional systems are derived from the generalized transport equation.

In this paper we consider the electric multipole moments of fractal distribution of charges. To describe fractal distribution, we use the fractional integrals. The fractional integrals are considered as approximations of integrals on fractals. In the paper we compute the electric multipole moments for homogeneous fractal distribution of charges.

In this paper, we use the generalized fractional derivative in order to study the fractional differential equation associated with a fractional Gaussian model. Moreover, we propose new properties of generalized differential and integral operators. As a practical application, we estimate the order of the derivative of the fractional Gaussian models by solving an inverse problem involving real data on the dengue fever outbreak.

The Hermite–Hadamard (HH)-type inequality plays a very important role in the fields of basic mathematics and applied mathematics. In recent years, many scholars have expanded and improved it. Although we have achieved some research results about HH-type inequality, the research on discrete HH-type inequalities has just begun, and a lot of work needs to be improved. In this paper, we introduce $(s,m)$-convex function and present discrete HH-type inequalities on time scale with discrete substitution method. In addition, the Hermite–Hadamard–Fejér(HHF)-type inequalities on time scale will be obtained, where the integrand is $\varphi \phi$, $\varphi$ is $(s,m)$-convex function on $[a,b]$ and $\phi$ is symmetric with respect to $\frac{a+mb}{2}$, our results in some special cases yield the well-known classic HHF-type inequalities. Finally, through the discrete substitution method, we get discrete fractional HH-type inequality and discrete fractional HHF-type inequality for $(s,m)$-convex function.

In this paper, we introduce $(s,m)$-convex function, and obtain a new identity by the method called integrating by parts. Based on the identity, many Ostrowski type inequalities are presented through the Hölder’s inequality and the well-known power-mean inequality. Under certain conditions, the results we obtained can be transformed into the classical results. Of course, at the end of the paper, some examples are given to support the main results.

Based on the pseudo-order relation, we introduce the concept of left and right $\hslash$-preinvex interval-valued functions (LR-$\hslash$-PIVFs). Further, we establish the Hermite–Hadamard and Hermite–Hadamard–Fejér-type estimates for LR-$\hslash$-PIVFs using generalized fractional integrals. Finally, an example of interval-valued fractional integrals is provided to illustrate the validity of the results derived herein. Our results not only extend some existing inequalities for Hadamard, Riemann–Liouville, and Katugampola fractional integrals, but also provide new insights for future research on generalized convexity and IVFs, among others.

In this paper, we first obtain a new identity for time scales by using a weighted kernel. Then, by using this equality, we prove a weighted Čeby šev inequality. Moreover, we establish a weighted Ostrowski-type inequality by using a method which is different from among in the literature.

The authors prove the Hardy-Littlewood-Sobolev theorems for generalized fractional integrals L^-α/2 for 0<α<n/m, where L is a complex elliptic operator of arbitrary order 2m on ℝⁿ.

Some experiments for determining transport properties are based on distributions data of paths integrals of particles undergoing displacement. Unfortunately occurrence of abnormally long immobile periods renders the celebrated Feynman-Kac equation inappropriate to rule the joint density of positions and path integrals of walkers. Nevertheless, a natural partition of the distribution of particles (mobile and immobile) may then be reunified by a fractional integral operator. An adequate Feynman-Kac type's equation follows from mass conservation principle.