Related papers: Two-weight, two-parameter fractional integration w…

Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the…

Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex plane such that $\omega$ admits the doubling property $\sup_{0\le r<1}\frac{\int_r^1 \omega(s)\,ds}{\int_{\frac{1+r}{2}}^1 \omega(s)\,ds}<\infty$. Consider the one…

We study a two-component asymmetric simple exclusion process (ASEP) that is equivalent to the ASEP with second-class particles. We prove self-duality with respect to a family of duality functions which are shown to arise from the reversible…

For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where…

In this paper, by using the decomposition theorem for weak Hardy spaces, we will obtain the boundedness properties of some integral operators with variable kernels on these spaces, under some Dini type conditions imposed on the variable…

Let $v,~\omega_1, ~\omega_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(\omega_1, \omega_2)\in RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in martingale…

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the…

In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to…

Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx, \quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is…

We consider the fractional Laplacian operator $(-\Delta)^s$ (let $ s \in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar product between a…

We study trace inequalities of the type $$ \| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $d\sigma$ and…

We establish a general identity (Theorem 1.2) that implies both the $L^{p}$-Hardy identities and the $L^{p}$-Caffarelli-Kohn-Nirenberg identities (Theorems 1.3 and 1.4) and $L^{p}$-Hardy inequalities and the…

For a limited range of indices $p$, we obtain $L^p(\mathbb{R}^n)$ boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical H\"ormander smoothness estimate. These operators are assumed to be…

There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…

Two classes of fractional type variable weights are established in this paper. The first kind of weights ${A_{\vec p( \cdot ),q( \cdot )}}$ are variable multiple weights, which are characterized by the weighted variable boundedness of…

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a…

Bounded and compact generalized weighted composition operators acting from the weighted Bergman space $A^p_\omega$, where $0<p<\infty$ and $\omega$ belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided doubling…

In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with…

This article investigates the Fourier extension operator associated with the fractional surface $(\xi,|\xi|^{\alpha})$ for $\alpha\geq 2$. We show that the relevant $L^p\to L^q$ Fourier extension inequality possesses extremals for all…

The Lp-Liouville property of a non-local operator A is investigated via the associated Dirichlet form. We will show that any non-negative continuous Lp E-subharmonic functions are constant under a quite mild assumption on the kernel of E if…