Related papers: On good-$\lambda$ inequalities for couples of meas…
We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in…
For $c\in(1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{\lambda \in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x)…
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the…
This paper aims to characterize the function appearing in the weighted Hermite-Hadamard inequality. We provide improved inequalities for the weighted means as applications of the obtained results. Modifications of the weighted…
This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator…
In this paper we will establish necessary and sufficient conditions for a Laplace-Carleson embedding to be bounded for certain spaces of functions on the positive half-line. We will use these results to characterise weighted (infinite-time)…
A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.
We define a generalized dyadic maximal operator involving the infinite product and discuss weighted inequalities for the operator. A formulation of the Carleson embedding theorem is proved. Our results depend heavily on a generalized…
New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex.…
We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari anc C. Fefferman are proved.
The $\lambda$-superposition calculus is a successful approach to proving higher-order formulas. However, some parts of the calculus are extremely explosive, notably due to the higher-order unifier enumeration and the functional…
We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties…
We prove fractional Leibniz rules and related commutator estimates in the settings of weighted and variable Lebesgue spaces. Our main tools are uniform weighted estimates for sequences of square-function-type operators and a bilinear…
We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic…
Motivated by previous work leveraging factorizations of second- and fourth-order differential operators, a general integral inequality involving higher order derivatives is proven by elementary means. It is then shown how this framework…
This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of…
We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the…
In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure spaces. We give the characterization of weights for the bilinear Hardy inequality to hold on general metric measure spaces having polar…
This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative…
We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic…