相关论文: Square functions of fractional homogeneity and Wol…

In this work we obtain a geometric characterization of the measures $\mu$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}\mu (x) = \int…

We characterize the non-atomic measures $\mu$ for which all Calder\'{o}n-Zygmund operators with antisymmetric kernels of a fixed non-integer dimension $s$ are bounded in $L^2(\mu)$ in terms of a positive quantity, the Wolff energy.

Let $f\in\mathbb R[x,y,z]$ be a fixed non-degenerate quadratic polynomial. Given an $\alpha$-Frostman probability measure $\mu$ supported on $[0,1]$ with $\alpha\in(0,1)$, consider the pushforward measure $\nu=f_{\#}(\mu\times\mu\times\mu)$…

Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\mu$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu)$ in terms of the Wolff energy. This…

We establish a good lambda inequality relating to the distribution function of Riesz potential and fractional maximal function on $\left(\mathbb{R}^n, d\mu\right)$ where $\mu$ is a positive Radon measure which doesn't necessarily satisfy a…

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth…

In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ implies that a nonlinear potential of exponential type…

In this paper it is shown that if $\mu$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty \beta_{\mu,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $\mu$-a.e. $x\in\mathbb R^d$,}$$…

Let $(W,H,\mu)$ be the classical Wiener space where $H$ is the Cameron-Martin space which consists of the primitives of the elements of $L^2([0,1],\,dt)\otimes \R^d$, we denote by $L^2_a(\mu,H)$ the equivalence classes w.r.t. $dt\times…

We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the…

\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $\varphi$ in a weighted Sobolev space and for weight functions $\mu$ of a quite general type \begin{equation*} c_{N,\mu}…

Given a Young function $A$, $n\geq 1$ and $s\in(0,1)$ we consider the energy functional $$ \mathcal{J}_s(u)=(1-s)\iint_{\mathbb{R}^n\times \mathbb{R}^n} A\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right)\frac{dxdy}{|x-y|^n}. $$ Without assuming the…

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{\varphi^2} {|x|^2}d\mu\leq\int_{\mathbb{R}^N}|\nabla \varphi |^2 d\mu+C\int_{\mathbb{R}^N} \varphi^2d\mu,\quad \varphi\in…

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad…

We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials…

We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, where $J_{\nu}$ denotes the Bessel function of the first kind of order $\nu$. As shown by Walther (2002,…

Recently, Kr\"uger and Vlugt [Phys. Rev. E 97, 051301(R) (2018)] have proposed a method to approximate an improper integral $\int_0^\infty \text{d}r\, F(r)$, where $F(r)$ is a given oscillatory function, by a finite-range integral $\int_0^L…

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds:…

We consider the minimizers of the energy $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well…

This paper is motivated by a gauged Schr\"odinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \Delta u(x) + \left(\omega +…