Related papers: The measures with $L^2$-bounded Riesz transform sa…

In this work we provide 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 $R\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}\,d\mu(y)$…

This work provides 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 $R\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}\,d\mu(y)$ belongs…

In this paper we show that if $\mu$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_\mu$ is bounded in $L^2(\mu)$, and $B\subset\mathbb R^{n+1}$ is a ball with $\mu(B)\approx…

In this paper we explore the connection between quantitative rectifiability of measures and the $L^2$ boundedness of the codimension one Riesz transform. Among other things, we prove the following. Let $\mu$ be a Radon measure in $\mathbb…

Let $\mu$ be a Radon measure on the $n$-th Heisenberg group $\mathbb{H}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on $\mathbb{H}^n$ is $L^2(\mu)$-bounded, and if $\mu(F)=0$ for all Borel sets…

Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $\Omega$ satisfies…

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…

In this paper, we prove that for $s\in(1,2)$ there exists no totally lower irregular finite positive Borel measure $\mu$ in $\R^2$ with\break $\mathcal H^s(\supp\mu)<+\infty$ such that $\|R\mu\|\ci{L^\infty(m_2)}<+\infty$, where…

We prove that, for totally irregular measures $\mu$ on $\mathbb{R}^{d}$ with $d\geq3$, the $(d-1)$-dimensional Riesz transform $$ T_{A,\mu}^{V}f(x) = \int_{\mathbb{R}^d} \nabla_{1}\mathcal{E}_{A}^{V}(x,y) f(y) \, d \mu(y) $$ adapted to the…

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix $A$ with real, merely bounded, and possibly non-symmetric coefficients. If $$\omega_A(r)=\sup_{x\in \mathbb{R}^{n+1}}…

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium…

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density…

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 for anymeasure $\mu$ in $\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\displaystyle{\iint_0^\infty(\frac{\mu(B(x,r))}{r^s})^2\,\frac{dr}{r}d\mu(x)}$ is comparable to…

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…

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 $\nu=f_{\#}(\mu\times\mu\times\mu)$…

In this note it is shown that if $\mu$ is an $n$-Ahlfors regular measure in $\mathbb R^{n+1}$ such that the $n$-dimensional Riesz transform is bounded in $L^2(\mu)$ and the so-called BAUPP (bilateral approximation by unions of parallel…

In this note, we study both the Riesz and reverse Riesz transforms on broken line. This model can be described by $(-\infty, -1] \cup [1,\infty)$ equipped with the measure $d\mu = |r|^{d_{1}-1}dr$ for $r \le -1$ and $d\mu = r^{d_{2}-1}dr$…

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform $d\Delta ^{-\frac{1}{2}}$ on…

We construct an example of a purely unrectifiable measure $\mu$ in $\mathbb{R}^d$ for which the singular integrals associated to the kernels $\displaystyle{K(x)=\frac{P_{2k+1}(x)}{|x|^{2k+d}}}$, with $k\geq 1$ and $P_{2k+1}$ a homogeneous…