Search results for "Riesz transform"

Integration of functions ranging in complex Riesz space and some applications in harmonic analysis

2015

The theory of Henstock—Kurzweil integral is generalized to the case of functions ranging in complex Riesz space R and defined on any zero-dimensional compact Abelian group. The constructed integral is used to solve the problem of recovering the R-valued coefficients of series in systems of characters of these groups by using generalized Fourier formulas.

Gradient estimates for heat kernels and harmonic functions

2020

Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carr\'e du champ". Assume that $(X,d,\mu,\E)$ supports a scale-invariant $L^2$-Poincar\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse H\"older inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalised Bakry-\'Emery condition. We show that, for $p\in (2,\infty)$, (i), (ii) (iii) are equivalent, wh…

The boundedness of Riesz 𝑠-transforms of measures in ℝⁿ

1996

Let μ \mu be a finite nonzero Borel measure in R n \mathbb {R}^{n} satisfying 0 &gt; c − 1 r s ≤ μ B ( x , r ) ≤ c r s &gt; ∞ 0 &gt;c^{-1}r^{s}\le \mu B(x,r)\le cr^{s} &gt;\infty for all x ∈ spt ⁡ μ x\in \operatorname {spt}\mu and 0 &gt; r ≤ 1 0 &gt; r\le 1 and some c &gt; 0 c &gt;0 . If the Riesz s s -transform C s , μ ( x ) = ∫ y − x | y − x | s + 1 d μ y \begin{equation*}{\mathcal {C}}_{s,\mu }(x)=\int \frac {y-x}{|y-x|^{s+ 1}}\, d\mu y \end{equation*} is essentially bounded, then s s is an integer. We also give a related result on the L 2 L^{2} -boundedness.

Exudate Segmentation on Retinal Atlas Space

2013

International audience; Diabetic macular edema is characterized by hard exudates. Presence of such exudates cause vision loss in the affected areas. We present a novel approach of segmenting exudates for screening and follow-ups by building an ethnicity based statistical atlas. The chromatic distribution in such an atlas gives a good measure of probability of the pixels belonging to the healthy retinal pigments or to the abnormalities (like lesions, imaging artifacts etc.) in the retinal fundus image. Post-processing schemes are introduced in this paper for the enhancement of the edges of such exudates for final segmentation and to separate lesion from false positives. A sensitivity(recall)…

Riesz transform and vertical oscillation in the Heisenberg group

2023

We study the $L^{2}$-boundedness of the $3$-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. Inspired by the notion of vertical perimeter, recently defined and studied by Lafforgue, Naor, and Young, we first introduce new scale and translation invariant coefficients $\operatorname{osc}_{\Omega}(B(q,r))$. These coefficients quantify the vertical oscillation of a domain $\Omega \subset \mathbb{H}$ around a point $q \in \partial \Omega$, at scale $r > 0$. We then proceed to show that if $\Omega$ is a domain bounded by an intrinsic Lipschitz graph $\Gamma$, and $$\int_{0}^{\infty} \operatorname{osc}_{\Omega}(B(q,r)) \, \frac{dr}{…

The Bochner and Riesz integral representations for the Radon transform

1984