Search results for "singular integrals"
showing 5 items of 5 documents
Singular integrals on regular curves in the Heisenberg group
2019
Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all …
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}{…
A symmetric Galerkin BEM for plate bending analysis
2009
Abstract The Symmetric Galerkin Boundary Element Method is employed in thin plate bending analysis in accordance with the Love–Kirchhoff kinematical assumption. The equations are obtained through the stationary conditions of the total potential energy, written for a plate whose boundary is discretized in boundary elements. Since the matrix coefficients are made up as double integrals with high order singularities, a strategy is shown to compute these coefficients in closed form. Furthermore, in order to model the kinematical discontinuities and to weight the mechanical quantities along the boundary elements, the Lagrangian quadratic shape functions, rather than C 1 type (spline, Hermitian),…
Off-diagonal estimates for bi-parameter commutators
2022
Tutkimme kahden parametrin singulaari-integraalien kommutaattoreiden rajoit-tuneisuutta iteroitujen avaruuksien[b, T]:Lp1Lp2→Lq1Lq2välillä kunqi, pi∈(1,∞)ja erityisesti fraktionaalisessa tilanteessaqi6=pi. Yhteensä seitsemässä ta-pauksessa yhdeksästä saavutamme täyden karaterisoinnin rajoittuneisuudella symboliabkoskevillaoskillatorisilla testiehdoilla. We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces [b,T]:Lp1Lp2→Lq1Lq2 in the off-diagonal situation qi,pi∈(1,∞) where we allow qi≠pi. Boundedness is fully characterized in terms of oscillatory testing conditions on the function b for a total seven out of the nine possible arrangements of the inte…