Related papers: L^p-summability of Riesz means for the sublaplacia…

Let $\h_n$ be the $(2n+1)$-dimensional Heisenberg group. and let ${\cal L}_\alpha$ be the sublaplacian of the Lie algebra of $\h_n$ A new spherical harmonics with its orthogonal polynomial properties is presented for the group.

In this paper, we study the almost everywhere convergence problem for the Bochner--Riesz means $S_t^\delta f$ for $f\in L^p(\mathbb R^d)$ in the subcritical range \[ 0\le \delta < \delta(d,p):=d\Big(\frac12-\frac1p\Big)-\frac12, \qquad…

We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is…

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann…

We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$,…

Let $M$ be a complete non-compact manifold satisfying the volume doubling condition, with doubling index $N$ and reverse doubling index $n$, $n\le N$, both for large balls. Assume a Gaussian upper bound for the heat kernel, and an…

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$…

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order $\alpha$ using classical Lobatto-type polynomials when $\alpha \in (0,1)$ and generalized Jacobi functions (GJF) for arbitrary $\alpha…

Let $L_1$ be a nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the Davies-Gaffney estimates and $L_2$ a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of…

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform operators on metric measure spaces with Riemannian Ricci curvature bounded from below, without any restriction on their dimension. This large class of spaces include…

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both…

We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $\Omega\times \R^+$, where…

We establish improved and sharp $L^p$ estimates for the maximal bilinear Bochner-Riesz means in all dimensions $n\geq 1$. This work extends the results proved by Jeong and Lee \cite{JL}. We also recover the known results for the bilinear…

We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if…

In this work, we study the rate of convergence of the finite element method for the $p(x)$Laplacian ($1< p_1\leq p(x)\leq p_2\leq 2$) in two dimensional convex domains.

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds…

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical…

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Ces\`aro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence…