We give a necessary and sufficient triangular condition characterizing the toroidal pseudodifferential operators on $\mathbb T^d$ that preserve the positive-frequency cone $\mathbb N_0^d$ and hence act on holomorphic boundary values on the polydisk. For symbols of type $(1,0)$, we prove boundedness on holomorphic Besov and Triebel--Lizorkin spaces throughout the quasi-Banach range. For $S^m_{\rho,\delta}$, $0\leq\delta<\rho\leq1$, we obtain the critical loss $d(1-\rho)|1/p-1/2|$, together with endpoint estimates. A positive-cone oscillatory multiplier proves sharpness of the loss and necessity of the Besov endpoint condition $q\leq t$. As an application, we prove well-posedness for a first-order holomorphic differential operator on these scales.
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpr...
We establish a sharp well-posedness and norm inflation theory for the Camassa--Holm equation in critical Triebel--Lizorkin $F^{1+1/p}_{p,q}(\mathbb{T})$. At the endpoint $p=1$, we prove local Hadamard well-posedness for $1\le q<\infty$. In contrast, we prove norm inflation for $1<p<\infty$ and $1\le q\le\infty$. We als...
We introduce the non-symmetric heat and Poisson semigroups associated with the Heckman-Opdam Laplacian in the compact setting. Based on the Poisson semigroup, we study several Littlewood-Paley $g$-functions in the spirit of Stein's work for compact Lie groups and prove their $L^{p}$-boundedness for $1<p\leq 2$ and for...
In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if...
R. Alvarado, Efstathios Konstantinos Chrontsios Garitsis· 0 citations
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1<p<\infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available,...
M. Mišur· 0 citations
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