arXiv subjects
Publications and source records attributed to Tiago Picon.
In this work, we present a nonhomogeneous version of the classical div-curl type estimates in the setup of elliptic system of complex vector fields with constant coefficients on local Hardy space $h^1$. As an application, we obtain a decomposition of the local $bmo$ space via a family of vector fields depending on div-curl terms.
Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ \mu $ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $\mu$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=\mu$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $\mu$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields.
In this work, we establish higher-order div-curl type estimates in the sense of Coifman, Lions, Meyer & Semmes, in a local setting for elliptic homogeneous linear differential operators with smooth coefficients acting on localizable Hardy spaces. Our results imply and extend previously known estimates for first-order operators associated with elliptic systems and complexes of vector fields. As tools of independent interest, we develop a new smooth atomic decomposition for localizable Hardy-Sobolev spaces and prove a Poincar\'e-type inequality in this framework.
In this work, we present sufficient cancellation conditions for the boundedness of inhomogeneous Calder\'on-Zygmund type operators on local Hardy spaces defined over spaces of homogeneous type in the sense of Coifman & Weiss for $ 0<p\leq 1 $. A new approach to atoms and molecules for local Hardy spaces in this setting are introduced with special moment conditions.
In this note we give an upper bound on the Hausdorff dimension of removable setsfor elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of Frostman's lemma in Euclidean space with an additional power decay.
In this note we address the continuity of strongly singular Calder\'on-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^{\lambda}(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples that falls into this scope are pseudodifferential operators on the H\"ormander classes $OpS^{m}_{\sigma,\mu}(\mathbb{R}^n)$ with $0<\sigma \leq 1$, $0 \leq \mu <1$, $\mu \leq \sigma$ and $m\leq -n(1-\sigma)/2$.
In this work, we present new results on solvability of the equation $A^{*}(D)f=\mu$ for $f \in L^{p}$ and positive measure data $\mu$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $\mu$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.
In this work we present necessary cancellation conditions for the continuity of linear operators in $h^p(\mathbb{R}^n)$, $0<p\leq 1$, that map atoms into pseudo-molecules. Our necessary condition, expressed in terms of the $T^{\ast}$ condition, is the same as the one recently proved sufficient in [3], thus providing a necessary and sufficient cancellation condition for the boundedness of inhomogeneous Calder\'on--Zygmund type operators
In this work we present a new approach to molecules on Goldberg's local Hardy spaces $h^p(\mathbb{R}^n)$, $0<p\leq1$, assuming an appropriate cancellation condition. As applications, we prove a version of Hardy's inequality and improved continuity results for inhomogeneous Calder\'on-Zygmund operators on these spaces.
In this work, we establish results on the continuity of strongly singular Calder\'on-Zygmund operators of type $\sigma$ on Hardy spaces $H^p(\mathbb{R}^n)$ for $0<p\leq 1$ assuming a weaker $L^{s}-$type H\"ormander condition on the kernel. Operators of this type include appropriated classes of pseudodifferential operators $OpS^{m}_{\sigma,b}(\mathbb{R}^n)$ and operators associated to standard $\delta$-kernels of type $\sigma$ introduced by \'Alvarez and Milman. As application, we show that strongly singular Calder\'on-Zygmund operators are bounded from $H^{p}_{w}(\mathbb{R}^n)$ to $L^{p}_{w}(\mathbb{R}^n)$, where $w$ belongs to a special class of Muckenhoupt weight.
In this work, we investigate the limit case $p=1$ of the classical Stein--Weiss inequality for the Riesz potential.We present a characterization for a special class of vector fields associated to cocanceling operators introduced by Van Schaftingen in arXiv:1104.0192. As an application, we recover some div-curl inequalities found in the literature. In addition, we discuss a two-weight inequality with general weights in the scalar case, extending the previous result of Sawyer to this case.
Consider $A(x,D):C^{\infty}(\Omega,E) \rightarrow C^\infty(\Omega,F)$ an elliptic and canceling linear differential operator of order $\nu$ with smooth complex coefficients in $\Omega \subset \mathbb{R}^{N}$ from a finite dimension complex vector space $E$ to a finite dimension complex vector space $F$ and $A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation $A^{*}(x,D)v=f$ (in the distribution sense) for a given distribution $f$; more precisely we show that any $x_0\in\Omega$ is contained in a neighborhood $U\subset \Omega$ in which its continuous solvability is characterized by the following condition on $f$: for every $\epsilon>0$ and any compact set $K \subset \subset U$, there exists $\theta=\theta(K,\epsilon)>0$ such that the following holds for all smooth function $\varphi$ supported in $K$: \begin{equation}\nonumber \left| f(\varphi) \right| \leq \theta\|\varphi\|_{W^{\nu-1,1}} + \epsilon\|A(x,D) \varphi\|_{L^{1}}, \end{equation} where $W^{\nu-1,1}$ stands for the homogenous Sobolev space of all $L^1$ functions whose derivatives of order $\nu-1$ belongs to $L^{1}(U)$. This characterization implies and extends results obtained before for operators associated to elliptic complex of vector fields (see \cite{MP}); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator in [4] and [9].
Let $A(D)$ be an elliptic homogeneous linear differential operator of order $\nu$ on $\mathbb{R}^{N}$, $N \geq 2$, from a complex vector space E to a complex vector space F. In this paper we show that if $\ell\in \mathbb{R}$ satisfies $0< \ell <N$ and $\ell \leq \nu$, then the estimate \begin{equation}\nonumber \left(\int_{\mathbb{R}^{N}}| (-\Delta)^{(\nu-\ell)/2}u(x)|^{q}|x|^{-N+(N-\ell)q}\,dx\right)^{1/q}\leq C \|A(D)u\|_{L^{1}} \end{equation} holds for every $u \in C_{c}^{\infty}(\mathbb{R}^{N};E)$ and $1\le q<\frac{N}{N-\ell}$ if and only if $A(D)$ is canceling in the sense of V. Schaftingen [VS]. Here $(-\Delta)^{a/2}u$ is the fractional Laplacian defined as a Fourier multiplier. This estimate extends, implies and unifies a series of classical inequalities discussed by P. Bousquet and V. Schaftingen in [BVS]. We also present a local version of the previous inequality for operators with smooth variables coefficients.}
We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces $\dot{H}^{1,p}(\mathbb{R}^d)$ when $1/p < 1+1/d$. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces $\dot{h}^{1,p}(\mathbb{R}^d)$ in the same range of exponents.
In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset Ω$) to the divergence-type equation $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F,$$ where $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $Ω\subset \mathbb{R}^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\mathbb{R}^N$, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer.