arXiv subjects

Marcelo F. de Almeida

Publications and source records attributed to Marcelo F. de Almeida.

5 recordsLinked to original sources

arXiv2026

Nonlinear parabolic thin sets and parabolic Wolff inequalities

We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling $δ_c(x,t)=(cx,c^2t)$ and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic $(α,q)$-thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of $(α,2)$-thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points $z_0\in\partialΩ$ for the heat operator $\partial_t-Δ$ and for the degenerate operator $\mathscr{L}a=\partial_t(|y|^a\cdot)-\operatorname{div}(|y|^a\nabla\cdot)$ in $Ω\subset\mathbb{R}^{d+1}$ are negligible with respect to the thermal capacity $\mathrm{cap}^{\mathcal T}$ and the parabolic Bessel capacity $C_{α,2}$, respectively.

math.AP↗

arXiv2023

Fourier transform decay of distributions in Hardy-Morrey spaces

In this paper we establish decay estimates for Fourier transform on Hardy-Morrey spaces and its localizable version. Our work include some aspects to these spaces linked up with pointwise Fourier estimates, in particular a natural approach on cancellation moment conditions. As application, we discuss the optimality for continuity of Fourier multipliers and pseudodifferential operators in Hardy-Morrey spaces.

math.AP↗

arXiv2021

Adams' trace principle on Morrey-Lorentz spaces over $β$-Hausdorff dimensional surfaces

In this paper we strengthen to Morrey-Lorentz spaces the famous trace principle introduced by Adams. More precisely, we show that Riesz potential $I_α$ is continuous \begin{equation} \Vert I_αf\Vert_{\mathcal{M}_{q, \infty}^{λ_{\ast}}(dμ)}\lesssim \Arrowvertμ\Arrowvert_β^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^λ(dν)}\nonumber\\[0.02in] \end{equation} if and only if the Radon measure $dμ$ supported in $Ω\subset \mathbb{R}^n$ is controlled by $$\Arrowvertμ\Arrowvert_β=\sup_{x\in\mathbb{R}^n,\,r>0}r^{-β}μ(B(x,r))<\infty$$ provided that $1<p<q<\infty$ satisfies $n-αp<β\leq n,\; α=\frac{n}λ-\fracβ{λ_\ast}\; \text{ and }\;\frac{λ_\ast}{q}\leq \fracλ{p}\nonumber\,$. Our result provide a new class of functions spaces which is larger than previous ones, since we have strict continuous inclusions $\dot{B}_{p,\infty}^{s}\hookrightarrow L^{λ, \infty}\hookrightarrow \mathcal{M}_{p}^λ\hookrightarrow\mathcal{M}_{p, \infty}^λ \nonumber $ as $1<p<λ<\infty$ and $s\in\mathbb{R}$ satisfies $\frac{1}{p}-\frac{s}{n}=\frac{1}λ$. If $dμ$ is concentrated on $\partial\mathbb{R}^n_+$, as a byproduct we get Sobolev-Morrey trace inequality on half-spaces $\mathbb{R}^n_+$ which recovers the well-known Sobolev-trace inequality in $L^p(\mathbb{R}^n_+)$. Also, by a suitable analysis on non-doubling Caderón-Zygmund decomposition we show that \begin{equation} \Vert M_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\,\sim\, \Vert I_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\nonumber \end{equation} provided that $μ(B_r(x))\sim r^β$ on support $\text{spt}(μ)$ and $n-α<β\leq n$ with $0<α<n$. This result extends the previous ones.

math.AP↗

arXiv2021

Nonlinear boundary problem for Harmonic functions in higher dimensional Euclidean half-spaces

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-1}u+f \; & \text{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $ \mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t>0\}$ for $n\geq 2$. More precisely, inspired at \cite{deAlmeida1} and \cite{Quittner} we introduce a new functional space based in weak-Morrey spaces and we shown existence of positive solutions $u$ to the above problem when inhomogeneous term $f\in\text{weak-}\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and potential $V\in \text{week-}\mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ are sufficiently small in the natural $n/(n-1)<ρ<\infty$. Our theorems recover the range $(n+1)/(n-1)\leq ρ<\infty$ and immediately imply in solvability of the equivalent nonlocal half-Laplacian problem $(-Δ)^{{1}/{2}}u=Vu+b\vert u\vert^{ρ-1}u+ f (x)$ for $f$ and potential $V$ rough than previous ones, in view of strictly inclusions $L^λ\varsubsetneq\mathcal{M}^λ_{p} \varsubsetneq \text{week-}\mathcal{M}^λ_{p}$ for $1<p<λ<\infty$. Also, from Campanato's lemma we conclude that $u\in C^{0,α}_{loc}( \overline{\mathbb{R}^{n+1}_+})$ is locally Hölder continuous, for $f\in\mathcal{M}_{p}^{n{(ρ-1)}/ρ}(\mathbb{R}^{n})$ and $V\in \mathcal{M}^{n}_{\ell}(\mathbb{R}^{n})$ in Morrey spaces.

math.AP↗

arXiv2014

On the heat equation with nonlinearity and singular anisotropic potential on the boundary

This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate qualitative properties of solutions like self-similarity, positivity and symmetry around the axis $\overrightarrow{Ox_{n}}$.

math.AP↗