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Tan Duc Do

Publications and source records attributed to Tan Duc Do.

9 recordsLinked to original sources
arXiv2026

A Weighted Bossel-Daners Transfer Principle and a Pure-Power Robin Faber-Krahn Inequality

We prove a weighted Bossel-Daners transfer principle for the first Robin eigenvalue of the $p$-Laplacian under $w=m^{1/p'}$, where $p'=p/(p-1)$. The argument combines double-density isoperimetry with spectral admissibility for singular weights and normalized-flux monotonicity. It uses an exact $BV$ zero-extension formula, an $L^{p'}(m\,dx)$ selection lemma, and a nonatomic rank map that remains well defined on positive-measure level sets. For the singular pair \[ m_b(x)=|x|^b, \qquad w_b(x)=|x|^{b/p'}, \] known power-weight isoperimetry provides the geometric input. We establish spectral admissibility up to the critical exponent $p=N$ and, for the positive radial first eigenfunction $z=z(r)$ on the centered ball $B_R$, derive the integrated singular radial equation and center asymptotics and prove directly that \[ \theta_R'(r)>0 \quad(0 0$. The range $1<p<N$ is complementary to the previously known $p\ge N$ weighted Talenti theory; at the shared endpoint $p=N$, the present proof also permits the singular contact $0\in\partial\Omega$.

math.AP↗
arXiv2026

Mixed Poincar\'e and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces

We prove a fixed-outer-domain content--capacity estimate for every positive codimensional gain on an unbounded complete $2$-PI space. As its principal measure-theoretic consequence, a one-sided ball-growth condition for an arbitrary positive Radon measure $\pi$ yields \[ \pi(K)\lesssim\operatorname{cap}_{2,\omega}(K;\Lambda_0B) \] uniformly for compact $K\subset B$. No doubling or lower-dimensional regularity is imposed on $\pi$. This capacitary domination gives a representative-independent mean-zero trace inequality. On normalized balls, it is equivalent, modulo the ambient Poincar\'e energy, to the mixed $d\omega\,d\pi$ oscillation required in Fefferman--Phong reductions. A standard bounded-overlap localization then yields two-sided global Fefferman--Phong inequalities, cover-independent energy norms, and a concrete realization of the associated homogeneous energy completion. The abstract results are verified for Euclidean $A_2$ weights with generalized Schr\"odinger measure potentials, reverse-H\"older function potentials, Carnot groups, and lower-dimensional singular measures. We compare the local trace conclusion with existing two-weighted Poincar\'e and Sobolev embedding theorems: those routes apply under additional doubling or dimension assumptions on the target measure, whereas our capacity conclusion also supplies absolute continuity with respect to variational capacity and a fixed outer domain. The Euclidean application yields form-domain equivalence, smooth form cores, self-adjoint realization, resolvent energy estimates, and local critical-multiplier bounds. Finally, the method supplies the mixed-measure step missing from a previously published $A_2$ generalized Schr\"odinger argument and gives a fixed-dilate finite-scale $A_2$ extension, for the naturally augmented measure, of a later $A_1$ theory.

math.FA↗
arXiv2026

Ambient Hardy--Littlewood Maximal Functions on Weighted Musielak--Orlicz Spaces over Domains

We study the ambient-domain Hardy--Littlewood maximal operator \[ \mathcal M_\Omega f(x) := \sup_{B\ni x}\frac1{|B|}\int_{B\cap\Omega}|f(y)|\,dy, \qquad x\in\Omega \] on weighted Musielak--Orlicz spaces over a general open set \(\Omega\subset\mathbb R^n\), where the supremum is taken over all Euclidean balls \(B\subset\mathbb R^n\). For a Musielak--Orlicz function \(\varphi\), we use the pointwise lower Matuszewska--Orlicz index \(p_\varphi(\cdot)\) and the lower-index normalization \[ \psi_\varphi(x,t)=\varphi(x,t)^{1/p_\varphi(x)}. \] This factorizes the modular as a weighted variable-exponent modular applied to \(\psi_\varphi(x,|f|)\). Under endpoint lower growth, normalized weighted generalized Orlicz \((A0)\)--\((A2)\) assumptions and an admissible whole-space extension hypothesis for the weight at the lower-index exponent, we prove the boundedness of \[ \mathcal M_\Omega:L^{\varphi(\cdot)}_\omega(\Omega)\to L^{\varphi(\cdot)}_\omega(\Omega). \] For the converse direction we use the natural K\"othe-associate ambient ball condition \(A_\varphi(\Omega)\). Under the local characteristic-function hypothesis, boundedness of \(\mathcal M_\Omega\) implies \(\omega\in A_\varphi(\Omega)\). On the whole space \(\mathbb R^n\), this framework provides a weighted characterization conditional on an associate-to-lower-index product reduction. In the present paper this reduction is verified for uniformly lower-index-power-equivalent models; it remains open for genuinely two-phase growth such as \(t^p+a(x)t^q\). As an application, we prove density of \(C_c^\infty(\mathbb R^n)\) in weighted Musielak--Orlicz--Sobolev spaces.

math.FA↗
arXiv2020

Up-to-boundary pointwise gradient estimates for very singular quasilinear elliptic equations with mixed data

This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad Ω\\ u= 0 \quad & \text{on} \ \partial Ω, \end{cases} where $Ω\subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.

math.AP↗
arXiv2020

Heat kernels of generalized degenerate Schrödinger operators and Hardy spaces

Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + μ$ be the generalized degenerate Schrödinger operator in $L^2_w(\mathbb{R}^d)$ with $d\ge 3$ with suitable weight $w$ and measure $μ$. The main aim of this paper is threefold. First, we obtain an upper bound for the fundamental solution of the operator $L$. Secondly, we prove some estimates for the heat kernel of $L$ including an upper bound, the Hölder continuity and a comparison estimate. Finally, we apply the results to study the maximal function characterization for the Hardy spaces associated to the critical function generated by the operator $L$.

math.FA↗
arXiv2019

Lusin characterisation of Hardy spaces associated with Hermite operators

Let $d \in \{3, 4, 5, \ldots\}$ and $p \in (0,1]$. We consider the Hermite operator $L = -Δ+ |x|^2$ on its maximal domain in $L^2(\mathbb{R}^d)$. Let $H_L^p(\mathbb{R}^d)$ be the completion of $ \{ f \in L^2(\mathbb{R}^d): \mathcal{M}_L f \in L^p(\mathbb{R}^d) \} $ with respect to the quasi-norm $ \|\cdot\|_{H_L^p} = \|\mathcal{M}\cdot\|_{L^p}, $ where $\mathcal{M}_L f(\cdot) = \sup_{t > 0} |e^{-tL} f(\cdot)|$ for all $f \in L^2(\mathbb{R}^d)$. We characterise $H_L^p(\mathbb{R}^d)$ in terms of Lusin integrals associated with Hermite operator.

math.FA↗
arXiv2016

On sectoriality of degenerate elliptic operators

Let $c_{kl} \in W^{1,\infty}(Ω, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$ and $Ω\subset \mathbb{R}^d$ be open with Lipschitz boundary. We consider the divergence form operator $ A_p = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $ in $L_p(Ω)$ when the coefficient matrix satisfies $(C(x) \, ξ, ξ) \in Σ_θ$ for all $x \in Ω$ and $ξ\in \mathbb{C}^d$, where $Σ_θ$ be the sector with vertex 0 and semi-angle $θ$ in the complex plane. We show that a sectorial estimate hold for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

math.AP↗
arXiv2016

Degenerate elliptic operators in $L_p$-spaces with complex $W^{2,\infty}$-coefficients

Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient matrix satisfies $(C(x) \, ξ, ξ) \in Σ_θ$ for all $x \in \mathbb{R}^d$ and $ξ\in \mathbb{C}^d$, where $Σ_θ$ be the sector with vertex 0 and semi-angle $θ$ in the complex plane. We show that for all $p$ in a suitable interval the contraction semigroup generated by $-A$ extends consistently to a contraction semigroup on $L_p(\mathbb{R}^d)$. For those values of $p$ we present a condition on the coefficients such that the space $C_c^\infty(\mathbb{R}^d)$ of test functions is a core for the generator on $L_p(\mathbb{R}^d)$. We also examine the operator $A$ separately in the more special Hilbert space $L_2(\mathbb{R}^d)$ setting and provide more sufficient conditions such that $C_c^\infty(\mathbb{R}^d)$ is a core.

math.AP↗