Related papers: Isocapacity Estimates for Hessian Operators

We present a Diaz--Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get the operator versions of P\'{o}lya-Szeg\"{o}'s, Greub-Rheinboldt's, Kantorovich's, Shisha-Mond's, Schweitzer's, Cassels' and…

We investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant…

We establish several Poincar\'e--Sobolev type inequalities for the Lapalce--Beltrami operator $\Delta_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincar\'e…

The Bessel operator, that is, the Schr\"odinger operator on the half-line with a potential proportional to $1/x^2$, is analyzed in the momentum representation. Many features of this analysis are parallel to the approach \`a la K. Wilson to…

In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…

In this paper, we prove the energy stable property for time-dependent (generalized) Schr\"{o}dinger operators by using Hardy inequality. Such property acts very important roles in quantum scattering theory and nonlinear problem. As an…

We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of…

We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one…

The four exactly-solvable models related to non-sinusoidal coordinates, namely, the Coulomb, Eckart, Rosen-Morse type I and II models are normally being treated separately, despite the similarity of the functional forms of the potentials,…

For a non-empty, bounded, open, and convex set of class $C^2$, we consider the Torsional Rigidity associated to the $k$-Hessian operator. We first prove P\'olya type lower bound for the $k$-Torsional Rigidity in any dimension; then, in…

We study the following Choquard type equation in the whole plane $(C) -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr\"odinger potential and the nonlinearity…

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and…

We consider the homogenization at second-order in $\varepsilon$ of $\mathbb{L}$-periodic Schr\"odinger operators with rapidly oscillating potentials of the form $H^\varepsilon =-\Delta + \varepsilon^{-1} v(x,\varepsilon^{-1}x ) + W(x)$ on…

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use…

Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group H_1 of the form L=P_n(X,Y)+Q(X,Y) where the P_n are certain homogeneous polynomials of order n greater than or equal to 2…

We prove interior weighted Hessian estimates in Orlicz spaces for nondivergence type elliptic equations with a lower order term which involves a nonnegative potential satisfying a reverse H\"older type condition.

We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded…

We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal L\'{e}vy process with the characteristic exponent $\psi$ satisfying some scaling condition. We show sharp…

We prove interior boundedness and H\"{o}lder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for…

We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, P\"oschl-Teller I and P\"oschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special…