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We introduce an $R$-sectoriality perturbation technique for non-commuting operators defined in Bochner spaces. Based on this and on bounded $H^{\infty}$-functional calculus results for the Laplacian on manifolds with conical singularities,…
We consider equations involving the truncated laplacians and having lower order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in…
We calculate arbitrarily tight upper and lower bounds on an unconstrained control, linear-quadratic, singularly perturbed optimal control problem whose exact solution is computationally intractable. It is well known that for the…
The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the…
This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing…
We prove that the $L^2$ bound of an oscillatory integral associated with a polynomial depends only on the number of monomials that this polynomial consists of.
We consider the integration of two-dimensional, piecewise constant functions with respect to copulas. By drawing a connection to linear assignment problems, we can give optimal upper and lower bounds for such integrals and construct the…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
We apply the Thom-Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges(Maxwell's problem on point charges), (2) critical points of SINR, (3) critical…
We calculate the threshold singularities in one-dimensional models using the universal low-energy formfactors obtained in the framework of the non-linear Luttinger liquid model. We find the reason why the simplified picture of the impurity…
We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for…
Let 0<n<d be integers and let H denote the n-dimensional Hausdorff measure restricted to an n-dimensional Lipschitz graph in R^d with slope strictly less than 1. For r>2, we prove that the r-variation and oscillation for Calder\'on-Zygmund…
The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a…
Enlightened by Lemma 1.7 in \cite{LiangLuo2021}, we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schr{\"o}dinger equation $${\rm i}\partial_t u = -\partial_x^2…
We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.
In this paper we develop a theory for oscillatory integrals with complex phases. When $f:{\mathbb C}^n \to {\mathbb C}$, we evaluate this phase function on the basic character ${\rm e}(z) := e^{2\pi i x} e^{2\pi i y}$ of ${\mathbb C} \simeq…
We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.
The one-dimensional Schr\"{o}dinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive…
We study bilinear rough singular integral operators $\mathcal{L}_{\Omega}$ associated with a function $\Omega$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slav\'ikov\'a (Math. Ann. 376: 431-455, 2020), they…
Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx, \quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is…