The Calderón problem for Hilbert couples

We prove that intermediate Banach spaces\(\mathcal{A}\) and\(\mathcal{B}\) with respect to arbitrary Hilbert couples\(\bar {H}\) and\(\bar {K}\) are exact interpolation if and onlyif they are exactK-monotonic, i.e. the condition\(f^0 \in \mathcal{A}\) and the inequality\(K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0\), implyg⁰∈B and ‖g⁰‖B≤‖f⁰‖ _A (K is Peetre’sK-functional). It is well known that this property is implied by the following: for each ?>1 there exists an operator\(T:\bar {H} \to \bar {K}\) such thatTf⁰=g⁰, and\(K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0\). Verifying the latter property, it suffices to consider the “diagonal case” where\(\bar {H} = \bar {K}\) is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ?=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.     

The Calderón problem for quasilinear elliptic equations

In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.     

The Calderón problem for variable coefficients nonlocal elliptic operators

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator (???(A(x)?))^s+q), for 0<s<1. We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension n≥2. Our results generalize the recent initiative [18 Ghosh, T., Salo, M., Uhlmann, G. (2016). The Calderón problem for the fractional Schrödinger equation. arXiv preprint arXiv:1609.09248. [Google Scholar]] of introducing and solving inverse problem for fractional Schrödinger operator ((?Δ)^s+q) for 0<s<1. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.     

A regularity result for Calderón commutators

We present a regularity result for the Calderon commutator $[u,\mathcal H ](v)$ where $u,v$ are moduli of continuity and $\mathcal H $ is the Hilbert transform.     

Bergman and Calderón Projectors for Dirac Operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$ , we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L ² harmonic spinors on $\bar{X}$ , and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$ . We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {?(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.     

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.     

Application of Calderón’s inverse problem in civil engineering

In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderón problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

Calderón constants of finite-dimensional couples

The Calderón constant æ( $\bar X$ ) is a numerical invariant of finite-dimensional Banach couple $\bar X = (X_0 ,X_1 )$ measuring its interpolation property with respect to linear operators acting in $\bar X$ . In the paper we prove the duality relation æ( $\bar X$ )≈ æ( $\bar X$ ^*)and calculate the asymptotic behavior of æ( $\bar X$ ) as dim $\bar X \to \infty $ for a few “classical” Banach couples.     

On the Marcinkiewicz–Calderón Interpolation Theorem for Integral Operators

Mathematical Notes - The inverse problem to the classical Marcinkiewicz–Calderón interpolation theorem is considered. Necessary conditions for the Marcinkiewicz–Calderón...     

The Calderón Problem with Partial Data for Less Smooth Conductivities

ABSTRACT

In this article we consider the inverse conductivity problem with partial data. We prove that in dimensions n ≥ 3 knowledge of the Dirichlet-to-Neumann map measured on particular subsets of the boundary determines uniquely a conductivity with essentially 3/2 derivatives.     

An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

Journal of Fourier Analysis and Applications - The purpose of this article is to provide an alternative proof of the weak-type $$\left( 1,\ldots ,1;\frac{1}{m}\right) $$ estimate for m-multilinear...     

The Calderón Projector for an elliptic operator in divergence form

The Calderón Projector, is one of the most important tools in the study of boundary value problems for elliptic operators. Its construction is well known for elliptic operators with C^∞ coefficients on C^∞ domains and even for the Laplacian operator on C¹ domains. The aim of this article is to extend the results for the Laplacian case to elliptic operators in divergence form with Lipschitz coefficients on C¹ domains.     

Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems

?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Weighted Norm Inequalities with General Weights for the Commutator of Calderón      Hu  Guo En  Zhu  Yue Ping 《数学学报(英文版)》2013,29(3):505-514 In this paper, by a sharp function estimate and an idea of Lerner, the authors establish someweighted estimates for the m-multilinear integral operator which is bounded from L1(Rn)×···×L1 (Rn)to L1/m,∞ (Rn),, and the associated kernel K(x; y1, . . . , ym)) enjoys a regularity on the variable x. As anapplication, weighted estimates with general weights are given for the commutator of Calderón.      17. Calderón-Zygmund theory for operator-valued kernels      《Advances in Mathematics》1986,62(1):7-48       18. Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators          下载免费PDF全文 Yan Lin  Guo Ming Zhang 《数学学报(英文版)》2016,32(11):1297-1311 In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.      19. Pointwise Convergence of the Calderón Reproducing Formula      Kangwei Li  Wenchang Sun 《Journal of Fourier Analysis and Applications》2012,18(3):439-455 In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every $f\in L_{w}^{p}(\mathbb {R}^{d})$ with an $\mathcal{A}_{p}$ weight w, 1??p<??, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.      20. The Linearized Calderón Problem on Complex Manifolds          下载免费PDF全文 Guillarmou  Colin  Salo  Mikko  Tzou  Leo 《数学学报(英文版)》2019,35(6):1043-1056 In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.

Weighted Norm Inequalities with General Weights for the Commutator of Calderón

In this paper, by a sharp function estimate and an idea of Lerner, the authors establish someweighted estimates for the m-multilinear integral operator which is bounded from L1(Rn)×···×L1 (Rn)to L1/m,∞ (Rn),, and the associated kernel K(x; y1, . . . , ym)) enjoys a regularity on the variable x. As anapplication, weighted estimates with general weights are given for the commutator of Calderón.     

Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators

In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.     

Pointwise Convergence of the Calderón Reproducing Formula

In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every $f\in L_{w}^{p}(\mathbb {R}^{d})$ with an $\mathcal{A}_{p}$ weight w, 1??p<??, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.     

The Linearized Calderón Problem on Complex Manifolds

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.