Browsing Analysis of Partial Differential Equations (APDE) by Title
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A quantitative approach to weighted Carleson condition
(20170505)Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{Q}\int_{Q}f(x)dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are ... 
Quantitative weighted estimates for rough homogeneous singular integrals
(20170311)We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound ... 
Quantitative weighted estimates for Rubio de Francia's LittlewoodPaley square function
(201912)We consider the Rubio de Francia's LittlewoodPaley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. ... 
Quantitative weighted estimates for singular integrals and commutators
(20180227)In this dissertation several quantitative weighted estimates for singular integral op erators, commutators and some vector valued extensions are obtained. In particular strong and weak type $(p, p)$ estimates, CoifmanFe ... 
Quantitative weighted mixed weaktype inequalities for classical operators
(20160630)We improve on several mixed weak type inequalities both for the HardyLittlewood maximal function and for CalderónZygmund operators. These type of inequalities were considered by Muckenhoupt and Wheeden and later on by ... 
Quasiinvariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS
(20220101)The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasiinvariance of the Gaussian measure on L2(T) with covariance [1+(−Δ)s]−1 ... 
Reconstruction from boundary measurements for less regular conductivities
(20161001)In this paper, following Nachman's idea [14] and Haberman and Tataru's idea [9], we reconstruct $C^1$ conductivity $\gamma$ or Lipchitz conductivity $\gamma$ with small enough value of $\nabla log\gamma$ in a Lipschitz ... 
Reconstruction of the Derivative of the Conductivity at the Boundary
(201908)We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. This boundary determination implies the uniqueness ... 
Regularity of fractional maximal functions through Fourier multipliers
(2018)We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function ... 
Regularity of maximal functions on Hardy–Sobolev spaces
(20181201)We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy–Sobolev spaces H1,p(Rd) when p > d/(d + 1). This range of exponents is sharp. As a byproduct of the ... 
Relativistic Hardy Inequalities in Magnetic Fields
(20141231)We deal with Dirac operators with external homogeneous magnetic fields. Hardytype inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality ... 
The relativistic spherical $\delta$shell interaction in $\mathbb{R}^3$: spectrum and approximation
(20170803)This note revolves on the free Dirac operator in $\mathbb{R}^3$ and its $\delta$shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by ... 
RESTRICTED TESTING FOR POSITIVE OPERATORS
(2020)We prove that for certain positive operators T, such as the HardyLittlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality ... 
Reverse Hölder Property for Strong Weights and General Measures
(20160630)We present dimensionfree reverse Hölder inequalities for strong $A^{\ast}_p$ weights, $1 \le p < \infty$. We also provide a proof for the full range of local integrability of $A^{\ast}_1$ weights. The common ingredient ... 
Riemann's nondifferentiable function and the binormal curvature flow
(20200714)We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
(2019)Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
(2018)\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad ... 
Rotational smoothing
(20220105)Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators ... 
A Scaling Limit from the Wave Map to the Heat Flow Into S2
(20190708)In this paper we study a limit connecting a scaled wave map with the heat flow into the unit sphere 𝕊2. We show quantitatively how the two equations are connected by means of an initial layer correction. This limit is ... 
Scattering with criticallysingular and δshell potentials
(2019)The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz ...