This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequality may fail to hold even in the presence of a lower sectional curvature bound. The rôle of a gradient estimate of the form $\| \nabla u \|_{L^{p}} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$, and its connections with the $L^{p}$-Hessian estimate, are also discussed.
Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.
Based on a construction due to B. Güneysu and S. Pigola (\textit{Adv. Math.} \textbf{281} (2015), pp.353--393), for each $p \in [1,\infty]$ and $m \in \mathbb{Z}_{\geq 2}$, we exhibit an $m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calderón--Zygmund estimate \begin{equation*} \|\nabla \nabla f\|^p_{L^p} \leq C_1 \|Δf\|_{L^p}^p + C_2 \| f\|_{L^p}^p \qquad \text{ for all } f \in C^\infty_c(\mathcal{M}) \end{equation*} is false for any $C_1, C_2$ depending on $m$ and $p$. Therefore, one must impose further geometric conditions on the manifold to ensure the validity of…
We construct, for $p>n$, a concrete example of a complete non-compact $n$-dimensional Riemannian manifold of positive sectional curvature which does not support any $L^p$-Calderón-Zygmund inequality: \[ \forall\,φ\in C^{\infty}_c(M),\qquad\|\operatorname{Hess} φ\|_{L^p}\le C(\|φ\|_{L^p}+\|Δφ\|_{L^p}). \] The proof proceeds by local deformations of an initial metric which (locally) Gromov-Hausdorff converge to an Alexandrov space. In particular, we develop on some recent interesting ideas by G. De Philippis and J. Núñez-Zimbron dealing with the case of compact manifolds. As a straightforward co…
In this note, we give a simple proof of the pointwise BMO estimate for Poisson's equation. Then the Calderón-Zygmund estimate follows by the interpolation and duality.
We show, in a borderline case which was not covered before, the validity of nonlinear Calderón-Zygmund estimates for a class of non-uniformly elliptic problems driven by double phase energies.
We discuss tangential Sobolev-estimates up to the boundary for solutions to the regional fractional laplacian on the upper half-plane. These estimates can be used to reduce the boundary Calderon-Zygmund theory of any dimension to a one-dimensional nonlocal problem.
In this paper, we consider a kind of singular integral which can be viewed as an extension of the classical Calder$\acute{\rm o}$n-Zygmund type singular integral. We establish an estimate of the singular integral in the $L^q$ space for $1<q<\infty$. In particular, the Calder$\acute{\rm o}$n-Zygmund estimate can be recovered from our obtained estimate. The proof of our main result is via the so called "geometric approach", which was applied in \cite{CP} on the $L^q$ estimate of the elliptic equations and in \cite{LW,Wang} on a new proof of the the Calder$\acute{\rm o}$n-Zygmund estimate.
We find necessary and sufficient conditions for the validity of weighted Rellich and Calderon-Zygmund inequalities in L^p, 1 \leq p \leq \infty, in the whole space and in the half-space with Dirichlet boundary conditions. General operators like L=Δ+c\frac{x}{|x|^2}\cdot\nabla-\frac{b}{|x|^2} are considered. We compute best constants in some situations.
Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calderón-Zygmund theory in terms of Calderón-Zygmund singular integrals, for the Laplacian. As a consequence, a flexible, comprehensive approach to gradient bounds for the $p$-Laplace system for a broad class of norms is derived. In particular, new gradient estimates are exhibited, and well-known results in customary function spaces are easily recovered.
We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
We obtain Musielak Orlicz bumps conditions on a pair of weights for the boundedness of Calderón Zygmund operators and their commutators between variable Lebesgue spaces with different weights. The symbols of the commutators belong to a wider class of functions. We also give Bloom type estimates for commutators of Calderón Zygmund and fractional integral operators in the variable Lebesgue context. The techniques involved in both type of results are related with the theory of sparse domination.
We consider local weak solutions of the Poisson equation for the p--Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate.
We deal with a global Calderón-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
We give a proof of a so-called "local $Tb$" Theorem for singular integrals whose kernels satisfy the standard Calderón-Zygmund conditions. The present theorem, which extends an earlier result of M. Christ \cite{Ch}, was proved in \cite{AHMTT} for "perfect dyadic" Calderón-Zygmund operators. The proof in \cite{AHMTT} essentially carries over to the case considered here, with some technical adjustments.
In this note we address the continuity of strongly singular Calderón-Zygmund operators on Hardy-Morrey spaces $\mathcal{HM}_{q}^λ(\mathbb{R}^n)$, assuming weaker integral conditions on the associated kernel. Important examples that falls into this scope are pseudodifferential operators on the Hörmander classes $OpS^{m}_{σ,μ}(\mathbb{R}^n)$ with $0<σ\leq 1$, $0 \leq μ<1$, $μ\leq σ$ and $m\leq -n(1-σ)/2$.
In this work, by recent work of Lerner and Ombrasi (J. Geom. Anal. 30(1): 1011-1027, 2020), we show a maximally modulated singular integral operator which its kernel satisfying $L^r$- Hörmander condition can be dominated by sparse operators. Also, the local exponential decay estimates for these operators are obtained.
We show that Petermichl's dyadic operator $\mathcal{P}$ (S. Petermichl (2000), Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol) is a Calderón-Zygmund type operator on an adequate metric normal space of homogeneous type. As a consequence of a general result on spaces of homogeneous type, we get weighted boundedness of the maximal operator $\mathcal{P}^*$ of truncations of the singular integral. We show that dyadic $A_p$ weights are the good weights for the maximal operator $\mathcal{P}^*$ of the scale truncations of $\mathcal{P}$.
We consider non-linear elliptic equations having a measure in the right hand side, of the type $ \divo a(x,Du)=μ, $ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.
We give necessary and sufficient conditions for inhomogeneous Calderón-Zgymund operators to be bounded on the local hardy spaces $h^p(\mathbb{R}^n)$. We then give applications to local and truncated Riesz transforms, as well as pseudo-differential operators defined by amplitudes.
In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $γ\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-λu+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ \|u_{t}\|_{L_{q}(\fR,L_{p})}+\|(-Δ)^{γ/2}u\|_{L_{q}(\fR,L_{p})} +λ\|u\|_{L_{q}(\fR,L_{p})}\leq N\|f\|_{L_{q}(\fR,L_{p})} $$ are obtained for any $λ> 0$ and $p,q\in (1,\infty)$.
We study general parabolic equations of the form $u_t = div A(x,t, u,D u) + div(|F|^{p-2} F)+ f$ whose principal part depends on the solution itself. The vector field $A$ is assumed to have small mean oscillation in $x$, measurable in $t$, Lipschitz continuous in $u$, and its growth in $Du$ is like the $p$-Laplace operator. We establish interior Calderón-Zygmund estimates for locally bounded weak solutions to the equations when $p>2n/(n+2)$. This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucia…
We study the properties of reflectionless measures for an $s$-dimensional Calderón-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $μ$ for which $T(μ)$ is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure theory.
The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~Ω\times (0,T)\subset \mathbb{R}^{n+1}, $$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any $q>\max\{p,\frac{n(2-p)}{2}\}$ and $p>1$. Acerbi and Mingione \cite{Acerbi07} proved this estimate in the case $p>\frac{2n}{n+2}$. In this article we settle the case $1<p\leq \frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm…
We prove that a large class of operators, which arise as the projections of martingale transforms of stochastic integrals with respect to Brownian motion, as well as other closely related operators, are in fact Calderón--Zygmund operators. Consequently, such operators are not only bounded on $L^p$, $1<p<\infty$, but also satisfy weak-type inequalities. Unlike the boundedness on $L^p,$ which can be obtained directly from the Burkholder martingale transform inequalities, the weak-type estimates do not follow from the corresponding martingale results.
We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with 3 or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1<p<\infty$ and satisfy weak 1-1 bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.
We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with…
We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $μ$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded i…
For weak solutions $u \in W^{m,1}(Ω;\R^N)$ of higher order systems of the type \int_Ω< A(x,D^m u),D^m ϕ> dx = \int_Ω< |F|^{p(x)-2}F,D^m ϕ> dx, for all $ϕ\in C^{\infty}_c(Ω;\R^N), m > 1$ with variable growth exponent $p:Ω\to (1,\infty)$ we prove that if $|F|^{p(\cdot)} \in L^q_{loc}(Ω)$ with $1 < q < \frac{n}{n-2} + δ$, then $|D^m u|^{p(\cdot)} \in L^q_{loc}(Ω)$. We should note that we prove this implication both in the non-degenerate and in the degenerate case.
Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_Ω \int_Ω K\left (x,|x-y|,\frac{x-y}{|x-y|}\right ) \frac{(u(x)-u(y))(φ(x)-φ(y))}{|x-y|^{n+2s}} dx dy = g[φ], \quad \forall ϕ\in C_c^{\infty}(Ω) \] where $Ω\subset \mathbb{R}^{n}$ is an open set. Here we assume $K$ is bounded, nonnegative and continuous in the first entry -- and ellipticity is ensured by assuming that $K$ is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on m…
Suppose $L=-Δ+V$ is a Schrödinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse Hölder class $RH_σ$ with $σ\geq n/2$. The main aim of this paper is to provide necessary and sufficient conditions in terms of $T1$ criteria for a generalised Calderón--Zygmund type operator with respect to $L$ to be bounded on Hardy spaces $H^p_L(\mathbb{R}^n)$ and on BMO type spaces BMO$_L^α(\mathbb{R}^n)$ associated with $L$. As applications, we prove the boundedness for several singular integral operators associated to $L$. Our approach is flexible enough to prove the boundedness …
We continue our study of the reflectionless measures associated to an $s$-dimensional Calderón-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $μ$ for which $T$ is a bounded operator in $L^2(μ)$. In this way, we shall reduce certain well-known problems at the interface of harmonic analysis and geometric measure theo…
The paper is concerned with higher order Calderon-Zygmund estimates for the $p$-Laplace equation $$ -\textrm{div}(A(\nabla u)) := -\textrm{div}{(|\nabla u|^{p-2}\nabla u)}=-\textrm{div} F, \qquad 1<p<\infty. $$ We are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the force term $F$ to the flux $A(\nabla u)$. For $p\geq 2$ we show that $F \in B^s_{ρ,q}$ implies $A(\nabla u) \in B^s_{ρ,q}$ for any $s \in (0,1)$ and all reasonable $ρ,q \in (0,\infty]$ in the planar case. The result fails for $p<2$. In case of higher dimensions and systems…
In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establihes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respec- tively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.
We prove that if a pair of weights $(u,v)$ satisfies a sharp $A_p$-bump condition in the scale of log bumps and certain loglog bumps, then Haar shifts map $L^p(v)$ into $L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón-Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to p…
