We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and continuously. The same result holds on $\mathbb{R}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.
The aim of this paper is to propose weak assumptions to prove maximal L^q regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require "off-diagonal" estimates on the real semigroup (e^{tL})_{t>0} to obtain maximal L^q regularity. The main idea is to use a one kind of Hardy space H^1 adapted to this problem and then use interpolation results. These techniques permit us to prove weighted maximal regularity too.
We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the l_infty-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.
We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that \emph{if} the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt, we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.
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 maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.
We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.
We prove that maximal operators of convolution type associated to smooth kernels are bounded in the homogeneous Hardy-Sobolev spaces $\dot{H}^{1,p}(\mathbb{R}^d)$ when $1/p < 1+1/d$. This range of exponents is sharp. As a by-product of the proof, we obtain similar results for the local Hardy-Sobolev spaces $\dot{h}^{1,p}(\mathbb{R}^d)$ in the same range of exponents.
This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators, $$\mathfrak{M}_{α,Ω}(\vec{f})(x)=\sup\limits_{0<r<{\rm dist}(x,Ω^c)}\frac{r^α}{|B(x,r)|^m}\prod\limits_{i=1}^m\int_{B(x,r)}|f_i(y)|dy,\quad \hbox{for \ }0\leqα<mn,$$ where $Ω$ is a subdomain in $\mathbb{R}^n$, $Ω^c=\mathbb{R}^n\backslashΩ$ and $B(x,r)$ is the ball in $\mathbb{R}^n$ centered at $x$ with radius $r$. Several new pointwise estimates for the derivative of the local multilinear maximal function $\mathfrak{M}_{0,Ω}$ and the fractional maxi…
Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^{2}$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^{p}$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p,2}$ for $p$ in a certain open range. We also study the case $p=\infty$.
We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof of maximal regularity for closed and maximal accretive operators following from Kato's inequality for fractional powers and almost orthogonality arguments.
We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cosθ)\cosϕ,\,(t+s\cosθ)\sinϕ,\, s\sinθ): θ, ϕ\in [0,2π) \}$ with $c_0t>s>0$ for some $c_0\in (0,1)$. We prove that the associated (two-parameter) maximal function is bounded on $L^p$ if and only if $p>2$. We also obtain $L^p$--$L^q$ estimates for the local maximal operator on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operator…
We prove that for a finite type curve in $\mathbb R^3$ the maximal operator generated by dilations is bounded on $L^p$ for sufficiently large $p$. We also show the endpoint $L^p \to L^{p}_{1/p}$ regularity result for the averaging operators for large $p$. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.
In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.
This is an expository paper on the regularity theory of maximal operators, when these act on Sobolev and BV functions, with a special focus on some of the current open problems in the topic. Overall, a list of fifteen research problems is presented. It summarizes the contents of a talk delivered by the author at the CIMPA 2017 Research School - Harmonic Analysis, Geometric Measure Theory and Applications, in Buenos Aires, Argentina.
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $$Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \barΩ_r} \big|f(\vec{n} + \vec{m})\big|,$$ where $\big\{\barΩ_r\big\}_{r \geq 0}$ are dilations of a convex set $Ω$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside $\barΩ_r$. We prove here that the operator $f \mapsto \nabla M f$ is bounded and continuous from $l^1(\Z^d)$ to $l^1(\Z^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.
We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on $L^p(\R^N)$ with Lebesgue measure.
The aim of this article is to deepen the understanding of the derivation of $\mathrm{L}^p$-estimates of non-local operators. We review the $\mathrm{L}^p$-extrapolation theorem of Shen which builds on a real variable argument of Caffarelli and Peral and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear for example in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire. As an applic…
In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order c…
Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence form elliptic operator on $\R^n$ with bounded measurable real-valued coefficients and let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our main result implies that the stochastic convolution process $$ u(t) = \int_0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\ge 0,$$ satisfies, for all $1\le p<\infty$, a conical maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{ T_2^{p,2}(\R_+\times\R^n)}^p \le C_p^p \E \n g \n_{ T_2^{p,2}(\R_+\times\R^n;H)}^p.$$ Here, $T_2^{p,2}(\R_+\times\R^n)$ and $T_2^{p,2}(\R_+\times\R^n;H)$ ar…
