We consider the $H^{s}$--$L^q$ maximal estimates associated to the wave operator \begin{equation*}
e^{ it\sqrt{-Δ}}f(x) = \frac{1}{(2π)^d}\int_{\mathbb{R}^d} e^{i(x \cdot ξ\, + t|ξ|)} \widehat{f}(ξ\,) dξ. \end{equation*} Rogers--Villarroya proved $H^{s}$--$L^q$ estimates for the maximal operator $f\mapsto$ $\sup_{t} |e^{ it\sqrt{-Δ}}f|$ up to the critical Sobolev exponents $s_c(q,d)$. However, the endpoint case estimates for the critical exponent $s=s_c(q,d)$ have remained open so far. We obtain the endpoint $H^{s_c(q,d)}$--$L^q$ bounds on the maximal operator $f\mapsto \sup_{t} |e^{ it\sqrt{-Δ}}f|$. We also prove that several different forms of the maximal estimates considered by Rogers--Villarroya are basically equivalent to each other.
Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates
University of Alabama（阿拉巴马大学）；Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学理学院）；IMAS-CONICET（阿根廷国家科学技术研究委员会数学研究所）；University of the Basque Country（巴斯克大学）；Ikerbasque（伊克拉巴斯科基金会）；BCAM（巴斯克数学中心）；IMAS- CONICET-UBA（阿根廷国家科学技术研究委员会数学研究所-布宜诺斯艾利斯大学）
We provide off-diagonal estimates for a maximal operator arising from a geometric problem of estimating the size of certain geometric configuration of k- skeletons in $\mathbb{R}^n$. This is achieved by interpolating a weak-type endpoint estimate with the known diagonal bounds. The endpoint estimate is proved by combining a geometric result about k-skeletons and adapting an argument used to prove off-diagonal estimates for the circular maximal function in the plane.
Let $0<α<d$ and $1\leq p<d/α$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_αf$ are weakly differentiable and $ \|\nabla\mathcal M_αf\|_{p^*} \leq C_{d,α,p} \|\nabla f\|_p , $ where $ p^* = (p^{-1}-α/d)^{-1} . $ In particular it covers the endpoint case $p=1$ for $0<α<1$ where the bound was previously unknown. For $p=1$ we can replace $W^{1,1}(\mathbb{R}^d)$ by $\mathrm{BV}(\mathbb{R}^d)$. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function…
An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators
One of the purposes of this paper is to prove that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L^{2,1}(G) \ast L^{2,1}({G}) \subseteq L^{2, \infty }({G}). Let {K} be a maximal compact subgroup of {G} and X=G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator { \cal M}_2f(z) = \sup_{z\in B} \frac{1}{|B|} \int_{B} |f(z')|\,dz' is bounded from L^{2,1}(X) to L^{2, \infty }(X) and from L^p(X) to L^p(X) in the sharp range of exponents p \in (2, \infty ]. The supremum in the definition of { \cal M}_2f(z) is …
We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.
In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.
We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.
