Michael Christ
The Riesz-Sobolev inequality provides an upper bound, in integral form, for the convolution of indicator functions of subsets of Euclidean space. We formulate and prove a sharper form of the inequality. This can be equivalently phrased as a stability result, quantifying an inverse theorem of Burchard that characterizes cases of equality.
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The Riesz-Sobolev inequality provides an upper bound for a trilinear expression involving convolution of indicator functions of sets. It is known that equality holds only for homothetic ordered triples of appropriately situated ellipsoids. We characterize ordered triples of subsets of Euclidean space $R^d$ that nearly realize equality, for arbitrary dimensions $d$, extending a result already known for $d=1$.
We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are $L^p$-boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fou…
The Riesz-Sobolev inequality provides a sharp upper bound for a trilinear expression involving convolution of indicator functions of sets. Equality is known to hold only for indicator functions of appropriately situated intervals. We characterize ordered triples of subsets of the real line that nearly realize equality, with quantitative bounds of power law form with the optimal exponent. This improves on an earlier manuscript by the author in at least two respects. An excessively strong hypothesis has been replaced by the natural assumption, and the conclusion has been strengthened from a "lit…
Sobolev meets Riesz: a characterization of weighted Sobolev spaces via weighted Riesz bounded variation space
本文引入加权 Riesz 有界变差空间，在权重属于 Muckenhoupt 类时刻画加权 Sobolev 空间，并利用 Rubio de Francia 外推理论推广到变指数情形。
Two weight Sobolev norm inequalities for fractional vector Riesz transforms and doubling weights
本文建立分数阶向量 Riesz 变换在两加权 Sobolev 空间间的 T1 定理，以 A_2 条件和两个对偶测试条件刻画有界性，并证明倍增测度下多种加权 Sobolev 范数等价。
The Riesz-Sobolev inequality relates the convolution of nonnegative functions on Euclidean space to the convolution of their symmetric nonincreasing rearrangements. We show that for dimension one, for indicator functions of sets, if the inequality is sufficiently close to an equality then the sets in question must nearly coincide with intervals.
A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group $\mathbb{R}/\mathbb{Z}$, a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing.
Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs
本文针对Xavier Tolsa提出的正无反射测度刚性猜想，在多项式增长和单点大尺度下质量界条件下，证明了两类平坦性结果，并建立了Lipschitz图上Riesz变换的Sobolev估计。
A sharpened energy-Strichartz inequality for the wave equation
本研究针对Bez和Rogers提出的1+5维能量空间波动方程精确Strichartz估计，借鉴经典精细化Sobolev估计思路，添加与极值元集距离成正比的项实现了估计的改进。
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
If a pair of functions nearly extremizes Young's convolution inequality for R^d, with all three exponents finite and strictly greater than 1, then each function is close in norm to a Gaussian. The proof relies on the Riesz-Sobolev rearrangement inequality and in particular, on an approximate inverse Riesz-Sobolev inequality established in a companion paper.
We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1, L^q)$ estimate. We find a sharp constant in the weighted $L^p$-inequality, generalizing the results of W. Beckner and S. Samko.
We consider the Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. We give a complete characterization of indices $\vec p$ and $\vec q$ such that the Riesz potential is bounded from $L^{\vec p}$ to $L^{\vec q}$, including all the endpoint cases. As a result, we get the mixed-norm Hardy-Littlewood-Sobolev inequality.
We disprove a conjecture of Foschi, regarding extremizers for the Strichartz inequality with data in the Sobolev space $\dot{H}^{1/2}\times\dot{H}^{-1/2}(\mathbb R^d)$, for even $d\ge 2$. On the other hand, we provide evidence to support the conjecture in odd dimensions, and refine his sharp inequality in $\mathbb R^{1+3}$, adding a term proportional to the distance of the initial data from the set of extremizers. The proofs use the conformal compactification of the Minkowski space-time given by the Penrose transform.
We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz $s$-energy on the sphere $\mathbb S^d.$ Our results are based in bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in $\mathbb S^2$ were obtained. Our result improves previously known bounds for $0\le s<2$ and $s\neq 1$ in $\mathbb S^2,$ where $s=0$ is Wolff's result, and for $d-t_0<s<d$ with $t_0\approx 2.5$ when $d\ge 3$ and $s\neq d-1.$
We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's conjecture fails) or to the word length of free groups. Lust-Piquard's work for discrete laplacians on LCA groups is also generalized in several ways. In th…
New pointwise bounds by Riesz potential type operators
本文研究了一类含参数的粗糙积分算子 $T_{Ω,α}$ 的逐点界，证明其受限于依赖于符号大小的稀疏位势算子，并由此导出了若干新的 Sobolev 映射。
Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates
本文基于Meyers-Ziemer框架，建立新的加权Sobolev不等式，扩展到加权变分函数，并提供端点估计和等周不等式，同时揭示新的双权Sobolev不等式。
Sparse domination modulo polynomials
本文提出一种模多项式稀疏控制理论，通过多项式检验条件刻画Calderón-Zygmund算子，并导出加权Hardy-Sobolev不等式及Riesz势点态界，揭示了BMO到L^∞检验条件的定量转变。
Sharp Poincaré--Sobolev Inequalities of Choquet--Lorentz Integrals with Respect to Hausdorff Contents on Bounded John Domains
本文证明了有界 John 域上关于 Hausdorff 容度的 Choquet--Lorentz 积分具有精确指数范围的 Poincaré--Sobolev 不等式，发展了分数阶极大不等式等新工具，并给出了临界情形的端点估计及 Riesz 位势的精确有界性。
Let $\mathcal{W}$ be a closed dilation and translation invariant subspace of the space of $\mathbb{R}^\ell$-valued Schwartz distributions in $d$ variables. We show that if the space $\mathcal{W}$ does not contain distributions of the type $a\otimes δ_0$, $δ_0$ being the Dirac delta, then the inequality $\|\mathbb{I}_α[f]\|_{L_{p,1}}\lesssim \|f\|_{L_1}$, $\frac{p-1}{p} = \fracα{d}$, holds true for functions $f\in\mathcal{W}\cap L_1$ with a uniform constant; here $\mathbb{I}_α$ is the Riesz potential of order $α$ and $L_{p,1}$ is the Lorentz space. This result implies as a particular case the i…
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω…
$\mathrm{L}^p$ bounds for parabolic Riesz transforms with rough coefficients: The case $1<p \leq 2$
研究非自治二阶抛物型微分算子相关里斯变换的$\mathrm{L}^p$界，通过$\mathrm{L}^p$预解式界确定复系数下$1<p \leq2$的最大开区间，实系数时证明可外推到全范围，基于两种几何的时空非对角界论证。
We consider a version of M. Riesz fractional integral operator on a space of homogeneous type and show an analogue of the well-known Hardy--Littlewood--Sobolev theorem in this context. In our main result, we investigate the dependence of the operator norm on weighted spaces on the weight constant, and find the relationship between these two quantities. It it shown that the estimate obtained is sharp in any given space of homogeneous type with infinitely many points. Our result generalizes the recent Euclidean result by Lacey, Moen, P{é}rez and Torres.
Sobolev--Morrey Spaces and Divergence-Form Degenerate Second-Order Elliptic Equations on Domains with Higher Co-Dimensional Boundaries
研究\(\mathbb{R}^n\)中高余维边界区域上加权齐次Sobolev - Morrey空间，建立实变理论，应用迹和延拓算子有界性，获散度型退化二阶椭圆方程Dirichlet问题加权先验估计。
A variation on the Pólya-Segő principle in one dimension
证明了Riesz $(p,\alpha)$-变差的Pólya-Szegő原理，该变差在有界$p$-变差和Sobolev空间$W^{1,p}$之间插值，结果对某些具有分数光滑性的不可微函数也成立。
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