Lineia Schutz, Juliana S. Ziebel, Janaina P. Zingano, Paulo R. Zingano
We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u \,\|_{\dot{H}^{s}(\mathbb{R}^{n})}^{n/(2s)} $ where $ s > n/2 $.
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We show that the first order Sobolev spaces on cuspidal symmetric domains can be characterized via pointwise inequalities. In particular, they coincide with the Hajlasz-Sobolev spaces.
Under certain restrictions we describe the set of all pointwise multipliers in case of Sobolev and Besov spaces of dominating mixed smoothness. In addition we shall give necessary and sufficient conditions for the case that these spaces form algebras with respect to pointwise multiplication.
New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates
本文提出新的加权里斯型点wise不等式，并将其应用于广义索伯列夫估计中。
We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure), and provide pointwise description of them.
We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators
We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.
We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations
We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.
A new pointwise inequality for rough operators and applications
本文提出了一种针对粗糙奇异积分算子的新点估计，并应用该估计推导出多种泛函空间中的Sobolev型不等式。
We provide a proof of the sharp log-Sobolev inequality on a compact interval.
In the Euclidean setting the Sobolev spaces $W^{α,p}\cap L^\infty$ are algebras for the pointwise product when $α>0$ and $p\in(1,\infty)$. This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices $α,p$.
We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it depends on $p$ for $p>2$.
In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order derivatives and seem to be new in the context of the Lorentz--Sobolev spaces defined in the hyperbolic spaces.
We establish sharp affine weighted $L^p$ Sobolev type inequalities by using the $L_p$ Busemann-Petty centroid inequality proved by Lutwak, Yang and Zhang. Our approach consists in combining in a convenient way the latter one with a suitable family of sharp weighted $L^p$ Sobolev type inequalities obtained by Nguyen and allows to characterize all extremizers in some cases. The new inequalities don't rely on any euclidean geometric structure.
We give in this short paper the exact value for norms of two operators of Hardy-Sobolev type acting between two weight Grand Lebesgue Space (GLS) based on the whole multidimensional Euclidean space.
For a pointwise multiplier $φ$ of the Hardy-Sobolev space $H^2_β$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_φ: H^2_β\to H^2_β$. In particular, we compute the spectrum and essential spectrum of $M_φ$ and develop the Fredholm theory for these operators.
Sparse pointwise bounds for maximal truncations of rough singular integrals and Sobolev-type inequalities
本文证明粗糙奇异积分极大截断的稀疏逐点控制，导出双权Sobolev不等式及Hedberg型估计，推广至多种加权空间。
A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$ Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine $L^p$ Sobolev inequality of Lutwak, Yang, and Zhang. When $p = 1$, the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extrem…
A sharp higher order Sobolev embedding
本文研究了高阶Sobolev空间的嵌入问题，获得了从$W^{k,2}_0(-1,1)$到$L^1(-1,1)$的锐化嵌入并确定极值函数，改进了Kalyabin先前的锐化常数估计。
We investigate asymptotically sharp upper and lower bounds for the approximation numbers of the compact Sobolev embeddings $\overset{\circ}{W}{^m}(Ω)\hookrightarrow L_2(Ω)$ and $ W^m(Ω)\hookrightarrow L_2(Ω)$, defined on a bounded domain $Ω\subset\mathbb{R}^d$, involving explicit constants depending on $m$ and $d$. The key of proof is to relate the approximation problems to certain Dirichlet and Neumann eigenvalue problems.
We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.
A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a genera…
On Choquet integrals and pointwise estimates
本文研究关于Hausdorff内容的Choquet积分不等式，在开连通集上具有足够正则性的连续可微函数满足逐点估计条件，证明了Hardy不等式和Poincare-Sobolev不等式在该背景下成立。
Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p$ which generalizes the result in \cite{NgoNguyenAMV} to the setting of Lorentz-Sobolev spaces. Second, we prove several sharp Poincaré-Sobolev type inequalities in $W^1L^{p,q}(\mathbb H^n)$ with $1\leq q \leq p < n$ which generalize the results in \cite{NguyenPS2018} to the setting of Lorentz-Sobolev spaces. Finally, …
We prove the existence of minimizers for some constrained variational problems on $BV(Ω)$, under subcritical and critical restrictions, involving the affine energy introduced by Zhang in \cite{Z}. Related functionals have non-coercive geometry and properties like semicontinuity and affine compactness are deeper in the weak* topology. As a by-product of the theory, extremal functions are shown to exist for various affine Poincaré-Sobolev type inequalities.
By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$ equipped with the weight $ω(x) = x_n^a$, $a\geq 0$. It amounts to work with the fractional dimension $n_a = n+a$. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del …
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
Oscillation inequalities, lower ball growth and Sobolev embeddings on metric measure spaces
研究度量测度空间上球函数下界、重排振荡与Sobolev型嵌入的关系，证明下界估计等价于点态对称化不等式，并导出赋范振荡不等式和Sobolev嵌入。
A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
本文研究分数阶变分的渐近行为，证明分数阶α-变分在α趋于1时收敛于标准De Giorgi变分，且β-变分在β趋于α时收敛于α-变分，均逐点及Γ-极限意义下成立。
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe…
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.
Sobolev regularity for a class of local fractional new maximal operators
本文研究局部新极大算子及其分数版本的正则性，给出逐点梯度估计，证明其在Sobolev空间上的有界性，并推广到零边值Sobolev空间。
Sharp pointwise gradient estimates for Riesz potentials with a bounded density
本文在$\mathbb{R}^{n}$中建立了具有有界密度的Riesz位势及其梯度的锐点态不等式，并说明其在位势分析、矩理论等领域的应用价值。
Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert, we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo--Nirenberg--Sobolev inequalities due to Del Pino and Dolbeault \cite{DelPino}.
The embedding relations between Besov-Triebel-Sobolev spaces and modulation spaces are determined explicitly. We extend the results of Sugimoto[2007]; Wang[2007] and Kobayashi[2011] to the most general cases. And we give the sharp embedding between Fourier $L^{p}$ spaces and modulation spaces.
This paper is a second one following our work [CLZ13] in series, considering sharp Hardy- Littlewood-Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made by Frank and Lieb in [FL12]. In this paper, analogous results are obtained for octonionic Heisenberg group.
We study the pointwise convergence and the $Γ$-convergence of a family of non-local, non-convex functionals $Λ_δ$ in $L^p(Ω)$ for $p>1$. We show that the limits are multiples of $\int_Ω |\nabla u|^p$. This is a continuation of our previous work where the case $p=1$ was considered.
A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set.
Let $p\in(1,\infty)$ and $q\in[1,\infty)$. In this article, the authors characterize the Triebel-Lizorkin space ${F}^α_{p,q}(\mathbb{R}^n)$ with smoothness order $α\in(0,2)$ via the Lusin-area function and the $g_λ^*$-function in terms of difference between $f(x)$ and its average $B_tf(x):=\frac1{|B(x,t)|}\int_{B(x,t)}f(y)\,dy$ over a ball $B(x,t)$ centered at $x\in\mathbb{R}^n$ with radius $t\in(0,1)$. As an application, the authors obtain a series of characterizations of $F^α_{p,\infty}(\mathbb{R}^n)$ via pointwise inequalities, involving ball averages, in spirit close to Hajłasz gradients, …
We establish new approximation results in the sense of Lusin for Sobolev functions $f$ with $|\nabla f| \in L\log L$ on infinite-dimensional spaces equipped with Gaussian measures. The proof relies on some new pointwise estimate for the approximations based on the corresponding semigroup which can be of independent interest.
We establish several Poincaré--Sobolev type inequalities for the Lapalce--Beltrami operator $Δ_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or sharp Sobolev inequality in $\mathbb H^n$. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for $Δ_g$ in $\mathbb H^n$. As a consequence, we obtain the Poincaré--Sobolev inequality for the seco…
In this note, we establish a $L^p-$version of the Poincaré--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in $\mathbb H^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré--Gagliardo--Nirenberg and Poincaré--…
Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy-Littlewood-Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the $σ_2$-curvature. Our argument relies on nice commutator identities deduce…
Sharp Hardy inequalities for Sobolev-Bregman forms
本文研究了Sobolev-Bregman形式中半空间和凸域的精确分数Hardy不等式，扩展了Bogdan和Dyda以及Loss和Sloane的结果。
Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality
本文利用 $\Lp$ Busemann-Petty 质心不等式，为 log-Sobolev、Sobolev 和 Gagliardo-Nirenberg 等尖锐仿射泛函不等式提供了统一且有力的证明方法，并可直接刻画等号成立情形。
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.
Sharp logarithmic Sobolev inequalities on free groups by variational comparison
本文通过变分比较方法，将循环群的Lagrange方法推广至有限迹冯·诺依曼代数，证明了自由群上词长Poisson半群常数为2的尖锐对数Sobolev不等式。
Sharp Complete Modified Log-Sobolev Inequalities on Classical and Quantum Tori
该研究证明圆周热半群的最优完全修正对数索伯列夫常数为1，借助矩阵值Wirtinger不等式得到Fisher信息收缩，进而通过张量化与转移原理推广到经典、量子环面热半群，确定其尖锐常数。
We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
本文证明了具有显式常数且大维数极限行为正确的Sobolev不等式尖锐定量稳定性，并据此推导出显式与维数无关常数的Gaussian log-Sobolev不等式最优定量稳定性估计。
The sharp log Sobolev inequality on finite cycles
通过证明尖锐常数等于谱隙的一半，解决了所有 n≥4 的 n-圈上对数 Sobolev 不等式的尖锐常数问题，该结果源自最优三次 Sobolev 不等式。
We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.
Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond
本文研究了周期性穿孔集的极值问题，证明了精确皮卡德-索波列夫不等式常数的存在性，并扩展到其他类型的集。
Sharp defective log-Sobolev inequalities on H-type groups
本文在H型群上建立尖锐缺陷对数Sobolev不等式，并借此证明Lipschitz函数关于热核测度的指数可积性，同时给出高斯型测度关于次黎曼距离的缺陷对数Sobolev型不等式。
In this paper, we got several sharp Hardy-Littlewood-Sobolev-type inequalities on quaternionic Heisenberg groups (a general form due to Folland and Stein [FS74]), using the symmetrization-free method in a paper of Frank and Lieb [FL12], where they considered the analogues on classical Heisenberg group. First, we give the sharp Hardy-Littlewood-Sobolev inequalities, both on quaternionic Heisenberg group and its equivalent on quaternionic sphere for exponent bigger than 4. The extremizer, as we guess, is almost uniquely constant function on sphere. Then their dual form, sharp conformally-invaria…
In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\ma…
The sharp affine $L^2$ Sobolev trace inequality and variants
本文利用 $L_p$ Busemann-Petty 质心不等式建立尖锐仿射 $L^p$ Sobolev 迹不等式，其在 $p=2$ 时强于 Escobar 和 Beckner 的经典结果，并刻画了所有取等号情形。
We prove pointwise estimates to the modified Riesz potential. We show the boundedness of its Luxemburg norm. As an application we obtain Orlicz embedding results. We study the sharpness of the results.
We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.
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