arXiv subjects
Publications and source records attributed to Yinqin Li.
We prove stability for the affine Sobolev inequality for exponents $p\geq 2$ with best possible norm and best possible stability exponent. We also show a corresponding result for critical points of the functional in the absence of bubbling. An important ingredient in our proof is the classification of positive energy solutions to the critical affine $p$-Laplace equation.
We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo--Nirenberg target spaces are precisely those given by the Calderón--Lozanovski\uı space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo--Nirenberg inequalities hold, thereby answering an open question posed by K. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.
While the theory of matrix-weighted function spaces is well established, the majority of previous results in the infinite-dimensional operator-valued setting deal with "no go" theorems, showing the impossibility of some prospective generalizations. However, we show that a complete real-variable theory of Besov and Triebel-Lizorkin spaces with operator-valued Muckenhoupt $A_p$ weights can still be developed, once correctly formulated. This covers operator-weighted extensions of results like the $φ$-transform characterization in terms of discrete sequence spaces, the boundedness of almost diagonal operators, and applications to the $T(1)$ theorem and trace/extension theorems. A key tool is a version of the reverse Hölder inequality, which is weak enough to follow from the operator-valued $A_p$ condition (unlike a variant that had to be imposed as an additional assumption in some previous works), yet strong enough to be used much like its classical counterpart. In contrast to the established scalar and matrix-weighted theories, our approach cannot build on operator-weighted $L^p$ results, as these fail in a very definite way. We also strengthen the existing "no go" statements in Hilbert spaces, showing (among other counterexamples) that every infinite-dimensional Banach space has an operator-valued $A_p$ weight $V$ for which the Hilbert transform is unbounded on $L^p(V)$. This is nontrivial, since the lack of Hilbert space structure also complicates the construction of $A_p$ weights. Building on results from Banach space theory, we achieve this unboundedness by combining two distinct methods in two different classes of spaces (so-called $K$-convex ones and those that are {\em not} UMD), whose union covers all Banach spaces.
In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist.
In this article, we establish a quantitative weighted variant of a far-reaching inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore in 2003, whose dependence on the $A_p$-weight constant for any $p\in[1,\infty)$ is sharp. As applications, we obtain the almost characterization of the critical weighted Sobolev space in terms of wavelets, a sharp real interpolation between this weighted Sobolev space and weighted Besov spaces, and three new Gagliardo--Nirenberg type inequalities in the framework of ball Banach function spaces. Moreover, we apply this sharp weighted inequality to extend the famous Brezis--Seeger--Van Schaftingen--Yung formula in ball Banach function spaces, which gives an affirmative answer to the question in page 29 of [Calc. Var. Partial Differential Equations 62 (2023), Paper No. 234]. Notably, we further establish two new characterizations of Muckenhoupt weights related to the inequality of Cohen et al.\ and the formula of Brezis et al. The most novelty of this article exists in applying and further developing the good cube method introduced by Cohen et al.\ to trace the sharp dependences on weight constants.
Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $Δ^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that, for both parameters $q$ and $γ$ in \emph{sharp} ranges which are related to $X$ and for any locally integrable function $f$ on ${\mathbb{R}^n}$ satisfying $|\nabla^k f|\in X$, $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{h\in\mathbb{R}^n:\ |Δ_h^k f(\cdot)|>λ|h|^{k+\fracγ{q}}\}} \left|h\right|^{γ-n}\,dh\right]^\frac{1}{q}\right\|_X \sim \left\|\,\left|\nabla^k f\right|\,\right\|_{X} $$ with the positive equivalence constants independent of $f$. As applications, the authors establish the Brezis--Seeger--Van Schaftingen--Yung (for short, BSVY) characterization of higher-order homogeneous ball Banach Sobolev spaces and higher-order fractional Gagliardo--Nirenberg and Sobolev type inequalities in critical cases. All these results are of quite wide generality and can be applied to various specific function spaces; moreover, even when $X:= L^{q}$, these results when $k=1$ coincide with the best known results and when $k\ge 2$ are completely new. The first novelty is to establish a sparse characterization of dyadic cubes in level sets related to the higher-order local approximation, which, together with the well-known Whitney inequality in approximation theory, further induces a higher-order weighted variant of the remarkable inequality obtained by A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore; the second novelty is to combine this weighted inequality neatly with a variant higher-order Poincaré inequality to establish the desired upper estimate of BSVY formulae in weighted Lebesgue spaces.
We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.
In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincaré-Ponce, Gaussian Sobolev, and John-Nirenberg. Our method is based on the K-functionals and allows one to derive self-improving type inequalities. We show the optimality of the method by obtaining new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limiting formulas. In particular, we derive these formulas for fractional powers of infinitesimal generators of operator semigroups on Banach spaces.
This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, local Herz-Hardy spaces, and weak Herz-Hardy spaces, and develop a complete real-variable theory of these Herz-Hardy spaces, including their various maximal function, atomic, finite atomic, molecular as well as various Littlewood-Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz-Hardy spaces. Finally, the inhomogeneous Herz-Hardy spaces and their complete real-variable theory are also investigated. Due to the deficiency of the associate space of the global Herz space, the known real-variable characterizations about Hardy-type spaces associated with ball quasi-Banach function spaces are not applicable to Hardy spaces associated with global generalized Herz spaces which need an improved generalization of the existing one, done by the authors also in this book and having more additional anticipating applications. The authors should also point out that, with the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, the exponents in all the obtained results of this book are sharp. Moreover, all of these results in this book are new and have never been published before.