Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.
We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of $\mathbb{R}^n$ with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of $\ell^2$ endowed with a nondegenerate Gaussian measure. To the best of our knowledge, this is the first such equivalence theorem on a proper open subset of an infinite-dimensional Hilbert space, beyond the earlier whole-space results. We also prove the Malliavin--Sobolev norm equivalence for all $p\in[1,\infty)$ and $k\ge2$, including the case $p=1$, $k\ge3$ left open by Addona--Muratori--Rossi in \cite{AddonaMuratoriRossi}.
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u{\i}\ operators to construct such approximations, thereby extending the approximation theory developed for smooth bounded domains and the simultaneous approximation framework introduced by \emph{Fefferman, Hajduk, and Robinson, {Proc. Lond. Math. Soc.} (3) \ {125} (2022), no.~4, 759-777}. As an application, we establish energy equality for Leray-Hopf weak solutions of the incompressible convective Brinkman-Forchheimer (CBF) equations on $\mathbb{R}^d$, $d\in\{2,3\}$ covering both the critical and supercritical regimes. For general unbounded domains, we employ the resolvent operator associated with the Stokes operator, developed by \emph{Farwig, Kozono and Sohr, {Acta Math.}, {195} (2005), 21-53}, to obtain simultaneous approximation results. We also establish a generalized version of the classical Lions-Magenes lemma, which is of independent interest. Finally, by combining this result with the simultaneous approximation framework for unbounded domains, we establish energy equality for weak solutions of the CBF equations on general unbounded domains.
Akram Khan, Sagar Gautam, Manil T. Mohan· 0 citations
Under small-ball bounds for the Malliavin determinants of two $\mathbb R^k$-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich--Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an $\mathbb R^k$-valued mapping has components in $W^{2,p}(\gamma)$ and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order $\varkappa\in(0,1]$, then the law of the mapping has fractional regularity of order \[ \frac{\varkappa}{1+(2k-1)\varkappa p^{-1}}. \] In particular, for large $p$, this gives regularity of order $\varkappa$ up to an $O(p^{-1})$ loss.
In this paper, we prove the $L^p(p>1)$-uncertainty principles for any $1<p<n$, including the classical Heisenberg-Pauli-Weyl inequality, Caffarelli-Kohn-Nirenberg interpolation inequality and Hardy inequality in $\mathbb R^n$ as special cases, on $n(\geq 2)$-dimensional forward complete and noncompact Finsler measure spaces $(M, F, \mathfrak{m})$ with curvatures bounded from above or below by constants. Further, we characterize the sharpness of $L^p$-uncertainty principles in terms of the reversibility of $F$ and the bounds of flag (or Ricci) curvature and S-curvature induced by the measure $\mathfrak m$ and obtained some rigidity results, which generalize the related ones in [HKZ] in Finslerian case and [KKPZ] in Riemannian case.
The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of finiteness is dense in $L^p$ for any $1\le p<\infty$. For any function with finite Willmore energy, we show that almost all of its level sets admit a mean curvature vector satisfying the natural integration by parts formula with respect to the tangential divergence. As a main application, we show that in ${\rm RCD}(0,N)$ spaces with Euclidean volume growth, almost every level set of the electrostatic potential possesses a mean curvature vector in the above sense. Furthermore, we prove that this vector satisfies the same sharp Willmore inequality as in the smooth setting, alongside rigidity and almost-rigidity statements. Finally, as a technical tool, we generalize the sharp isocapacitary inequality to the non-smooth setting.
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 Calder\'on--Lozanovski\u{\i} 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. Le\'snik, T. Roskovec, and F. Soudsk\'y. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.
Pingxu Hu, Yinqin Li, Da-Chun Yang et al.· 0 citations
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