arXiv subjects
Publications and source records attributed to Daniel Spector.
We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these non-additive integrals, e.g. Fatou's lemma and Lebesgue's dominated convergence theorem.
In this paper, we define a notion of $β$-dimensional mean oscillation of functions $u: Q_0 \subset \mathbb{R}^d \to \mathbb{R}$ which are integrable on $β$-dimensional subsets of the cube $Q_0$: \begin{align*} \|u\|_{BMO^β(Q_0)}:= \sup_{Q \subset Q_0} \inf_{c \in \mathbb{R}} \frac{1}{l(Q)^β} \int_{Q} |u-c| \;d\mathcal{H}^β_\infty, \end{align*} where the supremum is taken over all finite subcubes $Q$ parallel to $Q_0$, $l(Q)$ is the length of the side of the cube $Q$, and $\mathcal{H}^β_\infty$ is the Hausdorff content. In the case $β=d$ we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every $β\in (0,d]$ one has a dimensionally appropriate analogue of the John-Nirenberg inequality for functions with bounded $β$-dimensional mean oscillation: There exist constants $c,C>0$ such that \begin{align*} \mathcal{H}^β_\infty \left(\{x\in Q:|u(x)-c_Q|>t\}\right) \leq C l(Q)^β\exp(-ct/\|u\|_{BMO^β(Q_0)}) \end{align*} for every $t>0$, $u \in BMO^β(Q_0)$, $Q\subset Q_0$, and suitable $c_Q \in \mathbb{R}$. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space $BV^{α,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $α\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{α,p}$ function.
In this paper we give an affirmative answer to the Euclidean analogue of a question of Bourgain and Brezis concerning the optimal Lorentz estimate for a Div-Curl system: The function $Z=\operatorname*{curl} (-Δ)^{-1} F$ satisfies \begin{align*} \operatorname*{curl} Z = F \newline \operatorname*{div} Z = 0 \end{align*} and there exists a constant $C>0$ such that \begin{align*} \| Z\|_{L^{3/2,1}(\mathbb{R}^3;\mathbb{R}^3)} \leq C\| F\|_{L^{1}(\mathbb{R}^3;\mathbb{R}^3)}. \end{align*} Our proof relies on a new endpoint Hardy-Littlewood-Sobolev inequality for divergence free measures which we obtain via a result of independent interest, an atomic decomposition of such objects.
We prove that for $α\in (d-1,d]$, one has the trace inequality \begin{align*} \int_{\mathbb{R}^d} |I_αF| \;dν\leq C |F|(\mathbb{R}^d)\|ν\|_{\mathcal{M}^{d-α}(\mathbb{R}^d)} \end{align*} for all solenoidal vector measures $F$, i.e., $F\in M_b(\mathbb{R}^d,\mathbb{R}^d)$ and $\operatorname{div}F=0$. Here $I_α$ denotes the Riesz potential of order $α$ and $\mathcal M^{d-α}(\mathbb{R}^d)$ the Morrey space of $(d-α)$-dimensional measures on $\mathbb{R}^d$.
In this paper we give a simple proof of the endpoint Besov-Lorentz estimate $$ \|I_αF\|_{\dot{B}^{0,1}_{d/(d-α),1}(\mathbb{R}^d;\mathbb{R}^k)} \leq C \|F \|_{L^1(\mathbb{R}^d;\mathbb{R}^k)} $$ for all $F \in L^1(\mathbb{R}^d;\mathbb{R}^k)$ which satisfy a first order cocancelling differential constraint. We show how this implies endpoint Besov-Lorentz estimates for Hodge systems with $L^1$ data via fractional integration for exterior derivatives.
We establish a family of inequalities that allow one to estimate the $\mathrm{L}^{q}$-norm of a matrix-valued field by the $\mathrm{L}^{q}$-norm of an elliptic part and the $\mathrm{L}^{p}$-norm of the matrix-valued curl. This particularly extends previous work by Neff et al. and, as a main novelty, is applicable in the regime $p=1$.
Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>λ\}|\leq C\left(\frac{1}λ\|f\|_{L^1(\mathbb{R}^n)}+\sup_ν |\{|R_jν|>λ\}|\right) \end{align*} for any $λ>0$ and $f \in L^1(\mathbb{R}^n)$, where the above supremum is taken over measures of the form $ν=\sum_{k=1}^Na_kδ_{c_k}$ for $N \in \mathbb{N}$, $c_k \in \mathbb{R}^n$, and $a_k \in \mathbb{R}^+$ with $\sum_{k=1}^N a_k \leq 16\|f\|_{L^1(\mathbb{R}^n)}$. This shows that to establish dimensional estimates for the weak-type $(1,1)$ inequality for the Riesz tranforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón-Zygmund operators.
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 $Ω\subset \mathbb{R}^N$, $\mathcal{H}^β_{\infty}$ is the Hausdorff content, $L^{N/α,q}(Ω)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the Hölder conjugate to $q$, and $I_αf$ denotes the Riesz potential of $f$ of order $α\in (0,N)$.
The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we don't understand about the action of integral operators on functions. This is no more apparent than in the $L^1$ setting, where only recently have optimal inequalities been obtained on the Lebesgue and Lorentz scale for scalar functions, while the full resolution of similar estimates for vector-valued functions is incomplete. The purpose of this paper is to discuss how some often overlooked estimates for the classical Poisson equation give an entry into these questions, to the present state of the art of what is known, and to survey some open problems on the frontier of research in the area.
In this paper we give a streamlined proof of an inequality recently obtained by the author: For every $α\in (0,1)$ there exists a constant $C=C(α,d)>0$ such that \begin{align*} \|u\|_{L^{d/(d-α),1}(\mathbb{R}^d)} \leq C \| D^αu\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \end{align*} for all $u \in L^q(\mathbb{R}^d)$ for some $1 \leq q<d/(1-α)$ such that $D^αu:=\nabla I_{1-α} u \in L^1(\mathbb{R}^d;\mathbb{R}^d)$. We also give a counterexample which shows that in contrast to the case $α=1$, the fractional gradient does not admit an $L^1$ trace inequality, i.e. $\| D^αu\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)}$ cannot control the integral of $u$ with respect to the Hausdorff content $\mathcal{H}^{d-α}_\infty$. The main substance of this counterexample is a result of interest in its own right, that even a weak-type estimate for the Riesz transforms fails on the space $L^1(\mathcal{H}^{d-β}_\infty)$, $β\in [1,d)$. It is an open question whether this failure of a weak-type estimate for the Riesz transforms extends to $β\in (0,1)$.
In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ be the homogeneous dimension of $G$ and $\mathcal{I}_α$ denote the Riesz potential of order $α$ on $G$. Then, for every $α\in (0,Q)$, there exists a constant $C=C(α,Q)>0$ such that \begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align} for distributions $f$ such that $X \mathcal{I}_1 f \in L^1(G)$, where $X$ denotes the horizontal gradient.
In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*} \mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} to satisfy the estimates \begin{align*} \|D^{k-j}u\|_{L^{\frac{n}{n-j}}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} for $j\in \{1,\ldots,\min\{k,n-1\}\}$ and \begin{align*} \|D^{k-n}u\|_{L^{\infty}(\mathbb{R}^n)}\leq c\|f\|_{L^1(\mathbb{R}^n)} \end{align*} when $k\geq n$.
The uniqueness of equilibrium for a compressible, hyperelastic body subject to dead-load boundary conditions is considered. It is shown, for both the displacement and mixed problems, that there cannot be two solutions of the equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains are uniformly close to each other. This result is analogous to the result of Fritz John (Comm. Pure Appl. Math. 25, 617-634, 1972) who proved that, for the displacement problem, there is a most one equilibrium solution with uniformly small strains. The proof in this manuscript utilizes Geometric Rigidity; a new straightforward extension of the Fefferman-Stein inequality to bounded domains; and, an appropriate adaptation, for Elasticity, of a result from the Calculus of Variations. Specifically, it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close, in $BMO\cap\, L^1$, to the gradient of the equilibrium solution.
We prove a family of Sobolev inequalities of the form $$ \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} \le \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} $$ where $A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-order homogeneous linear differential operator with constant coefficients, $u$ is a vector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo-Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn-Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis--Whitney inequality and Gagliardo's lemma.
In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(α,d)>0$ such that \[ \|I_αF \|_{L^{d/(d-α),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F \in L^1(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.
We show how a strong capacitary inequality can be used to give a decomposition of any function in the Sobolev space $W^{k,1}(\mathbb{R}^d)$ as the difference of two non-negative functions in the same space with control of their norms.
In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into $L^\infty(\mathbb{R}^N)$ and those that embed into slightly larger target spaces. Concerning the former, we show that for $k \in \{1,\ldots, N-1\}$, $N-k$ even, one has an optimal constant $c_k>0$ such that \[ \|u\|_{L^\infty} \leq c_k \int |\nabla^k (-Δ)^{(N-k)/2} u|\] for all $u \in C^\infty_c(\mathbb{R}^N)$ (the case $k=N$ was handled in a recent paper by Shafrir). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For $Ω\subset \mathbb{R}^N$, $m \in \mathbb{N}$ and $p=\frac{N}{m}$, there exists $A>0$ and optimal constant $β_0>0$ such that \[ \int_Ω \exp (β_0 |u|^{p^\prime}) \leq A |Ω| \] for all $u$ such that $\|\nabla^m u\|_{L^p(Ω)} \leq 1$, where $\|\nabla^m u\|_{L^p(Ω)}$ is the traditional semi-norm on the space $W^{m,p}(Ω)$.