Let $d\geq 4$ and let $R>0$. When $d=4$, assume that $R^2\in\mathbb{N}\setminus 4\mathbb{N}$; when $d\geq 5$, let $R^2\in\mathbb{N}$ be arbitrary. We prove the fixed-radius estimate $$\|A_R f\|_{\ell^{p'}(\mathbb{Z}^d)}\leq C_{d,p,\varepsilon}R^{-d(2/p-1)+\varepsilon}\|f\|_{\ell^p(\mathbb{Z}^d)}$$ for $(d+2)/d\leq p\leq 2$, where $p'$ is the H\"older conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.
For $d \geq 2$, $p \geq 1$ and $\epsilon>0$, let $N_p(d,\epsilon)$ be the smallest integer $N$ such that for every integer $n$ and every $A\in\mathbb{R}^{n\times d}$, there exists a matrix $\Phi\in\mathbb{R}^{N\times n}$ satisfying $(1-\epsilon)\lVert Ax\rVert_p\leq \lVert\Phi A x\rVert_p\leq (1+\epsilon)\lVert Ax\rVert_p$ for all $x\in\mathbb{R}^d$. For every constant $p\geq 1$ with $p\not\in 2\mathbb{Z}$, when $d\gtrsim_p \log(1/\epsilon)$, the bound \[ N_p(d,\epsilon) \gtrsim_{p} \frac{d}{\epsilon^2 \operatorname{polylog}(d/\epsilon)} \] is established. This improves the previous lower bound $\Omega(1/(\epsilon^2\operatorname{polylog}(1/\epsilon)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\leq p<2$. The central technical idea originated from ChatGPT 5.6 Sol.
We prove P\'olya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2}, $$ where $\omega_d$ is the volume of the unit $d$-ball and $N_{B_R^d}^{<}(E)$ counts Neumann eigenvalues strictly below $E$. Combined with the Dirichlet theorem for balls, this settles both P\'olya inequalities for Euclidean balls in every dimension $d\ge2$. In the disk case, the proof replaces a computer-assisted finite-frequency step by explicit Rayleigh--Ritz estimates. In dimensions $d\ge3$, the radial Neumann condition is a Dini condition rather than a derivative-zero Bessel condition. A strict comparison with an auxiliary Robin problem transfers a derivative-zero Bessel phase estimate to the physical Neumann spectrum. The problem then becomes a comparison between a multiplicity-weighted phase staircase and an integral equal to the Weyl term. Variational trial spaces control low frequencies; finitely many radial levels and beta-integral estimates cover the intermediate range; and a uniform phase estimate treats high frequencies. All finite computations for $2\le d\le6$ are printed in the paper. For $d\ge7$, one compact two-parameter estimate is verified in exact rational arithmetic by the ancillary program.
Let $M=\mathbb{H}^n\times\mathbb{S}^m$, where $n\geq 2$, $m\geq 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\leq k<N/2$, and assume that $\Lambda_0=\inf\sigma_{L^2(M)}(P_k)>0$. We study$$P_kU-\lambda U=|U|^{q-2}U,\qquad q=\frac{2N}{N-2k},\qquad 0<\lambda\leq\Lambda_0,$$and attainment of the associated critical quotient $S_{\lambda,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<\lambda<\Lambda_0$, the inequality $S_{\lambda,k}(M)<S_{N,k}$ implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when $N\geq4k$, or when $2k+2\leq N<4k$ and $\lambda>\Lambda_{\mathrm{loc}}$, where $\Lambda_{\mathrm{loc}}$ is explicit. If $N\geq2k+2$ and $S_{\Lambda_0,k}(M)<S_{N,k}$, attainment also holds at $\lambda=\Lambda_0$ in the threshold form completion. At the threshold, $L^2$-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving $\mathcal{A}=\mathbb{R}^{n-1}\times{0}$. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping $\mathcal{A}$ from normalized minimizing sequences. If $r_j^{(J)}$ denotes the remainder after the first $J$ extracted profiles, then$$\lim_{J\to\infty}\limsup_{j\to\infty}|r_j^{(J)}|_{L^q(\mathbb{R}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.
We investigate $L^p$ estimates for the $\bar{\partial}$-problem on rational Hartogs triangles $\mathbb{H}_{m/n} = \{ (z_1, z_2) \in \mathbb{C}^2 : |z_1|^m<|z_2|^n<1 \}$. For $p \in (1, \infty)$, we establish the existence of a solution operator that is bounded on $L^p(\mathbb{H}_{m/n})$. Our approach avoid the need for any {\it a priori} condition on the data. We also show that the canonical solution $K_{\mathbb{H}_{m/n}}$ is bounded on $L^p(\mathbb{H}_{m/n})$ for $p \in (p_0, p_2)$, where $p_0=\frac{2m+2n}{m+n+1+\min\{m, n\}}$ and $p_2=\frac{2m+2n}{m+n-1}$. For classical Hartogs triangle, $\mathbb{H}_1$, this establishes boundedness for $p \in (1, 4)$.
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert>t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.
We prove a pointwise gradient estimate for the fractional $p$-Laplace equation with measure data. Let $n\ge 2$, $p>1$ and $\max\{n/p,1/p'\}<s<1$, where $p'=p/(p-1)$. Suppose $u\in W^{s,p}(\mathbb R^n)$ is a weak solution of \[ (-\Delta_p)^s u=\mu \quad\text{in }\Omega, \] with $\mu\in\mathcal M_{\mathrm{loc}}(\Omega)$. Set $\gamma=s-(p-1)/p$. If $B_{2R}(x_0)\Subset\Omega$ and $\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)<\infty$, then $u$ is Fr\'echet differentiable at $x_0$, and \[|\nabla u(x_0)|\le C\bigl[\mathcal A(u;x_0,2R)+\mathcal{W}_{\gamma,p}^{|\mu|}(x_0,2R)\bigr], \] where $\mathcal{W}_{\gamma,p}^{|\mu|}$ denotes the truncated Wolff potential, and $\mathcal A$ depends on the local oscillation of $u$ and its nonlocal tail. The proof uses a comparison with fractional $p$-harmonic replacements and an affine decay estimate for homogeneous solutions, with constants independent of the affine slope. In the large slope regime, the affine decay estimate follows from a Schauder estimate for the linear nonlocal equation satisfied by the affine remainder.
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