Papers

Let $d\ge5$, and let $A_\lambda$ denote the normalized discrete spherical average on $\mathbb{Z}^d$ with squared radius $\lambda$. For a strictly increasing sequence $(\mu_k)$ of positive integers, set $\lambda_k=\mu_k!$ and consider the lacunary maximal operator $A_\star f:=\sup_k|A_{\lambda_k}f|$. For each fixed $\gamma>0$, we prove that the estimate \[ \#\{x:A_\star f(x)>\alpha\} \le K\sum_x\frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right)^\gamma \] holds with some finite $K$ for all $\alpha>0$ and finitely supported $f$ if and only if \[ \sup_{N\ge2}\frac{\#\{k:\mu_k\le N\}}{(1+\log N)^\gamma}<\infty. \] The summand is interpreted as zero when $f(x)=0$. The case $\gamma=1$ answers the $\ell\log\ell$ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For $\mu_k=2^{2^k}$, these estimates hold for every fixed $\gamma>0$, but $A_\star$ is not of weak type $(1,1)$. For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type $(p,p)$, $1<p\le d/(d-1)$, by a uniform covering condition. In particular, for each $1<p_*\le d/(d-1)$, we construct a set of squared radii whose maximal operator is of restricted weak type $(p_*,p_*)$, of strong type $(p,p)$ for $p>p_*$, and unbounded on $\ell^p$ for $1\le p<p_*$.

Continuing the study of mother-body measures with algebraic Cauchy transform, we associate with a positive algebraic germ $f$, its irreducible equation $P(z,w)=0$, and a compact convex set $K$ the functional \[ \mathfrak D_{P,K}(\mu)= \int_K\left|P\bigl(z,\mathcal C_\mu(z)\bigr)\right|^{1/d}\,dA(z), \qquad d=\text{deg}_wP, \] on the set of positive measures supported in $K$ and whose Cauchy transform has germ $f$ at infinity. We prove continuity of $\mathfrak D_{P,K}(\mu)$ and attainment of its minimum, characterize zero defect, and obtain the estimate \[ \mu(D(a,r))\le C_1r+C_2\mathfrak D_{P,K}(\mu)/r \] away from the zero set of the leading coefficient of $P$, where $D(a,r)$ is the disk of radius $r$ centered at $a$. We establish a dual formula for the defect in the rational case and construct positive algebraic Cauchy transforms by Herglotz theory, Fuss–Catalan and Raney laws, positive sums, polynomial pushforwards, and branch graphs. The passage from zero defect to a mother body is made under the planar-null hypothesis of the support.

We prove a Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces. The admissible directions are those for which the deterministic orbit is square integrable with values in the Hilbertian range of the noise coefficient. Under this square function condition, the derivative of the semigroup is represented by a stochastic integral against an explicit deterministic control. The result is a concrete control realisation of the Cameron–Martin formula for the fixed-time Gaussian transition measures.

We establish global $L^p$-Poincaré inequalities, for $ p\in[1,\infty)$, on a class of nondoubling hyperbolic-type metric measure spaces. The proof relies on a discretisation of the space, which gives rise to a Gromov hyperbolic graph, called spiderweb, quasi-isometric to the original space. We prove global Poincaré inequalities for spiderwebs endowed with suitable measures and develop a general transference principle from discrete graphs to metric measure spaces. Combining these results yields global Poincaré inequalities under natural geometric and measure assumptions on the base space.

We establish two pointwise endpoint limits for nonlocal operators: one based on normalized integrals and a distributional limit based on weighted weak-type norms. As applications, the limits yield pointwise Bourgain–Brezis–Mironescu, Maz'ya–Shaposhnikova, Brezis–Seeger–Van Schaftingen–Yung and Gu–Yung formulas, together with higher-order and mean-oscillation variants. We then investigate endpoint limits for fractional powers generated by semigroups and for approximation processes, and derive sharp strong and weak endpoint estimates for operators in harmonic analysis. Finally, while Domínguez and Milman obtained weak-type estimates on product spaces for $p>1$ and left the endpoint $p=1$ open (see [Adv.~Math.~411 (2022), Paper~No.~108774, p.~22]), we prove a sharp pointwise limit in the parameter variable and obtain two–sided iterated weak-type estimates at $p=1$.

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.

We study how sparsely a nonzero discrete harmonic function on the standard lattice $\mathbb{Z}^d$ can be supported. Let $Q_n^{(d)}=\{-n,\cdots,n\}^d$, and let $m_d(n)$ denote the least possible value of $|\mathrm{supp}(u)\cap Q_n^{(d)}|$ among discrete harmonic functions $u:\mathbb{Z}^d\to\mathbb{C}$ with $u(0)\neq0$. For all $n\geq1$, we prove \begin{equation*} m_3(n)\asymp n^2, \quad c_dn^{d^2/(2d-1)} \leq m_d(n)\leq (2n+1)^{\lfloor d/2\rfloor+1} \quad d\ge 4. \end{equation*} These estimates extend the two-dimensional support estimate of Buhovsky, Logunov, Malinnikova, and Sodin [Duke Math. J. 171 (2022), 1349–1378] to higher dimensions and obtain sharpness in dimension three. For $d\geq4$, the lower exponent and the upper one differ by less than $3/4$ in even dimensions and $1/4$ in odd dimensions. The proof combines Hilbert functions of finite support sets with a position-translation uncertainty principle. The sharp three-dimensional bound additionally uses Cayley–Bacharach relations and rigidity of algebraic curves. Finally, for every nonzero lattice eigenfunction with eigenvalue $\lambda$, the Zariski closure of its full support has dimension at least $\lceil d/2\rceil$, and at least $\lfloor d/2\rfloor+1$ when $\lambda\neq0$. Both bounds are optimal. All proofs resulted from human-guided exploration by GPT-5.6 Sol in Ultra mode and checked by the author.