arXiv subjects

Shengzhao Hou

Publications and source records attributed to Shengzhao Hou.

5 recordsLinked to original sources

arXiv2026

Canonical analytic realizations of hyperbolic determinantal processes

Krishnapur asked whether the invariant hyperbolic determinantal point processes on the disk admit a random analytic zero-set interpretation at noninteger parameters. We construct such a realization for every positive real parameter as the full compact-open limit in distribution of normalized finite Blaschke products. The zeros determine the modulus and normalized analytic shape, leaving one independent uniform phase. We prove exact Möbius covariance and classify all realizations with this covariance and square-integrable logarithmic modulus at the origin: they are precisely independent positive random multiples of the canonical function. Within this covariant class, matching the canonical logarithmic mean and variance uniquely determines the canonical function law. The family is weakly continuous in the parameter and agrees at positive integers with determinants of matrix-valued Gaussian power series. An explicit Barnes $G$-function Mellin transform determines the basepoint normalization.

math.PR↗

arXiv2026

The Local Embedding Problem for Hardy Spaces of Dirichlet Series

We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{θ\in\mathbb{R}}\int_θ^{θ+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\le p<\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.

math.FA↗

arXiv2026

Hardy-Szegő Point Processes: Large Deviations and Strong Szegő Asymptotics

We study exponential-scale fluctuations of the Hardy-Szegő zero process, investigated by Peres and Virág in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szegő expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.

math.PR↗

arXiv2026

Two Regularity Problems on Analytic Tent Spaces

We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,α}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann--Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel--Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,α}$, and we also obtain the Triebel--Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,α}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy--Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.

math.FA↗

arXiv2026

Two Problems in Bergman Spaces with Non-radial Weights

This paper investigates two problems unified by the study of the uniform boundedness of the dilation operators (UBD) T_r f(z)=f(rz), 0<r<1, acting on weighted Bergman spaces A^p_omega with not necessarily radial weights. We first characterize the random symbol space for A^p_omega under a mild admissible condition (Theorem 1.2). This extends the main result of [7] from radial weights to non-radial weights. We then introduce two new notions, namely non-radial mixed norm spaces M(p,q;omega) and analytic tent spaces A(p,q;omega), and we characterize their corresponding symbol spaces as well (Theorem 1.8, Theorem 1.12). The novelty here is to employ a measure-disintegration framework to prove a general Littlewood-type theorem (M(p,q;omega))* = H(2,q;omega_r), from which the Bergman space result (A^p_omega)* = H(2,p;omega_r) follows as the special case p=q. Among other things, UBD plays a pivotal role in the proofs of the preceding theorems. The second main problem addressed in this paper is to establish a local-to-global criterion for UBD, which remains largely unexplored for non-radial weights. Our principle result in this part (Theorem 1.16) asserts that UBD is guaranteed by two local geometric conditions: bounded hyperbolic oscillation (BHO) and a reverse-Carleson tail condition (RC). This is the technical heart of the paper. In the course of our investigation, three new types of problems arise naturally, each of independent interest: a two-weight top-maximal operator (Theorem 1.17); a truncated maximal operator over hyperbolic balls (Theorem 1.18); and a single-testing Carleson embedding problem (Theorem 1.19).

math.CV↗