Classical Analysis and ODEs Papers of the Week

We study phase retrieval for multi-window short-time Fourier transforms with windows given by complex linear combinations of the first two normalized Hermite functions.

We prove that $\int|\Delta\mathbf{U}|^2dx\int|x|^2|\nabla\mathbf{U}|^2dx\ge C_N^2(\int|\nabla\mathbf{U}|^2dx)^2$, where $C_N=1+\frac12\sqrt{N^2+8}$ for $N\ge3$ and $C_2=1+\sqrt2$.

We prove a pointwise domination of Poisson variances for Hörmander--Mikhlin multipliers on the real line.

This area function defines a global Hardy space. We prove its $\Delta$-atomic description, its injective embedding into $L^1$, its characterization by the restricted maximal function, and its real interpolation identity with $L^2$.

In this article, we prove two compactness parabolic extrapolation theorems for linear operators and their commutators on parabolic weighted Lebesgue spaces.

We prove uniqueness, up to common translations and dilations, of the normalized nonnegative extremizing pairs for the corresponding Hardy-Littlewood-Sobolev inequality.

We study intersections of $K$ with rectifiable curves and chord-arc curves.

We allow the dispersal kernel to be heterogeneous and non-symmetric, and remove the standard global upper bound on the kernel (thereby covering the case of heavy-tailed dispersal kernels). To overcome the analytical difficulties brought by this more general setting, we establish a new Harnack-type inequality for non-compactly supported kernels, and apply homogenization and cut-off approximation techniques to derive two-sided bounds for the domain approximation limit of GPEs.

Our approach treats the boundary-induced interactions as a perturbation of the classical self-similar Gröbli collapse in the full plane.

This paper establishes sharp variational estimates for families of averaging operators over smooth hypersurfaces in $\mathbb{R}^{n+1}$.

We study the truncated Dirichlet functional \[ \Phi_n (f) = \sum_{k = 1}^n k |a_k|^2, \quad f(z) = \sum_{k = 0}^\infty a_k z^k, \] when $f$ belongs to the Bloch space.

We introduce the concept of Markov-driven self-similar measures, a notion that simultaneously generalizes classical self-similar measures, self-conformal measures, dynamically driven self-similar measures and many random measures.

For $1<q<p<\infty$ we study the $L^p\to L^q$ contractivity of the two-point operator $T_\omega(a+bx)=a+\omega bx$ with a complex parameter $\omega$, extending the Weissler two-point condition, usually discussed in the range $1<p\le q<\infty$, to the reversed order of exponents.

For parabolic Ahlfors regular sets, we show that the bilateral weak geometric lemma (BWGL), bilateral approximation by unions of planes (BAUP), and corona decompositions by $Lip$(1,1/2) graphs (whose defining functions need not have half-order time derivatives in BMO) are equivalent.

As an application, for every $s\in(0,1]$, there exists a bounded integer-based homogeneous Moran set of uniqueness $K$ such that $\dim_H K=s$ and the self-product $K\cdot K$ is an $M_0$-set. In sharp contrast, for every $s\in[0,1/2]$ we can construct an unbounded integer-based homogeneous Moran $U$-set $E$ with $\dim_H E=s$ such that both $E$ and $E\cdot E$ are $U$-sets, and $1/2$ is the optimal threshold for $U$-sets to be closed under the arithmetic self-product.

We give a direct proof on the discrete hypercube.

The determinants in question are those of cyclic tridiagonal matrices; their explicit evaluation is carried out, in addition to the inclusion-exclusion principle, conceptionally using combinatorial arguments based on monomer-dimer systems. This explicit evaluation serves as a core component in the study of arithmetic properties such as the modularity of the special values of the spectral zeta function.

We study multiple orthogonal polynomial systems obtained from a Nikishin system of two measures by applying a Christoffel transform to one of the measures at a point $x=x_0$.

We show that any uniformly $d$-rectifiable subset of $\mathbb{R}^n$ is contained in an Ahlfors $d$-regular set in $\mathbb{R}^{n+1}$ which supports a $(1,1)$-Poincaré inequality.