Volume 25, Issue 2 - Special Issue: In honor of Robert Fefferman

In this paper, we establish the existence of a bounded, linear extension operator T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 < p < 2 and E is a finite subset of R 2 ${\mathbb{R}}^{2}$ contained in a line.

We survey recent results on soliton resolution for the energy critical nonlinear wave equation.

We build an example of a simple sparse operator for which its norm with scalar A 2 weight has linear estimate in [ w ] A 2 ${\left[w\right]}_{{A}_{2}}$ , but whose norm in matrix setting grows at least as [ W ] A 2 3 / 2 ${\left[W\right]}_{{\mathbf{A}}_{2}}^{3/2}$ for specially constructed weights from A 2 . We also demonstrate a matrix weighted maximal function with the same estimate from below.

This article is intended as an introduction to the distorted Fourier transform associated with a Schrödinger operator on the line or the half-line. This versatile tool has seen numerous applications in nonlinear PDE in recent years. It typically arises in the asymptotic stability analysis of topological solitons in classical field theory, such as kinks in the sine-Gordon or ϕ 4 models. The distorted Fourier transform is also a natural technique in the analysis of dispersive equations on manifolds with symmetries. Such models appear in general relativity, for example in the study of waves on a black-hole background. While microlocal methods have proven to be powerful in such applications, the more classical Weyl–Titchmarsh spectral theory and with it, the distorted Fourier transform, continue to play an essential role in the analysis of evolution PDEs. This article explain how it can be derived from Stone’s formula, which also establishes the Plancherel and inversion theorems.

We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in which L p solvability of the Neumann problem is shown to yield solvability in L q for 1 < q < p , and in the Hardy space H 1 , in 2-sided chord-arc domains, under suitable background hypotheses.

In this paper, we prove the existence of a bounded linear extension operator T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T:{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 < p < 2, where E ⊂ R 2 $E\subset {\mathbb{R}}^{2}$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” Adv. Nonlinear Stud. , vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral gaps for an asymptotic expansion, with two leading terms, of Dirichlet eigenvalues. We also establish the convergence rates for the corresponding eigenfunctions. Our approach uses a known reduction to a degenerate elliptic eigenvalue problem for which a quantitative analysis is carried out.

Discussed here is a regularized version (0.1) u t + u x + u u x + A u 2 u x − u xxt = 0 , $${u}_{t}+{u}_{x}+u{u}_{x}+A{u}^{2}{u}_{x}-{u}_{\mathit{xxt}}=0,$$ of the classical Gardner equation u t + u x + u u x + A u 2 u x + u xxx = 0 , $${u}_{t}+{u}_{x}+u{u}_{x}+A{u}^{2}{u}_{x}+{u}_{\mathit{xxx}}=0,$$ that arises in hydrodynamics and plasma physics. Considered here are well-posedness issues for the initial-value problem for the regularized equation posed on all of R $\mathbb{R}$ .

We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the x , v variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato’s approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.

It is well-understood that partial sums of Fourier series for L 1 functions don’t necessarily converge to the original function in the pointwise-almost everywhere sense. Estimates for averages of partial sum operators demonstrate that the divergence can be quantified. We establish a bilinear maximal estimate for averages of partial sums projectors.

The main result in this paper is that, for singular integral operators associated with standard kernels, local L 1 -estimates imply global L p -estimates for every p ∈ (1, ∞ ). When combined with the result of Melnikov-Verdera, this yields a complete and self-contained proof of the L p -boundedness of the Cauchy operator on Lipschitz curves and chord-arc curves in the plane.

The Halo Conjecture presents one of the outstanding open problems in the theory of differentiation of integrals. In this paper we discuss the Halo Conjecture and its connections with topics of current interest in harmonic analysis.

Inequalities are established for certain trilinear scalar-valued functionals. These functionals act on measurable functions of one real variable, are defined by integration over two- or three-dimensional spaces, and are controlled in terms of Lebesgue space norms of the functions, and in terms of negative powers of large parameters describing a degree of oscillation. Related sublevel set inequalities are a central element of the analysis.