Use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results, which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove known sparse bounds for pseudodifferential operators with symbols in $S^0_{1,\delta}$ for $\delta \lt 1$.

We characterize weighted modulation spaces (data space) for which the heat semigroup $e^{-tL}f$ converges pointwise to the initial data f as time t tends to zero. Here L stands for the standard Laplacian $-\Delta $ or Hermite operator $H=-\Delta +|x|^2$ on the Euclidean space. This is the first result on pointwise convergence with data in a weighted modulation spaces (which do not coincide with weighted Lebesgue spaces). We also prove that the Hardy–Littlewood maximal operator operates on certain modulation spaces. This may be of independent interest. We have highlighted several open questions that arise naturally from our findings.

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus $\mathbb {T}^d$, with respect to arbitrary Borel measures $\mu $. Specifically, we characterize the measures $\mu $ for which the inequalities

$$ \begin{align*} \int |u|^2 \mathop{}\!\mathrm{d}\mu \lesssim \int |u|^2 \mathop{}\!\mathrm{d} x \quad \text{(trace)}, \qquad \int |u|^2 \mathop{}\!\mathrm{d}\mu \gtrsim \int |u|^2 \mathop{}\!\mathrm{d} x \quad \text{(observability)} \end{align*} $$

hold uniformly for all eigenfunctions u of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\mu $, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain–Rudnick to higher dimensions. Applications include results in the spirit of Cantor–Lebesgue theorems, constraints on quantum limits, and control theory for the Schrödinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.

We study the class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition and show that such functions are $\varepsilon$-approximable. As a consequence, we obtain the quantitative Fatou theorem in the spirit of works, for example, by Garnett [6] and Bortz–Hofmann [1]. Such a class contains harmonic functions, as well as non-harmonic ones, for example, nonnegative subharmonic functions whose gradient norm is quasi-nearly subharmonic, as illustrated by our discussion.

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $\alpha>0$. Let $T_{\varphi }$ be the Toeplitz operator on the Fock space $F^2_{\alpha }$ over $\mathbb {C}^n$ with symbol $\varphi \in L^{\infty }$. For $1<p<\infty $ and any finite sum T of finite products of Toeplitz operators $T_{\varphi }$’s, we show that T is compact on the weighted Fock space $F^p_{\alpha ,w}$ if and only if its Berezin transform vanishes at infinity, where w is a restricted $A_p$-weight on $\mathbb {C}^n$. Concerning boundedness, for $1\leq p<\infty $, we characterize the r-doubling weights w such that $T_{\varphi }$ is bounded on the weighted spaces $L^p_{\alpha ,w}$ via a $\varphi $-adapted $A_p$-type condition. Our method also establishes a two-weight inequality for the Fock projections in the case of r-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman–Toeplitz operators, which answers a question raised by Stockdale and Wagner (2023, Math. Z. 305, Article no. 10).

A characterization of Muckenhoupt weights in terms of a Harnack-type property is presented along with several applications.

We show sharp well-posedness with analytic data-to-solution mapping in the semilinear regime for dispersion-generalized KP-I equations on $\mathbb {R}^2$ and $\mathbb {R} \times \mathbb {T}$. On $\mathbb {R}^2$ we cover the full subcritical range, whereas on $\mathbb {R} \times \mathbb {T}$ the sharp well-posedness is strictly subcritical. We rely on linear and bilinear Strichartz estimates which are proved using decoupling techniques and square function estimates. Nonlinear Loomis-Whitney inequalities are a further ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb {R}^2$, the cylinder $\mathbb {R} \times \mathbb {T}$, and the torus $\mathbb {T}^2$. Finally, we point out that on tori $\mathbb {T}^2_\gamma $, KP-I equations are never semilinear.

Given a Fourier transformable measure in two dimensions, we find a formula for the intensity of its Fourier transform along circles. In particular, we obtain a formula for the diffraction measure along a circle in terms of the autocorrelation measure. We then look at some applications of this formula.

In this note, the author recalls the Calderon–Zygmund theory on the unit ball and derives the weak (1,1) boundedness of the projection for $\mathcal {H}$-harmonic Bergman space.

In this paper, we establish the $L^p$ bounds for partial polynomial Carleson operators along polynomial curves for $p \gt 1$, which depend only on $p$ and the number of monomials in the defining polynomial. Additionally, we study two classes of oscillatory integral operators of Radon type and derive uniform $L^2$ bounds.

We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power $\operatorname{dist}(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt A1 class of weights in X. To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest E-free holes, these sets characterize the mentioned A1-property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in ${\mathbb R}^n$ that do not belong to $A_\infty.$

In this article, we introduce Orlicz spaces on $ \mathbb Z^n \times \mathbb T^n $ and Orlicz modulation spaces on $\mathbb Z^n$, and study inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{\Phi }(\mathbb Z^n)$ is close to the modulation space $M^{2}(\mathbb Z^n)$ for some particular Young function $\Phi $. Then, we study localization operators on $\mathbb Z^n$. In particular, using appropriate classes for symbols, we prove that these operators are bounded on Orlicz modulation spaces on $\mathbb Z^n$, compact and in the Schatten–von Neumann classes.

The purpose of this paper is to expose and investigate natural phase-space formulations of two longstanding problems in the restriction theory of the Fourier transform. These problems, often referred to as the Stein and Mizohata–Takeuchi conjectures, assert that Fourier extension operators associated with rather general (codimension 1) submanifolds of Euclidean space may be effectively controlled by the classical X-ray transform via weighted $L^2$ inequalities. Our phase-space formulations, which have their origins in recent work of Dendrinos, Mustata and Vitturi expose close connections with a conjecture of Flandrin from time-frequency analysis, and rest on the identification of an explicit ‘geometric’ Wigner transform associated with an arbitrary (smooth strictly convex) submanifold S of $\mathbb {R}^n$. Our main results are certain natural ‘Sobolev variants’ of the Stein and Mizohata–Takeuchi conjectures and involve estimating the Sobolev norms of such Wigner transforms by geometric forms of classical bilinear fractional integrals. Our broad geometric framework allows us to explore the role of the curvature of the submanifold in these problems, and in particular we obtain bounds that are independent of any lower bound on the curvature; a feature that is uncommon in the wider restriction theory of the Fourier transform. Finally, we provide a further illustration of the effectiveness of our analysis by establishing a form of Flandrin’s conjecture in the plane with an $\varepsilon $-loss. While our perspective comes primarily from Euclidean harmonic analysis, the procedure used for constructing phase-space representations of extension operators is well-known in optics.

Let $M$ denote the centered Hardy–Littlewood operator on $\mathbb{R}$. We prove that

\begin{equation*}{\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big|,\end{equation*}

for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt [3] proved for indicator functions of bounded variation vanishing at $\pm\infty$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.

We prove new weighted decoupling estimates. As an application, we give an improved sufficient condition for almost everywhere convergence of the Bochner-Riesz means of arbitrary $L^p$ functions for $1<p<2$ in dimensions 2 and 3.

The $L^p$ boundedness of the commutator $[b, T]$ has been intensively studied in recent decades in part because it has important connections and applications to partial differential equations. Inspired by these works, we study the boundedness and compactness of the Riesz transform commutator in a general setting, namely, in the scale of Lorentz spaces and on stratified Lie groups. In this article, we provide a complete characterization between the space of the symbol b and the Lorentz estimates of $[b, R_j]$.

Let $M_\mu $ be the uncentered Hardy–Littlewood maximal operator with a Borel measure $\mu $ on $\mathbb {R}$. In this note, we verify that the norm of $M_\mu $ on $L^p(\mathbb {R},\mu )$ with $p\in (1,\infty )$ is just the upper bound $\theta _p$ obtained by Grafakos and Kinnunen and reobtain the norm of $M_\mu $ from $L^1(\mathbb {R},\mu )$ to $L^{1,\infty }(\mathbb {R},\mu )$. Moreover, the norm of the “strong” maximal operator $N_{\vec {\mu }}^{n}$ on $L^p(\mathbb {R}^n, \vec {\mu })$ is also given.

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As two applications, we first obtain the upcrossing inequalities with exponential decay of ergodic averages and then provide an explicit bound on the convergence rate such that the ergodic averages with strongly continuous regular group actions are metastable (or locally stable) on a large interval. Before exploiting the transference techniques, we actually obtain a stronger result—the jump estimates on a metric space with a measure not necessarily doubling. The ideas or techniques involve martingale theory, non-doubling Calderón–Zygmund theory, almost orthogonality argument, and some delicate geometric argument involving the balls and the cubes on a group equipped with a not necessarily doubling measure.

In this article, the authors establish an extension-restriction theorem between homogeneous weighted Besov spaces and weighted mixed-Riesz potential spaces. This general frame covers both the classical Besov spaces and their logarithmic analogs.