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Wednesday, 4 November 2026: 8 new papers, 5 cross-lists and 7 replacements in Classical Analysis and ODEs (math.CA), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study the boundedness of the right-sided Hardy--Littlewood maximal operator on variable Lebesgue spaces over the real line. Motivated by the $K_0$ condition for the classical Hardy--Littlewood maximal operator, we introduce a directional condition $K_0^+$ adapted to the geometry of the one-sided maximal operator. We characterize $K_0^+$ in terms of the uniform boundedness of a family of one-sided averaging operators and prove that it is a necessary condition for the boundedness of $M^+$ on $L^{p(\cdot)}(\mathbb{R})$. We also establish one-sided analogues of the modular and reverse H\"older estimates associated with the classical $K_0$ condition. These estimates constitute the main local ingredients in the proof of boundedness of $M^+$ under the combined assumptions $K_0^+$ and $N_\infty$.
For trigonometric and polynomial orthogonal expansions, we present coefficient bounds with an explicit numerator determined by the total variation of the function and the largest cumulative signed area of the weighted basis element. This area, denoted by $M_n$, is the sharp constant in the corresponding weighted integral estimate. We calculate or bound $M_n$ for several classical systems, thereby obtaining quantitative inequalities beyond an asymptotic statement of coefficient decay. Integration between consecutive zeros gives a geometric interpretation of the estimate. The comparison includes Chebyshev polynomials of the first three kinds, Legendre, Jacobi, Gegenbauer, and ordinary Laguerre polynomials, with normalization and boundary conditions stated explicitly. Classical estimates and their elementary consequences are organized as Fourier analysis in inequalities. In the controlled distortion application, ideal curves have selected coefficients absent, and restrictions on the variation, corridor width, and number of extrema of admissible departures give explicit guarantees for those coefficients.
Let $M=\mathbb R^m\sharp(\mathbb R^n\times\mathbb S^{m-n})$, where $m>n\geq3$. The full-cone Poisson area and non-tangential maximal functions fail to be of weak type $(p,p)$ for $1<p<2$. Let $K$ be the compact part joining the two ends and put $\rho(x)=1+d(x,K)$. For fixed small $c>0$, we restrict the horizontal distance from the vertex to $c\rho(x)$ and retain every positive time. The resulting maximal function is of weak type $(1,1)$ and bounded on $L^p$ for $1<p\leq\infty$. The corresponding area function uses the full spatial and time gradient. It satisfies a Fefferman--Stein distribution inequality. Its $L^p$ norm is equivalent to that of $f$ for $1<p\leq2$. 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 $L^2$. Direct estimates for spectral-band kernels then give a Mihlin multiplier theorem from this Hardy space to $L^1$. Interpolation and duality give $L^p$ boundedness for every $1<p<\infty$.
Let $\eta\geq 0$ and $\delta>0$, and let $(r_j)_{j\geq 0}$ be a sequence of real numbers satisfying $\eta \leq r_0 < r_1 < r_2 < \cdots$ and $r_{j+1}-r_j \geq \delta$ for every $j\geq 0$. We prove that, for $\delta\leq 1$ and every $x>0$, \[ \sum_{j=0}^\infty \frac{x^{r_j}}{\G(1+r_j)} \leq \sum_{j=0}^\infty \frac{x^{\eta+j\delta}}{\G(1+\eta+j\delta)} = x^\eta E_{\delta,1+\eta}(x^\delta), \] where $E_{\delta,1+\eta}$ denotes the two-parameter Mittag-Leffler function. Moreover, equality holds for some $x>0$ if and only if $r_j = \eta + j\delta$ for every $j$. The proof combines a finite compression argument for unimodal functions with a shift inequality for the Mittag-Leffler function. The upper bound $\delta \leq 1$ is sharp, since counterexamples exist for every $1 < \delta < 2$.
We introduce the bilinear Kor\'anyi spherical means on the Heisenberg group $\mathbb{H}^n,$ and derive $L^{p_1}(\mathbb{H}^n) \times L^{p_2}(\mathbb{H}^n) \to L^{p}(\mathbb{H}^n)$ estimates for the associated bilinear Kor\'anyi spherical maximal function on $\mathbb{H}^n.$ We also address multilinear extensions of our results.
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. Assume that the family of window coefficient vectors does phase retrieval in $\bbC^2$. We prove that the multi-window magnitude samples uniquely determine every signal in $L^2(\bbR)$ up to a global phase whenever the sampling set is separated with lower Beurling density greater than $2$, or is a translated lattice with density at least $2$. Moreover, we show that the sampling density is the best possible, which improves a result in [P.~Grohs, L.~Liehr, and M.~Rathmair. Multi-window {STFT} phase retrieval: lattice uniqueness, J. Funct. Anal., 288(3): Paper No. 110733, 23, 2025].
Varj\'u and Yu have given a complete description of non-singleton self-similar sets of uniqueness on the real line $\mathbb R$. In this paper, we first prove an arithmetic product theorem for self-similar $U$-sets $E:$ if $F\subset\mathbb R$ is compact, then $F\cdot E$ is an $M_0$-set if and only if $F$ is uncountable. We next show that this conclusion remains valid for bounded integer-based homogeneous Moran sets. 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. Moreover, we show that if $E\subset[0,1]$ is a compact set whose self-product $E\cdot E$ is contained in an $H$-set, then $\underline\dim_B E\le1/2$.
This survey concerns Stein's analytic family of generalized spherical means and the associated maximal operator. We review Stein's original theorem, the sharp necessary conditions for boundedness, and the complete theory for radial functions. We then explain the connection between positive results and local smoothing estimates for the half-wave propagator and, more generally, for Fourier integral operators. This point of view yields the sharp two-dimensional result through the work of Guth, H. Wang, and R. Zhang and, when combined with the recent local smoothing theorems, an improved higher-dimensional range. As a byproduct, the maximal operator associated with the first radial derivative of spherical means is bounded on $L^p(\R^4)$ for $2<p<3$, a case left open in recent work of Kwa\'snicki. We keep track of the relation between the local smoothing exponents and the resulting maximal estimates, and identify the region that remains open.
In this paper we consider cycles of singular algebraic foliations on the real projective plane. Our first result shows that there exists a limit cycle of singular algebraic foliations such that the limit cycle cannot be realized in any affine chart, that is, it fails to appear in any affine plane via the usual dehomogenization. On the other hand, under an appropriate convexity assumption, we verify that a cycle becomes fully visible, lying entirely in some affine chart. We then analyze the accumulation behavior of limit cycles of singular algebraic foliations on the real projective plane. Last we prove that algebraic foliations of degree at most two on the real projective plane have only finitely many limit cycles, and the maximum number of limit cycle is zero, one and at least six for algebraic foliations of degree zero, one and two on the real projective plane, respectively. This result extends Bamon's finiteness theorem for quadratic affine vector fields in [Publ. Math. IH\'ES, (64)1986, pp. 111--142].
For a bounded domain $\Omega\subset\mathbb R^d$, the spherical average $E_\Omega(R)=\int_{\mathbb S^{d-1}}|\widehat{\mathbb{1}_\Omega}(R\omega)|^2d\omega$ is never $O(R^{-a})$ with $a>d+1$, and for smooth boundaries it is $O(R^{-d-1})$, as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with $C^{1,\frac{1}{2}}$ boundary, and this regularity condition has not been improved since. How much regularity does the decay $R^{-d-1}$ require? We show that among H\"older classes the answer is $C^{1,\frac{1}{4}}$. Every bounded domain with $C^{1,\frac{1}{4}}$ boundary satisfies $E_\Omega(R)\leq C_\Omega R^{-d-1}$, while for every $0<s<\frac{1}{4}$ any smooth domain can be altered near a single boundary point to produce a $C^{1,s}$ domain for which this bound fails along a sequence of radii. More generally, when the boundary has fractional Sobolev regularity $W^{1+s,p}$, we determine the optimal decay exponent under a Dini condition on the normal, and in a large range of $s$ and $p$ without it. The upper bounds follow from $L^2$ estimates for nondegenerate oscillatory integral operators whose phases are distances between points of two graphs with Dini or fractional Sobolev gradients, and hence barely better than $C^1$. These are far too rough for the integrations by parts techniques from the smooth theory. We replace derivatives by finite differences adapted to the phase, a device that we expect to be useful in a variety of related problems.
We establish that any nontrivial anharmonic Fourier series with exponents quadratically close to the positive integers changes sign on each interval of length 2$\pi$.
Let $\alpha\in\mathbb R$, $0<p<\infty$, $0<q,r\le\infty$, and $W$ be a matrix $\mathcal A_p$ weight on ${\mathbb R}^n$. We introduce the matrix-weighted $Z$-spaces $Z^{p,q,r}_\alpha(W)$ and tent spaces $T^{p,q,r}_\alpha(W)$ on the upper half-space and establish their characterizations using reducing operators. We then characterize homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces in terms of the corresponding $Z$-spaces and tent spaces, respectively, using Schwartz kernels satisfying Tauberian and finite moment conditions. Furthermore, we establish duality for the matrix-weighted $Z$-spaces and tent spaces and obtain real and complex interpolation results for these spaces and the associated homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces.
We study the distortion of the lower dimension and lower spectrum of sets under quasiconformal mappings. As an application, we verify weighted Hardy--Sobolev inequalities in Euclidean domains arising as quasiconformal images of domains with sufficiently regular boundaries. The regularity in question is phrased in terms of estimates for either the Assouad or the lower dimension of the complement. We also provide dilatation-dependent classification results for certain planar open sets.