Papers

In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. The main purpose of this paper is to provide a singular Friedrichs finite-data variational framework valid in every angular-momentum sector, thereby extending the existing finite-data variational theories concerning either regular one-dimensional operators or the radial sector $\ell=0$. By means of a Volterra representation of the Friedrichs branch, we prove weak continuity and continuous Fréchet differentiability of nodal radii, exact realization of compatible same-mode nodal data, existence of optimal potentials, and finite-codimensional constraint geometry. The same framework also incorporates mixed angular-momentum and spectral–nodal observations through finite-dimensional transversality. Remarkably, a global uniqueness theorem is established for inward displacements of the unique interior node of the second mode in the radial sector \(\ell=0\). For a constant reference potential and \(p>(d+2)/2\), every such displacement admits a unique global optimizer. Unlike local inverse-mapping or one-dimensional integrability arguments, the proof first selects the admissible critical sign globally and then reduces every minimizer to a scalar mass-balance equation between a focusing ball branch and a logistic annulus branch. The strict opposite monotonicity of the two weighted masses makes the balance parameter unique, providing a global rigidity mechanism over the entire inward-displacement regime.

We provide a complete algebraic characterization of homogeneous self-affine measures whose Fourier transform does not tend to $0$ at infinity. Moreover, we prove that an integral self-affine measure has power Fourier decay if and only if it is absolutely continuous. For a given integral affine IFS, we also provide an algorithm for determining those probability vectors for which the associated self-affine measure is absolutely continuous.

In this work, we present a constructive method to derive homogeneous linear differential equations of arbitrary order for Laguerre–Hahn orthogonal polynomials. The approach is based on a hierarchy of structure relations, built recursively from the fundamental structure relation of the Laguerre–Hahn forms. The fourth-order differential equation, which has been extensively studied in the literature, is naturally recovered as a particular case of this general framework. Special attention is devoted to the semiclassical case. The method is fully algorithmic and has been implemented in {\it Mathematica$^{\circledR}$}. We apply it to two Laguerre–Hahn families of class zero analogous to Hermite, providing explicit structure relations and differential equations of orders five and ten. We also examine the classical Hermite sequence, which is recovered as a limiting case of the first family.

We extend the theory of $s$-energy to $\mathbb{Q}_p^n$. We give an exact representation for the $s$-energy in terms of the Fourier coefficients of a finite Borel measure with support in $\mathbb{Z}_p^n$, which is stronger than the approximate representation in $\mathbb{R}^n$. We also compute the Fourier dimension of the $p$-adic Riesz product measure, even in the case $p=2$, which has never been studied before.

For all $0< s \le 1$, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to $s$. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis $\{e^{2\pi i \lambda x}: \lambda\in \Lambda\}$ for its $L^2$ space. This gives the first examples of singular Salem spectral measures in ${\mathbb R}^1$.

We study the sharp $L^2$-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp $L^2$-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of $L^2$-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of $L^2$-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps $\mu_j-\mu_{j-1}$, and the coefficients of the second sum are $\mu_1$ times the gaps of the reciprocal spectrum, $\mu_1\left(\mu_{j-1}^{-1}-\mu_j^{-1}\right)$. On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when $N=2,3$ and by the degree three modes when $N\geq4$. In the Gaussian case, all the constants are rational numbers and the $L^2$-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.

In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $\Lambda(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\varepsilon<\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$ preserves, after normalization by $\sqrt m$, the squared norm of every complex vector $y$ with $\|y\|_1\leq\sqrt r\|y\|_2$ within a factor $1\pm\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\widehat f$ the condition is exactly $\operatorname{FR}(f)^2\leq r$, where $\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\log n$ by $\log\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$, and stable sparse recovery from $Cs\log\frac{2en}{s}\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.

We study variation functions of some lacunary series along the sequence of $b$-adic partitions, where $b>1$ is an integer. By such a lacunary series, we mean that in the definition of the well-known Weierstrass function, the Lipschitz continuous cosine and sine functions are replaced by a continuous and piecewise continuously differentiable periodic function whose Fourier series has zero $(jb)^{\mathrm{th}}$-Fourier coefficients for all non-zero integers $j$. In the proofs, we use a probabilistic approach applying a martingale central limit theorem. Our result extends some recent results of Han, Schied and Zhang (2021), and can be applied when one chooses the periodic function in question, for example, as the Takagi function, a Weierstrass function or a certain lacunary series itself.

Let $n,d\geq1$, and suppose that $E\subset \mathbb{Q}_p^n$ contains a degree-$d$ polynomial image of $\mathbb{Z}_p$ in every leading direction of a nonempty set $V\subset\mathbb{P}^{n-1}(\mathbb{Q}_p)$. We prove the sharp bound $\dim_H E\geq\dim_H V+1$. For joint block-valued polynomial evaluations, the total incidence dimension is at least one plus the supremum of the image dimensions. For analytic coefficient sets, Haar-almost every evaluation attains this supremum, and we bound the Hausdorff dimension of the parameters where the image dimension drops. For bounded coefficient sets and arbitrary fixed anisotropic block scales, we prove a quantitative covering comparison with a common Haar-null exceptional set. Outside this set, each evaluation's covering number dominates the maximum over any compact reference ball, up to any positive power loss at all sufficiently small scales. The proofs use the polynomial-to-line lifting map due to Nadjimzadah, together with Dhar's prime-power Kakeya set estimate and a tube maximal inequality derived from it.

Let \(\lambda\succ\gamma\succ\mu\) be equal-degree exponent partitions with at most three parts, and let \begin{equation*} P_{\lambda,\gamma,\mu} = J_\lambda+J_\mu-2J_\gamma, \end{equation*} where \(J_\nu\) denotes the symmetric monomial orbit sum associated with \(\nu\). We prove a necessary and sufficient collision criterion for the positivity of this three-point majorization difference in three variables. For nonnegative integer exponent partitions, \begin{equation*} P_{\lambda,\gamma,\mu}(x,y,z)\ge0 \qquad(x,y,z>0) \end{equation*} if and only if \begin{equation*} P_{\lambda,\gamma,\mu}(t,1,1)\ge0 \qquad(t>0). \end{equation*} Thus the positivity of a genuinely three-variable symmetric polynomial of this form is completely determined by its one-variable restriction to the locus where two variables coincide. The theorem gives a uniform and effectively checkable criterion for an infinite class of symmetric polynomial inequalities. For a fixed integer chain, the global three-variable problem is reduced to a single univariate polynomial inequality, which can often be verified exactly by factorization, Sturm's theorem, or other one-variable methods. The criterion extends well beyond the classical Schur family, produces explicit inequalities outside the Schur pattern, and yields genuine strengthenings and refinements of Schur's inequality.

Weighted Hardy inequalities on a domain $\Omega\subset\R^n$ can be deduced from lower Hausdorff-content bounds on suitable visible parts of the boundary, and it is natural to look for visible sets described by straight segments, which are easier to verify than the curve-based visual boundary of Koskela and Lehrbäck. We show that this approach is subject to an intrinsic obstruction. If a boundary point $\xi$ is seen from $x\in\Omega$ along a segment satisfying a uniform thickness condition, then $\Omega$ contains a truncated cone with vertex $\xi$; consequently the quantitatively straight visible boundary $\partial^{\mathrm{str}}_{x,\alpha}\Omega\cap B(x,C_0d_\Omega(x))$ is a finite union of Lipschitz graphs, is $(n-1)$-rectifiable, and carries no $\lambda$-Hausdorff content for any $\lambda>n-1$. Since content conditions with $\lambda\le n-1$ already imply the $(p,\beta)$-Hardy inequality for $\beta<p-n+\lambda$ without any visibility assumption, segment-based criteria are either unsatisfiable or redundant: they never reach the regime $\beta\ge p-1$ in which visibility is genuinely needed. We show that the exponent $n-1$ is attained, so that the rectifiability theorem is sharp; we exhibit a boundary point lying in the visual boundary of Koskela and Lehrbäck but in no $\partial^{\mathrm{str}}_{x,\alpha}\Omega$; and we observe that raw straight visibility, without the thickness condition, escapes the obstruction, is sufficient for $\lambda\le n-1$, and remains open for $\lambda>n-1$.

We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.

We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of $C(K)$, where $K$ is a compact homogeneous $G$-space. Within this framework, we characterize the finite-dimensional $G$-invariant subspaces for which the $G$-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter–Weyl decomposition of $L_2(K)$. For such subspaces, averaging shows that this unique $G$-equivariant projection is minimal; it is the restriction to $C(K)$ of the $L_2(K)$-orthogonal projection. Consequently, their projection constants are given by the $L_1$-norm of an explicit reproducing-kernel slice. We apply this method to spaces of $d$-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.

We study incidence problems for sticky sets of tubes. In the first part, we prove a sharp Furstenberg set estimate for sticky sets of tubes in $\mathbb R^3$ satisfying convex Wolff axioms. In the second part, we prove the Kakeya set conjecture in $\mathbb R^4$ for sticky sets of tubes satisfying convex Wolff axioms.

An Askey-type confluence scheme is developed for Jacobi multiple orthogonal systems of mixed type, with $q$ row weights and $p$ column weights. Rescaling the Jacobi variable near $0$ and near $1$ produces Laguerre systems of the first and second kinds. A Hermite system with one Gaussian–gamma row and $q-1$ exponential convolutions is obtained from Laguerre of the first kind, directly from the second kind, and directly from Jacobi. Explicit limiting weights and orthogonal forms are given, together with convergence of their mixed moments. For each fixed index, the recurrence coefficients and the entries of the bidiagonal factors also converge in the limit to Laguerre of the first kind, under nonvanishing conditions. The Laguerre family of the second kind is described by additive gamma convolutions. A closed product formula for its moment determinants holds for arbitrary positive real shape parameters and near-diagonal row multi-indices, yielding explicit normality conditions. Its forms admit contour and Rodrigues representations with terminating hypergeometric components: a multiple Kampé de Fériet series for $A$, and at most $q+1$ Srivastava–Daoust series for each $B$-component when $q\ge2$. For the Hermite family, normality holds at every Laguerre-admissible index under the stated separation conditions, and these indices form an infinite set. Its components have finite Hermite expansions and terminating Srivastava–Daoust representations. The three routes give the same limiting moments and normalized components, the Jacobi one being a single limit and the other two iterated. An appendix treats several derivative rows, where the connection determinant vanishes beyond a finite index range, with exact rank loss and diverging components. The one-row reductions recover the classical multiple second-kind Laguerre and Hermite systems.

In this paper, we prove a boundedness result for parabolic Riesz transforms in the range $p \le 2$ associated with degenerate parabolic operators whose elliptic part is in divergence form with complex coefficients depending measurably on space and time. The degeneracy is determined by a spatial weight in the Muckenhoupt class $A_2(\mathbb{R}^n)$. The argument relies on off-diagonal estimates for the parabolic gradient of the resolvent family, together with weighted embeddings. We obtain boundedness in a range of exponents strictly below $2$, with explicit ranges quantified by the reverse Hölder class of the weight. For real coefficients, this quantification is sharp and extends down to $1$.

Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $\beta$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $\beta$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,\beta)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $\beta$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais–Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.

Let $L_v=-\Delta-2v\cdot\nabla$ on $L^2(\mathbb R^n,e^{2v\cdot x}\,dx)$, where $v\in\mathbb R^n\setminus\{0\}$ is a constant drift vector. We prove that the full vector Riesz transform $\nabla L_v^{-1/2}$ satisfies a weak-type $(1,1)$ estimate for real-valued functions with constant at most $2$, uniformly in both the dimension and the drift vector. Our proof extends the fractional obstacle method of Ouyang, Spector and Stockdale (https://arxiv.org/abs/2608.18068) to the exponentially weighted setting. Interpolation then yields strong $L^p$ bounds for $1<p\le 2$, with constants depending only on $p$.

We establish a complete characterization of local $C^1$ linearizability for planar diffeomorphisms. For every prescribed hyperbolic real Jordan form, we obtain necessary and sufficient integrability conditions on the modulus of continuity of the derivative for local $C^1$ linearizability. These criteria reduce to three canonical thresholds: the classical, polynomially weighted, and squared-logarithmically weighted Dini conditions. Each threshold is optimal: whenever the corresponding condition fails, we construct a diffeomorphism with the identical linear part that is not locally $C^1$ linearizable. For every non-hyperbolic linear part, explicit polynomial counterexamples show that smoothness alone cannot guarantee even local topological linearization. Beyond $C^1$ existence, we determine the optimal regularity for the derivative of the linearizing conjugacy in every hyperbolic case, with the optimality established by counterexamples.

In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel $\mathcal{F}(\Delta,z) = {}_2F_1(\Delta,\Delta;2\Delta;z)$ is proved to be strictly totally positive of order infinity for $\Delta > 0$ and $z \in (0,1)$. Second, the sharp lower $\Delta$-parameter threshold for the associated one-dimensional $SL(2,\mathbb{R})$ conformal block $G_{\Delta}(z) = z^{\Delta}\mathcal{F}(\Delta,z)$ is shown to be $1/2$: the conformal-block kernel $G_{\Delta}(z)$ is strictly totally positive of order infinity for $\Delta \geq 1/2$. For every $\tau \in (0,1/2)$, there exists a strictly negative odd-order minor of $G_{\Delta}(z)$ such that all its $\Delta$-values are in $(\tau,1/2)$ and all its $z$-values can be chosen arbitrarily close to one. Consequently, no restriction $z > z_0$ with $z_0 < 1$ can restore total positivity of order infinity over all $\Delta > 0$.

Let $q_m=2m/(m+1)$ and put \[ \beta_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satisfies \[ \left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{\beta_0+\varepsilon}\norm{f}_\infty. \] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{\mu_r/r}r^B\min\{r,L_M\}^8$, $L_M\asymp_B \log (M+1)$, with $\mu_r=\lceil3r/2\rceil$ for $r\ge2$. A parity-compatible central window handles $r<L_M$, while a shrinking balanced window handles $r\ge L_M\asymp_B\log(M+1)$, no parity is used in this regime. The leading high-degree loss–gain factor is $\ee\,2^{-B}$; careful uniform bounds close the bootstrap for every $B>1/\log2$. The formula for $\mu_r$ is explained below.

We use the series representation of the Bessel functions of type $B_N$ in terms of Jack polynomials and show that $$\lim_{k_1\to\infty} J_{(k_1,k_2)}^B(x,k_1y)^{1/k_1}= 2^N\prod\limits_{i=1}^N \Bigg( e^{\sqrt{1+x_i^2 y_i^2}-1}\cdot \frac{1}{\sqrt{1+x_i^2 y_i^2}+1} \Bigg)$$ for $x,y\in\mathbb R^N$ and $k_2\ge0$. Moreover, the known Laplace-type integral representations for $N\ge1$ and $k_2=0,1/2,1,2$ and for $N=2$ and $k_2>0$ by Rösler and Demni respectively lead to related limits for $$\lim_{k_1 \to \infty} \partial_{x_j} J_{(k_1,k_2)}^B(x,k_1y) /(k_1 \cdot J_{(k_1,k_2)}^B(x,k_1 y)) \quad (j=1,\ldots,N).$$ These limits lead to weak limit results for the associated Bessel processes with drift. For $k_2=1/2,1,2$, these limit results have applications to radial parts of Brownian motions with drift on the $M\times N$-dimensional matrices over $\mathbb R,\mathbb C$, and the quaternions for $M\to\infty$. We also discuss these limits in the Dunkl case $N=1$.

Unlike ordinary time, a Stieltjes clock may advance continuously, remain constant over intervals, or jump. For an ordinary continuous clock, Hölder regularity with exponent $\alpha>1$ forces constancy, whereas jumps of a Stieltjes clock may allow nonconstant behavior. We refer to Hölder regularity of exponent $\alpha>1$ measured relative to g as super-g-Hölder regularity, and show that, in finite dimensions, such functions admit an atomic representation determined entirely by their jumps, with their g-Hölder seminorms given exactly by the corresponding normalized jump sizes. This atomic representation yields a linear isometric description of the associated function space, together with its Banach and separability properties, an exact total-variation formula with an optimal bound, and finite-jump approximation results. We then apply this structure to Stieltjes differential equations. Nonzero state jumps require positive clock jumps, leading to quantitative finite-jump bounds and, for affine dynamics, exact thresholds for the number of jumps and extremal terminal distances under a prescribed total Stieltjes mass. Finally, we study recovery of the Stieltjes clock from observed state jumps. For known single-valued dynamics, clock jumps can be recovered under a natural local identifiability condition, while for nonconvex differential inclusions the super-g-Hölder bound can reduce, and in some cases remove, nonuniqueness.

We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets $\Lambda \subset \mathbb{R}$, of uniform density one, such that the exponential system $\{e^{2\pi i\lambda x} : \lambda \in \Lambda\}$ is complete in $L^p(S)$ for every $1 \leq p < \infty$ and every measurable set $S \subset \mathbb{R}$ with $|S| < 1$. We also show that no set that is asymptotically integer can have this universality property. Additionally, for every $v \in (0,1)$, we construct a set of integer frequencies and uniform density $v$ whose exponential system is complete in $L^p(S)$ for every $1 \leq p < \infty$ and every measurable set $S \subset [0,1]$ with $|S| < v$. Finally, we prove that the Sobolev regularity condition $\alpha > \frac12$ for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.

We prove mixed $A_p$-$A_\infty$ estimates for sparse operators in two non-doubling settings. In the first setting, we consider sparse operators defined using stopping times in continuous time. We obtain both strong- and weak-type bounds with the same powers of the weight characteristics as in the classical setting. Some of our weak-type bounds are even new for the dyadic filtration on $\mathbb R^d$ and, in particular, imply a sharp weak-type $(2,2)$ estimate for Rubio de Francia square functions, solving a problem left open by Garg, Roncal and Shrivastava [J. Geom. Anal., 31:748-771, 2021]. In the second setting, we consider dyadic sparse forms in which distinct cubes may interact, provided their dyadic distance is bounded. We obtain strong-type bounds with the same powers of the weight characteristics as in the classical setting. As a one-dimensional application, we obtain strong-type bounds for Haar shifts over balanced non-doubling measures, answering a quantitative question posed by Conde-Alonso, Pipher, and Wagner [Math. Ann., 391:2209-2253, 2025].

We prove a generalization of the version, due to A. Gabrielov, of the Łojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the Łojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.

In this paper, we investigate weighted $L^p\rightarrow L^q$ restriction estimates for the Fourier extension operator associated with planar curves of non-vanishing curvature. More precisely, we establish estimates of the form \begin{equation} \|Ef\|_{L^q(X)}\leq C_{\epsilon}R^\epsilon\|f\|_p, \end{equation} where $X$ is an $\alpha$-dimensional set. As an application of this result, we derive a new $L^p$ decay estimate for circular means of Fourier transforms of fractal measures in $\mathbb{R}^2$. We also construct examples that give several upper bounds for the decay exponent when $p\in[1,2]$.

It is shown that if $u$ solves the wave equation in $\mathbb{R}^{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H^s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H^{s-1}$, where $0.5 < s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}^{n+1}$ is obtained for $n \geq 4$ and $1/2 < s < n/4$.

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón–Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-\Delta+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.

This work focuses on range characterization of spherical mean transform (SMT) in odd-dimension Euclidean space. In a recent work, a simpler description characterizing the range of SMT in odd dimensions was derived. This involved symmetry relations involving certain linear ordinary differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range. The sufficiency part of this characterization was shown using an existing range characterization result due to Agranovsky-Kuchment-Quinto. In the current work, we provide a direct and constructive proof showing the sufficiency of the aforementioned symmetry conditions and we obtain this without invoking any of the existing range characterization results. As an added advantage, given a function $g$ satisfying the range conditions, our approach provides an explicit construction of a function $f$ such that the SMT of $f$ is $g$.