Papers

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

We study the zeros of functions in the Dirichlet space. Using extremal functions, we produce a necessary and sufficient condition for a sequence of points in the unit disk to be a zero set of the classical Dirichlet space. This Shapiro-Shields type condition involves kernels of the harmonic Dirichlet space associated with measures depending on the zero sequence.

For the fractional powers of operators, the classical formulation by V. Balakrishnan inherently requires positivity or specific sectorial conditions of the generator (i.e., the generation of analytic semigroups). To overcome these limitations, this paper presents a novel approach to constructing fractional powers of operators $D^k$ through the $\delta$-regularized logarithmic representation of infinitesimal generators $D$, based on the framework of $C^0$-semigroup theory for abstract evolution equations in Banach spaces. The proposed method bypasses the conventional geometric constraints by employing an algebraic $\delta$-regularization for bounded operators. Specifically, by applying a complex-analytic logarithmic representation to a family of bounded operators derived from the resolvent of the semigroup, we mathematically prove that, under the minimal algebraic assumption of invertibility, the fractional powers for generators of more general $C^0$-semigroups are uniquely and rigorously well-defined, independent of the choice of the regularization parameter $\delta$. The theoretical framework established in this study offers extensive potential for applications, including regularity estimates for non-analytic semigroups and non-autonomous systems where the infinitesimal generators depend explicitly on the time variable.

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $\Omega \subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(\Omega) \le \frac{\pi^{s-\frac{1}{2}}\Gamma\left(\frac{3-s}{2}\right)} {s(1-s)\Gamma\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partial\Omega)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $\Gamma$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

Let $N\in\mathbb N$ and $\gamma\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $\gamma\in[-1,0)$, $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy–Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy–Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$, and neither $\dot{W}^{1,1}(\gamma)/\mathbb R$ nor $\dot{BV}(\gamma)/\mathbb R$ is normable, which answers the above two questions.

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg–Landau equation $$ -\Delta u = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as Open Problem 2.5 in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.

We develop a general framework for $B$-multiplier spaces; these are vector spaces $M = M(V,W)$ obtained from a bilinear operator $B \colon M \times V \longrightarrow W$, where $V$ and $W$ are Banach spaces. We focus on their normability and completeness, mainly in the setting of spaces consisting of functions with values in a Banach space $X$, particularly sequences. Our approach relies on the underlying Banach spaces satisfying the BK property, that is, having continuous evaluations. Classical multiplier spaces arise when $B$ is a pointwise product of scalar functions and we make the point for the case when $V$ or $W$ consists of vector functions. Special attention is given to the sequence spaces $\Sigma \ell_{\infty}(X)$ (bounded partial sums), $\Sigma c(X)$ (summable), and $\ell_u(X)$ (unconditionally summable). Given a BK scalar sequence space $V$, we introduce the multiplier space $M_{\Sigma}(V,X)$ and establish conditions under which it determines a closed subspace of bounded linear operators from $V$ into $X$. The notion of associate space is precised for BK-spaces, linking this construction with classical Köthe duality. We consider what we named strong vectorialization $Y(X)$ and weak vectorialization $Y_w(X)$ of a Banach sequence ideal $Y$. The weak vectorialization is obtained as a multiplier space and employed to describe classical sequence spaces as $\ell_{p,w}(X)$, $1 \leq p \leq \infty$. We introduce the $b$-ideal part of a space and show that the $b$-ideal part of $\Sigma c(X)$ is $\ell_u(X)$ and that of $\Sigma \ell_{\infty}(X)$ is $\ell_{1,w}(X)$. We also study the sequence space $bv(X)$ (bounded variation), proving that $M(\Sigma c(X),\Sigma c(X)) = bv(\mathbb K)$.

We characterize all bounded complex symmetric weighted composition operators on the Fock space \(\mathcal F_{\alpha}^{2}\), without prescribing a conjugation a priori. Our approach uses reproducing kernels and two generating functions. The Taylor coefficients of these functions are eigenvectors of the operator and its adjoint, respectively. The conjugations arising from our construction are anti-linear Gaussian integral operators. Some of them are weighted composition conjugations, whereas others are not.

We determine the dimension-free threshold for the comparison between the Bombieri–Weyl norm and the supremum norm of complex homogeneous polynomials on $\ell_p^n$. For $m$-homogeneous polynomials, the critical scale is $p=2m$: below this threshold no dimension-free comparison is possible, while at $p=2m$ we obtain $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy–Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.

We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,\alpha}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann–Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel–Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,\alpha}$, and we also obtain the Triebel–Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,\alpha}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy–Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.

Let $\mu$ be a positive Borel measure on $\mathbb C^n$. We prove that, for every $0<p<\infty$, the Toeplitz operator $T_\mu^{\mathrm{ph}}$ induced by $\mu$ on the pluriharmonic Fock space belongs to the Schatten class $S_p$ if and only if the local mass function $z\mapsto\mu(B(z,r))$ belongs to $L^p(\mathbb C^n)$ for one, or equivalently every, $r>0$. For $n\geq2$, this resolves a conjecture of Jaguzovi'c and Vujadinovi'c, and the range $0<p<1$ is new in every dimension. Writing $T_\mu$ for the corresponding holomorphic Toeplitz operator, we obtain the sharp estimates $$\|T_\mu\|_{S_p}^p \leq \|T_\mu^{\mathrm{ph}}\|_{S_p}^p \leq 2^{\max\{1,p\}} \|T_\mu\|_{S_p}^p. $$ We also prove a sharp comparison with constant two in every symmetrically normed ideal and an exact trace formula, using positivity and a $2\times2$ block decomposition whose diagonal blocks are $T_\mu$ and an antiunitary copy of its compression to the functions orthogonal to constants. We then consider generalized Fock weights satisfying $m \, dd^c|z|^2\le dd^c\phi\le M \, dd^c|z|^2$. For the canonical holomorphic and antiholomorphic direct sum norm, the same Schatten and symmetrically normed ideal estimates hold. For the norm inherited from $L^2(\mathbb C^n,e^{-2\phi}dV)$, the local mass criterion also holds whenever $e^{-2\phi}$ is comparable to a generalized Fock weight invariant under the scalar circle action. Without further assumptions, the local mass criterion can fail for the inherited norm: in one complex dimension, we construct a weight of the form $\phi(z)=|z|^2/2+\operatorname{Re}q(z)$, with $q$ entire, and a finite positive measure whose local masses belong to every $L^p$, although the Toeplitz form on the inherited pluriharmonic space is unbounded.

We build on the recent breakthrough of Faulhuber, Petersen, van Velthoven, and Voigtlaender that disproved the HRT conjecture with a Schwartz function and a $12$-point configuration. We give a human-readable treatment of their mechanism and find HRT counterexample functions with exponential (or faster) decay. By a result of Bownik and Speegle, this is the fastest possible decay for an HRT counterexample, up to a logarithmic factor in the exponent.

Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1<n<\dim X$, are all linearly isometric must be a Hilbert space. Gromov proved the conjecture for even $n$, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd $n$, including all previously unresolved cases. Together with Gromov's result for even $n$, this completes Banach's isometric conjecture in the real case. The proof combines principal bundle theory with a Brouwer degree argument.

We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\ge0}$ on a Hilbert space satisfies \[ \|T_t\|\le C(1+t)^{1-\varepsilon_K}, \qquad t\ge0, \] where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. The same conclusion is obtained for positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^p$-spaces. We further obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces. Finally we prove that every Kreiss bounded operator on a UMD Banach space has a polynomial gap below linear growth

We develop an intrinsic theory of weighted, inhomogeneous Besov spaces on general homogeneous Lie groups without recourse to group-specific arguments. Starting from a definition in terms of localised test functions, we establish an equivalent, wavelet-like, multiscale characterisation. This provides a unified mechanism for deriving Besov embeddings, a Taylor-remainder characterisation, Young-type product estimates, Schauder estimates for convolution semigroups, and a weighted Kolmogorov criterion for random distributions. We apply this framework to parabolic Anderson-type equations associated with positive Rockland operators and with singular initial data. For space-time noise, our results cover the Young regime. For purely spatial noise, we also establish well-posedness in the first singular regime using a variant of the Cole-Hopf transform for Rockland operators. Both applications rely on a mild sewing lemma that accommodates time-dependent Banach spaces and increments with a singularity at the initial time.

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent~$\frac{2}{d+2}$ obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbf{\Theta}_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbf{\Theta}_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbf{\Theta}_n$-contractions. Finally, we show that for a $\mathbf{\Theta}_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbf{\Theta}_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbf{\Theta}_n$.

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded $H^\infty$-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].

For any given time lag $\gamma\in(0,1)$, we prove that the one-sided parabolic BMO space $\mathrm{BMO}^+(\gamma)$ coincides with the parabolic BMO space $\mathrm{PBMO}^-(\gamma)$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty}^+(\gamma)$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^+(\gamma)$ with $r\in[1,\infty)$, and the parabolic reverse Hölder classes $\bigcup_{q\in(1,\infty]}RH_q^+$ coincide with the parabolic Muckenhoupt classes $\bigcup_{r\in[1,\infty)}A_r^+(\gamma)$, and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016)]. To show them, we establish the uniform parabolic space-time shifting property for parabolic reverse Hölder weights, and develop the one-sided stopping time argument which yields a new parabolic John–Nirenberg inequality for $\mathrm{BMO}^+(\gamma)$. As applications, we obtain John–Nirenberg and exponential integrability characterizations of $\mathrm{BMO}^+(\gamma)$, prove that $\mathrm{BMO}^+(\gamma)$ is independent of the positive time lag, and identify its null space.

In this paper, we establish new fixed point theorems for multivalued mappings satisfying Hardy-Rogers type contractive conditions in complete $b$-metric spaces. Motivated by the constant-coefficient framework of Chaib et al. (Fixed Point Theory 27 (2026), 209-228), we develop a distance-dependent formulation in which the contractive coefficients are nonnegative functions of the interpoint distance. We introduce a piecewise contraction factor corresponding to the two cases determined by the relative sizes of the cross terms in the Hardy-Rogers inequality, thereby providing a formulation adapted to the structure of the contractive condition. Under appropriate assumptions on the coefficient functions, we establish fixed point existence through either $h$-upper semicontinuity or asymptotic limsup conditions. The resulting framework yields corresponding fixed point theorems for multivalued Reich, Kannan, and Chatterjea-type contractions. We further investigate strict fixed points and obtain conditions under which the fixed point and strict fixed point sets coincide and consist of a unique point. In addition, we establish a fixed point transfer result for suitable finite iterates, showing that the contractive condition may be imposed on $T^p$ rather than on the original mapping, provided that the latter has a strict fixed point. Corresponding single-valued consequences are also derived. Several examples illustrate the applicability of the results and demonstrate situations in which the proposed conditions apply beyond the corresponding existing formulations.

For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ \Sigma_0(M_k)=\frac{\pi^2}{6},\qquad \Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.

We prove a Hilbert space-valued analogue of Pego's compactness theorem on compact groups. For square-integrable functions taking values in a Hilbert space rather than in the complex numbers, we show that a bounded family is precompact exactly when it is simultaneously well behaved in two complementary senses: its members do not change much under small translations of the group, and their Fourier coefficients decay uniformly across the family. This equivalence holds without restriction when the Hilbert space is finite-dimensional, and it specializes to the known scalar-valued theorem when the Hilbert space is just the complex numbers. We then construct an explicit example showing that the equivalence genuinely breaks down once the Hilbert space is allowed to be infinite-dimensional. To repair this, we introduce a uniform tightness condition and we show that under this extra hypothesis the equivalence is restored regardless of the dimension of the Hilbert space. Along the way we establish the Plancherel isometry, the Hausdorff-Young inequality and its inverse for this vector-valued Fourier transform.

We prove that the sharp $\frac{1}{2}$ lower bound for the Haraux function holds for every maximally monotone operator of type~(NI) on an arbitrary real Banach space. This extends the result established in reflexive Banach spaces to arbitrary real Banach spaces. We also establish an exact decomposition at each graph point, showing that the local contribution to the Haraux function and a nonnegative residual sum to one half of the weighted squared displacement. Due to the equivalence between type~(NI) and quasidensity, this decomposition also yields the sharp bound without requiring a graph point at which the residual vanishes. Moreover, for every operator with a nonempty graph, this decomposition yields a lower bound involving the residual infimum, and for maximally monotone operators it further yields a new characterization of type~(NI) in terms of the Haraux function. Finally, on $c_0$, we give a maximally monotone operator of type~(NI) for which the residual infimum is zero at some target but is not attained. This shows that the existence of a graph point at which the residual vanishes is strictly stronger than the vanishing of the residual infimum required in our proof.

We show the existence of ultralimits of sequences of Hajlasz-Sobolev mappings $f_i:X_i\to Y_i$ when $X_i$ converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the mappings pmG-converge to the ultralimit in a suitable sense, and obtain Arzela-Ascoli theorems in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for Hajlasz-Sobolev maps $f:(X,\mu)\to V$ into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda \subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $\xi$ in the dual lattice $\Lambda^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K \xi,\xi\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ (equivalently, the length of the shortest nonzero dual vector $\lambda_1(\Lambda^*)$) in terms of the maximal covariance of $K$: \[ \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda) = 4\pi^2 \lambda_1(\Lambda^*)^2 \geq \frac{\pi^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_\Lambda$ of a lattice with $\det \Lambda = 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\quad \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\quad \lambda_1(\Lambda^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan–Lovász–Simonovits conjecture for all flat tori. While there are lattices $\Lambda$ and corresponding Voronoi cells $K = K_\Lambda$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).

We establish exact anisotropic $L^p$-Hardy and Caffarelli-Kohn-Nirenberg identities associated with the Minkowski functional $\|\cdot\|_K$ and the anisotropic radial derivative $\mathcal R_K$, where $1<p<\infty$ and $K\subset\RN$ is a smooth and strictly convex body containing the origin in its interior, not necessarily origin-symmetric. These identities contain explicit nonnegative $R_p$-remainders and yield sharp anisotropic radial $L^p$-Hardy and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. In the noncritical parameter regions, we characterize the full family of extremal functions in the natural weighted completion space, while in the critical case the sharp constant is not attained by any nonzero function in that space. As applications, we establish sharp anisotropic $L^p$-Heisenberg uncertainty principles and obtain explicit anisotropic Gaussian-type optimizers. Finally, letting $K^*$ denote the polar body of $K$, we derive norm-based anisotropic gradient inequalities associated with the origin-symmetric body $K^*\cap(-K^*)$. When $K$ is origin-symmetric, the corresponding gradient constants are sharp, including those in the critical case.

We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz $H^1$-extension domains, which includes domains with fractal boundaries such as the Koch snowflake. We define and study these operators based on the surjectivity of the trace operator for elements of $H^1(\Omega)$ and the Hilbert structure of the associated trace space. The normal trace operator is defined on $\boldsymbol{H}({\rm div},\Omega)$ in $\mathbb{R}^n$. A generalized Stokes formula allows us to introduce the tangential trace operator on $\boldsymbol{H}({\bf curl},\Omega)$ and $\boldsymbol{H}^1(\Omega)$ in two and three dimensions. Following the approach of Buffa, Costabel and Sheen (2002) for Lipschitz domains, we define two abstract tangential boundary spaces as images of these trace operators, establish their Hilbert structure, and construct an abstract rotation operator linking them, in place of the geometric rotation based on the normal vector. We also extend Costabel's (1991) coercive bilinear form approach for Maxwell's equations to non-Lipschitz domains, yielding a theory that supports the treatment of the Hodge-Dirac operator and Green's formulas and enables the solution of boundary-value problems for the $\operatorname{\textbf{curl}}\operatorname{\textbf{curl}}+\textbf{I}$ operator within the $H^1$-extension domains framework.

We prove a linear operator of a finite-dimensional Banach space has the topological shadowing property, as introduced in \cite{lny}, if and only if its spectrum lies either within the open unit complex disk or outside the closure of that disk. Moreover, the spectrum of every uniformly expansive linear operator with the topological shadowing property lies either within the open unit complex disk or outside the closure of that disk. Lastly, we prove that a normal operator with the topological shadowing property on a Hilbert space does not have any nonzero nonwandering points.

We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point $\mathbf{p}$ in a map $\mathcal{M} \subset \mathbb{R}^d$, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a $\mathcal{C}^\infty$ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in $L^2(\mathbb{S}^{d-1})$, establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm $|f|_{\mathcal{D},n} = \|P_n f\|_{L^2}$, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity $|f|_{\mathcal{D},n} \to \|f\|_{L^2(\mathbb{S}^{d-1})}$ as $n \to \infty$. Complementary results include Fréchet differentiability, lower semicontinuity under occlusion, and explicit Lipschitz stability bounds with constants depending on the kernel and temperature. These properties underpin a localization theory in which the descriptor grid enables nearest-neighbor position recovery with a certifiable static error bound, while robot motion generates an observability Gramian whose smallest eigenvalue controls a dynamic error bound and generically resolves symmetry-induced ambiguities that persist under single-observation matching.

The inverse spectral problem for the radial Schrödinger operators on the finite interval is investigated. The potential is recovered from the given eigenvalues plus its information on a smaller interval. The method is to discuss the connections between the Paley-Wiener spaces and the potentials, from which we completely determine the potential on the whole interval from the potential on a smaller interval and a set of eigenvalues in terms of the complete exponential systems.