37,053 papers in this slice of arXiv.
Sam Looi
Let μ be a positive Borel measure on ⁿ. For every 0<p<∞, we prove that the Toeplitz operator
John Jasper, Dustin G. Mixon
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.
Xinbao Lu, Kaiwen Yang
Banach asked in 1932 whether a real Banach space X whose n-dimensional subspaces, for some fixed 1<n<dimX, are all 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 even-dimensional result, this completes Banach's isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.
Loris Arnold
We prove that every Kreiss bounded C0-semigroup (Tt)t≥0 on a Hilbert space satisfies ∥Tt∥≤C(1+t)1−εK,t≥0, where εK>0 depends explicitly only on the Kreiss constant. This improves the previously known estimate O(t/log(t+1)) and shows that every Kreiss bounded C0-semigroup has a genuine polynomial gap below linear growth. In view of the examples of Eisner and Zwart with growth arbitrarily close to linear, no universal positive exponent can hold for the whole class of Kreiss bounded C0 semigroups on Hilbert spaces.
Harprit Singh
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.
Jonathan Niles-Weed, Jacob Shkrob
We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent d+22 obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every d≥2, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if ν is a discrete measure and the optimal coupling between μ and ν transports each point to a nearest atom of ν, then Wp(μ,ν)≤CdKSWp,1(μ,ν) 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.
Aparna Gupta, Shubhankar Mandal Avijit Pal, Bhaskar Paul
In this paper, we introduce the notion of a distinguished variety in the domain Θn. One of the main results of the paper is a determinantal representation for every distinguished variety in Θn. 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 Θn
Floris Roodenburg
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∞-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].
Weiyi Kong, Dachun Yang, Wen Yuan
We prove that, for any given time lag γ∈(0,1), itemize the one-sided parabolic BMO space BMO+(γ) coincides with the parabolic BMO space PBMO−(γ), i.e., PBMO−(γ)=BMO+(γ) with equivalent norms; the parabolic Muckenhoupt class A∞+(γ) defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes Ar+(γ) with r∈[1,∞), i.e., A∞+(γ)=⋃r∈[1,∞)Ar+(γ); the parabolic reverse Hölder classes ⋃q∈(1,∞]RHq+ coincide with the parabolic Muckenhoupt classes ⋃r∈[1,∞)Ar+(γ), i.e., ⋃q∈(1,∞]RHq+=⋃r∈[1,∞)Ar+(γ), itemize and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016), 289--299]. The main ingredients are a uniform parabolic space-time shifting property for parabolic reverse Hölder weights and a one-sided stopping-time argument, which yields a new parabolic John--Nirenberg inequality for BMO+(γ). As consequences, we obtain John--Nirenberg and exponential integrability characterizations of BMO+(γ), prove that BMO+(γ) is independent of the positive time lag, and identify its null space.
Hezekiah Seun Adewinbi, Oluwatosin Temitope Mewomo
This paper establishes new existence and uniqueness theorems for p-Hardy-Rogers type multivalued mappings in complete b-metric spaces. Our approach replaces the traditional constant coefficients in the contractive condition with distance-dependent control functions and incorporates a p-order iteration scheme, thereby providing a more general and flexible structure that encompasses a wider class of multivalued operators. Our findings significantly improve and generalize the Hardy-Rogers type fixed point theorems obtained by Chaib et al.(2026) by relaxing the required assumptions, incorporating the h-upper semicontinuity into the hypotheses and considering the p-th iterate of the mapping. Several examples are provided to demonstrate the applicability of our results.
Yin Liu, Yufeng Lu, Chao Zu
For k≥2, let Mk=[zk−wk]
Anaté Kodjovi Lakmon, Yaogan Mensah
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.
Weifeng Yang
Recent work established a sharp graph-distance lower bound for the Haraux function of maximally monotone operators on reflexive Banach spaces and raised the question of its extension beyond reflexivity. We prove that the sharp constant 1/2 remains valid for every maximally monotone operator of type (NI) on an arbitrary real Banach space and for every positive weight. The result is generated by an exact energy decomposition: the Haraux contribution of a graph point and its weighted quasidensity residual are complementary terms whose sum is one half of the weighted squared graph displacement. This decomposition yields a defect-corrected Haraux bound for every operator with nonempty graph and an exact Haraux-side characterization of the zero-defect regime. For maximally monotone operators, the known equivalence between type (NI) and quasidensity identifies this regime precisely with type (NI), so approximate certificates recover the sharp geometry without attainment. We further formalize the hierarchy from attained zero residual to zero residual infimum and then to the sharp bound. A nonreflexive example separates its first two levels, and a Fenchel--Young graph-distance consequence illustrates the same mechanism.
Milica Caković, Elefterios Soultanis
We show the existence of ultralimits of sequences of Hajlasz-Sobolev maps fi:Xi→Yi when Xi 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 graphs G(fi) of the mappings pmG-converge to the graph of the ultralimit in a suitable sense. The latter fact stems from a suitable Arzela-Ascoli theorem in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for mapping packages f:(X,μ)→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.
Emanuel Milman
We record several elementary relations between spectral and isoperimetric parameters of a flat torus TΛ=Rn/Λ and the covariance structure of a fundamental domain K for the lattice Λ⊂Rn. For every measurable fundamental domain K and nonzero vector ξ in the dual lattice Λ∗, we observe the sharp directional variance estimate ⟨CovKξ,ξ⟩≥121. This yields a lower bound on the torus spectral gap λSG(TΛ) (equivalently, the length of the shortest nonzero dual vector λ1(Λ∗)) in terms of the maximal covariance of K: λSG(TΛ)=4π2λ1(Λ∗)2≥3∥CovK∥opπ2. Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant DChe(TΛ) using an old argument of Hadwiger. In particular, when the Voronoi cell KΛ of a lattice with detΛ=1 is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that DChe(TΛ),λSG(TΛ),λ1(Λ∗)≥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 Λ and corresponding Voronoi cells K=KΛ 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).
Zhenzhen Wei
We establish exact anisotropic Lp-Hardy and Caffarelli-Kohn-Nirenberg identities associated with the Minkowski functional ∥⋅∥K and the anisotropic radial derivative RK, where 1<p<∞ and K⊂ is a smooth and strictly convex body containing the origin in its interior, not necessarily origin-symmetric. These identities contain explicit nonnegative Rp-remainders and yield sharp anisotropic radial Lp-Hardy and Lp-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 Lp-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∗∩(−K∗). When K is origin-symmetric, the corresponding gradient constants are sharp, including those in the critical case.
Romain Cervera, Patrick Ciarlet, Anna Rozanova-Pierrat +1
We generalize the classical vector-valued tangential and normal trace theory on Lipschitz domains to the setting of non-Lipschitz H1-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 H1(Ω) and the Hilbert structure of the associated trace space. The normal trace operator is defined on in Rn. A generalized Stokes formula allows us to introduce the tangential trace operator on and ¹(Ω) 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 +1 operator within the H1-extension domains framework.
Y. Yang, C. A. Morales
We prove a linear operator of a finite-dimensional Banach space has the topological shadowing property, as introduced in 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.
Phan Thanh An, Phiet DauThe
We develop a functional-analytic framework for smooth directional point descriptors in GPS-free map-based localization. Given a query point p in a map M⊂Rd, the descriptor integrates an environment signal against a partition-of-unity weight family built from a softmax kernel, yielding a C∞ alternative to hard angular binning. Our main contributions are: (i) a totality theorem showing that the associated linear functionals form a total family in L2(Sd−1), establishing asymptotic injectivity of the descriptor map; and (ii) a descriptor-induced seminorm ∣f∣D,n=∥Pnf∥L2, identified via the Gram matrix of the weights, which satisfies a Parseval-type identity ∣f∣D,n→∥f∥L2(Sd−1) as n→∞. 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.
Yongjiang Duan, Yufei Li, Zeguang Liu
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.