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Electronic ordering is prevalent in correlated systems, which commonly exhibit competing interactions. Here, we use x-ray diffraction to show that the charge density wave transition temperature of FeGe increases with pressure and evolves towards a $\sqrt{3}\times\sqrt{3}$ periodic lattice modulation, $\mathbf{q}$$^*$=$\left(\frac{1}{3}\ \frac{1}{3}\ \frac{1}{2}\right)$. In the pressure interval between 4$<$$p$$<$12 GPa both orders coexist and the spatial extent of the $\sqrt{3}\times\sqrt{3}$ order at high pressure becomes nearly long-range, $\sim$30 unit cells, while the correlation length of the 2$\times$2 phase remains shorter-ranged. The $\sqrt{3}\times\sqrt{3}$ phase is the ground state above 15 GPa, consistent with harmonic DFT calculations that predict a dimerization induced $\sqrt{3}\times\sqrt{3}$ order without phonon softening. The pressure dependence of the integrated intensities of $\mathbf{q}$$_\mathrm{CDW}=\left(\frac{1}{2}\ 0\ \frac{1}{2}\right)$ and $\mathbf{q}$$^*$ indicates a competition between the 2$\times$2 and $\sqrt{3}\times\sqrt{3}$ and demonstrates that the ground state of FeGe is characterized by a rich landscape of metastable/fragile phases. We discuss possible scenarios based on an order-disorder transformation and the formation of Friedel oscillations.

The intrinsic frustrated nature of a kagome lattice is amenable to the realization of exotic phases of matter, such as quantum spin liquids or spin ices, and more recently the multiple-$\mathrm{\textbf{q}}$ charge density waves (CDW) in the kagome metals. Despite intense efforts to understand the mechanism driving the electronic modulations, its origin is still unknown and hindered by competing interactions and intertwined orders. Here, we identify a dimerization-driven 2D hexagonal charge-diffuse precursor in the antiferromagnetic kagome metal FeGe and demonstrate that the fraction of dimerized/undimerized states is the relevant order parameter of the multiple-$\mathrm{\textbf{q}}$ CDW of a continuous phase transition. The pretransitional charge fluctuations with propagation vector $\mathrm{\textbf{q}=\textbf{q}_M}$ at T$_{\mathrm{CDW}}$$<$T$<$T$^*$(125 K) are anisotropic, hence holding a quasi-long-range bond-orientational order. The broken translational symmetry emerges from the anisotropic diffuse precursor, akin to the Ising scenario of antiferromagnetic triangular lattices. The temperature and momentum dependence of the critical scattering show parallels to the stacked hexatic $\mathrm{B}$-phases reported in liquid crystals and transient states of CDWs and highlight the key role of the topological defect-mediated melting of the CDW in FeGe.

A one-line proof of a minimax theorem due to Steinerberger is given.

In this article we derive an explicit diameter bound for graphs satisfying the so-called curvature dimension conditions $CD(K,n)$. This refines a recent result due to Liu, M\"unch and Peyerimhoff when the dimension $n$ is finite.

In this paper we extend Galbis' elegant norm bounds for self-adjoint Toeplitz operators on the Fock space to bounded and integrable symbols which are non-radial. The main ingredients are a transplantation of the remarkable Nicola-Tilli isoperimetric inequality to the realm of Fock-Toeplitz operator theory and a two-dimensional adaption of Galbis' integration and approximation arguments.

We provide in this Letter a two-point generalisation of the Agmon estimate for Schr\"odinger operators on graphs recently established by S. Steinerberger. It reduces to his estimate when the two points belong to different sets separated by the potential and the energy, i.e., the allowed and forbidden regions.

Sukochev and Zanin resolved an open problem due to B. Simon concerning optimal constants in H\"older inequality for the weak Schatten classes of compact operators. In this note we observe that these constants, by introducing the modified weak Schatten quasi-norms, can be renormalised so that the original Simon's conjecture (with optimal constant 1) does hold. We also provide an unexpectedly simple proof for the modified H\"older inequality and its sharpness.

The properties of kagome metals are governed by the interdependence of band topology and electronic correlations resulting in remarkably rich phase diagrams. Here, we study the temperature evolution of the bulk electronic structure of the antiferromagnetic kagome metal FeGe using infrared spectroscopy. We uncover drastic changes in the low-energy interband absorption at the 100 K structural phase transition that has been linked to a charge-density-wave (CDW) instability. We explain this effect by the minuscule Fe displacement in the kagome plane, which results in parallel bands in the vicinity of the Fermi level. In contrast to conventional CDW materials, however, the spectral weight shifts to low energies, ruling out the opening of a CDW gap in FeGe.

We simplify in this note Galbis' proof of certain norm estimates for self-adjoint Toeplitz operators on the Fock space. This relies on an extension (and a unification) of his integration lemmas, yet with a simpler proof in the same spirit.

In this Short Note we complement the intriguing harmonic analytic perspective due to P. Auscher and A. Axelsson for the abstract evolution equations. This concerns a unified approach to temporally weighted estimates for the forward and backward maximal regularity operators in presence of quadratic estimates and functional calculi. In particular we provide several invariance properties for the maximal regularity operators either in evolution form or in balayage form.

In this paper we present a complete proof of a conjecture due to V. V. Prelov in 2010 about an information inequality for the binary entropy function.

Let $\omega>0$. Given a strongly continuous semigroup $\{e^{tA}\}$ on a Banach space and an element $f\in\mathbf{D}(A^2)$ satisfying the exponential orbital estimates $$\|e^{tA}f\|\leq e^{-\omega t}\|f\| \quad\text{and}\quad \|e^{tA}A^2f\|\leq e^{-\omega t}\|A^2f\|,\quad t\geq0,$$ a dynamical inequality for $\|Af\|$ in terms of $\|f\|$ and $\|A^2f\|$ was derived by Herzog and Kunstmann (Studia Math., 2014). Here we provide an improvement of their result by relaxing the exponential decay to quadratic, together with a simple and direct way recovering the usual Landau inequality. Herzog and Kunstmann also demanded an analogue, again via semigroup orbits, for the Kato type inequality on Hilbert spaces. We provide such a result by using Hayashi-Ozawa machinery [Proc. Amer. Math. Soc., (2017)] which in turn relies on Hilbertian geometry., Comment: added a section on Kato inequality in the Hilbertian case, hence the title changed

The number-phase uncertainty result of Luo via the Hardy space on unit disc (Phys Lett A, 2000) is extended in this paper to the scale of weighted Bergman spaces. The minimum uncertainty states are thereby explicitly identified., Comment: a note written in the winter vacation of 2022-2023 and for fun

For absolutely convergent series we state explicitly a one-sided summation estimate that can be viewed as the discrete analogue of the change of variable formula on the half line. This estimate is implicit in Pascal Lef\`evre's recent elegant proof of the classical discrete Hardy inequality. Here we remove a superfluous irrationality condition therein and point out the change of variable character of his approach. This leads to a simpler, shorter and \textit{bona fide} Ingham type proof of the discrete Hardy inequality, and also provides the optimal constant., Comment: This is a companion note of ''A first order proof of the improved discrete Hardy inequality. Arch. Math. 117, 671-674 (2021)'' and was written in 08/2021. In light of the supersolution language of Pinchover et al., our motivation is to unify, at least conceptually, the proofs for both the discrete and the continuous Hardy inequalities

We give a natural convexity proof of an elementary inequality used by Ozawa and Rogers in proving their bilinear estimate for the one-dimensional Klein-Gordon equation. This robust approach also enables us to derive the optimality of Ozawa-Rogers estimate and establish a new bilinear estimate. Our estimate is in sharp analogy with the bilinear estimate of Bez and Rogers for the wave equation. We also present an alternative proof of some lower Jacobian estimates used crucially by Ozawa and Rogers in proving their Fourier restriction results on the whole hyperbola. Based on interpolation, our proof is purely non-trigonometric., Comment: added a section which presents an alternative non-trigonometric proof of some intriguing lower Jacobian estimates

In proving Rellich inequalities in the framework of equalities, N. Bez, S. Machihara, and T. Ozawa obtained some interesting norm inequalities in the spirit of Evans and Lewis that compare the standard Laplacian with its radial and spherical components. In this paper we give a simple unified proof and a strict improvement of these Evans-Lewis inequalities in the subtle dimension three case. Our approach is robust and explains clearly the occurrence of the sharp constant., Comment: added a section which extends our machinery to the abstract framework; added more references

About ten years ago, Schmuckenschl\"ager proved that the lowest eigenvalue of Dirichlet Laplacian for the intersection of two balls (i.e., convex, symmetric and compact subsets of $\mathbb{R}^n$ with non-empty interior) is less than the sum of the lowest eigenvalue for each. His arguments rely on Kac's formula, the log-concavity of Gaussian measures, the symmetry of balls and Lieb's classical result for the intersection of two domains. In this note we revisit Schmuckenschl\"ager's proof and propose a direct and elementary Lieb-free approach to these inequalities.

Given a generator of a bounded analytic semigroup on a Hilbert space, we show that the corresponding forward maximal regularity operator commutes with the backward. In particular, for self-adjoint generators the images under the two maximal regularity operators have equal unweighted Hilbert space norms., Comment: fixed a typo: "apply Theorem 2.1 or solely (3.2)" to "apply Theorem 2.1 or solely (2.2)"

We give a conceptually simple and essentially one-dimensional approach to Rellich inequality in Euclidean space $\mathbb{R}^n$. In particular, we show that the radial part and the spherical part of the standard Laplacian form an angle in $[0,\pi/2]$ when $n\geq4$, a property known in the works of Evans-Lewis, Machihara-Ozawa-Wadade, and Bez-Machihara-Ozawa. Our proof here is direct and short.

We simplify the proof of a Gaussian correlation inequality for plurisubharmonic functions found by Barthe and Cordero-Erausquin. The new observation is a second-order integration-by-parts formula in the complex Gaussian setting.

The long range electronic modulations recently discovered in the geometrically frustrated kagome lattice have opened new avenues to explore the effect of correlations in materials with topological electron flat bands. The observation of the lattice response to the emergent new phases of matter, a soft phonon mode, has remained elusive and the microscopic origin of charge density waves (CDWs) is still unknown. Here, we show, for the first time, a complete melting of the ScV$_ 6$Sn$_ 6$ (166) kagome lattice. The low energy phonon with propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ collapses at 98 K, without the emergence of long-range charge order, which sets in with a propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{3}$. The CDW is driven (but locks at a different vector) by the softening of an overdamped phonon flat plane at k$_z$=$\pi$. We observe broad phonon anomalies in momentum space, pointing to (1) the existence of approximately flat phonon bands which gain some dispersion due to electron renormalization, and (2) the effects of the momentum dependent electron-phonon interaction in the CDW formation. Ab initio and analytical calculations corroborate the experimental findings to indicate that the weak leading order phonon instability is located at the wave vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ of a rather flat collapsed mode. We analytically compute the phonon frequency renormalization from high temperatures to the soft mode, and relate it to a peak in the orbital-resolved susceptibility, obtaining an excellent match with both ab initio and experimental results, and explaining the origin of the approximately flat phonon dispersion. Our data report the first example of the collapse of a softening of a flat phonon plane and promote the 166 compounds of the kagome family as primary candidates to explore correlated flat phonon-topological flat electron physics., Comment: 10 pages, 4 figures

We establish in this paper some weighted Hardy and Rellich type inequalities on the half line in the framework of equalities, extending recent results proved by Machihara-Ozawa-Wadade and Bez-Machihara-Ozawa. In particular, the one-dimensional classical Rellich inequality is proved in the framework of equality.

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation $$\frac{\partial h}{\partial t}=\frac{\partial^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta},\quad\beta>0$$ subject to initial boundary value conditions $$h(\cdot,0)=h_0>0,\quad h(\pm1,t)=1.$$ We derive the intermediate extinction profile with refined asymptotics, and with extinction time $T$ and extinction point $0$, the gradient profile behaves as $x\rightarrow0$ like $$\lim_{t\rightarrow T}\,(\nabla h)(x,t)\quad\sim\quad\frac{1}{\sqrt{2\beta}}\frac{x}{|x|}\frac{1}{\sqrt{|\log|x||}}\left[\frac{(\beta+1)^2}{8\beta}\frac{|x|^2}{|\log|x||}\right]^{\frac{1}{\beta+1}-\frac12},$$ agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions., Comment: 54 pages

A spin-triplet superconductor can harbor Majorana bound states that can be used in topological quantum computing. Recently, K2Cr3As3 and its variants with critical temperature Tc as high as 8 K have emerged as a new class of superconductors with ferromagnetic spin fluctuations. Here we report a discovery in K2Cr3As3 single crystal that, the spin susceptibility measured by 75As Knight shift below Tc is unchanged with the magnetic field H0 applied in the ab plane, but vanishes toward zero temperature when H0 is along the c axis, which unambiguously establishes this compound as a spin-triplet superconductor described by a vector order-parameter d parallel to the c axis. Combining with points-nodal gap we show that K2Cr3As3 is a new platform for the study of topological superconductivity and its possible technical application.