Journal of Functional Analysis最新文献

英文中文

Sharp stability of the Heisenberg Uncertainty Principle: Second-order and curl-free field cases 海森堡测不准原理的锐稳定性：二阶和无旋场情况

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-22 DOI: 10.1016/j.jfa.2025.111321

Anh Xuan Do , Nguyen Lam , Guozhen Lu

Using techniques from harmonic analysis, we derive several sharp stability estimates for the second order Heisenberg Uncertainty Principle. We also present the explicit lower and upper bounds for the sharp stability constants and compute their exact limits when the dimension

N \to \infty

. Our proofs rely on spherical harmonics decomposition and Fourier analysis, differing significantly from existing approaches in the literature. Our results substantially improve the stability constants of the second order Heisenberg Uncertainty Principle recently obtained in [27]. As direct consequences of our main results, we also establish the sharp stability, with exact asymptotic behavior of the stability constants, of the Heisenberg Uncertainty Principle with curl-free vector fields and a sharp version of the second order Poincaré type inequality with Gaussian measure.

利用谐波分析技术，我们得到了二阶海森堡测不准原理的几个尖锐稳定性估计。我们还给出了尖锐稳定常数的显式下界和上界，并计算了它们在维数N→∞时的精确极限。我们的证明依赖于球谐波分解和傅立叶分析，与文献中现有的方法有很大的不同。我们的结果大大提高了最近在[27]中得到的二阶海森堡测不准原理的稳定常数。作为我们主要结果的直接结果，我们还建立了具有无旋度向量场的海森堡测不准原理的尖锐稳定性，具有稳定常数的精确渐近行为，以及具有高斯测度的二阶poincar型不等式的尖锐版本。

引用次数: 0

Fluctuation exponents of the half-space KPZ at stationarity 平稳时半空间KPZ的波动指数

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111315

Yu Gu, Ran Tao

We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. Utilizing this identity, we obtain estimates for the polymer endpoints, leading to optimal fluctuation exponents for the height function in both the subcritical and critical regimes, as well as an optimal upper bound for the fluctuation exponents in the extended critical regime. We also compute the average growth rate as a function of the boundary parameter.

从平稳布朗初始数据出发，研究了具有诺伊曼边界条件的半空间KPZ方程。我们推导了一个方差恒等式，将高度函数的波动与半空间聚合物模型的横向波动联系起来。利用这一恒等式，我们得到了聚合物端点的估计，得到了亚临界和临界状态下高度函数的最优波动指数，以及扩展临界状态下波动指数的最优上界。我们还计算了平均增长率作为边界参数的函数。

引用次数: 0

Discrete Triebel-Lizorkin spaces and expansive matrices 离散triiebel - lizorkin空间与扩展矩阵

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111316

Jordy Timo van Velthoven , Felix Voigtlaender

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices A and B, and arbitrary

α \in R

and

p, q \in (0, \infty]

, it is shown that

{\dot{f}}_{p, q}^{α} (A) = {\dot{f}}_{p, q}^{α} (B)

if and only if the set

{A^{j} B^{- j} : j \in Z}

is finite, or in the trivial case when

| \det (A) |^{α + 1 / 2 - 1 / p} = | \det (B) |^{α + 1 / 2 - 1 / p}

and

p = q

. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.

我们给出了两个膨胀膨胀矩阵产生相等离散各向异性triiebel - lizorkin空间的表征。对于两个这样的矩阵A和B，任意α∈R和p，q∈(0,∞)，证明了f˙p，qα(A)=f˙p,qα(B)当且仅当集合{AjB−j:j∈Z}是有限的，或者当|det (A)|α+1/2−1/p=|det (B)|α+1/2−1/p和p=q时。这将triiebel对角扩张的结果推广到任意扩张矩阵。得到的膨胀分类与各向异性triiebel - lizorkin函数空间的相应结果不同。

引用次数: 0

Sharp concentration phenomena in high-dimensional Orlicz balls 高维奥利兹球中的尖锐集中现象

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111322

Lorenz Frühwirth, Joscha Prochno

In this article, we present a precise deviation formula for the intersection of two Orlicz balls generated by Orlicz functions V and W. Additionally, we establish a (quantitative) central limit theorem in the critical case and a strong law of large numbers for the “W-norm” of the uniform distribution on

B^{(n, V)}

. Our techniques also enable us to derive a precise formula for the thin-shell concentration of uniformly distributed random vectors in high-dimensional Orlicz balls. In our approach we establish an Edgeworth-expansion using methods from harmonic analysis together with an exponential change of measure argument.

本文给出了由Orlicz函数V和w生成的两个Orlicz球相交的精确偏差公式，并建立了临界情况下的（定量）中心极限定理和B（n，V）上均匀分布的“w -范数”的强大数定律。我们的技术还使我们能够推导出高维Orlicz球中均匀分布的随机向量的薄壳浓度的精确公式。在我们的方法中，我们利用调和分析的方法和测度的指数变化论证建立了一个埃奇沃斯展开式。

引用次数: 0

Nonlocal operators in divergence form and existence theory for integrable data 发散形式的非局部算子与可积数据的存在性理论

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111317

David Arcoya , Serena Dipierro , Edoardo Proietti Lippi , Caterina Sportelli , Enrico Valdinoci

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to

L^{1} (Ω)

and to be suitably dominated.

We also prove that the solution that we find converges, as

s ↗ 1

, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in

L^{1} (Ω)

and therefore the usual regularity theory cannot be leveraged to our benefit in this framework.

The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as

s ↗ 1

, every classical operator in divergence form.

我们给出了由散度形式的非局部算子控制的Dirichlet边值问题的弱解的存在唯一性结果，且存在一个假定只属于L1（Ω）并被适当支配的基准。我们还证明了我们找到的解收敛于局部对应问题的解，作为极限情况恢复了经典结果。这需要一些非平凡的自定义统一估计和表示公式，因为数据仅在L1中（Ω），因此通常的规则理论无法在此框架中为我们所用。极限过程使用一个非局部算子，得到一个齐次核的仿射变换，在极限s × 1下恢复所有经典算子的散度形式。

{"title":"Nonlocal operators in divergence form and existence theory for integrable data","authors":"David Arcoya , Serena Dipierro , Edoardo Proietti Lippi , Caterina Sportelli , Enrico Valdinoci","doi":"10.1016/j.jfa.2025.111317","DOIUrl":"10.1016/j.jfa.2025.111317","url":null,"abstract":"<div><div>We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> and to be suitably dominated.</div><div>We also prove that the solution that we find converges, as <span><math><mi>s</mi><mo>↗</mo><mn>1</mn></math></span>, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> and therefore the usual regularity theory cannot be leveraged to our benefit in this framework.</div><div>The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as <span><math><mi>s</mi><mo>↗</mo><mn>1</mn></math></span>, every classical operator in divergence form.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111317"},"PeriodicalIF":1.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Closed BV-extension and W1,1-extension sets 闭bv -可拓集和w1,1 -可拓集

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111319

Emanuele Caputo , Jesse Koivu , Danka Lučić , Tapio Rajala

This paper studies the relations between extendability of different classes of Sobolev

W^{1, 1}

and BV functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak

(1, 1)

-Poincaré inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected BV-extension sets are always also

W^{1, 1}

-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set.

本文研究了广义度量测度空间中闭集Sobolev W1、1和BV函数的不同类的可扩展性之间的关系。在度量测度空间满足弱(1,1)- poincar不等式和测度加倍的假设下，进一步证明了扩展集的性质。在欧氏平面上，我们证明了紧有限连通的bv -扩展集总是w1,1 -扩展集。这是通过扩展集补的局部拟凸性结果来证明的。

引用次数: 0

Weighted Korenblum-Roberts theory 加权Korenblum-Roberts理论

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111324

Bartosz Malman

The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk

D

as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a

P^{t} (μ)

space, the closure of analytic polynomials in a Lebesgue space

L^{t} (μ)

corresponding to a measure of the form

d A_{α} + w d m

, with

d A_{α}

being the standard weighted area measure on

D

, dm the Lebesgue measure on the unit circle

T

, and w a general weight on

T

. We characterize when

P^{t} (μ)

of this form is a space of analytic functions on

D

by computing the Thomson decomposition of the measure μ. The structure of the decomposition is expressed in terms of what we call the family of associated Beurling-Carleson sets. We characterize the cyclic singular inner functions in the analytic

P^{t} (μ)

spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols

b : D \to D

which generate a de Branges-Rovnyak space with a dense subset of functions smooth on

T

. The characterization is given explicitly in terms of the modulus of b on

T

and the singular measure corresponding to the singular inner factor of b. Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained

L^{1}

-optimization tools.

经典的Korenblum-Roberts定理将单位盘D的Bergman空间中的循环奇异内函数刻画为其对应的奇异测度在Lebesgue测度为0的Beurling-Carleson集合上消失的内函数。我们解决问题的加权变异的伯格曼空间取代了Pt(μ)空间,关闭分析多项式在勒贝格空间Lt(μ)对应的表单dAα+ wdm与dAα标准加权面积测量在D, dm单位圆上的勒贝格测度T)和w一般体重T .我们描述当Pt(μ)的这种形式是空间分析功能在D的汤姆森分解通过计算测量μ。分解的结构用我们所说的相关Beurling-Carleson集合族来表示。我们将解析Pt（μ）空间中的循环奇异内函数刻画为相应的奇异测度在相关的Beurling-Carleson集合族上消失的循环奇异内函数。与经典的设定不同，在我们的描述中出现了零和正勒贝格测度的Beurling-Carleson集。作为我们的结果的应用，我们完成了符号b:D→D的表征，它产生了一个具有T上光滑函数的密集子集的de Branges-Rovnyak空间。我们的表征明确地给出了b在T上的模和对应于b的奇异内因子的奇异测度。我们的证明涉及赫鲁晓夫的多项式近似技术和Korenblum的线性规划思想。结合最近建立的约束l1优化工具。

{"title":"Weighted Korenblum-Roberts theory","authors":"Bartosz Malman","doi":"10.1016/j.jfa.2025.111324","DOIUrl":"10.1016/j.jfa.2025.111324","url":null,"abstract":"<div><div>The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk <span><math><mi>D</mi></math></span> as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> space, the closure of analytic polynomials in a Lebesgue space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> corresponding to a measure of the form <span><math><mi>d</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>+</mo><mi>w</mi><mspace></mspace><mi>d</mi><mtext>m</mtext></math></span>, with <span><math><mi>d</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> being the standard weighted area measure on <span><math><mi>D</mi></math></span>, <em>dm</em> the Lebesgue measure on the unit circle <span><math><mi>T</mi></math></span>, and <em>w</em> a general weight on <span><math><mi>T</mi></math></span>. We characterize when <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> of this form is a space of analytic functions on <span><math><mi>D</mi></math></span> by computing the Thomson decomposition of the measure <em>μ</em>. The structure of the decomposition is expressed in terms of what we call the family of <em>associated Beurling-Carleson sets</em>. We characterize the cyclic singular inner functions in the analytic <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo></math></span> spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols <span><math><mi>b</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>D</mi></math></span> which generate a de Branges-Rovnyak space with a dense subset of functions smooth on <span><math><mi>T</mi></math></span>. The characterization is given explicitly in terms of the modulus of <em>b</em> on <span><math><mi>T</mi></math></span> and the singular measure corresponding to the singular inner factor of <em>b</em>. Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-optimization tools.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"290 7","pages":"Article 111324"},"PeriodicalIF":1.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Schauder frames of discrete translates in L2(R) L2(R)中离散平移的Schauder坐标系

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111318

Nir Lev , Anton Tselishchev

We construct a uniformly discrete sequence

{λ_{1} < λ_{2} < \dots} \subset R

and functions g and

{g_{n}^{⁎}}

L^{2} (R)

, such that every

f \in L^{2} (R)

admits a series expansion

f (x) = \sum_{n = 1}^{\infty} 〈 f, g_{n}^{⁎} 〉 g (x - λ_{n})

convergent in the

L^{2} (R)

norm.

我们构造了一个一致离散序列{λ1<；λ2<；⋯}∧R以及函数g和{gn}在L2(R)中，使得每个f∈L2(R)允许一个级数展开f(x)=∑n=1∞< f,gn > g（x−λn）收敛于L2(R)范数。

引用次数: 0

On Hermite pseudo–multipliers with non-smooth kernels 非光滑核的Hermite伪乘子

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-19 DOI: 10.1016/j.jfa.2025.111323

The Anh Bui , Xuan Thinh Duong , Fu Ken Ly

Let

H

be the Hermite operator on

R^{n}

. For a bounded function

m : R^{n} \times R \to C

, we can define the Hermite pseudo-multipliers

m (x, H)

formally by setting

m (x, H) = \sum_{k = 0}^{\infty} m (x, 2 k + n) P_{k},

where

P_{k}

is the orthogonal projection of

L^{2} (R^{n})

onto the k-th eigenspace of

H

corresponding to the eigenvalue

2 k + n

. In this paper, we consider new conditions on m for which

m (x, H)

may not possess any kernel regularity. For such pseudo-multipliers we establish their boundedness on various function spaces including weighted Lebesgue spaces, BMO and Hardy spaces associated to

H

. In the scale of the weighted Lebesgue spaces, our results improve those in [Bagchi & Thangavelu, J. Funct. Anal. 2015].

设H是Rn上的厄米算子。对于有界函数m:Rn×R→C，我们可以通过设m(x，H)=∑k=0∞m(x,2k+n)Pk来正式定义Hermite伪乘子m(x，H)，其中Pk是L2（Rn）在H的第k个特征空间上对应于特征值2k+n的正交投影。本文考虑m上m（x，H）不具有核正则性的新条件。对于这些伪乘子，我们建立了它们在各种函数空间上的有界性，包括加权Lebesgue空间、BMO和与h相关的Hardy空间。在加权Lebesgue空间的尺度上，我们的结果改进了[Bagchi &； Thangavelu， J. Funct]中的结果。肛交,2015]。

引用次数: 0

Estimates of Green and Martin integrals of Schrödinger equations and a semilinear boundary value problem Schrödinger方程的Green和Martin积分的估计及半线性边值问题

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-17 DOI: 10.1016/j.jfa.2025.111307

Moshe Marcus

Consider Schrödinger operators

L^{V} : = Δ + V

in a bounded Lipschitz domain

Ω \subset R^{N}

. Assume that

V \in C^{1} (Ω)

satisfies

V (x) \leq \bar{a} dist {(x, \partial Ω)}^{- 2}

and a subcriticality condition that guarantees the existence of a ground state

Φ_{V}

. We derive sharp estimates of signed

L_{V}

superharmonic functions that possess an

L_{V}

boundary trace, i.e., a measure boundary trace associated with

L_{V}

. Using these estimates we derive a-priori estimates of positive solutions of a related semilinear boundary value problem.

考虑有界Lipschitz域中的Schrödinger算子LV:=Δ+V Ω∧RN。假设V∈C1（Ω）满足V(x)≤a¯dist（x,∂Ω）−2和保证基态存在的亚临界条件ΦV。我们得到了具有LV边界迹的有符号LV超调和函数的尖锐估计，即与LV相关的测量边界迹。利用这些估计，我们得到了一个相关的半线性边值问题正解的先验估计。

引用次数: 0

首页上一页

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Journal of Functional Analysis

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀