On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).     

Aromatic compounds produced by endophytic fungi isolated from red alga Asparagopsis taxiformis - Falkenbergia stage

Endophytic fungi were isolated from red alga Asparagopsis taxiformis - Falkenbergia stage, collected from the Brazilian coast, and were identified as Annulohypoxylon stygium (AT-03) and A. yungensis (AT-06) based on their macro/micromorphological and molecular features. Bioassay-guided fractionation of the EtOAc extract from laboratory cultures of both strains yielded known compounds pyrogallol from A. stygium, (3R)-scytalone and (3R,4R)-4-hydroxy-scytalone from A. yungensis. Pyrogallol was active against methicillin-resistant Staphylococcus aureus (MRSA) and Escherichia coli strains. An inactive fraction from A. stygium afforded two additional compounds, (3R,4R)-3,4,5-trihydroxy-1-tetralone and tyrosol. Optically active compounds had their stereochemistry determined by circular dichroism (CD) spectroscopy.     

PtII-Catalyzed Hydroxylation of Terminal Aliphatic C(sp3)−H Bonds with Molecular Oxygen

The practical application of Shilov-type Pt catalysis to the selective hydroxylation of terminal aliphatic C−H bonds remains a formidable challenge, due to difficulties in replacing Pt^IV with a more economically viable oxidant, particularly O₂. We report the potential of employing FeCl₂ as a suitable redox mediator to overcome the kinetic hurdles related to the direct use of O₂ in the Pt reoxidation. For the selective conversion of butyric acid to γ-hydroxybutyric acid (GHB), a significantly enhanced catalyst activity and stability (turnover numbers (TON)>30) were achieved under 20 bar O₂ in comparison to current state-of-the-art systems (TON<10). In this regard, essential reaction parameters affecting the overall activity were identified, along with specific additives to attain catalyst stability at longer reaction times. Notably, deactivation by reduction to Pt⁰ was prevented by the addition of monodentate pyridine derivatives, such as 2-fluoropyridine, but also by introducing varying partial pressures of N₂ in the gaseous atmosphere. Finally, stability tests revealed the involvement of Pt^II and FeCl₂ in catalyzing the non-selective overoxidation of GHB. Accordingly, in situ esterification with boric acid proved to be a suitable strategy to maintain enhanced selectivities at much higher conversions (TON>60). Altogether, a useful catalytic system for the selective hydroxylation of primary aliphatic C−H bonds with O₂ is presented.     

Asymmetric Reactions involving Lewis Base Catalyst Tethered Dearomatized Intermediates

Numerous protocols have been developed for the functionalization of aromatic substances. Among them, the strategy by which aromatic substrates are activated in situ to generate dearomatized intermediates is highly efficient but challenging, especially in the field of asymmetric catalysis. In this Concept article, the application of some well-established chiral Lewis base catalysis, including primary/secondary amines and N-heterocyclic carbenes, that can covalently form catalyst-tethered dearomatized ortho/para-quinodimethane species with diverse heteroaryl and aryl carbonyl substrates is summarized in a number of asymmetric cycloaddition and addition reactions with diverse reagents generally having electrophilic properties. As a result, a variety of enantioenriched aromatic products with higher molecular complexity are constructed effectively through a rearomatization process.     

Photoinduced C(sp3)−N Bond Cleavage Leading to the Stereoselective Syntheses of Alkenes

Herein we report a versatile Mizoroki–Heck-type photoinduced C(sp³)−N bond cleavage reaction. Under visible-light irradiation (455 nm, blue LEDs) at room temperature, alkyl Katritzky salts react smoothly with alkenes in a 1:1 molar ratio in the presence of 1.0 mol % of commercially available photoredox catalyst without the need for any base, affording the corresponding alkyl-substituted alkenes in good yields with broad functional-group compatibility. Notably, the E/Z-selectivity of the alkene products can be controlled by an appropriate choice of photoredox catalyst.     

Mono-Aryl/Alkylthio-Substituted (Hetero)acenes of Exceptional Thermal and Photochemical Stability by the Thio-Friedel–Crafts/Bradsher Cyclization Reaction

The highly substituted mono-aryl/alkylthio-(hetero)acenes prepared in this study have been found to be thermally more stable (T_decomp.=331–354 °C) than the known di-aryl/alkylthio-substituted acenes by an average of 25 °C. They are also much more photostable at 254 and 365 nm (in both argon and air) than the parent anthracene and other reported anthracenes. The most photostable aryl/alkylthio-anthracenes at 254 nm were found to be 60–70 (in air) and 130 (in argon) times more stable in solution than the unsubstituted anthracene, and much more stable than known EDG/EWG-substituted anthracenes (EDG=electron-donating group, EWG=electron-withdrawing group) with an extended aromatic core. Furthermore, the acenes showed significantly higher photostability at 365 nm in both air and argon. The anthracenes were obtained by the novel thio-Friedel–Crafts/Bradsher cyclization reaction of hitherto unknown [o-(1,3-dithian-2-yl)aryl](aryl)methyl thioethers. The developed approach provides a general access to mono-aryl/alkylthio-substituted (hetero)acene frameworks containing at least three fused (hetero)aromatic rings. The characteristic feature of this approach, which leads to highly substituted acenes, is that the substituents, unlike in other methods, may be introduced at an early stage of the synthesis. DFT and TD-DFT calculations confirmed the stabilizing role of the aryl/alkylthio substituent in the mono-aryl/alkylthio-substituted anthracenes, which are the most stable anthracenes prepared to date. Their high photostability is mainly due to the quenching of singlet oxygen by the acene and the quenching of the acene S₁ state by molecular oxygen.     

Two weight commutators in the Dirichlet and Neumann Laplacian settings

In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian ${\mathrm{\Delta }}_{N_{+}}$ on the upper half space ${\mathbb{R}}_{+}^n$ and the reflection Neumann Laplacian ${\mathrm{\Delta }}_N$ on ${\mathbb{R}}^n$ with respect to the weights associated to ${\mathrm{\Delta }}_{N_{+}}$ and ${\mathrm{\Delta }}_N$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to ${\mathrm{\Delta }}_N$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt–Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via the area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on ${\mathbb{R}}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,\frac{n+1}{n}]$. We also point out that this two weight commutator theorem might not be true in the setting of general operators L, and in particular we show that it is not true when L is the Dirichlet Laplacian ${\mathrm{\Delta }}_{D_{+}}$ on ${\mathbb{R}}_{+}^n$.     

The character and the wave front set correspondence in the stable range

We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.     

Enantioselective Organocatalytic Reduction of β‐Trifluoromethyl Nitroalkenes: An Efficient Strategy for the Synthesis of Chiral β‐Trifluoromethyl Amines

An efficient organocatalytic stereoselective reduction of β‐trifluoromethyl‐substituted nitroalkenes, mediated by 3,5‐dicarboxylic ester‐dihydropyridines (Hantzsch ester type), has been successfully developed. A multifunctional thiourea‐based (S)‐valine derivative was found to be the catalyst of choice, promoting the reaction in up to 97 % ee. The methodology has been applied to a wide variety of substrates, leading to the formation of differently substituted precursors of enantiomerically enriched β‐trifluoromethyl amines. The mechanism of the reaction and the mode of action of the metal‐free catalytic species were computationally investigated; on the basis of DFT transition‐state (TS) analysis, a model of stereoselection was also proposed.