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We study commutators with the Riesz transforms on the Heisenberg group$${\mathbb {H}}^{n}$$$H^n$. The Schatten norm of these commutators is characterized in terms of Besov norms of the symbol. This generalizes the classical Euclidean results of Peller, Janson–Wolff and Rochberg–Semmes. The method in proof bypasses the use of Fourier analysis, allowing us to address not just the Riesz transforms, but also the Cauchy–Szegő projection and second order Riesz transforms on$${\mathbb {H}}^{n}$$$H^n$among other settings.

 

We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten classS_presult for dyadic paraproducts, where$1< p < \infty $$1<p<\infty$. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.

 

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space. 

In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel–Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure. 

Abstract Niche signals maintain stem cells in a prolonged quiescence or transiently activate them for proper regeneration 1 . Altering balanced niche signalling can lead to regenerative disorders. Melanocytic skin nevi in human often display excessive hair growth, suggesting hair stem cell hyperactivity. Here, using genetic mouse models of nevi 2,3 , we show that dermal clusters of senescent melanocytes drive epithelial hair stem cells to exit quiescence and change their transcriptome and composition, potently enhancing hair renewal. Nevus melanocytes activate a distinct secretome, enriched for signalling factors. Osteopontin, the leading nevus signalling factor, is both necessary and sufficient to induce hair growth. Injection of osteopontin or its genetic overexpression is sufficient to induce robust hair growth in mice, whereas germline and conditional deletions of either osteopontin or CD44, its cognate receptor on epithelial hair cells, rescue enhanced hair growth induced by dermal nevus melanocytes. Osteopontin is overexpressed in human hairy nevi, and it stimulates new growth of human hair follicles. Although broad accumulation of senescent cells, such as upon ageing or genotoxic stress, is detrimental for the regenerative capacity of tissue 4 , we show that signalling by senescent cell clusters can potently enhance the activity of adjacent intact stem cells and stimulate tissue renewal. This finding identifies senescent cells and their secretome as an attractive therapeutic target in regenerative disorders. 

Metamorphic devolatilization of subducted slabs generates aqueous fluids that ascend into the mantle wedge, driving the partial melting that produces arc magmas. These magmas have oxygen fugacities some 10–1,000 times higher than magmas generated at mid-ocean ridges. Whether this oxidized magmatic character is imparted by slab fluids or is acquired during ascent and interaction with the surrounding mantle or crust is debated. Here we study the petrology of metasedimentary rocks from two Tertiary Aegean subduction complexes in combination with reactive transport modelling to investigate the oxidative potential of the sedimentary rocks that cover slabs. We find that the metasedimentary rocks preserve evidence for fluid-mediated redox reactions and could be highly oxidized. Furthermore, the modelling demonstrates that layers of these oxidized rocks less than about 200 m thick have the capacity to oxidize the ascending slab dehydration flux via redox reactions that remove H₂, CH₄and/or H₂S from the fluids. These fluids can then oxidize the overlying mantle wedge at rates comparable to arc magma generation rates, primarily via reactions involving sulfur species. Oxidized metasedimentary rocks need not generate large amounts of fluid themselves but could instead oxidize slab dehydration fluids ascending through them. Proposed Phanerozoic increases in arc magma oxygen fugacity may reflect the recycling of oxidative weathering products following Neoproterozoic–Palaeozoic marine and atmospheric oxygenation.