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We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $$(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$$ in $$\mathbb{R}^3$$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates. 

Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $$\operatorname{UMD}$$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $$\operatorname{UMD}$$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $$\textrm{UMD}$$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $$\textrm{UMD}$$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors. 

Abstract We investigated the photosensitizing properties of secondary organic aerosol (SOA) formed during the hydroxyl radical (OH) initiated oxidation of naphthalene. This SOA was injected into an aerosol flow tube and exposed to UV radiation and gaseous volatile organic compounds or sulfur dioxide (SO₂). The aerosol particles were observed to grow in size by photosensitized uptake of d‐limonene and β‐pinene. In the presence of SO₂, a photosensitized production (0.2–0.3 µg m⁻³ h⁻¹) of sulfate was observed at all relative humidity (RH) levels. Some sulfate also formed on particles in the dark, probably due to the presence of organic peroxides. The dark and photochemical pathways exhibited different trends with RH, unraveling different contributions from bulk and surface chemistry. As naphthalene and other polycyclic aromatics are important SOA precursors in the urban and suburban areas, these dark and photosensitized reactions are likely to play an important role in sulfate and SOA formation.