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Abstract This paper first defines operators that are “well‐localized” with respect to a pair of accretive functions and establishes a global two‐weight theorem for such operators. Then it defines operators that are “well‐localized” with respect to a pair of accretive systems and establishes a local two‐weight theorem for them. The proofs combine recent proof techniques with arguments used to prove earlierT1 theorems for well‐localized operators. 

We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities. 

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions. 

Abstract We study almost surely separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0–1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov–Sobolev spaces $$B_{2}^{\sigma }\left( \mathbb {B}_{d}\right) $$ B 2 σ B d , in the range $$0 < \sigma \le 1 / 2$$ 0 < σ ≤ 1 / 2 . For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) and its multiplier algebra $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) : in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) -interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $$0-1$$ 0 - 1 law for random interpolating sequences for $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) .