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Many techniques in harmonic analysis use the fact that a continuous object can be written as a sum (or an intersection) of dyadic counterparts, as long as those counterparts belong to a distinct dyadic system. Here we generalize the notion of distinct dyadic system and explore when it occurs, leading to some new and perhaps surprising classifications.

The potential of GaAs‐based photonic crystals for fast all‐optical switching in the telecom spectral range is exploited by controlling the surface recombination and, thereby, the carrier relaxation dynamics. The structure is entirely coated with a layer of aluminium oxide using atomic layer deposition. This results in a carrier lifetime of about 10 ps, as determined by spectrally resolved pump–probe measurements. We show that the nonlinear response of the resonator is optimized when it is excited with a few‐picoseconds pulse. This dynamics is perfectly captured by our model accounting for the carrier diffusion with an impulse response function. Moreover, the suppression of photo‐induced oxidation is revealed to be crucial to demonstrate all‐optical operation at GHz rates with average coupled pump power of 0.5 mW (hence 100 fJ/bit). The switching window is 12 ps wide (1/e), as resolved by homodyne pump–probe measurements. The devices respond to a sequence of closely spaced pump pulses demonstrating a gating window close to 10 ps, with a contrast as high as 7 dB.

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The purpose of this paper is to study the sparse bound of the operator of the form f → ψ(x) f (γt(x))K(t)dt, where γt(x) is a C ∞ function defined on a neighborhood of the origin in (x, t) ∈ R n × R k , satisfying γ 0 (x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ R n and K is a Calderón-Zygmund kernel suppported on a small neighborhood of 0 ∈ R k . Christ, Nagel, Stein and Wainger gave conditions on γ under which T : L p → L p (1 < p < ∞) is bounded. Under the these same conditions, we prove sparse bounds for T , which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.