For Boolean functions computed by de Morgan formulas of subquadratic size or read-once de Morgan formulas, we prove a sharp concentration of the Fourier mass on ``small-degree'' coefficients. For a Boolean function f:01n1−1 computable by a de Morgan formula of size s, we show thats^{1/\Gamma + \epsilon}} \hat{f}(A)^2 \leq exp(-s^{\epsilon/3}), }">A[n]:As1+f(A)2exp(−s3) where is the shrinkage exponent for the corresponding class of formulas: =2 for de Morgan formulas, and =1log2(5−1)327 for read-once de Morgan formulas. We prove that this Fourier concentration is essentially optimal.

As an application, we get that subquadratic-size de Morgan formulas have negligible correlation with parity, and are learnable under the uniform distribution, and also lossily compressible, in subexponential time. We also prove that the average sensitivity of a read-once function f on n variables is at most n1+o(1) , and is at least (n1) .