On the Local Tb Theorem: A Direct Proof under the Duality Assumption

We give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1/p + 1/q ⩽ 1, the ‘dual case’ 1/p + 1/q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.

Projective Stochastic Equations and Nonlinear Long Memory

A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Projective Stochastic Equations and Nonlinear Long Memory

A projective moving average {X t , t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of X t on ‘intermediate’ lagged innovation subspaces with given coefficients α i and β i,j . The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution X t . We show that, under certain conditions on Q, α i , and β i,j , this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.