Boundedness of differential transforms for the heat semigroup generated by biharmonic operator

Abstract

In this paper we analyze the convergence of the following type seriesTNf(x)=∑j=N1N2vj(e−aj+1Δ2f(x)−e−ajΔ2f(x)),x∈Rn, where {e−tΔ2}t>0 is the heat semigroup of the biharmonic operator Δ2 with Δ being the classical laplacian, N=(N1,N2)∈Z2 with N1<N2, {vj}j∈Z is a bounded real sequences and {aj}j∈Z is a ρ-lacunary sequence of positive numbers, that is, 1<ρ≤aj+1/aj,for allj∈Z. Our analysis will consist in the boundedness, in Lp(Rn) and in BMO(Rn), of the operators TN and its maximal operator T⁎f(x)=supN⁡|TNf(x)|. The proofs of these results need the language of semigroups in an essential way.