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

Abstract In this paper we analyze the convergence of the following type series T N f ( x ) = ∑ j = N 1 N 2 v j ( e − a j + 1 Δ 2 f ( x ) − e − a j Δ 2 f ( x ) ) , x ∈ R n , where { e − t Δ 2 } t > 0 is the heat semigroup of the biharmonic operator Δ 2 with Δ being the classical laplacian, N = ( N 1 , N 2 ) ∈ Z 2 with N 1 N 2 , { v j } j ∈ Z is a bounded real sequences and { a j } j ∈ Z is a ρ-lacunary sequence of positive numbers, that is, 1 ρ ≤ a j + 1 / a j , for all j ∈ Z . Our analysis will consist in the boundedness, in L p ( R n ) and in B M O ( R n ) , of the operators T N and its maximal operator T ⁎ f ( x ) = sup N ⁡ | T N f ( x ) | . The proofs of these results need the language of semigroups in an essential way.