Maximal and singular integral operators via Fourier transform estimates

Singular integral and maximal operators associated to hypersurfaces:L p theory

“…The proof of Proposition 2.1 follows exactly along the lines in [5] together with the Calderón-Zygmund theory with respect to a general family of dilations {δ(t)} t>0 satisfying (15). In fact one fixes a nonnegative…”

“…Unfortunately we cannot use the simple pointwise factorization estimate (10), as we did for the maximal operator N , to separate the u and v integration defining each S j and reduce matters to an application of Theorem 4 in [14]. Instead we use Littlewood-Paley arguments as in [5] which rely on corresponding maximal function estimates where the above pointwise factorization (10) can be employed. In fact we will repeatedly use a generalization of Theorem D in [5] which we explicitly state for the convenience of the reader.…”

“…Instead we use Littlewood-Paley arguments as in [5] which rely on corresponding maximal function estimates where the above pointwise factorization (10) can be employed. In fact we will repeatedly use a generalization of Theorem D in [5] which we explicitly state for the convenience of the reader. On a fixed subspace of W ⊂ R d (in our situation d = N + 1) let {δ(t)} t>0 be a family of linear operators acting on W and satisfying the Rivière condition, namely for s ≤ t,…”

“…The first and third inequalities use the corresponding Littlewood-Paley theory (see [2] and [3]) and the second inequality uses the hypothesis that the maximal operator σ is bounded on all L p , p > 1, exactly as in [5].…”

On the commutators of singular integrals related to block spaces

“…Recently, to weaken the condition imposed on Ω, Hu Guoen et al employed the method of Littlewood-Paley theory and Fourier transform estimates from [7] to obtain the following results.…”