A maximal function for families of Hilbert transforms along homogeneous curves

Abstract

Let H (u) be the Hilbert transform along the parabola (t, ut 2 ) where u ∈ R. For a set U of positive numbers consider the maximal func-We obtain an (essentially) optimal result for the L p operator norm of H U when 2 < p < ∞. The results are proved for families of Hilbert transforms along more general nonflat homogeneous curves.

Maximal functions associated with families of homogeneous curves: Lp bounds for P ≤ 2

“…For a Schwartz function f on R 2 we let M (u) f (x) = sup For 2 < p < ∞ the operators M U are bounded on L p (R 2 ) for all U ; this was shown by Marletta and Ricci [8]. For the operators H U a corresponding satisfactory theorem was proved in a previous paper [6] of the authors. To describe the result let N(U ) = 1 + #{n ∈ Z : [2 n , 2 n+1 ] ∩ U = ∅}.…”

A Cotlar type maximal function associated with Fourier multipliers

“…Proof. The proof is essentially the same as that of an analogous result in [4], which is in turn a modification of the argument for the standard Cotlar inequality regarding the truncation of singular integrals (see [8], sec 1.7).…”

“…We wish to show that for l ≥ 0, the operator T l * is bounded on L p for 1 < p < ∞ with the corresponding operator norm p Bl −α+1 ( B for l = 0). Since our assumption (6) on m is much weaker than that in [4], a pointwise inequality like (5) for all r > 0 seems out of reach. However, by using similar ideas, we are able to establish a Cotlar type inequality…”

“…The following lemma is the main step in establishing the Cotlar type inequality. The ideas used are similar to Lemma A.1 in [4].…”

“…The above operator was studied by Guo, Roos, Seeger and Yung in [4], in connection with proving L p bounds for a maximal operator associated with families of Hilbert transforms along parabolas. The multiplier m in [4] was assumed to satisfy the condition…”