Weak-Type (1, 1) Estimates for Strongly Singular Operators

Abstract

Let ψ be a positive function defined near the origin such that lim_t→0+ ψ(t) = 0. We consider the operator

$${T_\theta }\;f(x) =\lim_{\varepsilon \rightarrow 0^0}\int_\varepsilon ^1 {{e^{i\gamma (t)}}} f(x - t)\frac{{dt}}{{{t^{\theta \psi {{(t)}^{1 - \theta }}}}}},$$

where γ is a real function with lim_t→0+ |γ(t)| = ∞ and 0 ≤ θ ≤ 1. Assuming certain regularity and growth conditions on ψ and γ, we show that T₁ is of weak type (1, 1).