The Hilbert transform and maximal function for approximately homogeneous curves

Abstract:

Let ${\mathcal {H}_\gamma }f(x) = {\text {p}}{\text {.v}}{\text {.}}\int _{ - 1}^1 {f(x - \gamma (t))dt/t}$ and ${\mathfrak {M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int _0^h {|f(x - \gamma (t))|dt}$. It is proved that for $f \in \mathcal {S}({{\mathbf {R}}^n})$, the Schwartz class, and for an approximately homogeneous curve $\gamma (t) \in {{\mathbf {R}}^n}$, ${\left \| {{\mathcal {H}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2}$, ${\left \| {{\mathfrak {M}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2}$. A homogeneous curve is one which satisfies a differential equation ${\gamma ’_1}(t) = (A/t){\gamma _1}(t)$, $0 < t < \infty$, where $A$ is a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curve $\gamma (t)$ has the form ${\gamma _1}(t) + {\gamma _2}(t)$, where ${\gamma _2}(t)$ is a carefully specified "error", such that $\gamma _2^{(j)}$ is also restricted for $j = 2, \ldots ,n + 1$. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.