Directional Hardy-Littlewood maximal function

Question

I am currently reading Calderón's paper ,,Estimates for singular integral operators in terms of maximal functions" and I am trying to understand the proof of Lemma 7. To be specific: the directional Hardy-Littlewood maximal operator for a given function $f\colon\mathbb{R}^{d}\to\mathbb{R}$ and a direction $\nu\in\mathbb{S}^{d-1}$ is defined by $$ M_{\nu}f(x)=\sup_{t>0}\frac{1}{t}\int_{0}^{t}|f(x+s\nu)|\,ds. $$ Is it true that this operator is smaller then the classical Hardy-Littlewood operator, that is $$ M_{\nu}f(x)\leq Mf(x) $$ almost everywhere (or maybe the inequality holds with some constant)? The passages in the afforementioned paper are vague to me.

Answer 1

This is a 1D maximal function going in one direction. So if you want to prove inequalities, factor $\mathbb R^n$ as $\mathbb R \times \mathbb R^{n-1}$, apply the 1D maximal function to $\mathbb R$, and then integrate over $\mathbb R^{n-1}$, and use Fubini's Theorem.