$\ell ^p\left( \mathbb {Z}^d\right)$-estimates for discrete operators of Radon type: variational estimates

Abstract

We prove $\ell ^p\left( {\mathbb {Z}}^d\right)$ bounds for $p\in (1, \infty )$, of r-variations $r\in (2, \infty )$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts.

Appendix: Variational estimates for the continuous analogues

This section is intended to provide r-variational estimates for averaging and truncated singular operators of Radon type in the continuous settings. These kinds of questions were extensively discussed in [10], see also the references given there. Here we propose a different approach. Firstly, we will discuss long variation estimates. We give a new proof of Lépingle’s inequality which will be very much in spirit of good-$\lambda$ inequalities. Secondly, we present a new approach to short variation estimates which is based on vector-valued bounds from [15]. This observation, as far as we know, has not been used in this context before. To fix notation let $\mathcal {P}=\left( \mathcal {P}_1,\ldots , \mathcal {P}_{d_0}\right) : \mathbb {R}^k \rightarrow \mathbb {R}^{d_0}$ be a polynomial mapping whose components $\mathcal {P}_j$ are real valued polynomials on $\mathbb {R}^k$ such that $\mathcal {P}_j(0) = 0$. One of the main objects of our interest will be

$$\begin{aligned} {\mathcal {M}}_t^{\mathcal {P}}f(x)=\frac{1}{|G_t|}\int _{G_t}f\left( x-\mathcal P(y)\right) {\, \mathrm{d}}y \end{aligned}$$

for $x\in \mathbb {R}^{d_0}$ where G is an open bounded convex subset of $\mathbb {R}^k$, containing the origin and

$$\begin{aligned} G_t = \left\{ x \in \mathbb {R}^k : t^{-1} x \in G \right\} . \end{aligned}$$

For any $r\in [1, \infty )$ the r-variational seminorm $V_r$ of complex-valued functions $(a_t(x): t>0)$ is defined by

$$\begin{aligned} V_r\left( a_t(x): t>0\right) =\sup _{0< t_0<\ldots <t_J} \left( \sum _{j=0}^J|a_{t_{j+1}}(x)-a_{t_j}(x)|^r\right) ^{1/r} \end{aligned}$$

where the supremum is taken over all finite increasing sequences. In order to avoid some problems with measureability of $V_r\left( a_t(x): t>0\right)$ we assume that $(0, \infty )\ni t\mapsto a_t(x)$ is always a continuous function for every $x\in \mathbb {R}^{d_0}$. The main result of this section is the following theorem.

Theorem 9.1

For every $1< p < \infty$ and $r\in (2, \infty )$ there is $C_{p, r} > 0$ such that for all $f \in L^p\left( \mathbb {R}^{d_0}\right)$

$$\begin{aligned} \left||V_r\left( {\mathcal {M}}_t^\mathcal {P}f: t>0\right) \right||_{L^p}\le C_{p, r}\Vert f\Vert _{L^p}. \end{aligned}$$

Moreover, the constant $C_{p, r}\le C_p\frac{r}{r-2}$ for some $C_p>0$ which is independent of the coefficients of the polynomial mapping $\mathcal {P}$.

Suppose that $K\in {\mathcal {C}}^1\left( \mathbb {R}^k{\setminus }\{0\}\right)$ is a Calderón–Zygmund kernel satisfying the differential inequality

$$\begin{aligned} {\left|y \right|}^k {|{K(y)} |} + {\left|y \right|}^{k+1} {\left|\nabla K(y) \right|} \le 1 \end{aligned}$$

(9.1)

for all $y \in \mathbb {R}^k{\setminus }\{0\}$ and the cancellation condition

$$\begin{aligned} \int _{G_t {\setminus } G_s}K(y){\, \mathrm{d}}y=0 \end{aligned}$$

for every $t> s > 0$. We consider a truncated singular Radon transform defined by

$$\begin{aligned} {\mathcal {T}}_t^{\mathcal {P}}f(x)=\int _{G_t^c}f\left( x-{\mathcal {P}}(y)\right) K(y){\, \mathrm{d}}y \end{aligned}$$

for $x\in \mathbb {R}^{d_0}$ and $t>0$. The second main result is the following theorem.

Theorem 9.2

For every $p \in (1, \infty )$ and $r\in (2, \infty )$ there is $C_{p, r} > 0$ such that for all $f \in L^p\left( \mathbb {R}^{d_0}\right)$

$$\begin{aligned} \left||V_r\left( {\mathcal {T}}_t^\mathcal {P}f: t>0\right) \right||_{L^p}\le C_{p, r}\Vert f\Vert _{L^p}. \end{aligned}$$

(9.2)

Moreover, the constant $C_{p, r}\le C_p\frac{r}{r-2}$ for some $C_p>0$ which is independent of the coefficients of the polynomial mapping $\mathcal {P}$.

We immediately see that (9.2) remains true for the operator

$$\begin{aligned} \tilde{{\mathcal {T}}}_t^{\mathcal {P}}f(x)={\mathrm{p.v.}}\int _{G_t}f\left( x-{\mathcal {P}}(y)\right) K(y) {\, \mathrm{d}}y. \end{aligned}$$

Let

$$\begin{aligned} \deg \mathcal {P}= \max \left\{ \deg \mathcal {P}_j : 1 \le j \le d_0\right\} . \end{aligned}$$

It is convenient to work with the set

$$\begin{aligned} \Gamma = \left\{ \gamma \in \mathbb {Z}^k {\setminus }\{0\} : 0 < |\gamma | \le \deg \mathcal {P}\right\} \end{aligned}$$

with the lexicographic order. Then each $\mathcal {P}_j$ can be expressed as

$$\begin{aligned} \mathcal {P}_j(x) = \sum _{\gamma \in \Gamma } c_j^\gamma x^\gamma \end{aligned}$$

for some $c_j^\gamma \in \mathbb {R}$. Let us denote by d the cardinality of the set $\Gamma$. We identify $\mathbb {R}^d$ with the space of all vectors whose coordinates are labeled by multi-indices $\gamma \in \Gamma$. Let A be the diagonal $d \times d$ matrix such that

$$\begin{aligned} (A v)_\gamma = {|{\gamma } |} v_\gamma . \end{aligned}$$

For $t > 0$ we set

$$\begin{aligned} t^{A}=\exp (A\log t) \end{aligned}$$

i.e. $t^A x=(t^{|\gamma |}x_{\gamma }: \gamma \in \Gamma )$ for any $x\in \mathbb {R}^d$. Next, we introduce the canonical polynomial mapping

$$\begin{aligned} \mathcal {Q}= \left( {\mathcal {Q}_\gamma }: {\gamma \in \Gamma }\right) : \mathbb {R}^k \rightarrow \mathbb {R}^d \end{aligned}$$

where $\mathcal {Q}_\gamma (x) = x^\gamma$ and $x^\gamma =x_1^{\gamma _1}\cdot \ldots \cdot x_k^{\gamma _k}$. The coefficients $\left( {c_j^\gamma }: {\gamma \in \Gamma , j \in \{1, \ldots , d_0\}}\right)$ define a linear transformation $L: \mathbb {R}^d \rightarrow \mathbb {R}^{d_0}$ such that $L\mathcal {Q}= \mathcal {P}$. Indeed, it is enough to set

$$\begin{aligned} (L v)_j = \sum _{\gamma \in \Gamma } c_j^\gamma v_\gamma \end{aligned}$$

for each $j \in \{1, \ldots , d_0\}$ and $v \in \mathbb {R}^d$. Now, proceeding as in Lemma 2.3 we can reduce the matters (see also [5] or [19, p. 515]) to the canonical polynomial mapping. To simplify the notation we will write ${\mathcal {M}}_t={\mathcal {M}}_t^{\mathcal {Q}}$ and ${\mathcal {T}}_t={\mathcal {T}}_t^{\mathcal {Q}}$.

1.1 Long variations

In this subsection we give a new proof of Lépingle’s inequality. Since we will appeal to the results from [10] we are going to follow their notation and therefore we will work with a more general setup than it is necessary for our further purposes.

We consider a slightly more general dilation structure

$$\begin{aligned} t^A=\exp (A\log t) \end{aligned}$$

for any $t>0$, where A is a $d\times d$ matrix whose eigenvalues have positive real parts. We say that any regular quasi-norm $\rho :\mathbb {R}^d\rightarrow [0, \infty )$ is homogeneous with respect to the dilations $(t^A: t>0)$ if $\rho (t^Ax)=t\rho (x)$ for any $x\in \mathbb {R}^d$ and $t>0$. Here $\mathbb {R}^d$ endowed with a quasi-norm $\rho$ and the Lebesgue measure will be considered as a space of homogeneous type with the quasi-metric induced by $\rho$.

In this setting let us recall Christ’s construction of dyadic cubes [3].

Lemma 9.1

([3]) There exists a collection of open sets $\{Q_{\alpha }^k: k\in \mathbb {Z}\text { and } \alpha \in I_k\}$ and constants $D>1$, $\delta , \eta >0$ and $0<C_1, C_2$ such that

• (i) $\left| \mathbb {R}^d{\setminus }\bigcup _{\alpha \in I_k}Q_{\alpha }^k\right| =0$ for all $k\in \mathbb {Z}$;

• (ii) if $l\le k$ then either $Q_{\beta }^l\subseteq Q_{\alpha }^k$ or $Q_{\beta }^l\cap Q_{\alpha }^k = \emptyset$;

• (iii) for each $(l, \beta )$ and $l\le k$, there exists a unique $\alpha$ such that $Q_{\beta }^l\subseteq Q_{\alpha }^k$;

• (iv) each $Q_{\alpha }^k$ contains some ball $B(z_{\alpha }^k, \delta D^k)$ and ${\text {diam}}(Q_{\alpha }^k)\le C_1 D^k$;

• (v) for each $(\alpha , k)$ and $t>0$ we have $|\{x\in Q_{\alpha }^k: {\mathrm{dist}}(x, \mathbb {R}^d{\setminus } Q_{\alpha }^k)\le tD^k\}|\le C_2 t^{\eta }|Q_{\alpha }^k|$.

Now two comments are in order. Firstly, each cube $Q_{\alpha }^k$ contains a ball and is contained in some ball, each with radius $\simeq D^k$. Secondly, the quasi-metric is translation invariant thus we see that for each $(\alpha , k)$ the measure of $Q_{\alpha }^k$ is $\simeq D^{{\mathrm{tr}}(A)k}$. In particular, there is $R > 0$ such that for all $k \in \mathbb {Z}$, $\alpha \in I_k$ and $\beta \in I_{k+1}$

$$\begin{aligned} |Q_\alpha ^{k+1}| \le R |Q_\beta ^k|. \end{aligned}$$

(9.3)

The collection $\{Q_{\alpha }^k: k\in \mathbb {Z}\text { and } \alpha \in I_k\}$ will be called the collection of dyadic cubes in $\mathbb {R}^d$ adapted to the dilation group $(t^A: t>0)$. In view of Lemma 9.1, it gives rise to an atomic filtration. Namely, for each $k\in \mathbb {Z}$ let $\mathcal {F}_k=\sigma (\{Q_{\alpha }^l: \alpha \in I_l \text { and } l \ge -k\})$ be the $\sigma$-algebra generated by the cubes at level at least $-k$. Then

$$\begin{aligned} \mathcal {F}_k \subset \mathcal {F}_{k+1}. \end{aligned}$$

For a localy integrable function f we set

$$\begin{aligned} \mathbb {E}_k f(x) = \mathbb {E}[f | \mathcal {F}_k] (x) = \frac{1}{|Q_{\alpha }^k|} \int _{Q_{\alpha }^k}f(y){\, \mathrm{d}}y \end{aligned}$$

provided $Q_{\alpha }^k$ is the unique dyadic cube containing $x\in \mathbb {R}^d$. Thanks to Lemma 9.1 (i), it is true for almost all x. We define a martingale difference by

$$\begin{aligned} \mathbb {D}_k f = \mathbb {E}_k f - \mathbb {E}_{k-1} f \end{aligned}$$

Finally, the maximal function and the square function are given by

$$\begin{aligned} Mf=\sup _{k\in \mathbb {Z}}|{\mathbb {E}}_kf| \quad \text { and } \quad Sf=\left( \sum _{k\in \mathbb {Z}}|{\mathbb {D}}_k f|^2\right) ^{1/2}, \end{aligned}$$

respectively.

The variational estimates for $\left( \mathbb {E}_k f : k \in \mathbb {Z}\right)$ follow from estimates on $\lambda$-jump function $J_{\lambda }$, see [1, 10, 18]. Recall, that for a sequence of complex numbers $(a_j: j\in \mathbb {Z})$ the function $J_{\lambda }(a_j: j\in \mathbb {Z})$ is equal to the supremum over all $J\in \mathbb {N}$ for which there exists a sequence of integers $t_1<t_2<\ldots <t_J$ so that

$$\begin{aligned} |a_{t_{j+1}}-a_{t_j}|>\lambda \end{aligned}$$

for every $j=1, 2,\ldots , N-1$. We immediately see that $J_{\lambda }(a_j: j\in \mathbb {Z})\le \lambda ^{-r}V_r(a_j: j\in \mathbb {Z})^r$.

Theorem 9.3

([1, 18]) For each $p \ge 1$ there exists $B_p > 0$ such that for all $f \in L^p\left( \mathbb {R}^d\right)$ and $\lambda > 0$, if $p\in (1, \infty )$

$$\begin{aligned} \left\| \lambda \left( J_\lambda ({\mathbb {E}}_kf: k\in \mathbb {Z})\right) ^{1/2} \right\| _{L^p} \le B_p \Vert f\Vert _{L^p}, \end{aligned}$$

and if $p=1$ then for any $t > 0$ we have

$$\begin{aligned} \left| \left\{ x\in \mathbb {R}^d: \lambda \left( J_\lambda ({\mathbb {E}}_kf(x): k\in \mathbb {Z})\right) ^{1/2} > t \right\} \right| \le B_1 t^{-1} \Vert f\Vert _{L^1}. \end{aligned}$$

The next theorem is a new ingredient in proving Lépingle’s inequality and is inspired by [7].

Theorem 9.4

For each $q \ge 2$ there is $C_q > 0$ such that for all $r > 2$ and $\lambda > 0$

$$\begin{aligned}&C_q \cdot \left| \left\{ x\in \mathbb {R}^d: V_r({\mathbb {E}}_kf(x): k\in \mathbb {Z})> \lambda \text { and } Mf(x) < \lambda /2\right\} \right| \nonumber \\&\quad \le \left| \left\{ x\in \mathbb {R}^d: Sf(x) > \lambda \right\} \right| + \lambda ^{-q} (r-2)^{-q/2} \int _{\{S(f) \le \lambda \}} Sf(x)^q {\, {\mathrm{d}}}x. \end{aligned}$$

(9.4)

Proof

By homogeneity, it suffices to prove the result with $\lambda = 1$. Let $B = \{x\in \mathbb {R}^d: Sf(x) > 1\}$, $B^* = \{x\in \mathbb {R}^d: M {\mathbbm {1}_{{B}}}(x) > 1/(2R) \}$ and $G = (B^*)^c$. By the maximal inequality, we have

$$\begin{aligned}&|B^*|=|\{x\in \mathbb {R}^d: M {\mathbbm {1}_{{B}}}(x)> 1/(2R) \}|\\&\quad \lesssim \int _{\mathbb {R}^d} {\mathbbm {1}_{{B}}}(x)^2 {\, \mathrm{d}}x = |\{x\in \mathbb {R}^d: Sf(x) > 1\}|. \end{aligned}$$

Therefore, it is enough to show that

$$\begin{aligned}&\left| \left\{ x\in G: V_r({\mathbb {E}}_kf(x): k\in \mathbb {Z}) > 1 \text { and } Mf(x) < 1/2\right\} \right| \\&\quad \lesssim (r-2)^{-q/2} \int _{B^c} Sf(x)^q {\, \mathrm{d}}x. \end{aligned}$$

We can pointwise dominate the variation (see [1])

$$\begin{aligned} V_r({\mathbb {E}}_kf: k\in \mathbb {Z})^r \le \sum _{l \in \mathbb {Z}} 2^{rl} J_{2^l}({\mathbb {E}}_kf: k\in \mathbb {Z}). \end{aligned}$$

Since $Mf < 1/2$, the above sum runs over $l \le 0$, which leads to the containment

$$\begin{aligned}&\left\{ x\in G: V_r({\mathbb {E}}_kf(x): k\in \mathbb {Z})> 1 \text { and } Mf(x) < 1/2 \right\} \nonumber \\&\quad \subseteq \left\{ x\in G: \sum _{l \le 0} 2^{rl} J_{2^l}({\mathbb {E}}_kf(x): k\in \mathbb {Z}) > 1 \right\} . \end{aligned}$$

(9.5)

For each $n \in \mathbb {Z}$ we define $U_n =\left\{ x\in \mathbb {R}^d: {\mathbb {E}}_n{\mathbbm {1}_{{B}}}(x) \le 1/2\right\}$. Notice that, if $x \in G$ then $x \in U_n$ for all $n \in \mathbb {Z}$. Let

$$\begin{aligned} g(x) = \sum _{n \in \mathbb {Z}} {\mathbb {D}}_nf(x) \cdot {\mathbbm {1}_{{U_{n-1}}}}(x). \end{aligned}$$

We observe that ${\mathbb {E}}_ng(x) = {\mathbb {E}}_nf(x)$ for all $x \in G$ and $n \in \mathbb {Z}$. Indeed, ${\mathbb {D}}_nf \cdot {\mathbbm {1}_{{U_{n-1}}}}$ is $\mathcal {F}_n$-measurable and

$$\begin{aligned} {\mathbb {E}}_m\left( {\mathbb {D}}_nf\cdot {\mathbbm {1}_{{U_{n-1}}}}\right) = 0 \end{aligned}$$

for every $m \le n-1$. Thus, for $x \in G$ we have

$$\begin{aligned} {\mathbb {E}}_mg(x) = \sum _{n \le m} {\mathbb {D}}_nf(x) \cdot {\mathbbm {1}_{{U_{n-1}}}}(x) ={\mathbb {E}}_m f(x). \end{aligned}$$

Hence, by (9.5) and Hölder’s inequality with $a=\frac{q}{2}$ and $a'=\frac{q}{q-2}$, we obtain

$$\begin{aligned}&\left\{ x\in G: V_r({\mathbb {E}}_kf(x): k\in \mathbb {Z})> 1 \text { and } Mf(x) < 1/2 \right\} \\&\qquad \subseteq \left\{ x\in \mathbb {R}^d: \sum _{l \le 0}2^{rl} J_{2^l}({\mathbb {E}}_kg(x): k\in \mathbb {Z})> 1 \right\} \\&\qquad \qquad \subseteq \left\{ x\in G: \left( \sum _{l\le 0}2^{\frac{1}{2}l(r-2)\frac{q}{q-2}}\right) ^{\frac{q-2}{q}}\right. \\&\qquad \qquad \quad \left. \left( \sum _{l\le 0}2^{\frac{1}{2}l(r-2)\frac{q}{2}} \left( 2^{l}J_{2^l}({\mathbb {E}}_kg(x): k\in \mathbb {Z})^{1/2}\right) ^q\right) ^{\frac{2}{q}}>1\right\} . \end{aligned}$$

Define

$$\begin{aligned} A_{q, r}=\left( \sum _{l\le 0}2^{\frac{1}{2}l(r-2)\frac{q}{q-2}}\right) ^{\frac{q-2}{q}} ={\mathcal {O}}\left( (r-2)^{-\frac{q-2}{q}}\right) . \end{aligned}$$

Now Theorem 9.3 immediately leads to the majorization

$$\begin{aligned}&\left| \left\{ x\in G: V_r({\mathbb {E}}_kf(x): k\in \mathbb {Z})> 1 \text { and } Mf(x) < 1/2 \right\} \right| \\&\quad \le \left| \left\{ x\in G: \sum _{l\le 0}2^{\frac{1}{2}l(r-2)\frac{q}{2}}\left( 2^{l}J_{2^l}({\mathbb {E}}_kg(x): k\in \mathbb {Z})^{1/2}\right) ^q>A_{q, r}^{-\frac{q}{2}}\right\} \right| \\&\quad \lesssim A_{q, r}^{\frac{q}{2}} \sum _{l\le 0}2^{\frac{1}{2}l(r-2)\frac{q}{2}} \int _{\mathbb {R}^d} |g(x)|^q {\, \mathrm{d}}x\\&\quad \lesssim (r-2)^{1-q/2-1} \int _{\mathbb {R}^d} {|{g(x)} |}^q {\, \mathrm{d}}x\lesssim (r-2)^{-q/2} \int _{\mathbb {R}^d} {|{g(x)} |}^q {\, \mathrm{d}}x. \end{aligned}$$

Next, the square function S is bounded from below on $L^q\left( \mathbb {R}^d\right)$, therefore

$$\begin{aligned} \int _{\mathbb {R}^d} {|{g(x)} |}^q {\, {\mathrm{d}}}x \lesssim \int _{\mathbb {R}^d} Sg(x)^q {\, {\mathrm{d}}}x = \int _{\mathbb {R}^d} \left( \sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_k f(x)} |}^2 \cdot {\mathbbm {1}_{{U_{k-1}}}}(x) \right) ^{q/2} {\, {\mathrm{d}}}x. \end{aligned}$$

Since for $x \in U_{k-1}$, we have ${\mathbb {E}}_{k-1}{\mathbbm {1}_{{B}}}(x) \le 1/(2R)$, by the doubling property (9.3) we get $\mathbb {E}_k{\mathbbm {1}_{{B}}}(x) \le 1/2$. Hence,

$$\begin{aligned} {\mathbbm {1}_{{U_{k-1}}}}(x) \le 2 \cdot \mathbb {E}_k{\mathbbm {1}_{{B^c}}}(x), \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}^d} {|{g(x)} |}^q {\, {\mathrm{d}}}x \lesssim \int _{\mathbb {R}^d} \left( \sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_kf(x)} |}^2 \cdot {\mathbb {E}}_k{\mathbbm {1}_{{B^c}}}(x) \right) ^{q/2} {\, {\mathrm{d}}}x. \end{aligned}$$

We observe that for $q = 2$ we have

$$\begin{aligned} \int _{\mathbb {R}^d}\sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_kf(x)} |}^2 \cdot {\mathbb {E}}_k{\mathbbm {1}_{{B^c}}}(x) {\, {\mathrm{d}}}x = \int _{B^c} Sf(x)^2 {\, {\mathrm{d}}}x. \end{aligned}$$

For $q > 2$ let $\tilde{q} = q/2 > 1$ and $\tilde{q}'$ be its dual exponent. Then for $h \in L^{\tilde{q}'}\left( \mathbb {R}^d\right)$

$$\begin{aligned}&\int _{\mathbb {R}^d}\left( \sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_k f(x)} |}^2 \cdot {\mathbb {E}}_k {\mathbbm {1}_{{B^c}}}(x)\right) h(x) {\, {\mathrm{d}}}x\\&\quad = \sum _{k \in \mathbb {Z}}\int _{\mathbb {R}^d} {|{{\mathbb {D}}_k f(x)} |}^2 \cdot {\mathbb {E}}_k {\mathbbm {1}_{{B^c}}}(x)h(x) {\, {\mathrm{d}}}x \\&\quad = \sum _{k \in \mathbb {Z}} \int _{B^c} {|{{\mathbb {D}}_k f(x)} |}^2 \cdot {\mathbb {E}}_k h(x) {\, {\mathrm{d}}}x. \end{aligned}$$

Therefore, by Hölder’s inequality, we get

$$\begin{aligned} \int _{\mathbb {R}^d}\left( \sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_kf(x)} |}^2 \cdot {\mathbb {E}}_k{\mathbbm {1}_{{B^c}}}(x)\right) h(x) {\, {\mathrm{d}}}x \le \left( \int _{B^c} Sf(x)^q {\, {\mathrm{d}}} x \right) ^{2/q} ||M h ||_{\tilde{q}'}. \end{aligned}$$

Taking the supremum over all $h \in L^{\tilde{q}'}\left( \mathbb {R}^d\right)$ we conclude

$$\begin{aligned} \int _{\mathbb {R}^d} \left( \sum _{k \in \mathbb {Z}} {|{{\mathbb {D}}_kf(x)} |}^2 \cdot {\mathbb {E}}_k{\mathbbm {1}_{{B^c}}}(x) \right) ^{q/2} {\, {\mathrm{d}}}x\lesssim \int _{B^c} Sf(x)^q {\, {\mathrm{d}}}x, \end{aligned}$$

which finishes the proof. $\square$

Now, using Theorem 9.1 we prove Lépingle’s inequality for the sequence $({\mathbb {E}}_kf: k\in \mathbb {Z})$.

Theorem 9.5

([13]) For each $p \in (1, \infty )$ there exists $C_p > 0$ such that for all $r\in (2, \infty )$ and $f \in L^p\left( \mathbb {R}^d\right)$ we have

$$\begin{aligned} \left\| V_r({\mathbb {E}}_kf: k\in \mathbb {Z}) \right\| _{L^p} \le C_p\frac{r}{r-2} \Vert f\Vert _{L^p}, \end{aligned}$$

Moreover, $V_r({\mathbb {E}}_kf: k\in \mathbb {Z})$ is also weak type (1, 1).

Proof

Given $p > 1$ we take $q=2p>2$. By (9.4), for $f\in L^p\left( \mathbb {R}^d\right) \cap L^q\left( \mathbb {R}^d\right)$ we have

$$\begin{aligned}&C_q \cdot \left| \left\{ x \in \mathbb {R}^d : V_r(\mathbb {E}_k f : k \in \mathbb {Z})> \lambda \right\} \right| \le \left| \left\{ x \in \mathbb {R}^d : S f(x)> \lambda \right\} \right| \\&\quad + \left| \left\{ x \in \mathbb {R}^d : M f(x) > \lambda /2 \right\} \right| \\&\quad + \lambda ^{-q} (r-2)^{-q/2} \int _{\{Sf \le \lambda \}} S f(x)^q {\, {\mathrm{d}}} x. \end{aligned}$$

Therefore, we get

$$\begin{aligned} \left\| V_r(\mathbb {E}_kf: k\in \mathbb {Z})\right\| _{L^p}^p= & {} p\int _{0}^{\infty }\lambda ^{p-1}\left| \left\{ x\in \mathbb {R}^d: V_r(\mathbb {E}_kf(x): k\in \mathbb {Z})>\lambda \right\} \right| {\, {\mathrm{d}}}\lambda \\\lesssim & {} {\left||{Sf} \right||}_{L^p}^p + {\left||{Mf} \right||}_{L^p}^p\\&+ (r-2)^{-p}\int _{0}^{\infty }\lambda ^{p-q-1}\int _{\{S(f) \le \lambda \}} Sf(x)^q {\mathrm{d}}x{\, \mathrm{d}}\lambda \\\lesssim & {} \Vert f\Vert _{L^p}^p+(r-2)^{-p} \int Sf(x)^q\int _{Sf(x)}^{\infty }\lambda ^{p-q-1} {\mathrm{d}}\lambda {\, \mathrm{d}}x\\\lesssim & {} \Vert f\Vert _{L^p}^p+(r-2)^{-p}\Vert Sf\Vert _{L^p}^p\lesssim \left( 1+(r-2)^{-p}\right) \Vert f\Vert _{L^p}^p. \end{aligned}$$

For $p=1$ it suffices to apply the Calderón–Zygmund decomposition and the desired claim follows. $\square$

Long variational bounds for $\mathcal {M}_t$ follows the same line as in [10]. For every $f\in L^p\left( \mathbb {R}^d\right)$ with $p\in (1, \infty )$ we obtain

$$\begin{aligned}&\left||V_r\left( {\mathcal {M}}_{2^n} f: n\in \mathbb {Z}\right) \right||_{L^p}\le \left||V_r\left( {\mathbb {E}}_nf: n\in \mathbb {Z}\right) \right||_{L^p}\nonumber \\&\quad +\left\| \left( \sum _{n\in \mathbb {N}} \left| {\mathcal {M}}_{2^n} f- {\mathbb {E}}_nf\right| ^2\right) ^{1/2}\right\| _{L^p}. \end{aligned}$$

(9.6)

The first term in (9.6) is bounded by Theorem 9.5, whereas the square function can be estimated as in [10, Proof of Theorem 1.1]. In the next theorem we consider long variational estimates for ${\mathcal {T}}_{t}$.

Theorem 9.6

For every $p \in (1, \infty )$ there is $C_p > 0$ such that for all $r\in (2, \infty )$ and $f \in L^p\left( \mathbb {R}^{d}\right)$

$$\begin{aligned} \left||V_r\left( {\mathcal {T}}_{2^n} f: n\in \mathbb {Z}\right) \right||_{L^p}\le C_p \frac{r}{r-2} {\left||{f} \right||}_{L^p}. \end{aligned}$$

Proof

Let $\Phi \in {\mathcal {C}}^{\infty }\left( \mathbb {R}^d\right)$ with compact support and integral one. Then we have the following decomposition (see [6])

$$\begin{aligned} \sum _{j\ge n}{\mathcal {T}}_{2^j}f&=\Phi _{2^n}*\left( {\mathcal {T}}f-\sum _{j<n}\mu _{2^j}*f\right) +(\delta _0-\Phi _{2^n})*\sum _{j\ge n}\mu _{2^j}*f\nonumber \\&=\Phi _{2^n}*{\mathcal {T}}f-\left( \Phi _{2^n}*\sum _{j< n}\mu _{2^j}\right) *f+\sum _{j\ge 0}(\delta _0-\Phi _{2^n})*\mu _{2^{j+n}}*f, \end{aligned}$$

(9.7)

where

$$\begin{aligned} \mu _{2^j}*f(x)=\int _{G_{2^{j+1}}{\setminus } G_{2^j}}f\left( x-\mathcal {Q}(y)\right) K(y){\, \mathrm{d}}y. \end{aligned}$$

The variational estimates for the first term in (9.7) follows by [10, Theorem 1.1, Lemma 2.1] and the boundedness for the operator ${\mathcal {T}}$, see [6, 19]. For the second term we apply the Littlewood–Paley theory since for each $n \in \mathbb {N}$

$$\begin{aligned} \Phi _{2^n} * \sum _{j < n} \mu _{2^j} \end{aligned}$$

is a Schwartz function with integral zero. Thus, by (2.3),

$$\begin{aligned} V_r\left( \Phi _{2^n} * \sum _{j< n} \mu _{2^j}*f : n \in \mathbb {Z}\right) \lesssim \left( \sum _{n \in \mathbb {Z}} \left|\Phi _{2^n} * \sum _{j < n} \mu _{2^j} * f \right|^2\right) ^{1/2}. \end{aligned}$$

For the last term we have

$$\begin{aligned}&\left\| V_r\left( \sum _{j\ge 0}(\delta _0-\Phi _{2^n})*\mu _{2^{j+n}}*f: n\in \mathbb {Z}\right) \right\| _{L^p}\\&\quad \le \sum _{j\ge 0} \left\| \left( \sum _{n\in \mathbb {Z}} \left| (\delta _0-\Phi _{2^n})*\mu _{2^{j+n}}*f\right| ^2\right) ^{1/2}\right\| _{L^p} \end{aligned}$$

and due to the Littlewood–Paley theory one can show that there is $C_p>0$ and $\delta _p>0$ such that

$$\begin{aligned} \left\| \left( \sum _{n\in \mathbb {Z}}\left| (\delta _0-\Phi _{2^n})*\mu _{2^{j+n}} *f\right| ^2\right) ^{1/2}\right\| _{L^p} \le C_p2^{-\delta _p j}\Vert f\Vert _{L^p}. \end{aligned}$$

This completes the proof of Theorem 9.6. $\square$

1.2 Short variations for averaging operators

To deal with short variations we need a counterpart of Lemma 2.2.

Lemma 9.2

Let $u < v$ be real numbers and $a:[u, v]\rightarrow \mathbb {C}$ be a differentiable function. For any $h \in \mathbb {N}$ and the sequence $\left( {s_j}: {0 \le j \le h}\right)$ with $s_j=u+h^{-1}(v-u)j$ we have for every $r\in [1, \infty )$

$$\begin{aligned} V_r\left( a(t): t\in [u, v)\right) \lesssim \left( \sum _{j=0}^h|a(s_j)|^r\right) ^{1/r} +\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^r\right) ^{1/r}. \end{aligned}$$

(9.8)

Moreover, if $p\ge r$ then

$$\begin{aligned}&\left( \sum _{j=0}^h|a(s_j)|^r\right) ^{1/r} +\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^r\right) ^{1/r}\\&\quad \lesssim h^{1/r-1/p}\left( \sum _{j=0}^h|a(s_j)|^p\right) ^{1/p} +h^{1/r-1}(v-u)^{1-1/p}\left( \int _{u}^{v}|a'(t)|^p{\, \mathrm{d}}t\right) ^{1/p}. \end{aligned}$$

Proof

Fix $h \in \mathbb {N}$ and consider the sequence $\left( {s_j}: {0 \le j \le h}\right)$ such that $s_j=u+h^{-1}(v-u)j$. Then

$$\begin{aligned} V_r\left( a(t): t\in [u, v)\right)&\lesssim \left( \sum _{j=0}^h|a(s_j)|^r\right) ^{1/r} +\left( \sum _{j=0}^{h-1}V_1\left( a(t): t\in [s_j, s_{j+1})\right) ^r\right) ^{1/r}\\&\lesssim \left( \sum _{j=0}^h|a(s_j)|^r\right) ^{1/r} +\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^r\right) ^{1/r}. \end{aligned}$$

If $p \ge r$, by Hölder’s inequality, we get

$$\begin{aligned}&\left( \sum _{j=0}^h|a(s_j)|^r\right) ^{1/r} +\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^r\right) ^{1/r}\\&\quad \le h^{1/r - 1/p} \left( \sum _{j=0}^h|a(s_j)|^p\right) ^{1/p} \\&\quad +h^{1/r-1/p}\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^p\right) ^{1/p}. \end{aligned}$$

For the second term we again use Hölder’s inequality to obtain

$$\begin{aligned}&h^{1/r-1/p}\left( \sum _{j=0}^{h-1}\left( \int _{s_j}^{s_{j+1}}|a'(t)|{\, \mathrm{d}}t\right) ^p\right) ^{1/p} \lesssim h^{1/r-1/p}\left( \sum _{j=0}^{h-1}(s_{j+1}\right. \\&\quad \left. -s_j)^{p(1-1/p)}\left( \int _{s_j}^{s_{j+1}}|a'(t)|^p{\, \mathrm{d}}t\right) \right) ^{1/p}\\&\quad \lesssim h^{1/r-1}(v-u)^{1-1/p}\left( \int _{u}^{v}|a'(t)|^p{\, \mathrm{d}}t\right) ^{1/p} \end{aligned}$$

where in the last estimate we have used $s_{j+1}-s_j=(v-u)/h$. This completes the proof of the lemma. $\square$

Now the task is to prove that for every $p\in (1, \infty )$ there are $C_p>0$ such that for every $f\in L^p\left( \mathbb {R}^d\right)$ we have

$$\begin{aligned} \left||\left( \sum _{n\in \mathbb {Z}} V_2\left( \left( {\mathcal {M}}_t-{\mathcal {M}}_{2^n}\right) f: t\in \left[ 2^n, 2^{n+1}\right) \right) ^2\right) ^{1/2} \right||_{L^p}\le C_p\Vert f\Vert _{L^p}. \end{aligned}$$

(9.9)

We may assume that f is a Schwartz function. Let $S_j$ be a Littlewood–Paley projection $\mathcal {F}(S_j g)(\xi )=\phi _j(\xi )\mathcal {F}g (\xi )$ associated with $\left( \phi _j: j\in \mathbb {Z}\right)$ a smooth partition of unity of $\mathbb {R}^d{\setminus }\{0\}$ such that for each $j \in \mathbb {Z}$ we have $0\le \phi _j\le 1$ and

$$\begin{aligned} {\text {supp}}\, \phi _j \subseteq \left\{ \xi \in \mathbb {R}^d: 2^{-1}< {|{2^{jA}\xi } |}_{\infty } < 2 \right\} \end{aligned}$$

and for $\xi \in \mathbb {R}^d{\setminus }\{0\}$

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\phi _j(\xi )=1. \end{aligned}$$

We are going to prove that for every $p\in (1, \infty )$ there are $C_p>0$ and $\delta _p>0$ such that for every $j\in \mathbb {Z}$ we have

$$\begin{aligned} \left||\left( \sum _{n\in \mathbb {Z}} V_2\left( \left( {\mathcal {M}}_t-{\mathcal {M}}_{2^n}\right) S_{j+n}f: t\in \left[ 2^n, 2^{n+1}\right) \right) ^2\right) ^{1/2} \right||_{L^p}\le C_p2^{-\delta _p|j|}\Vert f\Vert _{L^p}. \end{aligned}$$

(9.10)

Applying (9.8) with $h = 2^{\varepsilon {|{j} |}}$, $u=2^n$ and $v=2^{n+1}$ we obtain that

$$\begin{aligned}&\left||\left( \sum _{n\in \mathbb {Z}} V_2\left( \left( {\mathcal {M}}_t-{\mathcal {M}}_{2^n}\right) S_{j+n}f: t\in \left[ 2^n, 2^{n+1}\right) \right) ^2\right) ^{1/2} \right||_{L^p} \\&\quad \lesssim \left||\left( \sum _{n\in \mathbb {Z}}\sum _{l=0}^h \left| \left( {\mathcal {M}}_{s_l}-{\mathcal {M}}_{2^n}\right) S_{j+n}f\right| ^2\right) ^{1/2} \right||_{L^p} \\&\quad + \left||\left( \sum _{n\in \mathbb {Z}}\sum _{l=0}^{h-1}\left( \int _{s_l}^{s_{l+1}} \left| \frac{{\mathrm{d}}}{{\, \mathrm{d}}t}{\mathcal {M}}_{t} S_{j+n}f\right| {\, \mathrm{d}}t\right) ^2\right) ^{1/2} \right||_{L^p}=I_p^1+I_p^2. \end{aligned}$$

1.2.1 The estimates for $I_p^1$

First, using vector-valued estimates from [15, Theorem A.1] together with the Littlewood–Paley theory we get

$$\begin{aligned} \begin{aligned} I_p^1&\lesssim 2^{\varepsilon |j|/2} \left||\left( \sum _{n\in \mathbb {Z}} \sup _{t>0}\left| {\mathcal {M}}_{t}S_{j+n}f\right| ^2\right) ^{1/2} \right||_{L^p} \\&\lesssim 2^{\varepsilon |j|/2} \left\| \left( \sum _{n\in \mathbb {N}}|S_{j+n}f|^2\right) ^{1/2}\right\| _{L^p} \lesssim 2^{\varepsilon |j|/2}\Vert f\Vert _{L^p}. \end{aligned} \end{aligned}$$

(9.11)

Next, we are going to refine the estimate (9.11) for $p=2$. Let $m_t$ be the multiplier associated with the operator ${\mathcal {M}}_t$. By van der Corput’s lemma [21], for each $t\in [2^n, 2^{n+1})$ we have

$$\begin{aligned} |m_t(\xi )-m_{2^n}(\xi )|\lesssim \min \left\{ 1, |2^{nA}\xi |_\infty , |2^{nA}\xi |_\infty ^{-1/d}\right\} . \end{aligned}$$

Therefore, by Plancherel’s theorem

$$\begin{aligned} I_2^1= & {} \left( \sum _{l=0}^h\int _{\mathbb {R}^d}\sum _{n\in \mathbb {Z}}\left| (m_{s_l}(\xi )-m_{2^n} (\xi ))\phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, {\mathrm{d}}}\xi \right) ^{1/2}\nonumber \\\lesssim & {} 2^{-|j|/d+\varepsilon |j|/2}\left( \int _{\mathbb {R}^d}\sum _{n\in \mathbb {Z}}\left| \phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\mathrm{d}}\xi \right) ^{1/2} \lesssim 2^{-{|{j} |}/d + \varepsilon {|{j} |} /2}\Vert f\Vert _{L^2}.\nonumber \\ \end{aligned}$$

(9.12)

Interpolating (9.11) with (9.12) and choosing appropriate $\varepsilon < 2/d$ we get

$$\begin{aligned} I_p^1\lesssim 2^{-\delta _p|j|}\Vert f\Vert _{L^p}. \end{aligned}$$

1.2.2 The estimates for $I_p^2$

Since G is an open bounded convex set containing the origin, with the aid of the spherical coordinates we may write

$$\begin{aligned} \mathcal {M}_t g(x) = \frac{1}{t^k {|{G} |}} \int _{S^{k-1}} \int _0^{r(\omega ) t} g\left( x - \mathcal {Q}(r \omega ) \right) r^{k-1} {\, {\mathrm{d}}} r {\, {\mathrm{d}}}\sigma (\omega ) \end{aligned}$$

where $S^{k-1}$ is a unit sphere in $\mathbb {R}^k$ and $\sigma$ is the surface measure on $S^{k-1}$. We observe that if g is a Schwartz function

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}} t} \mathcal {M}_t g(x)&= -k \frac{1}{t^{k+1} {|{G} |}} \int _{S^{k-1}} \int _0^{r(\omega ) t} g\left( x - \mathcal {Q}(r \omega )\right) r^{k-1} {\, \mathrm{d}}r {\, \mathrm{d}}\sigma (\omega ) \nonumber \\&\quad + \frac{1}{t^k {|{G} |}} \int _{S^{k-1}} g\left( x - \mathcal {Q}(r(\omega ) t \omega )\right) r(\omega )^k t^{k-1} {\, \mathrm{d}}\sigma (\omega ). \end{aligned}$$

(9.13)

Change of the order of integration and differentiation is permited since g is bounded. Hence, if $t \in [s_l, s_{l+1})$ and $s_l, s_{l+1} \simeq 2^n$, by Tonnelli’s theorem, we get

$$\begin{aligned} \sum _{l=0}^{h-1}\int _{s_l}^{s_{l+1}} \left| \frac{{\mathrm{d}}}{{\mathrm{d}} t} \mathcal {M}_t g(x) \right| {\, \mathrm{d}} t&\lesssim \mathcal {M}_{2^{n+1}} {|{g} |}(x) + \frac{1}{2^{nk} |G|} \int _{2^n}^{2^{n+1}} \int _{S^{k-1}} \left| g\left( x \right. \right. \\&\quad \left. \left. - \mathcal {Q}(r(\omega ) t \omega )\right) \right| r(\omega )^k t^{k-1} {\, \mathrm{d}}\sigma (\omega ) {\, \mathrm{d}}t \\&\lesssim \mathcal {M}_{2^{n+1}} {|{g} |} (x). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} I_p^2\lesssim & {} \left||\left( \sum _{n\in \mathbb {Z}}\left( \mathcal {M}_{2^{n+1}} {|{S_{j+n}f} |} \right) ^2\right) ^{1/2} \right||_{L^p}\nonumber \\\lesssim & {} \left||\left( \sum _{n \in \mathbb {Z}}\sup _{t > 0} \left( \mathcal {M}_t {|{S_{j + n} f} |} \right) ^2 \right) ^{1/2} \right||_{L^p} \lesssim {\left||{f} \right||}_{L^p}, \end{aligned}$$

(9.14)

where the last inequality follows by the same line of reasoning as (9.11).

Next, we refine the estimates of $I_p^2$ for $p = 2$. Let $\tilde{m}_{t}$ be the multiplier associated with the operator $\frac{{\mathrm{d}}}{{\mathrm{d}}t}{\mathcal {M}}_{t}$. We have

$$\begin{aligned} \tilde{m}_t(\xi ) = - \frac{k}{t^{k+1} {|{G} |}} \int _{G_t} e^{2\pi i {\xi \cdot \mathcal {Q}(x)}} {\, \mathrm{d}}x + \frac{1}{t^k {|{G} |}} \int _{S^{k-1}} e^{2\pi i {\xi \cdot \mathcal {Q}(r(\omega ) t \omega )}} r(\omega )^k t^{k-1} {\, \mathrm{d}}\sigma (\omega ). \end{aligned}$$

(9.15)

Indeed, by (9.13), we have

$$\begin{aligned} \mathcal {F}\left( \frac{{\mathrm{d}}}{{\mathrm{d}} t} \mathcal {M}_t g\right) (\xi )&= -\frac{k}{t^{k+1} {|{G} |}} \int _{G_t} e^{2\pi i {\xi \cdot \mathcal {Q}(x)}} {\, \mathrm{d}} x \mathcal {F}g(\xi ) \\&\quad + \frac{1}{t^k {|{G} |}} \int _{\mathbb {R}^d} e^{2\pi i {\xi \cdot x}} \int _{S^{k-1}} g\left( x \right. \\&\quad \left. - \mathcal {Q}(r(\omega ) t \omega )\right) r(\omega )^k t^{k-1} {\, \mathrm{d}}\sigma (\omega ) {\, \mathrm{d}}x. \end{aligned}$$

Again, for the second term we need to justify the change of integrations. Let

$$\begin{aligned} R = \sup \left\{ {\left|\mathcal {Q}(y) \right|} : y \in G_t \right\} . \end{aligned}$$

Then for all ${\left|x \right|} \ge 2 R$ and $|y|\le R$ we have

$$\begin{aligned} {\left|x - y \right|} \ge \frac{{\left|x \right|}}{2}, \end{aligned}$$

thus

$$\begin{aligned} \left| g\left( x - \mathcal {Q}(y)\right) \right| \lesssim \left( 1 + {\left|x \right|}\right) ^{-2d}. \end{aligned}$$

(9.16)

By Fubini’s theorem we get the claim. Moreover, we see that for $t\simeq 2^n$ we obtain $|\tilde{m}_{t}(\xi )|\lesssim 2^{-n}$.

Now, using (9.15), by the Cauchy–Schwarz inequality and Plancherel’s theorem we get

$$\begin{aligned} I_2^2\le & {} \left||\left( \sum _{n\in \mathbb {Z}}\frac{2^n}{h}\int _{2^n}^{2^{n+1}} \left| \frac{{\mathrm{d}}}{{\mathrm{d}}t}{\mathcal {M}}_{t} S_{j+n}f\right| ^2{\, \mathrm{d}}t \right) ^{1/2} \right||_{L^2} \nonumber \\= & {} \left( \sum _{n\in \mathbb {Z}}\frac{2^n}{h}\int _{2^n}^{2^{n+1}}\int _{\mathbb {R}^d} \left| \tilde{m}_{t}(\xi ) \phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, \mathrm{d}}\xi {\, \mathrm{d}} t\right) ^{1/2}\nonumber \\\lesssim & {} 2^{-\varepsilon |j|/2} \left( \sum _{n\in \mathbb {Z}}\int _{\mathbb {R}^d} \left| \phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, \mathrm{d}}\xi \right) ^{1/2} \lesssim 2^{-\varepsilon |j|/2}\Vert f\Vert _{L^2}\qquad \qquad \end{aligned}$$

(9.17)

for $0<\varepsilon <1/d$. Thus interpolation of (9.14) with (9.17) gives

$$\begin{aligned} I_p^2\lesssim 2^{-\delta _p|j|}\Vert f\Vert _{L^p} \end{aligned}$$

and the proof of (9.10) is completed.

1.3 Short variations for truncated singular integral operators

We are going to show that for any $p\in (1, \infty )$ there are $C_p>0$ and $\delta _0 > 0$ such that for every $j \in \mathbb {Z}$ and $f\in L^p\left( \mathbb {R}^d\right)$ we have

$$\begin{aligned} \left||\left( \sum _{n\in \mathbb {Z}} V_2\left( \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right) S_{j +n} f: t\in \left[ 2^n, 2^{n+1}\right) \right) ^2 \right) ^{1/2} \right||_{L^p} \le C_p 2^{-\delta _p {|{j} |}} \Vert f\Vert _{L^p}. \end{aligned}$$

(9.18)

We may assume that f is a Schwartz function. The proof of (9.18) follows the same line as the one for the averaging operator. By Lemma 9.2, for $h = 2^{\varepsilon {|{j} |}}$, $u=2^n$ and $v=2^{n+1}$ we obtain

$$\begin{aligned}&\left||\left( \sum _{n\in \mathbb {Z}} V_2\left( \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right) S_{j+n}f: t\in \left[ 2^n, 2^{n+1}\right) \right) ^2\right) ^{1/2} \right||_{L^p}\\&\quad \lesssim \left||\left( \sum _{n\in \mathbb {Z}}\sum _{l=0}^h \left| \left( {\mathcal {T}}_{s_l}-{\mathcal {T}}_{2^n}\right) S_{j+n}f\right| ^2\right) ^{1/2} \right||_{L^p} \\&\quad + \left||\left( \sum _{n\in \mathbb {Z}}\sum _{l=0}^{h-1}\left( \int _{s_l}^{s_{l+1}} \left| \frac{{\mathrm{d}}}{{\, \mathrm{d}}t}\left( {\mathcal {T}}_{t}-{\mathcal {T}}_{2^n}\right) S_{j+n}f\right| {\, \mathrm{d}}t\right) ^2\right) ^{1/2} \right||_{L^p} =I_p^1+I_p^2. \end{aligned}$$

1.3.1 The estimates for $I_p^1$

By the vector-valued estimates from [15, Theorem A.1] and the Littlewood–Paley theory we get

$$\begin{aligned} \begin{aligned} I_p^1&\lesssim 2^{\varepsilon |j|/2} \left||\left( \sum _{n\in \mathbb {Z}} \sup _{t>0}\left( {\mathcal {M}}_{t}|S_{j+n}f|\right) ^2\right) ^{1/2} \right||_{L^p} \\&\lesssim 2^{\varepsilon |j|/2} \left\| \left( \sum _{n\in \mathbb {N}}|S_{j+n}f|^2\right) ^{1/2}\right\| _{L^p} \lesssim 2^{\varepsilon |j|/2}\Vert f\Vert _{L^p}. \end{aligned} \end{aligned}$$

(9.19)

For $p = 2$ we get better estimate. Let $m_{2^n, t}$ be the multiplier associated with the operator ${\mathcal {T}}_t - {\mathcal {T}}_{2^n}$ for $t \in [2^n, 2^{n+1})$. By van der Corput’s lemma [21] (or more precisely the method of proof of van der Corput lemma from [21]) we have

$$\begin{aligned} |m_{t, 2^n}(\xi )| \lesssim \min \left\{ 1, |2^{nA}\xi |_\infty , |2^{nA}\xi |_\infty ^{-1/d}\right\} . \end{aligned}$$

Therefore, by Plancherel’s theorem

$$\begin{aligned} I_2^1= & {} \left( \sum _{l=0}^h\int _{\mathbb {R}^d}\sum _{n\in \mathbb {Z}}\left| (m_{s_l, 2^n}(\xi )\phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, \mathrm{d}}\xi \right) ^{1/2}\nonumber \\\lesssim & {} 2^{-|j|/d+\varepsilon |j|/2} \left( \int _{\mathbb {R}^d}\sum _{n\in \mathbb {Z}}\left| \phi _{j+n}(\xi ) {\mathcal {F}}f(\xi )\right| ^2{\mathrm{d}}\xi \right) ^{1/2} \lesssim 2^{-{|{j} |}/d + \varepsilon {|{j} |} /2}\Vert f\Vert _{L^2}.\nonumber \\ \end{aligned}$$

(9.20)

Interpolating (9.19) with (9.20) and choosing appropriate $\varepsilon < 2/d$ we get

$$\begin{aligned} I_p^1\lesssim 2^{-\delta _p|j|}\Vert f\Vert _{L^p}. \end{aligned}$$

1.3.2 The estimates for $I_p^2$

Since G is an open bounded convex set containing the origin, with the aid of the spherical coordinates we may write

$$\begin{aligned} \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right) g(x) =- \int _{S^{k-1}} \int _{r(\omega ) 2^n}^{r(\omega ) t} g\left( x - \mathcal {Q}(r \omega ) \right) K(r \omega ) r^{k-1} {\, \mathrm{d}} r {\, \mathrm{d}}\sigma (\omega ) \end{aligned}$$

where $S^{k-1}$ is a unit sphere in $\mathbb {R}^k$ and $\sigma$ is the surface measure on $S^{k-1}$. We observe that if g is a Schwartz function

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}} t} \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n} \right) g(x)&=- t^{k-1} \int _{S^{k-1}} g\left( x - \mathcal {Q}(r(\omega ) t \omega )\right) K(r(\omega ) t \omega ) r(\omega )^k {\, \mathrm{d}}\sigma (\omega ). \end{aligned}$$

(9.21)

Change of the order of integration and differentiation is permited since g is bounded and the kernel K satifies (9.1). Hence, if $t \in [s_l, s_{l+1})$ and $s_l, s_{l+1} \simeq 2^n$, by Tonnelli’s theorem, we get

$$\begin{aligned} \sum _{l=0}^{h-1}\int _{s_l}^{s_{l+1}} \left| \frac{{\mathrm{d}}}{{\mathrm{d}} t} \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right) g(x) \right| {\, \mathrm{d}} t \lesssim \mathcal {M}_{2^{n+1}} {|{g} |}(x). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} I_p^2\lesssim & {} \left||\left( \sum _{n\in \mathbb {Z}}\left( \mathcal {M}_{2^{n+1}} {|{S_{j+n}f} |} \right) ^2\right) ^{1/2} \right||_{L^p}\nonumber \\\lesssim & {} \left||\left( \sum _{n \in \mathbb {Z}} \sup _{t > 0} \left( \mathcal {M}_t {|{S_{j + n} f} |} \right) ^2 \right) ^{1/2} \right||_{L^p} \lesssim {\left||{f} \right||}_{L^p}. \end{aligned}$$

(9.22)

Now we refine the estimate of $I_p^2$ for $p = 2$. Let $\tilde{m}_{t, 2^n}$ be the multiplier associated with the operator $\frac{{\mathrm{d}}}{{\mathrm{d}}t} \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right)$. We have

$$\begin{aligned} \tilde{m}_{t, 2^n}(\xi ) = - t^{k-1} \int _{S^{k-1}} e^{2\pi i {\xi \cdot \mathcal {Q}(r(\omega ) t \omega )}} r(\omega )^k K(r(\omega ) t \omega ) {\, \mathrm{d}}\sigma (\omega ). \end{aligned}$$

(9.23)

Indeed, by (9.21), we have

$$\begin{aligned} \mathcal {F}\left( \frac{{\mathrm{d}}}{{\mathrm{d}} t} \left( {\mathcal {T}}_t-{\mathcal {T}}_{2^n}\right) g\right) (\xi )&=- t^{k-1} \int _{\mathbb {R}^d} e^{2\pi i {\xi \cdot x}} \int _{S^{k-1}} g\left( x \right. \\&\quad \left. - \mathcal {Q}(r(\omega ) t \omega )\right) r(\omega )^k K(r(\omega ) t \omega ) {\, \mathrm{d}}\sigma (\omega ) {\, \mathrm{d}}x \end{aligned}$$

and thanks to estimates (9.1) and (9.16) we may change the order of integrations. Note that if $t\simeq 2^n$ we obtain $|\tilde{m}_{t, 2^n}(\xi )|\lesssim 2^{-n}$.

Now, using (9.23), by the Cauchy–Schwarz inequality and Plancherel’s theorem we get

$$\begin{aligned} I_2^2\le & {} \left||\left( \sum _{n\in \mathbb {Z}}\frac{2^n}{h}\int _{2^n}^{2^{n+1}} \left| \frac{{\mathrm{d}}}{{\mathrm{d}}t}\left( \mathcal {T}_{t} - \mathcal {T}_{2^n}\right) S_{j+n}f\right| ^2{\, \mathrm{d}}t \right) ^{1/2} \right||_{L^2} \nonumber \\= & {} \left( \sum _{n\in \mathbb {Z}}\frac{2^n}{h}\int _{2^n}^{2^{n+1}}\int _{\mathbb {R}^d} \left| \tilde{m}_{t,2^n}(\xi ) \phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, \mathrm{d}}\xi {\, \mathrm{d}} t \right) ^{1/2}\nonumber \\\lesssim & {} 2^{-\varepsilon |j|/2} \left( \sum _{n\in \mathbb {Z}}\int _{\mathbb {R}^d} \left| \phi _{j+n}(\xi ){\mathcal {F}}f(\xi )\right| ^2{\, \mathrm{d}}\xi \right) ^{1/2} \lesssim 2^{-\varepsilon |j|/2}\Vert f\Vert _{L^2}\nonumber \\ \end{aligned}$$

(9.24)

for $0<\varepsilon <1/d$. Thus interpolation of (9.22) with (9.24) gives

$$\begin{aligned} I_p^2\lesssim 2^{-\delta _p|j|}\Vert f\Vert _{L^p} \end{aligned}$$

and the proof of (9.18) is completed.