# Rectangular Summability of Multi-Dimensional Fourier Transforms

• Ferenc Weisz
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

In the last chapter, we deal with the rectangular summability of Fourier transforms defined by a function θ = θ1 ⊗⋯ ⊗θ d . We consider two types of convergence, the so-called restricted and unrestricted convergence. In the first case, T is in a cone and T while in the second case, min(T1, , T d ) → , which is called Pringsheim’s convergence. Analogously, we consider two types of maximal operators, the restricted and unrestricted ones. We prove similar results as in Chap.  For the restricted convergence, we use the Hardy space $$H_{p}^{\square }(\mathbb{R}^{d})$$ and for the unrestricted $$H_{p}(\mathbb{R}^{d})$$. We show that both maximal operators are bounded from the corresponding Hardy space to $$L_{p}(\mathbb{R}^{d})$$, which implies the almost everywhere convergence. In both cases, the set of convergence is characterized as two types of Lebesgue points.

In the last chapter, we deal with the rectangular summability of Fourier transforms defined by a function θ = θ1 ⊗⋯ ⊗θ d . We consider two types of convergence, the so-called restricted and unrestricted convergence. In the first case, T is in a cone and T while in the second case, min(T1, , T d ) → , which is called Pringsheim’s convergence. Analogously, we consider two types of maximal operators, the restricted and unrestricted ones. We prove similar results as in Chap.  For the restricted convergence, we use the Hardy space $$H_{p}^{\square }(\mathbb{R}^{d})$$ and for the unrestricted $$H_{p}(\mathbb{R}^{d})$$. We show that both maximal operators are bounded from the corresponding Hardy space to $$L_{p}(\mathbb{R}^{d})$$, which implies the almost everywhere convergence. In both cases, the set of convergence is characterized as two types of Lebesgue points.

## 6.1 Norm Convergence of Rectangular Summability Means

The $$L_{p}(\mathbb{R}^{d})$$-norm convergence of the rectangular Dirichlet integrals of the Fourier transforms was proved in Theorem  for 1 < p < . That theorem does not hold for p = 1 and p = , so we consider summability methods again. In this chapter we will always assume that
$$\displaystyle{ \theta =\theta _{1} \otimes \cdots \otimes \theta _{d},\qquad \theta _{j} \in L_{1}(\mathbb{R}) \cap C_{0}(\mathbb{R})\qquad \mbox{ and}\qquad \theta _{j}(0) = 1 }$$
(6.1.1)
for all j = 1, , d.

### Definition 6.1.1

The Tth rectangular θ-mean of the function $$f \in L_{p}(\mathbb{R}^{d})\ (1 \leq p \leq 2)$$ is given by
$$\displaystyle{\sigma _{T}^{\theta }f(x):= \frac{1} {(2\pi )^{d/2}}\int _{\mathbb{R}^{d}}\prod _{j=1}^{d}\theta _{ j}\left (\frac{-t_{j}} {T_{j}} \right )\widehat{f}(t)e^{\imath x\cdot t}\,dt\qquad (T \in \mathbb{R}_{ +}^{d}).}$$
As in the one-dimensional case, the integral is well defined. For an integrable function f,
$$\displaystyle{\sigma _{T}^{\theta }f(x) =\int _{ \mathbb{R}^{d}}f(x - t)K_{T}^{\theta }(t)\,dt = f {\ast} K_{ T}^{\theta }(x)\qquad (x \in \mathbb{R}^{d},T \in \mathbb{R}_{ +}^{d}).}$$

### Definition 6.1.2

The Tth rectangular θ-kernel is given by
$$\displaystyle{K_{T}^{\theta }(x):= \frac{1} {(2\pi )^{d}}\int _{\mathbb{R}^{d}}\prod _{j=1}^{d}\theta _{ j}\left (\frac{-t_{j}} {T_{j}} \right )e^{\imath x\cdot t}\,dt = \frac{1} {(2\pi )^{d/2}}\prod _{j=1}^{d}T_{ j}\widehat{\theta _{j}}(T_{j}x_{j}).}$$
From this it follows that
$$\displaystyle{\left \vert K_{T}^{\theta }\right \vert \leq C\prod _{ j=1}^{d}T_{ j}\qquad \left (T \in \mathbb{R}_{+}^{d}\right ).}$$
Moreover, the rectangular θ-means can be rewritten as
$$\displaystyle{\sigma _{T}^{\theta }f(x) = (2\pi )^{-d/2}\left (\prod _{ j=1}^{d}T_{ j}\right )\int _{\mathbb{R}^{d}}f(x - t)\prod _{j=1}^{d}\widehat{\theta _{ j}}(T_{j}t_{j})\,dt.}$$

For $$\widehat{\theta _{j}} \in L_{1}(\mathbb{R})\ (\,j = 1,\ldots,d)$$, we can extend the definition of the θ-means in the following way.

### Definition 6.1.3

If $$\widehat{\theta _{j}} \in L_{1}(\mathbb{R})$$ for all j = 1, , d, then we extend the Tth rectangular θ-mean to all $$f \in W(L_{1},\ell_{\infty })(\mathbb{R}^{d})$$ by
$$\displaystyle{\sigma _{T}^{\theta }f:= f {\ast} K_{ T}^{\theta }\qquad (T \in \mathbb{R}_{ +}^{d}).}$$
Since
$$\displaystyle{ K_{T}^{\theta } = K_{ T_{1}}^{\theta _{1} } \otimes \cdots \otimes K_{T_{d}}^{\theta _{d} }, }$$
(6.1.2)
Definition 6.1.2 implies that $$K_{T}^{\theta } \in L_{1}(\mathbb{R}^{d})$$. Here $$K_{T_{j}}^{\theta _{j}}$$ denotes the corresponding one-dimensional kernel. For even and differentiable functions θ j , the equality
$$\displaystyle{\sigma _{T}^{\theta }\,f(x) = \frac{-1} {\prod _{j=1}^{d}T_{j}}\int _{0}^{\infty }\prod _{ j=1}^{d}\theta _{ j}^{{\prime}}\left ( \frac{t_{j}} {T_{j}}\right )s_{t}f(x)\,dt}$$
can be shown as in the one-dimensional case. Hence, for the Fejér means , i.e. if each
$$\displaystyle{\theta _{j}(t):= \left \{\begin{array}{ll} 1 -\vert t\vert,&\mbox{ if }\vert t\vert \leq 1;\\ 0, &\mbox{ if } \vert t\vert> 1, \end{array} \right.}$$
we obtain
$$\displaystyle{\sigma _{T}^{\theta }\,f(x) = \frac{1} {\prod _{j=1}^{d}T_{j}}\int _{0}^{T_{1} }\ldots \int _{0}^{T_{d} }s_{t}f(x)\,dt.}$$
For the two-dimensional rectangular Fejér kernel see Fig. 6.1.

By (6.1.2), the following theorem can be proved exactly as Theorem .

### Theorem 6.1.4

Assume that B is a homogeneous Banach space on $$\mathbb{R}^{d}$$. If $$\theta _{j} \in L_{1}(\mathbb{R})$$ and $$\widehat{\theta _{j}} \in L_{1}(\mathbb{R})$$ for all j = 1, , d, then
$$\displaystyle{\left \|\sigma _{T}^{\theta }\,f\right \|_{ B} \leq C\left \|\,f\right \|_{B}\qquad \left (\,f \in B,T \in \mathbb{R}_{+}^{d}\right )}$$
and
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }\,f = f\qquad \mathit{\mbox{ in the }}B\mathit{\text{-norm for all }}f \in B.}$$

Besides the usual spaces $$C_{0}(\mathbb{R}^{d})$$, $$L_{p}(\mathbb{R}^{d})$$, $$W(L_{p},\ell_{q})(\mathbb{R}^{d})$$, $$W(L_{p},c_{0})(\mathbb{R}^{d})$$, $$W(C,\ell_{q})(\mathbb{R}^{d})\ (1 \leq p,q <\infty )$$, $$C_{u}(\mathbb{R}^{d})$$, the space $$W_{I}(L_{p},c_{0})(\mathbb{R}^{d})$$ is also a homogeneous Banach space. Indeed, it is easy to see that $$C_{c}(\mathbb{R}^{d})$$ is dense in $$W_{I}(L_{p},c_{0})(\mathbb{R}^{d})$$. Note that the iterated Wiener amalgam spaces $$W_{I}(L_{p},\ell_{q})(\mathbb{R}^{d})$$ were introduced in Sect. .

### Corollary 6.1.5

If f is a uniformly continuous and bounded function, $$\theta _{j} \in L_{1}(\mathbb{R})$$ and $$\widehat{\theta _{j}} \in L_{1}(\mathbb{R})$$ for all j = 1, , d, then
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }\,f = f\qquad \mathit{\mbox{ uniformly.}}}$$

## 6.2 Almost Everywhere Restricted Summability

In this section and in the following one, we assume that $$T \in \mathbb{R}^{d}$$ is in a cone. For a given ω ≥ 1, we define a cone by
$$\displaystyle{\mathbb{R}_{\omega }^{d}:=\{ x \in \mathbb{R}_{ +}^{d}:\omega ^{-1} \leq x_{ i}/x_{j} \leq \omega,i,j = 1,\ldots,d\}.}$$
The choice ω = 1 obviously yields the diagonal.

### Definition 6.2.1

The restricted maximal operator σ θ is defined by
$$\displaystyle{\sigma _{\square }^{\theta }\,f:=\sup _{ T\in \mathbb{R}_{\omega }^{d}}\left \vert \sigma _{T}^{\theta }\,f\right \vert.}$$
As we can see on Fig. 6.2, in the restricted maximal operator the supremum is taken on a cone only. Marcinkiewicz and Zygmund [244] were the first who considered the restricted convergence. Using the condition () with N = 0, we show that the restricted maximal operator is bounded from $$H_{p}^{\square }(\mathbb{R}^{d})$$ to $$L_{p}(\mathbb{R}^{d})$$ (Weisz [348, 355]).

### Theorem 6.2.2

For $$\theta _{j} \in L_{1}(\mathbb{R})$$ suppose that $$\widehat{\theta _{j}}\ (\,j = 1,\ldots,d)$$ is differentiable, $$\widehat{\theta _{j}}^{{\prime}}$$ is bounded and there exists 1 < β j < ∞ such that
$$\displaystyle{ \left \vert \widehat{\theta _{j}}(x)\right \vert,\left \vert \widehat{\theta _{j}}^{{\prime}}(x)\right \vert \leq C\vert x\vert ^{-\beta _{j} }\qquad (x\neq 0,j = 1,\ldots d). }$$
(6.2.1)
If
$$\displaystyle{p_{1}:=\max \left \{ \frac{d} {d + 1}, \frac{1} {\beta _{j} \wedge 2},j = 1,\ldots,d\right \} <p \leq \infty,}$$
then
$$\displaystyle{\left \|\sigma _{\square }^{\theta }\,f\right \|_{ p} \leq C_{p}\|\,f\|_{H_{p}^{\square }}\qquad (\,f \in H_{p}^{\square }(\mathbb{R}^{d})).}$$

### Proof

We will prove the result for d = 2, only. For d > 2, the verification is very similar. Instead of x and I1, I2 we will write (x, y) and I, J, respectively. Let a be an arbitrary cube p-atom with support I × J and
$$\displaystyle{[-2^{-K-2},2^{-K-2}] \subset I,J \subset [-2^{-K-1},2^{-K-1}]\qquad (K \in \mathbb{Z}).}$$
Choose $$s \in \mathbb{N}$$ such that 2 s−1 < ω ≤ 2 s . It is easy to see that if T1S or T2S, then we have T1, T2S2s . Indeed, since T is in a cone, T1S implies T2ω−1 T1S2s . Again, it is enough to prove that
$$\displaystyle{\int _{\mathbb{R}^{2}\setminus 4(I\times J)}\left \vert \sigma _{\square }^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy \leq C_{ p}.}$$
First suppose that 1 < β j ≤ 2 for both j = 1, 2 and let us integrate over $$(\mathbb{R}\setminus 4I) \times 4J$$. Obviously,
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}\setminus 4I}\int _{4J}\left \vert \sigma _{\square }^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \ \ \leq \sum _{\vert i\vert =1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\int _{4J}\sup _{T_{1},T_{2}\geq 2^{K-s}}\left \vert \sigma _{T}^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \qquad \ +\sum _{ \vert i\vert =1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\int _{4J}\sup _{T_{1},T_{2}<2^{K}}\left \vert \sigma _{T}^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \ \ =: (A) + (B). {}\\ \end{array}$$
We may suppose that i > 0. Since $$\widehat{\theta _{2}} \in L_{1}(\mathbb{R}^{d})$$, by (6.2.1),
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }a(x,y)\right \vert & =& CT_{ 1}T_{2}\left \vert \int _{I}\int _{J}a(t,u)\widehat{\theta _{1}}(T_{1}(x - t))\widehat{\theta _{2}}(T_{2}(\,y - u))\,dt\,du\right \vert \\ &\leq & C_{p}2^{2K/p}\int _{ I}T_{1}^{1-\beta _{1} }\vert x - t\vert ^{-\beta _{1} }\,dt. {}\end{array}$$
(6.2.2)
For x ∈ [i2K , (i + 1)2K ) (i ≥ 1) and tI, we have
$$\displaystyle{ \vert x - t\vert \geq \vert x\vert -\vert t\vert \geq i2^{-K} - 2^{-K-1} \geq Ci2^{-K}. }$$
(6.2.3)
From this, it follows that
$$\displaystyle{\left \vert \sigma _{T}^{\theta }a(x,y)\right \vert \leq C_{ p}2^{2K/p+K\beta _{1}-K}T_{ 1}^{1-\beta _{1} }i^{-\beta _{1} }.}$$
Since T1 ≥ 2 K 2s , we obtain
$$\displaystyle{(A) \leq C_{p}\sum _{i=1}^{\infty }2^{-2K}2^{2K+K\beta _{1}p-Kp}2^{Kp-K\beta _{1}p}i^{-\beta _{1}p} \leq C_{ p}\sum _{i=1}^{\infty }i^{-\beta _{1}p},}$$
which is a convergent series if p > 1∕β1.
To consider (B), let I = J = (−ν, ν) for some ν > 0 and
$$\displaystyle{A_{1}(x,v):=\int _{ -\nu }^{x}a(t,v)\,dt\qquad \mbox{ and}\qquad A_{ 2}(x,y):=\int _{ -\nu }^{y}A_{ 1}(x,t)\,dt.}$$
Then
$$\displaystyle{ \left \vert A_{k}(x,y)\right \vert \leq C_{p}2^{K(2/p-k)}. }$$
(6.2.4)
Integrating by parts, we get that
$$\displaystyle\begin{array}{rcl} & & \int _{I}a(t,u)\widehat{\theta _{1}}(T_{1}(x - t))\,dt \\ & & \ = A_{1}(\nu,u)\widehat{\theta _{1}}(T_{1}(x-\nu )) - T_{1}\int _{I}A_{1}(t,u)\widehat{\theta _{1}}^{{\prime}}(T_{ 1}(x - t))\,dt.{}\end{array}$$
(6.2.5)
Since $$\widehat{\theta _{2}}$$ is bounded, for x ∈ [i2K , (i + 1)2K ), (6.2.4) implies
$$\displaystyle\begin{array}{rcl} & & T_{1}T_{2}\left \vert \int _{J}A_{1}(\nu,u)\widehat{\theta _{1}}(T_{1}(x - t))\widehat{\theta _{2}}(T_{2}(\,y - u))\,du\right \vert {}\\ & &\qquad \qquad \qquad \quad \leq C_{p}T_{2}2^{2K/p-K}2^{-K}T_{ 1}^{1-\beta _{1} }\vert x -\nu \vert ^{-\beta _{1} } {}\\ & & \qquad \qquad \qquad \quad \leq C_{p}2^{2K/p+K\beta _{1}-2K}T_{ 1}^{2-\beta _{1} }i^{-\beta _{1} }. {}\\ \end{array}$$
Similarly,
$$\displaystyle\begin{array}{rcl} & & T_{1}^{2}T_{ 2}\left \vert \int _{J}\int _{I}A_{1}(t,u)\widehat{\theta _{1}}^{{\prime}}(T_{ 1}(x - t))\widehat{\theta _{2}}(T_{2}(\,y - u))\,du\,dt\right \vert {}\\ & &\qquad \qquad \qquad \quad \leq C_{p}2^{2K/p-K}\int _{ I}T_{1}^{2-\beta _{1} }\vert x - t\vert ^{-\beta _{1} }\,dt {}\\ & & \qquad \qquad \qquad \quad \leq C_{p}2^{2K/p+K\beta _{1}-2K}T_{ 1}^{2-\beta _{1} }i^{-\beta _{1} }. {}\\ \end{array}$$
Consequently,
$$\displaystyle{(B) \leq C_{p}\sum _{i=1}^{\infty }2^{-2K}2^{2K+K\beta _{1}p-2Kp}2^{K(2-\beta _{1})p}i^{-\beta _{1}p} \leq C_{ p}\sum _{i=1}^{\infty }i^{-\beta _{1}p} <\infty.}$$
Hence, we have proved that in this case
$$\displaystyle{\int _{\mathbb{R}\setminus 4I}\int _{4J}\left \vert \sigma _{\square }^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy \leq C_{ p}.}$$
Next, we integrate over $$(\mathbb{R}\setminus 4I) \times (\mathbb{R}\setminus 4J)$$,
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}\setminus 4I}\int _{\mathbb{R}\setminus 4J}\left \vert \sigma _{\square }^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \ \leq \sum _{\vert i\vert =1}^{\infty }\sum _{ \vert j\vert =1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\int _{j2^{-K}}^{(\,j+1)2^{-K} }\sup _{T_{1},T_{2}\geq 2^{K-s}}\left \vert \sigma _{T}^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \qquad +\sum _{ \vert i\vert =1}^{\infty }\sum _{ \vert j\vert =1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\int _{j2^{-K}}^{(\,j+1)2^{-K} }\sup _{T_{1},T_{2}<2^{K}}\left \vert \sigma _{T}^{\theta }a(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \ =: (C) + (D). {}\\ \end{array}$$
We may suppose again that i, j > 0.
For x ∈ [i2K , (i + 1)2K ) and y ∈ [j2K , ( j + 1)2K ), we have by (6.2.1) and (6.2.2) that
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }a(x,y)\right \vert & \leq & C_{ p}2^{2K/p}\int _{ I}T_{1}^{1-\beta _{1} }\vert x - t\vert ^{-\beta _{1} }\,dt\int _{J}T_{2}^{1-\beta _{2} }\vert y - u\vert ^{-\beta _{2} }\,du {}\\ & \leq & C_{p}2^{2K/p+K\beta _{1}+K\beta _{2}-2K}T_{ 1}^{1-\beta _{1} }T_{2}^{1-\beta _{2} }i^{-\beta _{1} }j^{-\beta _{2} }. {}\\ \end{array}$$
This implies that
$$\displaystyle\begin{array}{rcl} (C)& \leq & C_{p}\sum _{i=1}^{\infty }\sum _{ j=1}^{\infty }2^{-2K}2^{2K+K\beta _{1}p+K\beta _{2}p-2Kp}2^{Kp-K\beta _{1}p}2^{Kp-K\beta _{2}p}i^{-\beta _{1}p}j^{-\beta _{2}p} {}\\ & \leq & C_{p}\sum _{i=1}^{\infty }\sum _{ j=1}^{\infty }i^{-\beta _{1}p}j^{-\beta _{2}p} <\infty. {}\\ \end{array}$$
Using (6.2.5) and integrating by parts in both variables, we get that
$$\displaystyle\begin{array}{rcl} & & T_{1}T_{2}\int _{I}\int _{J}a(t,u)\widehat{\theta _{1}}(T_{1}(x - t))\widehat{\theta _{2}}(T_{2}(\,y - u))\,dt\,du {}\\ & & \ \ = -T_{1}T_{2}^{2}\int _{ J}A_{2}(\nu,u)\widehat{\theta _{1}}(T_{1}(x-\nu ))\widehat{\theta _{2}}^{{\prime}}(T_{ 2}(\,y - u))\,du {}\\ & & \qquad \quad + T_{1}^{2}T_{ 2}\int _{I}A_{2}(t,\nu )\widehat{\theta _{1}}^{{\prime}}(T_{ 1}(x - t))\widehat{\theta _{2}}(T_{2}(\,y-\nu ))\,dt {}\\ & & \qquad \quad - T_{1}^{2}T_{ 2}^{2}\int _{ I}\int _{J}A_{2}(t,u)\widehat{\theta _{1}}^{{\prime}}(T_{ 1}(x - t))\widehat{\theta _{2}}^{{\prime}}(T_{ 1}(\,y - u))\,dt\,du {}\\ & & \ \ =: E_{T}^{1}(x,y) + E_{ T}^{2}(x,y) + E_{ T}^{3}(x,y). {}\\ \end{array}$$
Note that A2(ν, −ν) = A2(ν, ν) = 0. Since $$\widehat{\theta _{j}}$$ is bounded as well, (6.2.1) implies
$$\displaystyle{\left \vert \widehat{\theta _{j}}(x)\right \vert = \left \vert \widehat{\theta _{j}}(x)\right \vert ^{\eta _{j}}\left \vert \widehat{\theta _{j}}(x)\right \vert ^{1-\eta _{j}} \leq C\vert x\vert ^{-\beta _{j}(1-\eta _{j})}}$$
for all 0 ≤ η j ≤ 1 and j = 1, 2. Clearly, the same is valid for $$\widehat{\theta _{j}}^{{\prime}}$$. Inequalities (6.2.3) and (6.2.4) imply
$$\displaystyle\begin{array}{rcl} & & \left \vert E_{T}^{1}(x,y)\right \vert \\ & &\ \leq C_{p}2^{2K/p-2K}T_{ 1}^{1-\beta _{1}(1-\eta _{1})}\vert x -\nu \vert ^{-\beta _{1}(1-\eta _{1})}\int _{ J}T_{2}^{2-\beta _{2}(1-\eta _{2})}\vert y - u\vert ^{-\beta _{2}(1-\eta _{2})}\,du \\ & & \ \leq C_{p}2^{2K/p-3K}T_{ 1}^{1-\beta _{1}(1-\eta _{1})}2^{K\beta _{1}(1-\eta _{1})}i^{-\beta _{1}(1-\eta _{1})}T_{ 2}^{2-\beta _{2}(1-\eta _{2})}2^{K\beta _{2}(1-\eta _{2})}j^{-\beta _{2}(1-\eta _{2})},{}\end{array}$$
(6.2.6)
whenever x ∈ [i2K , (i + 1)2K ), y ∈ [j2K , ( j + 1)2K ) and 0 ≤ η1, η2 ≤ 1. If
$$\displaystyle{1 -\beta _{1}(1 -\eta _{1}) + 2 -\beta _{2}(1 -\eta _{2}) \geq 0,}$$
then
$$\displaystyle{\sup _{n,m<2^{K}}\left \vert E_{T}^{1}(x,y)\right \vert \leq C_{ p}2^{2K/p}i^{-\beta _{1}(1-\eta _{1})}j^{-\beta _{2}(1-\eta _{2})}}$$
because T is in a cone. Choosing
$$\displaystyle{\eta _{j}:= \frac{\beta _{j} - 3/2} {\beta _{j}} \vee 0,}$$
we can see that
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}\setminus 4I}\int _{\mathbb{R}\setminus 4J\sup _{ T_{1},T_{2}<2^{K}}\left \vert E_{T}^{1}(x,y)\right \vert ^{p}\,dx\,dy} {}\\ & & \ \leq C_{p}\sum _{i=1}^{\infty }\sum _{ j=1}^{\infty }2^{-2K}2^{2K}i^{-(3/2\wedge \beta _{1})p}j^{-(3/2\wedge \beta _{2})p}, {}\\ \end{array}$$
which is a convergent series. The analogous estimate for $$\left \vert E_{T}^{2}(x,y)\right \vert$$ can be proved similarly.
For x ∈ [i2K , (i + 1)2K ) and y ∈ [j2K , ( j + 1)2K ), we conclude that
$$\displaystyle\begin{array}{rcl} \left \vert E_{T}^{3}(x,y)\right \vert & \leq & C_{ p}2^{2K/p-2K}\int _{ I}T_{1}^{2-\beta _{1} }\vert x - t\vert ^{-\beta _{1} }\,dt\int _{J}T_{2}^{2-\beta _{2} }\vert y - u\vert ^{-\beta _{2} }\,du {}\\ & \leq & C_{p}2^{2K/p-2K+K\beta _{1}+K\beta _{2}-2K}T_{ 1}^{2-\beta _{1} }T_{2}^{2-\beta _{2} }i^{-\beta _{1} }j^{-\beta _{2} }. {}\\ \end{array}$$
So
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}\setminus 4I}\int _{\mathbb{R}\setminus 4J}\sup _{T_{1},T_{2}<2^{K}}\left \vert E_{T}^{3}(x,y)\right \vert ^{p}\,dx\,dy {}\\ & & \ \leq C_{p}\sum _{i=1}^{\infty }\sum _{ j=1}^{\infty }2^{-2K}2^{2K-2Kp+K\beta _{1}p+K\beta _{2}p-2Kp}2^{4Kp-K\beta _{1}p-K\beta _{2}p}i^{-\beta _{1}p}j^{-\beta _{2}p} {}\\ & & \ \leq C_{p}\sum _{i=1}^{\infty }\sum _{ j=1}^{\infty }i^{-\beta _{1}p}j^{-\beta _{2}p} <\infty {}\\ \end{array}$$
by the hypothesis. The integration over $$4I \times (\mathbb{R}\setminus 4J)$$ can be done as above. This finishes the proof of the theorem when 1 < β j ≤ 2, j = 1, 2.
Now suppose that β j > 2 for some j = 1, 2. Since $$\widehat{\theta _{j}}$$ and $$\widehat{\theta _{j}}^{{\prime}}$$ are bounded and since $$\vert x\vert ^{-\beta _{j}} \leq \vert x\vert ^{-2}$$ if | x | ≥ 1, we conclude that
$$\displaystyle{ \left \vert \widehat{\theta _{j}}(x)\right \vert,\left \vert \widehat{\theta _{j}}^{{\prime}}(x)\right \vert \leq C\vert x\vert ^{-2}\qquad (x\neq 0), }$$
(6.2.7)
which finishes the proof.

### Remark 6.2.3

In the d-dimensional case, the constant d∕(d + 1) appears if we investigate the corresponding term to E T 1. More exactly, if we integrate the term
$$\displaystyle{\prod _{j=1}^{d}T_{ j}\int _{I_{d}}A(\nu,\cdots \,,\nu,u)\widehat{\theta _{1}}(T_{1}(x_{1}-\nu ))\cdots \widehat{\theta _{d-1}}(T_{d-1}(x_{d-1}-\nu ))\widehat{\theta _{d}}^{{\prime}}(T_{ d}(x_{d} - u))\,du}$$
over $$(\mathbb{R}\setminus 4I_{1}) \times \cdots \times (\mathbb{R}\setminus 4I_{d})$$ similar to (6.2.6), then we get that p > d∕(d + 1).
Since $$H_{p}^{\square }(\mathbb{R}^{d}) \sim L_{p}(\mathbb{R}^{d})$$ for 1 < p, we have
$$\displaystyle{\left \|\sigma _{\square }^{\theta }f\right \|_{ p} \leq C_{p}\left \|\,f\right \|_{p}\qquad (\,f \in L_{p}(\mathbb{R}^{d}),1 <p \leq \infty ).}$$
As we have seen in Theorem , in the one-dimensional case, the Fejér operator σ θ is not bounded from $$H_{p}^{\square }(\mathbb{R}^{d})$$ to $$L_{p}(\mathbb{R}^{d})$$ if 0 < p ≤ 1∕2. Using interpolation, we obtain the weak type (1, 1) inequality as usual.

### Corollary 6.2.4

Suppose the same conditions as in Theorem  6.2.2 . If $$f \in L_{1}(\mathbb{R}^{d})$$ , then
$$\displaystyle{\sup _{\rho>0}\rho \lambda (\sigma _{\square }^{\theta }\,f>\rho ) \leq C\|\,f\|_{ 1}.}$$

### Corollary 6.2.5

Suppose the same conditions as in Theorem  6.2.2 . If $$f \in L_{p}(\mathbb{R}^{d})$$ for some 1 ≤ p < ∞, then
$$\displaystyle{\lim _{T\rightarrow \infty,\,T\in \mathbb{R}_{\omega }^{d}}\sigma _{T}^{\theta }f = f\qquad \mathit{\mbox{ a.e.}}}$$

This result was proved by Marcinkiewicz and Zygmund [244] for the two-dimensional Fejér means. The general version of Corollary 6.2.5 is due to the author [348, 355]. It is easy to see that the summability methods given in Sect.  all satisfy the conditions of Theorem 6.2.2.

## 6.3 Restricted Convergence at Lebesgue Points

Introducing the weighted Lebesgue spaces , we briefly write $$L_{p}^{\tau }(\mathbb{R}^{d})$$ (τ ≥ 0) instead of the $$L_{p}^{\tau }(\mathbb{R}^{d},\lambda )$$ space equipped with the norm
$$\displaystyle{\left \|\,f\right \|_{L_{p}^{\tau }}:= \left (\int _{\mathbb{R}^{d}}\vert \,f(x)(1 + \left \vert x\right \vert )^{\tau }\vert ^{p}\,dx\right )^{1/p}\qquad (1 \leq p <\infty ),}$$
with the usual modification for p = . If τ = 0, then we get back the $$L_{p}(\mathbb{R}^{d})$$ spaces. Clearly, $$L_{p}(\mathbb{R}^{d}) \supset L_{p}^{\tau }(\mathbb{R}^{d})$$. We are generalizing the definition of the Herz spaces introduced in Definitions  and .

### Definition 6.3.1

A function $$f \in L_{q}^{loc}(\mathbb{R}^{d})$$ is in the Herz space $$E_{q}^{\tau }(\mathbb{R}^{d})$$ (1 ≤ q, τ ≥ 0) if
$$\displaystyle{\left \|\,f\right \|_{E_{q}^{\tau }}:=\sum _{ k_{1}=0}^{\infty }\cdots \sum _{ k_{d}=0}^{\infty }2^{\|k\|_{1}(\tau +1-1/q)}\left \|\,f1_{ Q_{k_{1}}\times \cdots \times Q_{k_{d}}}\right \|_{q} <\infty,}$$
where
$$\displaystyle{Q_{k}:= I(0,2^{k})\setminus I(0,2^{k-1})\qquad (k> 0),\qquad Q_{ 0}:= I(0,1).}$$
Recall that $$I(0,c) = \left \{x \in \mathbb{R}: \vert x\vert <c\right \}$$. It is clear that $$E_{q}^{0}(\mathbb{R}) = E_{q}(\mathbb{R})$$ and $$E_{q}^{\tau }(\mathbb{R}^{d}) \subset L_{q}(\mathbb{R}^{d})$$. Moreover,
$$\displaystyle{E_{q}^{\tau }(\mathbb{R}^{d}) \supset E_{ q}^{\tau ^{{\prime}} }(\mathbb{R}^{d})\qquad 0 \leq \tau <\tau ^{{\prime}} <\infty }$$
and
$$\displaystyle{L_{1}(\mathbb{R}^{d}) \supset L_{ 1}^{\tau }(\mathbb{R}^{d}) = E_{ 1}^{\tau }(\mathbb{R}^{d}) \supset E_{ q}^{\tau }(\mathbb{R}^{d}) \supset E_{ q^{{\prime}}}^{\tau }(\mathbb{R}^{d}) \supset E_{ \infty }^{\tau }(\mathbb{R}^{d})}$$
for any 1 < q < q < with continuous embeddings. We show that $$f \in E_{\infty }^{\tau }(\mathbb{R})$$ if and only if f has a decreasing majorant function belonging to $$L_{1}^{\tau }(\mathbb{R})$$.

### Theorem 6.3.2

Let $$\eta (x):=\sup _{\vert t\vert \geq \vert x\vert }\left \vert \,f(t)\right \vert$$. Then $$f \in E_{\infty }^{\tau }(\mathbb{R})$$ if and only if $$\eta \in L_{1}^{\tau }(\mathbb{R})$$ (τ ≥ 0) and
$$\displaystyle{C^{-1}\left \|\eta \right \|_{ L_{1}^{\tau }} \leq \left \|\,f\right \|_{E_{\infty }^{\tau }} \leq C\left \|\eta \right \|_{L_{1}^{\tau }} +\eta (0).}$$

### Proof

If $$\eta \in L_{1}^{\tau }(\mathbb{R})$$, then
$$\displaystyle{\left \|\,f\right \|_{E_{\infty }^{\tau }} \leq \sum _{k=0}^{\infty }2^{k(\tau +1)}\left \|\eta 1_{ Q_{k}}\right \|_{\infty } =\sum _{ k=1}^{\infty }2^{k(\tau +1)}\eta (2^{k-1}) +\eta (0) \leq C\left \|\eta \right \|_{ L_{1}^{\tau }} +\eta (0).}$$
For the converse we use the function ν introduced in the proof of Theorem  to obtain
$$\displaystyle\begin{array}{rcl} \|\eta \|_{L_{1}^{\tau }}& \leq & \|\nu \|_{L_{1}^{\tau }} =\sum _{ k=0}^{\infty }a_{ n_{k}}\int _{B(0,2^{n_{k}})\setminus B(0,2^{n_{k-1}})}(1 + x)^{\tau }\,dx {}\\ & =& C\sum _{k=0}^{\infty }2^{n_{k}\tau }\left (2^{n_{k} } - 2^{n_{k-1} }\right )a_{n_{k}} \leq C\left \|\,f\right \|_{E_{\infty }^{\tau }}, {}\\ \end{array}$$
which proves the theorem.

As in the previous section, let θ = θ1 ⊗⋯ ⊗θ d satisfy (6.1.1). It is easy to see that $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$ if and only if $$\widehat{\theta _{j}} \in E_{q}^{\tau }(\mathbb{R})$$ for all j = 1, , d. Note that all examples from Sect.  satisfy this condition.

### Theorem 6.3.3

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞ and 1∕p + 1∕q = 1. If $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$ for some τ > 0, then for all $$f \in L_{p}^{loc}(\mathbb{R}^{d})$$,
$$\displaystyle{\sigma _{\square }^{\theta }\,f(x) \leq C_{ p}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\mathcal{M}_{p}^{(1)}f(x)\qquad (x \in \mathbb{R}^{d}).}$$

### Proof

We have
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }\,f(x)\right \vert & =& (2\pi )^{-d/2}\left (\prod _{ j=1}^{d}T_{ j}\right )\left \vert \int _{\mathbb{R}^{d}}f(x - t)\widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\,dt\right \vert \\ &\leq & C\left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{1}=0}^{\infty }\cdots \sum _{ k_{d}=0}^{\infty } \\ & &\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\vert \,f(x - t)\vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt,{}\end{array}$$
(6.3.1)
where
$$\displaystyle{Q_{i}(T_{k}):= I(0,2^{i}/T_{ k})\setminus I(0,2^{i-1}/T_{ k})\quad (i \geq 1),\qquad Q_{0}(T_{k}):= I(0,1/T_{k})}$$
(k = 1, , d). By Hölder’s inequality,
$$\displaystyle\begin{array}{rcl} & & \left \vert \sigma _{T}^{\theta }\,f(x)\right \vert \\ & &\ \leq C\left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{1}=0}^{\infty }\cdots \sum _{ k_{d}=0}^{\infty }\left (\int _{ Q_{k_{1}}(T_{1})}\cdots \int _{Q_{k_{d}}(T_{d})}\vert \,f(x - t)\vert ^{p}\,dt\right )^{1/p} \\ & & \quad \left (\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert ^{q}\,dt\right )^{1/q} \\ & & \ = C\left (\prod _{j=1}^{d}T_{ j}\right )^{1-1/q}\sum _{ k_{1}=0}^{\infty }\cdots \sum _{ k_{d}=0}^{\infty }\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} \\ & & \quad \left (\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\vert \,f(x - t)\vert ^{p}\,dt\right )^{1/p}. {}\end{array}$$
(6.3.2)
Since $$T \in \mathbb{R}_{\omega }^{d}$$, we conclude
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }\,f(x)\right \vert & \leq & C_{ p}\sum _{k_{1}=0}^{\infty }\cdots \sum _{ k_{d}=0}^{\infty }2^{\|k\|_{1}(\tau +1/p)}\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & & 2^{-\tau \|k\|_{1} }\left ( \frac{T_{1}^{d}} {2^{\|k\|_{1}+d\omega }}\int _{-2^{k_{1}+\omega }/T_{1}}^{2^{k_{1}+\omega }/T_{ 1}}\cdots \int _{ -2^{k_{d}+\omega }/T_{1}}^{2^{k_{d}+\omega }/T_{ 1}}\vert \,f(x - t)\vert ^{p}\,dt\right )^{1/p} {}\\ & \leq & C_{p}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\mathcal{M}_{p}^{(1)}f(x), {}\\ \end{array}$$
which shows the theorem.

Theorems  and  imply the next two corollaries.

### Corollary 6.3.4

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, p < r∞, 1∕p + 1∕q = 1 and τ > 0. If $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$, then
$$\displaystyle{\sup _{\rho>0}\rho \lambda (\sigma _{\square }^{\theta }\,f>\rho )^{1/p} \leq C_{ p}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\left \|\,f\right \|_{p}\qquad (\,f \in L_{p}(\mathbb{R}^{d})),}$$
$$\displaystyle{\left \|\sigma _{\square }^{\theta }\,f\right \|_{ r} \leq C_{r}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\left \|\,f\right \|_{r}\qquad (\,f \in L_{r}(\mathbb{R}^{d}))}$$
and
$$\displaystyle{\left \|\sigma _{\square }^{\theta }\right \|_{ W(L_{p,\infty },\ell_{\infty })} \leq C_{p}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\|\,f\|_{W(L_{p},\ell_{\infty })}\qquad (\,f \in W(L_{p},\ell_{\infty })(\mathbb{R}^{d})),}$$
$$\displaystyle{\left \|\sigma _{\square }^{\theta }\,f\right \|_{ W(L_{r},\ell_{\infty })} \leq C_{r}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}\|\,f\|_{W(L_{r},\ell_{\infty })}\qquad (\,f \in W(L_{r},\ell_{\infty })(\mathbb{R}^{d})).}$$

### Corollary 6.3.5

If $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1, τ > 0 and $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$, then for all $$f \in W(L_{p},c_{0})(\mathbb{R}^{d})$$
$$\displaystyle{\lim _{T\rightarrow \infty,T\in \mathbb{R}_{\omega }^{d}}\sigma _{T}^{\theta }\,f = f\quad \mathit{\mbox{ a.e.}}}$$

Note that $$E_{q}^{\tau }(\mathbb{R}^{d}) \supset E_{q^{{\prime}}}^{\tau }(\mathbb{R}^{d})$$ whenever q < q . Now we are able to prove the convergence of σ T θ f at each modified Lebesgue point for all $$f \in L_{p}(\mathbb{R}^{d})$$ and even for all $$f \in W(L_{p},\ell_{\infty })(\mathbb{R}^{d})$$, whenever T is in a cone.

### Theorem 6.3.6

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1, τ > 0 and $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$. If $$f \in W(L_{p},\ell_{\infty })(\mathbb{R}^{d})$$, x is a modified p-Lebesgue point of f and $$\mathcal{M}_{p}^{(1)}f(x)$$ is finite, then
$$\displaystyle{\lim _{T\rightarrow \infty,T\in \mathbb{R}_{\omega }^{d}}\sigma _{T}^{\theta }\,f(x) = f(x).}$$

### Proof

Since x is a modified p-Lebesgue point of f, we can fix a number r < 1 such that U r, p (1) f(x) < ε. Recall that U r, p (1) f and the modified Lebesgue points were introduced in Sect. . Let us denote by r0 the largest number i, for which r∕2 ≤ 2 i T1 < r. It is easy to see that
$$\displaystyle{\sigma _{T}^{\theta }f(x) - f(x) = \left (\prod _{ j=1}^{d}T_{ j}\right )\int _{\mathbb{R}^{d}}\left (\,f(x - t) - f(x)\right )\widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\,dt}$$
and
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }f(x) - f(x)\right \vert & \leq & \left (\prod _{ j=1}^{d}T_{ j}\right )\int _{\mathbb{R}^{d}}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt {}\\ & \leq & A_{1}(x) + A_{2}(x), {}\\ \end{array}$$
where
$$\displaystyle\begin{array}{rcl} A_{1}(x)& =& \left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{1}=0}^{r_{0}-\omega }\cdots \sum _{ k_{d}=0}^{r_{0}-\omega } {}\\ & &\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt {}\\ \end{array}$$
and
$$\displaystyle\begin{array}{rcl} A_{2}(x)& =& \sum _{\pi _{1},\ldots,\pi _{d}}\left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{\pi _{1}}=r_{0}-\omega +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=r_{0}-\omega +1}^{\infty }\sum _{ k_{\pi _{j+1}}=0}^{\infty }\ldots \sum _{ k_{\pi _{d}}=0}^{\infty } {}\\ & &\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt, {}\\ \end{array}$$
where {π1, , π d } is a permutation of {1, , d} and 1 ≤ jd.
Similar to (6.3.2),
$$\displaystyle\begin{array}{rcl} & & \left \vert A_{1}(x)\right \vert {}\\ & &\ \leq \left (\prod _{j=1}^{d}T_{ j}^{1-1/q}\right )\sum _{ k_{1}=0}^{r_{0}-\omega }\cdots \sum _{ k_{d}=0}^{r_{0}-\omega }\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & & \quad \ \left (\int _{Q_{k_{ 1}}(T_{1})}\cdots \int _{Q_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert ^{p}\,dt\right )^{1/p} {}\\ & & \ \leq C_{p}\sum _{k_{1}=0}^{r_{0}-\omega }\cdots \sum _{ k_{d}=0}^{r_{0}-\omega }2^{\left \|k\right \|_{1} (\tau +1/p)}\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & & \quad \ 2^{-\tau \left \|k\right \|_{1} }\left ( \frac{T_{1}^{d}} {2^{\left \|k\right \|_{1}+d\omega }}\int _{-2^{k_{1}+\omega }/T_{1}}^{2^{k_{1}+\omega }/T_{ 1}}\cdots \int _{ -2^{k_{d}+\omega }/T_{1}}^{2^{k_{d}+\omega }/T_{ 1}}\vert \,f(x - t) - f(x)\vert ^{p}\,dt\right )^{1/p} {}\\ & & \ \leq C_{p}\left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}U_{r,p}^{(1)}f(x) \leq C_{ p}\epsilon \left \|\widehat{\theta }\right \|_{E_{q}^{\tau }}. {}\\ \end{array}$$
We can see in the same way that
$$\displaystyle\begin{array}{rcl} \left \vert A_{2}(x)\right \vert & \leq & C_{p}\sum _{\pi _{1},\ldots,\pi _{d}}\sum _{k_{\pi _{ 1}}=r_{0}-\omega +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=r_{0}-\omega +1}^{\infty }\sum _{ k_{\pi _{j+1}}=0}^{\infty }\ldots \sum _{ k_{\pi _{d}}=0}^{\infty } {}\\ & & 2^{\left \|k\right \|_{1} (\tau +1/p)}\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}2^{-\tau \left \|k\right \|_{1} } {}\\ & & \left ( \frac{T_{1}^{d}} {2^{\left \|k\right \|_{1}+d}}\int _{-2^{k_{1}+\omega }/T_{1}}^{2^{k_{1}+\omega }/T_{ 1}}\cdots \int _{ -2^{k_{d}+\omega }/T_{1}}^{2^{k_{d}+\omega }/T_{ 1}}\vert \,f(x - t) - f(x)\vert ^{p}\,dt\right )^{1/p}. {}\\ \end{array}$$
Since $$\mathcal{M}_{p}^{(1)}f(x)$$ is finite if x is a modified p-Lebesgue point of f, we have
$$\displaystyle\begin{array}{rcl} \left \vert A_{2}(x)\right \vert & \leq & C_{p}\sum _{\pi _{1},\ldots,\pi _{d}}\sum _{k_{\pi _{ 1}}=r_{0}-\omega +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=r_{0}-\omega +1}^{\infty }\sum _{ k_{\pi _{j+1}}=0}^{\infty }\ldots \sum _{ k_{\pi _{d}}=0}^{\infty } {}\\ & & 2^{\left \|k\right \|_{1} (\tau +1/p)}\left (\int _{ Q_{k_{1}}}\cdots \int _{Q_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}\Big(\mathcal{M}_{ p}^{(1)}f(x) + \vert \,f(x)\vert \Big). {}\\ \end{array}$$
Since r0 as T1 and $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$, we conclude that A2(x) → 0 as T, which finishes the proof.

### Corollary 6.3.7

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1 and τ > 0. If $$\widehat{\theta }\in E_{q}^{\tau }(\mathbb{R}^{d})$$ and $$f \in W(L_{p},\ell_{\infty })(\mathbb{R}^{d})$$ is continuous at a point x, then
$$\displaystyle{\lim _{T\rightarrow \infty,T\in \mathbb{R}_{\omega }^{d}}\sigma _{T}^{\theta }\,f(x) = f(x).}$$

## 6.4 Almost Everywhere Unrestricted Summability

### Definition 6.4.1

The unrestricted maximal operator σ θ is defined by
$$\displaystyle{\sigma _{{\ast}}^{\theta }\,f:=\sup _{ T\in \mathbb{R}_{+}^{d}}\left \vert \sigma _{T}^{\theta }\,f\right \vert.}$$
In this section, instead of the Hardy space $$H_{p}^{\square }(\mathbb{R}^{d})$$ we have to use the Hardy space $$H_{p}(\mathbb{R}^{d})$$. We will first prove that the operator σ θ is bounded from $$L_{p}(\mathbb{R}^{d})$$ to $$L_{p}(\mathbb{R}^{d})\ (1 <p \leq \infty )$$ and then that it is bounded from $$H_{p}(\mathbb{R}^{d})$$ to $$L_{p}(\mathbb{R}^{d})$$. To this end, we introduce the one-dimensional operators
$$\displaystyle{\tau _{T}^{\theta }\,f(x):=\int _{ \mathbb{R}}f(x - u)\vert K_{T}^{\theta }(u)\vert \,du = f {\ast}\left \vert K_{ T}^{\theta }\right \vert (x)\qquad (x \in \mathbb{R},T> 0)}$$
and
$$\displaystyle{\tau _{{\ast}}^{\theta }f:=\sup _{ T>0}\left \vert \tau _{T}^{\theta }f\right \vert.}$$
Obviously,
$$\displaystyle{\left \vert \sigma _{T}^{\theta }f\right \vert \leq \tau _{ T}^{\theta }\vert \,f\vert \quad (T> 0)\qquad \mbox{ and}\qquad \sigma _{ {\ast}}^{\theta }f \leq \tau _{ {\ast}}^{\theta }\vert \,f\vert.}$$
The next result can be proved as was Theorem .

### Theorem 6.4.2

Suppose that $$\theta \in L_{1}(\mathbb{R})$$ satisfies  (6.2.1) with the exponent β and that $$\widehat{\theta }^{{\prime}}$$ is bounded. If 1∕(β ∧ 2) < p∞, then
$$\displaystyle{\left \|\tau _{{\ast}}^{\theta }f\right \|_{ p} \leq C_{p}\|\,f\|_{H_{p}}\qquad (\,f \in H_{p}(\mathbb{R})).}$$

### Proof

It is easy to see that
$$\displaystyle{\left \|\tau _{{\ast}}^{\theta }f\right \|_{ \infty }\leq C\left \|\,f\right \|_{\infty }\qquad (\,f \in L_{\infty }(\mathbb{R})).}$$
Let 1 < β ≤ 2 and a be an arbitrary cube p-atom with support $$I \subset \mathbb{R}$$ and
$$\displaystyle{[-2^{-K-2},2^{-K-2}] \subset I \subset [-2^{-K-1},2^{-K-1}].}$$
We may assume that x > 0. Then
$$\displaystyle\begin{array}{rcl} \int _{\mathbb{R}\setminus 4I}\left \vert \tau _{{\ast}}^{\theta }a(x)\right \vert ^{p}\,dx& \leq & \sum _{ i=1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\sup _{T\geq 2^{K}}\left \vert \tau _{T}^{\theta }a(x)\right \vert ^{p}\,dx {}\\ & & +\sum _{ i=1}^{\infty }\int _{ i2^{-K}}^{(i+1)2^{-K} }\sup _{T<2^{K}}\left \vert \tau _{T}^{\theta }a(x)\right \vert ^{p}\,dx {}\\ & =:& (A) + (B). {}\\ \end{array}$$
The inequality
$$\displaystyle{(A) \leq C_{p}}$$
can be shown as in Theorem . To estimate (B), observe that
$$\displaystyle\begin{array}{rcl} \tau _{T}^{\theta }a(x)& =& T\int _{ I}a(t)\left \vert \widehat{\theta }(T(x - t))\right \vert \,dt {}\\ & =& T\int _{I}a(t)\left (\left \vert \widehat{\theta }(T(x - t))\right \vert -\left \vert \widehat{\theta }(Tx)\right \vert \right )\,dt. {}\\ \end{array}$$
Thus
$$\displaystyle{\left \vert \tau _{T}^{\theta }a(x)\right \vert \leq T\int _{ I}\vert a(t)\vert \left \vert \widehat{\theta }(T(x - t)) -\widehat{\theta } (Tx)\right \vert \,dt.}$$
Using Lagrange’s mean value theorem, we conclude
$$\displaystyle{\left \vert \widehat{\theta }(T(x - t)) -\widehat{\theta } (Tx)\right \vert = \left \vert \widehat{\theta }^{{\prime}}(T(x -\nu t))\right \vert \vert Tt\vert \leq T\vert T(x -\nu t)\vert ^{-\beta }\left \vert t\right \vert }$$
and the proof can be finished as in Theorem . Because of (6.2.7), the theorem is also valid for 2 < β < .
Since $$H_{p}(\mathbb{R}) \sim L_{p}(\mathbb{R})\ (1 <p \leq \infty )$$, Theorem 6.4.2 implies that
$$\displaystyle{\left \|\tau _{{\ast}}^{\theta }f\right \|_{ p} \leq C_{p}\|\,f\|_{p}\qquad (\,f \in L_{p}(\mathbb{R}))}$$
for 1 < p. Now, we return to the higher dimensional case and verify the $$L_{p}(\mathbb{R}^{d})$$ boundedness of σ θ .

### Theorem 6.4.3

Suppose that $$\theta _{j} \in L_{1}(\mathbb{R})$$ satisfies  (6.2.1) and that $$\widehat{\theta _{j}}^{{\prime}}$$ is bounded for each j = 1, , d. If 1 < p∞, then
$$\displaystyle{\left \|\sigma _{{\ast}}^{\theta }f\right \|_{ p} \leq C_{p}\left \|\,f\right \|_{p}\qquad (\,f \in L_{p}(\mathbb{R}^{d})).}$$

### Proof

We may suppose that d = 2. Applying Theorem 6.4.2, we have
$$\displaystyle\begin{array}{rcl} & & \int _{\mathbb{R}}\int _{\mathbb{R}}\sup _{T_{1},T_{2}>0}\left \vert \int _{\mathbb{R}}\int _{\mathbb{R}}f(t,u)K_{T_{1}}^{\theta _{1} }(x - t)K_{T_{2}}^{\theta _{2} }(\,y - u)\,dt\,du\right \vert ^{p}\,dx\,dy {}\\ & & \ \leq \int _{\mathbb{R}}\int _{\mathbb{R}}\sup _{T_{2}>0}\left (\int _{\mathbb{R}}\left (\sup _{T_{1}>0}\left \vert \int _{\mathbb{R}}f(t,u)K_{T_{1}}^{\theta _{1} }(x - t)\,dt\right \vert \right )\left \vert K_{T_{2}}^{\theta _{2} }(\,y - u)\right \vert \,du\right )^{p}\,dy\,dx {}\\ & & \ \leq C_{p}\int _{\mathbb{R}}\int _{\mathbb{R}}\sup _{T_{1}>0}\left \vert \int _{\mathbb{R}}f(t,y)K_{T_{1}}^{\theta _{1} }(x - t)\,dt\right \vert ^{p}\,dx\,dy {}\\ & & \ \leq C_{p}\int _{\mathbb{R}}\int _{\mathbb{R}}\vert \,f(x,y)\vert ^{p}\,dx\,dy, {}\\ \end{array}$$
which proves the theorem.

In the next theorem, besides the condition (6.2.1), we use its higher order version introduced first in ().

### Theorem 6.4.4

Suppose that $$\theta _{j} \in L_{1}(\mathbb{R})$$ satisfies  (6.2.1), $$\widehat{\theta _{j}}$$ is (N j + 1)-times differentiable for some $$N_{j} \in \mathbb{N}$$ and $$\widehat{\theta _{j}}^{{\prime}}$$ is bounded for each j = 1, , d. Moreover, assume that there exists N j + 1 < β j N j + 2 such that
$$\displaystyle{ \left \vert \left (\widehat{\theta _{j}}\right )^{(k_{j})}(x)\right \vert \leq C\vert x\vert ^{-\beta _{j} }\qquad (x\neq 0), }$$
(6.4.1)
whenever k j = N j and k j = N j + 1. If
$$\displaystyle{p_{2}:=\max \left \{\frac{1} {\beta _{j}},j = 1,\ldots,d\right \} <p \leq \infty,}$$
then
$$\displaystyle{ \left \|\sigma _{{\ast}}^{\theta }f\right \|_{ p} \leq C_{p}\|\,f\|_{H_{p}}\qquad (\,f \in H_{p}(\mathbb{R}^{d})). }$$
(6.4.2)

### Proof

By the first condition of the theorem and by Theorem 6.4.3, inequality (6.4.2) holds for 1 < p. We sketch the proof by giving only the main ideas. We will prove the theorem only for d = 3, because the proof is similar for larger d or for d = 2. Choose a simple p-atom a with support R = I1 × I2 × A where I1 and I2 are intervals with
$$\displaystyle{2^{-K_{i}-1} <\vert I_{ i}\vert \leq 2^{-K_{i} }\qquad (K_{i} \in \mathbb{Z},i = 1,2)}$$
and
$$\displaystyle{[-2^{-K_{i}-2},2^{-K_{i}-2}] \subset I_{ i} \subset [-2^{-K_{i}-1},2^{-K_{i}-1}].}$$
We assume that r i ≥ 2 are arbitrary integers. By Theorems  and , it is enough to show that
$$\displaystyle{ \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{A}\left \vert \sigma _{{\ast}}^{\theta }(x)\right \vert ^{p}\,dx \leq C_{ p}2^{-\eta _{1}r_{1} }2^{-\eta _{2}r_{2} }, }$$
(6.4.3)
and, if A = I3 is also an interval,
$$\displaystyle{ \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{(I_{3})^{c}}\left \vert \sigma _{{\ast}}^{\theta }(x)\right \vert ^{p}\,dx \leq C_{ p}2^{-\eta _{1}r_{1} }2^{-\eta _{2}r_{2} } }$$
(6.4.4)
for all p2 < p ≤ 1.
First, we decompose the supremum as
$$\displaystyle\begin{array}{rcl} \sigma _{{\ast}}^{\theta }a& \leq & \sup _{ \begin{array}{c}T_{1}<2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a\right \vert +\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a\right \vert \\ & & +\sup _{\begin{array}{c}T_{ 1}<2^{K_{1}},T_{ 2}\geq 2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a\right \vert +\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}\geq 2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a\right \vert.{}\end{array}$$
(6.4.5)
We will investigate only the second term. Obviously,
$$\displaystyle\begin{array}{rcl} & & \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{A}\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a(x)\right \vert ^{p}\,dx {}\\ & & \ \leq \sum _{\vert i_{1}\vert =2^{r_{1}-2}}^{\infty }\sum _{ \vert i_{2}\vert =2^{r_{2}-2}}^{\infty }\int _{ i_{1}2^{-K_{1}}}^{(i_{1}+1)2^{-K_{1}} }\int _{i_{2}2^{-K_{2}}}^{(i_{2}+1)2^{-K_{2}} }\int _{A}\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a(x)\right \vert ^{p}\,dx,{}\\ \end{array}$$
where we may suppose that i l > 0. As we noted in Sect. , we may suppose that the cancellation property (iii) of the definition of the simple atoms (Definition ) holds for all kM, where MM( p) is arbitrary. So assume that MN j + 2 for all j = 1, 2, 3. Using Taylor’s formula
$$\displaystyle{g(t) =\sum _{ k=0}^{N-1}\frac{g^{(k)}(0)} {k!} t^{k} + \frac{g^{(N)}(\nu t)} {N!} t^{N}}$$
for $$g_{j}(t) =\widehat{\theta _{j}}(T_{j}(x_{j} - t_{j}))$$, N = N1, N2 + 1, where 0 < ν j < 1, we conclude
$$\displaystyle\begin{array}{rcl} \sigma _{T}^{\theta }a(x)& =& \frac{T_{1}T_{2}T_{3}} {(2\pi )^{3/2}} \int _{R}a(t)\widehat{\theta _{1}}(T_{1}(x_{1} - t_{1}))\widehat{\theta _{2}}(T_{2}(x_{2} - t_{2}))\widehat{\theta _{3}}(T_{3}(x_{3} - t_{3}))\,dt {}\\ & =& \frac{T_{1}T_{2}T_{3}} {(2\pi )^{3/2}} \int _{R}a(t)\left (\widehat{\theta _{1}}(T_{1}(x_{1} - t_{1})) -\sum _{k=0}^{N_{1}-1}\frac{g_{1}^{(k)}(0)} {k!} t^{k}\right ) {}\\ & & \left (\widehat{\theta _{2}}(T_{2}(x_{2} - t_{2})) -\sum _{l=0}^{N_{2} } \frac{g_{2}^{(l)}(0)} {l!} t^{l}\right )\widehat{\theta _{ 3}}(T_{3}(x_{3} - t_{3}))\,dt {}\\ & =& CT_{1}T_{2}\int _{R}a(t)(-1)^{N_{1}+N_{2}+1}T_{ 1}^{N_{1} }\left (\widehat{\theta _{1}}\right )^{(N_{1})}(T_{ 1}(x_{1} -\nu _{1}t_{1}))t_{1}^{N_{1} } {}\\ & & T_{2}^{N_{2}+1}\left (\widehat{\theta _{ 2}}\right )^{(N_{2}+1)}(T_{ 2}(x_{2} -\nu _{2}t_{2}))t_{2}^{N_{2}+1}K_{ T_{3}}^{\theta _{3} }(x_{3} - t_{3})\,dt. {}\\ \end{array}$$
Then by (6.4.1),
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }a(x)\right \vert & \leq & CT_{ 1}^{N_{1}+1}T_{ 2}^{N_{2}+2}\int _{ I_{1}}\int _{I_{2}}\left \vert T_{1}(x_{1} -\nu _{1}t_{1})\right \vert ^{-\beta _{1} }\left \vert t_{1}\right \vert ^{N_{1} }\left \vert t_{2}\right \vert ^{N_{2}+1}\qquad {}\\ & & \left \vert T_{2}(x_{2} -\nu _{2}t_{2})\right \vert ^{-\beta _{2} }\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert \,dt_{1}\,dt_{2}. {}\\ \end{array}$$
Since $$T_{1} \geq 2^{K_{1}}$$, $$T_{2} <2^{K_{2}}$$ and for $$x_{j} \in [i_{j}2^{-K_{j}},(i_{j} + 1)2^{-K_{j}})$$ and $$t_{j} \in [-2^{-K_{l}-1},2^{-K_{j}-1})\ (\,j = 1,2)$$,
$$\displaystyle{\vert x_{j} -\nu _{j}t_{j}\vert \geq \vert x_{j}\vert -\vert t_{j}\vert \geq i_{j}2^{-K_{j} } - 2^{-K_{j}-1} \geq Ci_{ j}2^{-K_{j} },}$$
we have
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }a(x)\right \vert & \leq & CT_{ 1}^{N_{1}+1-\beta _{1} }T_{2}^{N_{2}+2-\beta _{2} }2^{-K_{1}(N_{1}-\beta _{1})}2^{-K_{2}(N_{2}+1-\beta _{2})}i_{ 1}^{-\beta _{1} }i_{2}^{-\beta _{2} } {}\\ & & \int _{I_{1}}\int _{I_{2}}\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert \,dt_{1}\,dt_{2} {}\\ & \leq & C2^{K_{1}+K_{2} }i_{1}^{-\beta _{1} }i_{2}^{-\beta _{2} }\int _{I_{1}}\int _{I_{2}}\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert \,dt_{1}\,dt_{2}. {}\\ \end{array}$$
Hence, by Hölder’s inequality,
$$\displaystyle\begin{array}{rcl} & & \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{A}\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a(x)\right \vert ^{p}\,dx {}\\ & & \ \leq C_{p}\sum _{i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }2^{-K_{1}-K_{2} }2^{K_{1}p+K_{2}p}i_{ 1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & & \quad \ \int _{A}\left (\int _{I_{1}}\int _{I_{2}}\sup _{T_{3}>0}\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert \,dt_{1}\,dt_{2}\right )^{p}\,dx_{ 3} {}\\ & & \ \leq C_{p}\vert A\vert ^{1-p}\sum _{ i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }2^{-K_{1}-K_{2} }2^{K_{1}p+K_{2}p}i_{ 1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & & \quad \ \left (\int _{A}\int _{I_{1}}\int _{I_{2}}\sup _{T_{3}>0}\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert \,dt_{1}\,dt_{2}\,dx_{3}\right )^{p}. {}\\ \end{array}$$
Using again Hölder’s inequality and the fact that σ θ is bounded on $$L_{2}(\mathbb{R}^{d})$$ for all d ≥ 1, we conclude
$$\displaystyle\begin{array}{rcl} & & \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{A}\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a(x)\right \vert ^{p}\,dx {}\\ & & \ \leq C_{p}\vert A\vert ^{1-p/2}\sum _{ i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }2^{-K_{1}-K_{2} }2^{K_{1}p+K_{2}p}i_{ 1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & & \quad \ \left (\int _{I_{1}}\int _{I_{2}}\left (\int _{\mathbb{R}}\sup _{T_{3}>0}\left \vert \int _{A}a(t)K_{T_{3}}^{\theta }(x_{3} - t_{3})\,dt_{3}\right \vert ^{2}\,dx_{ 3}\right )^{1/2}\,dt_{ 1}\,dt_{2}\right )^{p} {}\\ & & \ \leq C_{p}\vert A\vert ^{1-p/2}\sum _{ i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }2^{-K_{1}-K_{2} }2^{K_{1}p+K_{2}p}i_{ 1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & & \quad \ \left (\int _{I_{1}}\int _{I_{2}}\left (\int _{\mathbb{R}}\left \vert a(t_{1},t_{2},x_{3})\right \vert ^{2}\,dx_{ 3}\right )^{1/2}\,dt_{ 1}\,dt_{2}\right )^{p} {}\\ & & \ \leq C_{p}\vert A\vert ^{1-p/2}\sum _{ i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }2^{K_{1}p/2+K_{2}p/2}2^{-K_{1}-K_{2} }i_{1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & & \quad \ \left (\int _{I_{1}}\int _{I_{2}}\int _{\mathbb{R}}\left \vert a(t)\right \vert ^{2}\,dt\right )^{p/2}. {}\\ \end{array}$$
The following inequality follows from $$\|a\|_{2} \leq \left (2^{-K_{1}-K_{2}}\vert A\vert \right )^{1/2-1/p}$$:
$$\displaystyle\begin{array}{rcl} \int _{(I_{1}^{r_{1}})^{c}}\int _{(I_{2}^{r_{2}})^{c}}\int _{A}\sup _{ \begin{array}{c}T_{1}\geq 2^{K_{1}},T_{ 2}<2^{K_{2}} \\ T_{3}>0 \end{array}}\left \vert \sigma _{T}^{\theta }a(x)\right \vert ^{p}\,dx& \leq & C_{ p}\sum _{i_{1}=2^{r_{1}-2}}^{\infty }\sum _{ i_{2}=2^{r_{2}-2}}^{\infty }i_{ 1}^{-\beta _{1}p}i_{ 2}^{-\beta _{2}p} {}\\ & \leq & C_{p}2^{-r_{1}(\beta _{1}p-1)}2^{-r_{2}(\beta _{2}p-1)}. {}\\ \end{array}$$
The other terms of (6.4.5) can be handled in the same way, which shows (6.4.3). Obviously, the same ideas show (6.4.4).

The next corollary follows from Corollary  and Theorem  in the usual way.

### Corollary 6.4.5

Under the conditions of Theorem  6.4.4 , if $$f \in H_{1}^{i}(\mathbb{R}^{d})$$ for some i = 1, , d, then
$$\displaystyle{\sup _{\rho>0}\rho \lambda (\sigma _{{\ast}}^{\theta }f>\rho ) \leq C\|\,f\|_{ H_{1}^{i}}.}$$

By the density argument, we get here almost everywhere convergence for functions from the spaces $$H_{1}^{i}(\mathbb{R}^{d})$$ instead of $$L_{1}(\mathbb{R}^{d})$$. In some sense, the Hardy space $$H_{1}^{i}(\mathbb{R}^{d})$$ plays the role of $$L_{1}(\mathbb{R}^{d})$$ in higher dimensions.

### Corollary 6.4.6

Under the conditions of Theorem  6.4.4 , if $$f \in H_{1}^{i}(\mathbb{R}^{d})$$ for some i = 1, , d or $$f \in L_{p}(\mathbb{R}^{d})$$ for some 1 < p < ∞, then
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }f = f\qquad \mathit{\mbox{ a.e.}}}$$
The almost everywhere convergence is not true for all $$f \in L_{1}(\mathbb{R}^{d})$$.
A counterexample, which shows that the almost everywhere convergence is not true for all integrable functions, is due to Gát [132]. Recall that
$$\displaystyle{L_{1}(\mathbb{R}^{d}) \supset H_{ 1}^{i}(\mathbb{R}^{d}) \supset L(\log L)^{d-1}(\mathbb{R}^{d}) \supset L_{ p}(\mathbb{R}^{d})\qquad (1 <p \leq \infty ).}$$

## 6.5 Unrestricted Convergence at Lebesgue Points

First we introduce the homogeneous version of the Herz space of Definition 6.3.1.

### Definition 6.5.1

A function $$f \in L_{q}^{loc}(\mathbb{R}^{d})$$ is in the homogeneous Herz space $$\dot{E}_{q}^{\tau }(\mathbb{R}^{d})$$ (1 ≤ q, τ ≥ 0) if
$$\displaystyle{\left \|\,f\right \|_{\dot{E}_{q}^{\tau }}:=\sum _{ k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty }2^{\|k\|_{1}(\tau +1-1/q)}\left \|\,f1_{ P_{k_{1}}\times \cdots \times P_{k_{d}}}\right \|_{q} <\infty,}$$
where
$$\displaystyle{P_{k}:= I(0,2^{k})\setminus I(0,2^{k-1})\qquad (k \in \mathbb{Z}).}$$
It can be shown easily that $$\dot{E}_{q}^{0}(\mathbb{R}) =\dot{ E}_{q}(\mathbb{R})$$ and
$$\displaystyle{E_{q}^{\tau }(\mathbb{R}^{d}) \subset L_{ q}(\mathbb{R}^{d}) \cap \dot{ E}_{ q}^{\tau }(\mathbb{R}),\qquad \left \|\,f\right \|_{\dot{ E}_{q}^{\tau }} \leq C_{q}\left \|\,f\right \|_{E_{q}^{\tau }},\ \left \|\,f\right \|_{q} \leq C_{q}\left \|\,f\right \|_{E_{q}^{\tau }}.}$$
In the one-dimensional case, $$E_{q}^{\tau }(\mathbb{R}) = L_{q}(\mathbb{R}) \cap \dot{ E}_{q}^{\tau }(\mathbb{R})$$. Furthermore,
$$\displaystyle{L_{1}(\mathbb{R}) \supset L_{1}^{\tau }(\mathbb{R}) =\dot{ E}_{ 1}^{\tau }(\mathbb{R}) \supset \dot{ E}_{ q}^{\tau }(\mathbb{R}) \supset \dot{ E}_{ q^{{\prime}}}^{\tau }(\mathbb{R}) \supset \dot{ E}_{\infty }^{\tau }(\mathbb{R})}$$
for any 1 ≤ q < q with continuous embeddings. Similar to Theorem 6.3.2, $$f \in \dot{ E}_{\infty }^{\tau }(\mathbb{R})$$ if and only if f has a decreasing majorant function belonging to $$L_{1}^{\tau }(\mathbb{R})$$.

### Theorem 6.5.2

Let $$\eta (x):=\sup _{\vert t\vert \geq \vert x\vert }\left \vert \,f(t)\right \vert$$. Then $$f \in \dot{ E}_{\infty }^{\tau }(\mathbb{R})$$ if and only if $$\eta \in L_{1}^{\tau }(\mathbb{R})$$ (τ ≥ 0) and
$$\displaystyle{C^{-1}\left \|\eta \right \|_{ L_{1}^{\tau }} \leq \left \|\,f\right \|_{\dot{E}_{\infty }^{\tau }} \leq C\left \|\eta \right \|_{L_{1}^{\tau }}.}$$
For θ = θ1 ⊗⋯ ⊗θ d , $$\widehat{\theta }\in \dot{ E}_{q}^{\tau }(\mathbb{R}^{d})$$ if and only if $$\widehat{\theta _{j}} \in \dot{ E}_{q}^{\tau }(\mathbb{R})$$ for all j = 1, , d. In this section we use the strong maximal function which was introduced by
$$\displaystyle{M_{s,p}f(x):=\sup _{x\in I}\left ( \frac{1} {\vert I\vert }\int _{I}\vert \,f\vert ^{p}\,d\lambda \right )^{1/p}\qquad (x \in \mathbb{R}^{d})}$$
in Sect. , where the supremum is taken over all rectangles with sides parallel to the axes.

### Theorem 6.5.3

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞ and 1∕p + 1∕q = 1. If $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$, then for all $$f \in L_{p}^{loc}(\mathbb{R}^{d})$$,
$$\displaystyle{\sigma _{{\ast}}^{\theta }\,f(x) \leq C_{ p}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}M_{s,p}f(x).}$$

### Proof

Similar to (6.3.1) and (6.3.2),
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }f(x)\right \vert & \leq & C\left (\prod _{ j=1}^{d}T_{ j}\right )\sum _{k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty } \\ & &\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt \\ & \leq & C\left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty } \\ & &\left (\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert ^{q}\,dt\right )^{1/q} \\ & & \left (\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t)\right \vert ^{p}\,dt\right )^{1/p} \\ & =& C\left (\prod _{j=1}^{d}T_{ j}^{1-1/q}\right )\sum _{ k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty }\left (\int _{ P_{k_{1}}}\cdots \int _{P_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} \\ & & \left (\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t)\right \vert ^{p}\,dt\right )^{1/p}, {}\end{array}$$
(6.5.1)
where
$$\displaystyle{P_{i}(T_{j}):= I(0,2^{i}/T_{ j})\setminus I(0,2^{i-1}/T_{ j})\qquad (i \geq 1).}$$
If we define
$$\displaystyle{G(u):= \left (\int _{-u_{1}}^{u_{1} }\cdots \int _{-u_{d}}^{u_{d} }\left \vert \,f(x - t)\right \vert ^{p}\,dt\right )^{1/p}\qquad (u = (u_{ 1},\ldots,u_{d}) \in \mathbb{R}_{+}^{d}),}$$
then
$$\displaystyle{ \frac{G^{p}(u)} {2^{d}\prod _{j=1}^{d}u_{j}} \leq M_{s,p}^{p}f(x)\qquad (u = (u_{ 1},\ldots,u_{d}) \in \mathbb{R}_{+}^{d}).}$$
Hence
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }f(x)\right \vert & \leq & C\left (\prod _{ j=1}^{d}T_{ j}^{1/p}\right )\sum _{ k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty } {}\\ & &\left (\int _{P_{k_{ 1}}}\cdots \int _{P_{k_{ d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}G\left (\frac{2^{k_{1}}} {T_{1}},\ldots, \frac{2^{k_{d}}} {T_{d}} \right ) {}\\ & \leq & C_{p}\sum _{k_{1}=-\infty }^{\infty }\cdots \sum _{ k_{d}=-\infty }^{\infty }\left (\prod _{ j=1}^{d}2^{k_{j}/p}\right ) {}\\ & & \left (\int _{P_{k_{ 1}}}\cdots \int _{P_{k_{ d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}M_{ s,p}f(x) {}\\ & \leq & C_{p}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}M_{s,p}f(x). {}\\ \end{array}$$

Taking into account Corollary , we obtain

### Corollary 6.5.4

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1 and I = I1 × × I d with | I1 | = = | I d | = 1. If $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$, then
$$\displaystyle{\sup _{\rho>0}\rho \lambda (x:\sigma _{ {\ast}}^{\theta }(x)>\rho,x \in I)^{1/p} \leq C_{ p}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}\left (1 + \left \|\,f\right \|_{L_{p}(\log L)^{d-1}}\right ).}$$
For p < r∞,
$$\displaystyle{\left \|\sigma _{{\ast}}^{\theta }\,f\right \|_{ r} \leq C_{r}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}\left \|\,f\right \|_{r}\qquad (\,f \in L_{r}(\mathbb{R}^{d})).}$$
If $$f \in W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$, then
$$\displaystyle{\left \|\sigma _{{\ast}}^{\theta }\,f\right \|_{ W(L_{p,\infty },\ell_{\infty })} \leq C_{p}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}\left (1 + \left \|\,f\right \|_{W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })}\right )}$$
and, for p < r∞,
$$\displaystyle{\left \|\sigma _{{\ast}}^{\theta }\,f\right \|_{ W(L_{r},\ell_{\infty })} \leq C_{r}\left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}\left \|\,f\right \|_{W_{I}(L_{r},\ell_{\infty })}\qquad (\,f \in W_{I}(L_{r},\ell_{\infty })(\mathbb{R}^{d})).}$$

The next corollary follows from the density argument and from fact that $$C_{c}(\mathbb{R}^{d})$$ is dense in $$W_{I}(L_{p}(\log L)^{d-1},c_{0})(\mathbb{R}^{d})$$.

### Corollary 6.5.5

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞ and 1∕p + 1∕q = 1. If $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$, then
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }\,f = f\quad \mathit{\mbox{ a.e.}}}$$
for all $$f \in W_{I}(L_{p}(\log L)^{d-1},c_{0})(\mathbb{R}^{d})$$.
Note that
$$\displaystyle{C_{0}(\mathbb{R}^{d}) \subset W_{ I}(L_{p}(\log L)^{d-1},c_{ 0})(\mathbb{R}^{d}),}$$
$$\displaystyle{L_{r}(\mathbb{R}^{d}) \subset W_{ I}(L_{r},c_{0})(\mathbb{R}^{d}) \subset W_{ I}(L_{p}(\log L)^{d-1},c_{ 0})(\mathbb{R}^{d})}$$
for 1 ≤ p < r and
$$\displaystyle{L_{p}(\log L)^{d-1}(\mathbb{R}^{d}) \subset W_{ I}(L_{p}(\log L)^{d-1},c_{ 0})(\mathbb{R}^{d}) \subset W_{ I}(L_{p},c_{0})(\mathbb{R}^{d}),}$$
$$\displaystyle{L_{p}(\log L)^{d-1}(\mathbb{R}^{d}) \subset L_{ p}(\mathbb{R}^{d}) \subset W_{ I}(L_{p},c_{0})(\mathbb{R}^{d})}$$
for 1 ≤ p < . The last convergence result do not hold for all $$f \in L_{p}(\mathbb{R}^{d})$$. This was proved in a special case, for p = 1 and for the Fejér means of Fourier series by Gát [132] (for Walsh-Fourier series see [130]).
Theorem  implies
$$\displaystyle{\lim _{h\rightarrow 0} \frac{1} {\prod _{j=1}^{d}(2h_{j})}\int _{-h_{1}}^{h_{1} }\ldots \int _{-h_{d}}^{h_{d} }f(x - u)\,du = f(x)}$$
for almost every $$x \in \mathbb{R}^{d}$$, where $$f \in W_{I}(L_{1}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$.

### Definition 6.5.6

A point $$x \in \mathbb{R}^{d}$$ is called a strong p-Lebesgue point (1 ≤ p < ) of f if
$$\displaystyle{\lim _{h\rightarrow 0}\left ( \frac{1} {\prod _{j=1}^{d}(2h_{j})}\int _{-h_{1}}^{h_{1} }\ldots \int _{-h_{d}}^{h_{d} }\left \vert \,f(x - u) - f(x)\right \vert ^{p}\,du\right )^{1/p} = 0.}$$

The following facts can be proved in the usual way. If p < r, then all strong r-Lebesgue points are strong p-Lebesgue points.

### Theorem 6.5.7

Almost every point $$x \in \mathbb{R}^{d}$$ is a strong p-Lebesgue point of $$f \in W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})\ (1 \leq p <\infty )$$.

Now we can extend Corollary 6.5.5 to the space $$W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$ as follows. Note that for $$f \in W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$, M s, p f is almost everywhere finite by Corollary .

### Theorem 6.5.8

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1 and $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$. If $$f \in W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$, x is a strong Lebesgue point of f and M s, p f(x) is finite, then
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }\,f(x) = f(x).}$$

### Proof

Let
$$\displaystyle{G(u):= \left (\int _{-u_{1}}^{u_{1} }\cdots \int _{-u_{d}}^{u_{d} }\left \vert \,f(x - t) - f(x)\right \vert ^{p}\,dt\right )^{1/p},}$$
where $$u = (u_{1},\ldots,u_{d}) \in \mathbb{R}_{+}^{d}$$. Since x is a strong p-Lebesgue point of f, for all ε > 0 there exists $$m \in \mathbb{Z}$$ such that
$$\displaystyle{ \frac{G^{p}(u)} {2^{d}\prod _{j=1}^{d}u_{j}} \leq \epsilon ^{p}\qquad \mbox{ for }0 <u_{ j} \leq 2^{m},j = 1,\ldots,d. }$$
(6.5.2)
As we have seen before
$$\displaystyle\begin{array}{rcl} \left \vert \sigma _{T}^{\theta }f(x) - f(x)\right \vert & \leq & \left (\prod _{ j=1}^{d}T_{ j}\right )\int _{\mathbb{R}^{d}}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt {}\\ & \leq & A_{1}(x) + A_{2}(x), {}\\ \end{array}$$
where
$$\displaystyle\begin{array}{rcl} A_{1}(x)& =& \left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{1}=-\infty }^{m+\lfloor \log _{2}T_{1}\rfloor }\cdots \sum _{ k_{d}=-\infty }^{m+\lfloor \log _{2}T_{d}\rfloor } {}\\ & &\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt {}\\ \end{array}$$
and
$$\displaystyle\begin{array}{rcl} A_{2}(x)& =& \sum _{\pi _{1},\ldots,\pi _{d}}\left (\prod _{j=1}^{d}T_{ j}\right )\sum _{k_{\pi _{1}}=m+\lfloor \log _{2}T_{\pi _{1}}\rfloor +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=m+\lfloor \log _{2}T_{\pi _{j}}\rfloor +1}^{\infty } {}\\ & &\sum _{k_{\pi _{ j+1}}=-\infty }^{\infty }\ldots \sum _{ k_{\pi _{d}}=-\infty }^{\infty } {}\\ & &\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert \left \vert \widehat{\theta }(T_{1}t_{1},\ldots,T_{d}t_{d})\right \vert \,dt, {}\\ \end{array}$$
where {π1, , π d } is a permutation of {1, , d} and 1 ≤ jd.
As in (6.5.1),
$$\displaystyle\begin{array}{rcl} & & \left \vert A_{1}(x)\right \vert {}\\ & &\ \leq \left (\prod _{j=1}^{d}T_{ j}^{1-1/q}\right )\sum _{ k_{1}=-\infty }^{m+\lfloor \log _{2}T_{1}\rfloor }\cdots \sum _{ k_{d}=-\infty }^{m+\lfloor \log _{2}T_{d}\rfloor }\left (\int _{ P_{k_{1}}}\cdots \int _{P_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & & \quad \ \left (\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert ^{p}\,dt\right )^{1/p} {}\\ & & \ \leq \left (\prod _{j=1}^{d}T_{ j}^{1/p}\right )\sum _{ k_{1}=-\infty }^{m+\lfloor \log _{2}T_{1}\rfloor }\cdots \sum _{ k_{d}=-\infty }^{m+\lfloor \log _{2}T_{d}\rfloor } {}\\ & &\quad \ \left (\int _{P_{k_{ 1}}}\cdots \int _{P_{k_{ d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}G\left (\frac{2^{k_{1}}} {T_{1}},\ldots, \frac{2^{k_{d}}} {T_{d}} \right ). {}\\ \end{array}$$
Inequality (6.5.2) and $$2^{k_{j}}/T_{j} \leq 2^{m}T_{j}/T_{j} \leq 2^{m}$$ imply
$$\displaystyle\begin{array}{rcl} \left \vert A_{1}(x)\right \vert & \leq & C_{p}\epsilon \sum _{k_{1}=-\infty }^{m+\lfloor \log _{2}T_{1}\rfloor }\cdots \sum _{ k_{d}=-\infty }^{m+\lfloor \log _{2}T_{d}\rfloor } {}\\ & &\left (\prod _{j=1}^{d}2^{k_{j}/p}\right )\left (\int _{ P_{k_{1}}}\cdots \int _{P_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & \leq & C_{p}\epsilon \left \|\widehat{\theta }\right \|_{\dot{E}_{q}^{0}}. {}\\ \end{array}$$
Similarly,
$$\displaystyle\begin{array}{rcl} \left \vert A_{2}(x)\right \vert & \leq & \sum _{\pi _{1},\ldots,\pi _{d}}\left (\prod _{j=1}^{d}T_{ j}^{1/p}\right )\sum _{ k_{\pi _{1}}=m+\lfloor \log _{2}T_{\pi _{1}}\rfloor +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=m+\lfloor \log _{2}T_{\pi _{j}}\rfloor +1}^{\infty } {}\\ & &\sum _{k_{\pi _{ j+1}}=-\infty }^{\infty }\ldots \sum _{ k_{\pi _{d}}=-\infty }^{\infty }\left (\int _{ P_{k_{1}}}\cdots \int _{P_{k_{d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q} {}\\ & & \left (\int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert ^{p}\,dt\right )^{1/p}. {}\\ \end{array}$$
It is supposed that M s, p f(x) is finite and x is a strong p-Lebesgue point of f, so we have
$$\displaystyle\begin{array}{rcl} & & \int _{P_{k_{ 1}}(T_{1})}\cdots \int _{P_{k_{ d}}(T_{d})}\left \vert \,f(x - t) - f(x)\right \vert ^{p}\,dt\qquad {}\\ & & \qquad \quad \leq C_{p}\left (\prod _{j=1}^{d}\frac{2^{k_{j}}} {T_{j}} \right )\Big(M_{s,p}^{p}\,f(x) + \vert \,f(x)\vert ^{p}\Big), {}\\ \end{array}$$
thus
$$\displaystyle\begin{array}{rcl} \left \vert A_{2}(x)\right \vert & \leq & C_{p}\sum _{\pi _{1},\ldots,\pi _{d}}\,\sum _{k_{\pi _{ 1}}=m+\lfloor \log _{2}T_{\pi _{1}}\rfloor +1}^{\infty }\ldots \sum _{ k_{\pi _{j}}=m+\lfloor \log _{2}T_{\pi _{j}}\rfloor +1}^{\infty } {}\\ & &\sum _{k_{\pi _{ j+1}}=-\infty }^{\infty }\ldots \sum _{ k_{\pi _{d}}=-\infty }^{\infty }\left (\prod _{ j=1}^{d}2^{k_{j}/p}\right ) {}\\ & & \left (\int _{P_{k_{ 1}}}\cdots \int _{P_{k_{ d}}}\left \vert \widehat{\theta }(t)\right \vert ^{q}\,dt\right )^{1/q}\Big(M_{ s,p}f(x) + \vert \,f(x)\vert \Big). {}\\ \end{array}$$
Since $$\lfloor \log _{2}T_{\pi _{j}}\rfloor \rightarrow \infty$$ as T and $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$, we conclude that A2(x) → 0 as T.

### Corollary 6.5.9

Let $$\theta \in L_{1}(\mathbb{R}^{d})$$, 1 ≤ p < ∞, 1∕p + 1∕q = 1 and $$\widehat{\theta }\in \dot{ E}_{q}^{0}(\mathbb{R}^{d})$$. If $$f \in W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$ is continuous at a point x and M s, p f(x) is finite, then
$$\displaystyle{\lim _{T\rightarrow \infty }\sigma _{T}^{\theta }\,f(x) = f(x).}$$

Finally, we note again that $$W_{I}(L_{p}(\log L)^{d-1},\ell_{\infty })(\mathbb{R}^{d})$$ contains the spaces $$L_{p}(\log L)^{d-1}(\mathbb{R}^{d})$$, $$C_{0}(\mathbb{R}^{d})$$ and $$L_{r}(\mathbb{R}^{d})$$ for all 1 ≤ p < r.

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## Applied and Numerical Harmonic Analysis (80 Volumes)

1. 1.
A. Saichev and W.A. Woyczynski: Distributions in the Physical and Engineering Sciences (ISBN 978-0-8176-3924-2)Google Scholar
2. 2.
C.E. D’Attellis and E.M. Fernandez-Berdaguer: Wavelet Theory and Harmonic Analysis in Applied Sciences (ISBN 978-0-8176-3953-2)Google Scholar
3. 3.
H.G. Feichtinger and T. Strohmer: Gabor Analysis and Algorithms (ISBN 978-0-8176-3959-4)Google Scholar
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R. Tolimieri and M. An: Time-Frequency Representations (ISBN 978-0-8176-3918-1)Google Scholar
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T.M. Peters and J.C. Williams: The Fourier Transform in Biomedical Engineering (ISBN 978-0-8176-3941-9)Google Scholar
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G.T. Herman: Geometry of Digital Spaces (ISBN 978-0-8176-3897-9)Google Scholar
7. 7.
A. Teolis: Computational Signal Processing with Wavelets (ISBN 978-0-8176-3909-9)Google Scholar
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J. Ramanathan: Methods of Applied Fourier Analysis (ISBN 978-0-8176-3963-1)Google Scholar
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J.M. Cooper: Introduction to Partial Differential Equations with MATLAB (ISBN 978-0-8176-3967-9)Google Scholar
10. 10.
A. Procházka, N.G. Kingsbury, P.J. Payner, and J. Uhlir: Signal Analysis and Prediction (ISBN 978-0-8176-4042-2)Google Scholar
11. 11.
W. Bray and C. Stanojevic: Analysis of Divergence (ISBN 978-1-4612-7467-4)Google Scholar
12. 12.
G.T. Herman and A. Kuba: Discrete Tomography (ISBN 978-0-8176-4101-6)Google Scholar
13. 13.
K. Gröchenig: Foundations of Time-Frequency Analysis (ISBN 978-0-8176-4022-4)Google Scholar
14. 14.
L. Debnath: Wavelet Transforms and Time-Frequency Signal Analysis (ISBN 978-0-8176-4104-7)Google Scholar
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J.J. Benedetto and P.J.S.G. Ferreira: Modern Sampling Theory (ISBN 978-0-8176-4023-1)Google Scholar
16. 16.
D.F. Walnut: An Introduction to Wavelet Analysis (ISBN 978-0-8176-3962-4)Google Scholar
17. 17.
A. Abbate, C. DeCusatis, and P.K. Das: Wavelets and Subbands (ISBN 978-0-8176-4136-8)Google Scholar
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O. Bratteli, P. Jorgensen, and B. Treadway: Wavelets Through a Looking Glass (ISBN 978-0-8176-4280-80Google Scholar
19. 19.
H.G. Feichtinger and T. Strohmer: Advances in Gabor Analysis (ISBN 978-0-8176-4239-6)Google Scholar
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O. Christensen: An Introduction to Frames and Riesz Bases (ISBN 978-0-8176-4295-2)Google Scholar
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L. Debnath: Wavelets and Signal Processing (ISBN 978-0-8176-4235-8)Google Scholar
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G. Bi and Y. Zeng: Transforms and Fast Algorithms for Signal Analysis and Representations (ISBN 978-0-8176-4279-2)Google Scholar
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J.H. Davis: Methods of Applied Mathematics with a MATLAB Overview (ISBN 978-0-8176-4331-7)Google Scholar
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J.J. Benedetto and A.I. Zayed: Modern SamplingTheory (ISBN 978-0-8176-4023-1)Google Scholar
25. 25.
E. Prestini: The Evolution of Applied Harmonic Analysis (ISBN 978-0-8176-4125-2)Google Scholar
26. 26.
L. Brandolini, L. Colzani, A. Iosevich, and G. Travaglini: Fourier Analysis and Convexity (ISBN 978-0-8176-3263-2)Google Scholar
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W. Freeden and V. Michel: Multiscale Potential Theory (ISBN 978-0-8176-4105-4)Google Scholar
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O. Christensen and K.L. Christensen: Approximation Theory (ISBN 978-0-8176-3600-5)Google Scholar
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O. Calin and D.-C. Chang: Geometric Mechanics on Riemannian Manifolds (ISBN 978-0-8176-4354-6)Google Scholar
30. 30.
J.A. Hogan: Time?Frequency and Time?Scale Methods (ISBN 978-0-8176-4276-1)Google Scholar
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C. Heil: Harmonic Analysis and Applications (ISBN 978-0-8176-3778-1)Google Scholar
32. 32.
K. Borre, D.M. Akos, N. Bertelsen, P. Rinder, and S.H. Jensen: A Software-Defined GPS and Galileo Receiver (ISBN 978-0-8176-4390-4)Google Scholar
33. 33.
T. Qian, M.I. Vai, and Y. Xu: Wavelet Analysis and Applications (ISBN 978-3-7643-7777-9)Google Scholar
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G.T. Herman and A. Kuba: Advances in Discrete Tomography and Its Applications (ISBN 978-0-8176-3614-2)Google Scholar
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M.C. Fu, R.A. Jarrow, J.-Y. Yen, and R.J. Elliott: Advances in Mathematical Finance (ISBN 978-0-8176-4544-1)Google Scholar
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O. Christensen: Frames and Bases (ISBN 978-0-8176-4677-6)Google Scholar
37. 37.
P.E.T. Jorgensen, J.D. Merrill, and J.A. Packer: Representations, Wavelets, and Frames (ISBN 978-0-8176-4682-0)Google Scholar
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M. An, A.K. Brodzik, and R. Tolimieri: Ideal Sequence Design in Time-Frequency Space (ISBN 978-0-8176-4737-7)Google Scholar
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S.G. Krantz: Explorations in Harmonic Analysis (ISBN 978-0-8176-4668-4)Google Scholar
40. 40.
B. Luong: Fourier Analysis on Finite Abelian Groups (ISBN 978-0-8176-4915-9)Google Scholar
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G.S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 1 (ISBN 978-0-8176-4802-2)Google Scholar
42. 42.
C. Cabrelli and J.L. Torrea: Recent Developments in Real and Harmonic Analysis (ISBN 978-0-8176-4531-1)Google Scholar
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M.V. Wickerhauser: Mathematics for Multimedia (ISBN 978-0-8176-4879-4)Google Scholar
44. 44.
B. Forster, P. Massopust, O. Christensen, K. Gröchenig, D. Labate, P. Vandergheynst, G. Weiss, and Y. Wiaux: Four Short Courses on Harmonic Analysis (ISBN 978-0-8176-4890-9)Google Scholar
45. 45.
O. Christensen: Functions, Spaces, and Expansions (ISBN 978-0-8176-4979-1)Google Scholar
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J. Barral and S. Seuret: Recent Developments in Fractals and Related Fields (ISBN 978-0-8176-4887-9)Google Scholar
47. 47.
O. Calin, D.-C. Chang, and K. Furutani, and C. Iwasaki: Heat Kernels for Elliptic and Sub-elliptic Operators (ISBN 978-0-8176-4994-4)Google Scholar
48. 48.
C. Heil: A Basis Theory Primer (ISBN 978-0-8176-4686-8)Google Scholar
49. 49.
J.R. Klauder: A Modern Approach to Functional Integration (ISBN 978-0-8176-4790-2)Google Scholar
50. 50.
J. Cohen and A.I. Zayed: Wavelets and Multiscale Analysis (ISBN 978-0-8176-8094-7)Google Scholar
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D. Joyner and J.-L. Kim: Selected Unsolved Problems in Coding Theory (ISBN 978-0-8176-8255-2)Google Scholar
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G.S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 2 (ISBN 978-0-8176-4943-2)Google Scholar
53. 53.
J.A. Hogan and J.D. Lakey: Duration and Bandwidth Limiting (ISBN 978-0-8176-8306-1)Google Scholar
54. 54.
G. Kutyniok and D. Labate: Shearlets (ISBN 978-0-8176-8315-3)Google Scholar
55. 55.
P.G. Casazza and P. Kutyniok: Finite Frames (ISBN 978-0-8176-8372-6)Google Scholar
56. 56.
V. Michel: Lectures on Constructive Approximation (ISBN 978-0-8176-8402-0)Google Scholar
57. 57.
D. Mitrea, I. Mitrea, M. Mitrea, and S. Monniaux: Groupoid Metrization Theory (ISBN 978-0-8176-8396-2)Google Scholar
58. 58.
T.D. Andrews, R. Balan, J.J. Benedetto, W. Czaja, and K.A. Okoudjou: Excursions in Harmonic Analysis, Volume 1 (ISBN 978-0-8176-8375-7)Google Scholar
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T.D. Andrews, R. Balan, J.J. Benedetto, W. Czaja, and K.A. Okoudjou: Excursions in Harmonic Analysis, Volume 2 (ISBN 978-0-8176-8378-8)Google Scholar
60. 60.
D.V. Cruz-Uribe and A. Fiorenza: Variable Lebesgue Spaces (ISBN 978-3-0348-0547-6)Google Scholar
61. 61.
W. Freeden and M. Gutting: Special Functions of Mathematical (Geo-)Physics (ISBN 978-3-0348-0562-9)Google Scholar
62. 62.
A. Saichev and W.A. Woyczynski: Distributions in the Physical and Engineering Sciences, Volume 2: Linear and Nonlinear Dynamics of Continuous Media (ISBN 978-0-8176-3942-6)Google Scholar
63. 63.
S. Foucart and H. Rauhut: A Mathematical Introduction to Compressive Sensing (ISBN 978-0-8176-4947-0)Google Scholar
64. 64.
G. Herman and J. Frank: Computational Methods for Three-Dimensional Microscopy Reconstruction (ISBN 978-1-4614-9520-8)Google Scholar
65. 65.
A. Paprotny and M. Thess: Realtime Data Mining: Self-Learning Techniques for Recommendation Engines (ISBN 978-3-319-01320-6)Google Scholar
66. 66.
A. Zayed and G. Schmeisser: New Perspectives on Approximation and Sampling Theory: Festschrift in Honor of Paul Butzer’s 85 th Birthday (978-3-319-08800-6)Google Scholar
67. 67.
R. Balan, M. Begue, J. Benedetto, W. Czaja, and K.A Okoudjou: Excursions in Harmonic Analysis, Volume 3 (ISBN 978-3-319-13229-7)Google Scholar
68. 68.
H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral: Compressed Sensing and its Applications (ISBN 978-3-319-16041-2)Google Scholar
69. 69.
S. Dahlke, F. De Mari, P. Grohs, and D. Labate: Harmonic and Applied Analysis: From Groups to Signals (ISBN 978-3-319-18862-1)Google Scholar
70. 70.
G. Pfander: Sampling Theory, a Renaissance (ISBN 978-3-319-19748-7)Google Scholar
71. 71.
R. Balan, M. Begue, J. Benedetto, W. Czaja, and K.A Okoudjou: Excursions in Harmonic Analysis, Volume 4 (ISBN 978-3-319-20187-0)Google Scholar
72. 72.
O. Christensen: An Introduction to Frames andRieszBases, Second Edition (ISBN 978-3-319-25611-5)Google Scholar
73. 73.
E. Prestini: The Evolution of Applied Harmonic Analysis:Models of the Real World, Second Edition (ISBN 978-1-4899-7987-2)Google Scholar
74. 74.
J.H. Davis: Methods of Applied Mathematics with a Software Overview, Second Edition (ISBN 978-3-319-43369-1)Google Scholar
75. 75.
M. Gilman, E. M. Smith, S. M. Tsynkov: Transionospheric Synthetic Aperture Imaging (ISBN 978-3-319-52125-1)Google Scholar
76. 76.
S. Chanillo, B. Franchi, G. Lu, C. Perez, E.T. Sawyer: Harmonic Analysis, Partial Differential Equations and Applications (ISBN 978-3-319-52741-3)Google Scholar
77. 77.
R. Balan, J. Benedetto, W. Czaja, M. Dellatorre, and K.A Okoudjou: Excursions in Harmonic Analysis, Volume 5 (ISBN 978-3-319-54710-7)Google Scholar
78. 78.
I. Pesenson, Q.T. Le Gia, A. Mayeli, H. Mhaskar, D.X. Zhou: Frames and Other Bases in Abstract and Function Spaces: Novel Methods in Harmonic Analysis, Volume 1 (ISBN 978-3-319-55549-2)Google Scholar
79. 79.
I. Pesenson, Q.T. Le Gia, A. Mayeli, H. Mhaskar, D.X. Zhou: Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, Volume 2 (ISBN 978-3-319-55555-3)Google Scholar
80. 80.
F. Weisz: Convergence and Summability of Fourier Transforms and Hardy Spaces (ISBN 978-3-319-56813-3)Google Scholar
81. 81.
For an up-to-date list of ANHA titles, please visit http://www.springer.com/series/4968