Bochner–Riesz operators in grand lebesgue spaces

Abstract

We provide the conditions for the boundedness of the Bochner–Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate for the constant appearing in the Lebesgue–Riesz norm estimation of the Bochner–Riesz operator and we investigate the convergence of the Bochner–Riesz approximation in Lebesgue–Riesz spaces.

1 Introduction

The well-known (pseudo-differential) Bochner-Riesz linear operator $B_R^{\alpha }[f]$, acting on measurable functions $f: \mathbb {R}^n \rightarrow \mathbb {R}$ ($n>1$), is defined as follows

$$\begin{aligned} B_R^{\alpha }[f](t) {\mathop {=}\limits ^{def}} \int _{\mathbb {R}^n} e^{-i(t,y)} \ \left( \ 1 - \frac{|y|^2}{R^2} \ \right) ^{\alpha }_+ \ \tilde{f}(y) \ dy, \end{aligned}$$

(1.1)

where

$$\begin{aligned} R = \mathrm{const} > 0, \ \ \alpha = \mathrm{const}, \ \ z_+ = \max (z,0), \ \ z \in \mathbb {R}, \end{aligned}$$

(1.2)

(x, y) stands for the scalar (inner) product of two vectors $x,y\in \mathbb {R}^n$, $|y|^2 := (y,y)$ and $\tilde{f} \$ denotes, as ordinary, the Fourier transform

$$\begin{aligned} \tilde{f}(y) = \int _{\mathbb {R}^n}e^{i(x,y)} \, f(x) \, dx. \end{aligned}$$

(1.3)

The applications of these operators are described in particular in Functional Analysis (see [10, 23]) and in statistics of random processes and field (see [5, 41]).

The classical Lebesgue-Riesz norm $||f||_p, \ p \in [1,\infty ]$, for the function f, is denoted by

$$\begin{aligned} ||f||_p:= & {} \left[ \ \int _{\mathbb {R}^n} |f(x)|^p \ dx \ \right] ^{1/p}, \ \ \ 1 \le p < \infty ,\\ ||f||_{\infty }:= & {} {{\,\mathrm{ess\,sup}\,}}_{x \in \mathbb {R}^n} |f(x)|, \ \ \ p=\infty \end{aligned}$$

and the corresponding Banach space is, as usually,

$$\begin{aligned} L_p = L_p(\mathbb {R}^n) = \{ \ f: \ ||f||_p < \infty \ \}. \end{aligned}$$

Denote, for an arbitrary linear or quasi linear operator U, acting from $L_p$ to $L_q, \ p,q \ge 1$, its norm

$$\begin{aligned} ||U||_{p,q} = ||U||_{p \rightarrow q} := \sup _{0 \ne f \in L_p} \left[ \ \frac{||Uf||_q}{||f||_p} \ \right] . \end{aligned}$$

Of course, the operator U is bounded as the operator acting from the space $L_p$ into the space $L_q$ iff $||U||_{p,q} < \infty$.

More generally, for the operator U acting from some Banach space F equipped with the norm $||\cdot ||_F$ into another (in the general case) Banach space D, having the norm $||\cdot ||_D$, we denote as usually

$$\begin{aligned} ||U||_{F \rightarrow D} := \sup _{0 \ne f \in F} \left[ \ \frac{||Uf||_D}{||f||_F} \ \right] . \end{aligned}$$

There exists a huge numbers of works devoted to the $\ p,q \$ estimates for Bochner-Riesz operators $\ B_R^{\alpha }, \$ as a rule for the case $\ q = p, \$ see e.g. [23, chapter5], [9, 10, 13, 29, 35, 42, 43], etc. The boundedness of these operators in Morrey–Lorentz spaces and in $L^p$ spaces with variable exponent has been investigated in [25] and [8], respectively.

Our aim, in this paper, is to extend some results contained in the above mentioned works concerning upper estimates for the norm of the Bochner-Riesz operator, in the case of different Lebesgue–Riesz spaces (Sect. 2) and to extend these estimates to the so-called Grand Lebesgue Spaces (GLS), see Sect. 3.

We deduce also a non trivial quantitative lower estimate for the coefficient in the Lebesgue–Riesz norm estimation for the Bochner-Riesz operator (Sect. 4) and we study the convergence of the Bochner–Riesz approximation in Lebesgue-Riesz spaces (Sect. 5).

2 Norm estimates for the Bochner–Riesz operator acting between different Lebesgue-Riesz spaces

We clarify slightly the known results about the Lebesgue–Riesz p, r norms for the operators $B_R^{\alpha }$ defined in (1.1), see e.g. [10, 12, 23, 29, 43], ecc.

It is important to observe that the Bochner–Riesz operator $\ B_R^{\alpha }$ may be rewritten as a convolution, namely

$$\begin{aligned} B_R^{\alpha }[f](t) = f*K_{\lambda }^R(t) = \int _{\mathbb {R}^n} f(t-s) \ K_{\lambda }^R (s) \ ds, \end{aligned}$$

(2.1)

where $R = \mathrm{const} > 0$,

$$\begin{aligned} K_{\lambda }^R(z)= & {} C(\alpha ) \ R^n \ \frac{J_{\lambda }(R|z|)}{|Rz|^{\lambda }}, \ \ \ \lambda =\alpha + \frac{n}{2},\\ C(\alpha ):= & {} \frac{1}{\Gamma (\alpha + 1)}, \ \ \ \alpha > -1, \end{aligned}$$

$\Gamma (\cdot )$ is the Gamma-function, $J_{\lambda }(\cdot )$ is the Bessel function of order $\lambda$ and $|z| = \sqrt{(z,z)}$.

Notice that, under our restriction on $\alpha$,

$$\begin{aligned} \int _{\mathbb {R}^n} K_{\lambda }^R(t)\ dt = 1. \end{aligned}$$

(2.2)

Evidently,

$$\begin{aligned} K_{\lambda }^R(z) = R^n \ K_{\lambda }(Rz), \end{aligned}$$

(2.3)

where in turn

$$\begin{aligned} K_{\lambda }(z) {\mathop {=}\limits ^{def}} \ K_{\lambda }^1(z) = C(\alpha ) \frac{J_{\lambda }(|z|)}{|z|^{\lambda }}, \end{aligned}$$

(2.4)

see, e.g., [10, pp. 171–172].

Note that, from the identity (2.3), it follows

$$\begin{aligned} ||K_{\lambda }^R(z)||_q = R^{n - n/q} \ || K_{\lambda }||_q, \ \ \ q \ge 1, \end{aligned}$$

(2.5)

as long as $\ R > 0. \$

Assume here and in the sequel

$$\begin{aligned} -1 < \alpha \le \alpha _0, \alpha _0 := \frac{n-1}{2}, \end{aligned}$$

(2.6)

and introduce the value

$$\begin{aligned} q_0 := \frac{n}{(n+1)/2 + \alpha }, \end{aligned}$$

(2.7)

so that

$$\begin{aligned} q_0 \ge 1 \ \ \Leftrightarrow \ \ \frac{-(n+1)}{2} < \alpha \le \alpha _0, \end{aligned}$$

where the inequality on the left hand side is true due to the restriction $\alpha >-1$.

Moreover, using the well-known results about the behavior of the Bessel functions (see, e.g., [10, p. 172]),

$$\begin{aligned} K_{\lambda }(z)\asymp |z|^{-(\frac{n+1}{2}+\alpha )}, \ \ \ \mathrm{as} \ |z|\rightarrow \infty , \end{aligned}$$

it is easy to estimate

$$\begin{aligned} ||K_{\lambda }||_q \asymp (q - q_0)^{-1/q}, \ \ \ q > q_0, \end{aligned}$$

(2.8)

and $\ ||K_{\lambda }||_q = \infty \$ otherwise (see also, e.g., [23, pp. 339-341]).

Correspondingly,

$$\begin{aligned} ||K^R_{\lambda }||_q \asymp R^{n - n/q} \ (q - q_0)^{-1/q}, \ \ \ q > q_0. \end{aligned}$$

(2.9)

Furthermore, we recall the Beckner–Brascamp–Lieb–Young inequality for the convolution (see [3, 4])

$$\begin{aligned} ||f*g||_r \le \left( \frac{C_p C_q}{C_r}\right) ^n \ ||f||_p \ ||g||_q, \ \ \ \ 1 + 1/r=1/p + 1/q , \end{aligned}$$

(2.10)

where $p,q, r \ge 1$ and

$$\begin{aligned} C_m = \left( \frac{m^{1/m}}{(m')^{1/m'}}\right) ^{\frac{1}{2}}, \ \ \ m' = m/(m-1). \end{aligned}$$

We point out that the constant $C_m$ defined above, and considered in (2.10) for $m=p,q,r$, does not blow up when $m\rightarrow 1$ (which implies $m'\rightarrow \infty$), in fact $C_m\rightarrow 1$ in this case. Therefore p, q, r are allowed to be 1. Note that

$$\begin{aligned} \frac{C_p C_q}{C_r} \le 1. \end{aligned}$$

Recently in [22] has been given a generalization of the convolution inequality in the context of the Grand Lebesgue Spaces (see Sect. 3 for the definition), built on a unimodular locally compact topological group.

The estimate (2.10) is essentially non-improbable. Indeed, the equality in (2.10) is attained iff both functions f, g are proportional to Gaussian densities, namely there exists positive constants $c_1,c_2,c_3,c_4$ such that

$$\begin{aligned} f(x) = c_1 \exp (-c_2|x|^2), \ \ \ \ g(x) = c_3 \exp (-c_4|x|^2), \ \ \ \ x \in \mathbb R^n, \end{aligned}$$

(2.11)

where $|x|=\sqrt{(x,x)}$ is the euclidean norm. Then the convolution $f*g$ is also Gaussian and

$$\begin{aligned} ||f*g||_r = \left( \frac{C_p C_q}{C_r}\right) ^n \ ||f||_p \ ||g||_q, \ \ \ 1/p + 1/q = 1 + 1/r, \ \ \ p,q,r>1. \end{aligned}$$

(2.12)

The following result about the boundedness of the Bochner–Riesz operator, acting between different Lebesgue–Riesz spaces, holds.

Theorem 2.1

Let $p>1$ and $f\in L_p(\mathbb {R}^n)$. Let $\alpha$ be a constant such that

$$\begin{aligned} - 1 < \alpha \le \alpha _0 := \frac{n-1}{2} \end{aligned}$$

(2.13)

and

$$\begin{aligned} q_0 = \frac{n}{(n+1)/2 + \alpha }. \end{aligned}$$

(2.14)

Let $q,r\ge 1$ such that $1 + 1/r = 1/q +1/p$ and assume $q>q_0, \ r>r_0, \ p\le p_0$, where

$$\begin{aligned} r_0= & {} r_0(\alpha ,n; \, p) := \frac{p q_0}{p + q_0 - pq_0}, \end{aligned}$$

(2.15)

$$\begin{aligned} p_0= & {} q_0'= (1 - 1/q_0)^{-1} \end{aligned}$$

(2.16)

Define, for $R = \mathrm{const} > 0$,

$$\begin{aligned} W(\alpha ,n, R; \ p,r) {\mathop {=}\limits ^{def}} R^{n(1/p - 1/r)} \cdot [ \ q(p,r) - q_0(\alpha ,n) \ ]^{1/p - 1- 1/r }. \end{aligned}$$

(2.17)

Then the Bochner–Riesz operator satisfies

$$\begin{aligned} ||B_R^{\alpha } [f]||_r \le C(\alpha ,R) \ \cdot W(\alpha , n, R; \ p,r) \ \cdot ||f||_p. \end{aligned}$$

(2.18)

Proof

If $f \in L_p$ for some value $p>1$, then by (2.10) it follows

$$\begin{aligned} ||B_R^{\alpha }[f]||_r \le ||K_{\lambda }^R||_q \ ||f||_p, \end{aligned}$$

(2.19)

where $1 + 1/r = 1/q +1/p \$ and $\ p, \ q, \ r \ge 1, \ q > q_0$. On the other words,

$$\begin{aligned} q = q(p,r) {\mathop {=}\limits ^{def}} \frac{pr}{pr + p - r }, \ \ \ q> q_0, \ \ \ r> p. \end{aligned}$$

Moreover it is easy to verify that $r_0$, defined in (2.15), is such that $r_0> p$.

Under our restrictions $W(\alpha ,n, R; \ p,r)$ is finite and positive.

So we conclude that estimate (2.18) holds. $\square$

3 Main result: boundedness of Bochner–Riesz operators in Grand Lebesgue Spaces (GLS)

We recall here, for reader convenience, some known definitions and facts from the theory of Grand Lebesgue Spaces (GLS).

Definition 3.1

Let $\ \psi = \psi (p), \ p \in (a,b)$, $a,b = \mathrm{const}, \ 1 \le a < b \le \infty$, be a positive measurable numerical valued function, such that

$$\begin{aligned} \inf _{p \in (a,b)} \psi (p) > 0. \end{aligned}$$

(3.1)

The (Banach) Grand Lebesgue Space $G\psi =G\psi (a,b)$ consists of all the real (or complex) numerical valued measurable functions $f: \mathbb {R}^n \rightarrow \mathbb R$ having finite norm, defined by

$$\begin{aligned} || f ||_{G\psi } :=\sup _{p \in (a,b)} \left[ \frac{||f||_p}{\psi (p)} \right] . \end{aligned}$$

(3.2)

We agree to write $\ G\psi \$ in the case when $a = 1$ and $b = \infty$.

The function $\psi$ is named generating function for the space $G\psi$ and we denote by $\{\psi (\cdot )\}$ the set of all such functions.

For instance

$$\begin{aligned} \psi (p) := p^{1/m}, \ \ \ p \in [1,\infty ), \ \ \ m > 0, \end{aligned}$$

(3.3)

or

$$\begin{aligned} \psi (p) := (p-a)^{-\alpha } \ (b-p)^{-\beta }, \ \ \ p \in (a,b), \ \ \ 1\le a<b<\infty , \ \ \ \alpha ,\beta \ge 0, \end{aligned}$$

(3.4)

are generating functions.

If

$$\begin{aligned} \psi (p) = 1, \ \ p = r; \ \ \ \ \psi (p) = +\infty , \ \ p \ne r, \ \ \ \ r\in [1,\infty ), \end{aligned}$$

where $C/\infty := 0, \ C \in \mathbb R$ (extremal case), then the corresponding $G\psi$ space coincides with the classical Lebesgue-Riesz space $L_r = L_r(\mathbb {R}^d)$.

The Grand Lebesgue Spaces and several generalizations of them have been widely investigated, mainly in the case of GLS on sets of finite measure, (see, e.g., [6, 11, 14, 20, 26, 27, 34, 36], etc). They play an important role in the theory of Partial Differential Equations (PDEs) (see, e.g., [2, 15, 17, 24], etc.), in interpolation theory (see, e.g., [16, 19]), in the theory of Probability ([21, 38, 40]), in Statistics [36, chapter 5], in theory of random fields [30, 39], in Functional Analysis [36, 37, 39] and so one.

These spaces are rearrangement invariant (r.i.) Banach functional (complete) spaces; their fundamental functions have been considered in [39]. They do not coincide, in the general case, with the classical Banach rearrangement functional spaces: Orlicz, Lorentz, Marcinkiewicz etc., see [34, 37]. The belonging of a function $f: \mathbb {R}^n \rightarrow \mathbb {R}$ to some $G\psi$ space is closely related with its tail function behavior as $\ t \rightarrow 0+ \$ as well as when $\ t \rightarrow \infty , \$ see [30, 32].

The Grand Lebesgue Spaces can be considered not only on the Euclidean space $\mathbb {R}^n$ equipped with the Lebesgue measure, but also on an arbitrary measurable space with sigma-finite non-trivial measure.

In the following Theorem we investigate the boundedness of the Bochner-Riesz operator acting from some Grand Lebesgue Space $G\psi =G\psi (a,b)$ into another one $G\nu$. We will consider the same restrictions and quantities defined in Theorem 2.1.

Theorem 3.1

Let $1\le a<b\le \infty$ and $f\in G\psi (a,b)$. Let $\alpha$ be a constant such that

$$\begin{aligned} - 1 < \alpha \le \alpha _0 := \frac{n-1}{2}, \ \ \ n>1. \end{aligned}$$

Let $p>1$, $q, r\ge 1$, $p_0, q_0, r_0$ and $W(\alpha ,n, R; \ p,r)$ defined as in Theorem 2.1.

Denote

$$\begin{aligned} s = s(\alpha ,n;r) := \min \left\{ b, \ \frac{r q_0}{r q_0 + q_0 - r} \right\} , \end{aligned}$$

and

$$\begin{aligned} d := r_0(\alpha ,n; \, b)=\frac{bq_0 }{q_0 +b - b q_0}. \end{aligned}$$

For $r\in (d,\infty )$ let $\nu (r)$ be the following generating function

$$\begin{aligned} \nu (r) = \nu [\psi ](r) = \nu [\psi ](\alpha ,n,R; r) := \inf _{p \in (a,s)} [ \ W(\alpha ,n, R; \, p,r) \ \psi (p) \ ]. \end{aligned}$$

(3.5)

Then

$$\begin{aligned} || B_R^{\alpha }[f]||_{G\nu } \le C(\alpha ,n,R) \ ||f||_{G\psi }. \end{aligned}$$

(3.6)

Proof

The proof is simple and alike as the one in [40]. First we observe that $\nu (r)$ is finite. One can assume, without loss of generality, $||f||_{G\psi } = 1$, then $||f||_p \le \psi (p), \ p \in (a,b)$. Applying the inequality (2.18) we have

$$\begin{aligned} ||B_R^{\alpha }[f]||_r \le C(\alpha , n,R) \ \cdot W(\alpha ,n, R; \ p,r) \ \cdot \psi (p). \end{aligned}$$

(3.7)

Taking the minimum over p subject to our limitations, we get

$$\begin{aligned} ||B_R^{\alpha } [f]||_r\le & {} C(\alpha ,n,R) \, \cdot \inf _{p \in (a,s)} [ \ W(\alpha ,R; \, p,r) \ \cdot \psi (p) \ ]\nonumber \\= & {} C(\alpha ,n,R) \ \nu (r) = C(\alpha ,n,R) \ \nu (r) ||f||_{G\psi }, \end{aligned}$$

(3.8)

which is quite equivalent to our claim in (3.6). $\square$

Remark 3.1

Let us give the following generalization of Theorem 3.1 Let $\ \zeta = \zeta (r), \ r \in (d, \ \infty )$, be some another generating function from the set $\ G\Psi [d, \infty ], \$ such that a new function

$$\begin{aligned} \kappa (r) {\mathop {=}\limits ^{def}} \sup _{p \in (a,s)} \left\{ \ \frac{W(\alpha ,R; \ p,r) \ \psi (p)}{\zeta (r)} \ \right\} \end{aligned}$$

also belongs to this set. We deduce from (3.7)

$$\begin{aligned} \frac{ \ ||B_R^{\alpha }[f]||_r \ }{\zeta (r)} \le C(\alpha , n,R) \ \cdot \frac{W(\alpha ,n, R; \ p,r) \ \cdot \psi (p)}{\zeta (r)} \ \cdot ||f||G\psi , \end{aligned}$$

(3.9)

hence

$$\begin{aligned} \frac{ \ ||B_R^{\alpha }[f]||_r \ }{\zeta (r)} \le C(\alpha , n,R) \ \kappa (r) \ ||f||G\psi . \end{aligned}$$

Dividing on $\ \kappa (r) \$ and taking the supremum over $\ r \$, if we denote $\ \theta (r) = \zeta (r) \kappa (r) \$, we have

$$\begin{aligned} ||B_R^{\alpha }[f]||_{G\theta } \le C(\alpha , n,R) \ ||f||_{G\psi }. \end{aligned}$$

(3.10)

4 Lower bound for the coefficient in the Lebesgue–Riesz norm estimate for the Bochner–Riesz operator

Let $p,r>1$, $n>1$ and let us introduce the following variable

$$\begin{aligned} Q_n(p,r) {\mathop {=}\limits ^{def}} \sup _{\alpha> 0} \sup _{R > 0} W(\alpha ,n,R; \ p,r), \end{aligned}$$

where $W(\alpha ,n,R; \ p,r)$ is defined in Sect. 2. Our target in this Section is a lower bound for the above variable.

Theorem 4.1

Let $p,r> 1$, $n>1$ and

$$\begin{aligned} \Theta (n,p) := (2 \ \pi )^{ n(1 - p)/2p } \ p^{-n/2p}. \end{aligned}$$

Then

$$\begin{aligned} Q_n(p,r) \ge \Theta \left( n, \frac{pr}{pr + p - r} \ \right) , \ \ \ r> p. \end{aligned}$$

(4.1)

Remark 4.1

The possible case when $\ Q_n(p,r) = + \infty \$ can not be excluded.

Proof

We will apply equality (2.12), in which we choose the ordinary Gaussian density

$$\begin{aligned} f_0(x) := (2 \pi )^{-n/2} \exp (-|x|^2/2), \ \ x \in \mathbb {R}^n, \end{aligned}$$

and take $\ \alpha = R^2/2. \$ Obviously

$$\begin{aligned} Q_n(p,r) \ge \lim _{R \rightarrow \infty } W(R^2/2,n,R; \, p,r). \end{aligned}$$

We have

$$\begin{aligned} B_R^{R^2}[f](t) = \int _{\mathbb {R}^n} e^{-i(t,y)} \left( \ 1 - \frac{|y|^2}{R^2} \ \right) ^{R^2/2} I(|y| < R) \ \tilde{f}(y) \ dy , \end{aligned}$$

where $\ I(A) \$ denotes the indicator function of the (measurable) set $\ A, \ A \subset \mathbb {R}^n$.

Therefore, as $R \rightarrow \infty \$,

$$\begin{aligned} B_R^{R^2}[f](t)\rightarrow \int _{\mathbb {R}^n} e^{-i(t,y)} \ \exp \{- |y|^2/2 \ \} \,\tilde{f}(y) \ dy =[\sqrt{2 \pi } ]^n \ [f*f_0](t), \end{aligned}$$

(4.2)

by virtue of dominated convergence theorem.

If we take $\ f = f_0, \$ then in (4.2) the convolution of two Gaussian densities appears.

It remains to apply the relation (2.12); we omit some simple calculations. $\square$

5 Convergence of Bochner–Riesz operators

We investigate here the convergence, as $R \rightarrow \infty$, of the family of Bochner-Riesz approximations $\ B_R^{\alpha }[f] \$ to the source function f in the Lebesgue-Riesz norm $\ L_p(\mathbb {R}^n), \ p \in (1,\infty )$, in addition to the similar results in [13, 23, 29], etc.

For any function $f\in L_p(\mathbb {R}^n)$, its modulus of $\ L_p \$ continuity is defined alike as in approximation theory [1, chapter V] :

$$\begin{aligned} \omega _p[f](\delta ) {\mathop {=}\limits ^{def}} \sup _{h \, : \ |h| \le \delta } ||T_h[f] - f||_p, \ \ \ \delta \ge 0, \end{aligned}$$

where $\ T_h[f] \$ denotes the shift operator

$$\begin{aligned} T_h[f](t) =f(t - h), \ \ \ t,h \in \mathbb {R}^n. \end{aligned}$$

Obviously, $\ \omega _p[f](\delta ) \le 2 ||f||_p \$ and

$$\begin{aligned} \lim _{\delta \rightarrow 0+} \omega _p[f](\delta ) = 0, \ \ \ f \in L_p. \end{aligned}$$

(5.1)

Theorem 5.1

Let $p \in (1,\infty ), \ \alpha \le \alpha _0 = (n-1)/2$   and $\ f \in L_p(\mathbb {R}^n)$. Then

$$\begin{aligned} \lim _{R \rightarrow \infty } ||B_R^{\alpha }[f] - f||_p = 0. \end{aligned}$$

(5.2)

Proof

The difference $\ \Delta _R[f](t) = B_R^{\alpha } [f]-f \$ has the form

$$\begin{aligned} \Delta _R[f](t) = \int _{\mathbb {R}^n} \left\{ \ f \left( t - \frac{v}{R} \right) - f(t) \ \right\} \ K_{\lambda }(|v|) \ dv. \end{aligned}$$

We apply now the triangle inequality for the $\ L_p \$ norm in the integral form

$$\begin{aligned} ||\Delta _R[f]||_p\le & {} \int _{\mathbb {R}^n} ||T_{|v|/R}[f] - f||_p \ |K_{\lambda }(|v|)| \ dv \\\le & {} \int _{\mathbb {R}^n} \omega _p[f](|v|/R) \ |K_{\lambda }(|v|)| \ dv. \end{aligned}$$

Note that, under the above conditions,

$$\begin{aligned} \int _{\mathbb {R}^n} |K_{\lambda }(|v|)| \ dv < \infty , \end{aligned}$$

so that (5.2) follows again from the dominated convergence theorem. $\square$

Remark 5.1

As a slight consequence, under the above conditions, if $f \in L_p(\mathbb {R}^n)$, then

$$\begin{aligned} ||B_R^{\alpha }[f]||_p \le ||f||_p + \int _{\mathbb {R}^n} \omega _p[f](|v|/R) \ |K_{\lambda }(|v|)| \ dv \end{aligned}$$

and, consequently,

$$\begin{aligned} \forall f \in L_p(\mathbb {R}^n) \ \ \Rightarrow \ \ \sup _{R \ge 1} ||B_R^{\alpha }[f]||_p \le 3 ||f||_p\,. \end{aligned}$$

The case $\ p = \infty \$ requires a separate consideration. Introduce the Banach space $\ C_0(\mathbb {R}^n) \$ as a collection of all bounded and uniformly continuous functions $\ f: \mathbb {R}^n \rightarrow \mathbb {R}\$, equipped with the ordinary norm,

$$\begin{aligned} ||f||_{\infty } {\mathop {=}\limits ^{def}} \sup _{t \in \mathbb {R}^n}|f(t)|. \end{aligned}$$

As above

$$\begin{aligned} \omega _{\infty }[f](\delta ) {\mathop {=}\limits ^{def}} \sup _{h \,: \ |h| \le \delta } ||T_h[f] - f||_{\infty }, \ \ \ \delta \ge 0. \end{aligned}$$

Evidently, $\ \omega _{\infty }[f](\delta ) \le 2 ||f||_{\infty } \$ and

$$\begin{aligned} \lim _{\delta \rightarrow 0+} \omega _{\infty }[f](\delta ) = 0, \ \ \ f \in L_p. \end{aligned}$$

(5.3)

The assertion of Theorem 5.1 under the same conditions remains true in the case $\ p = \infty . \$

Theorem 5.2

Under the same conditions of Theorem 5.1, for any function $\ f: \mathbb {R}^n \rightarrow \mathbb {R} \$ from the space $\ C_0(\mathbb {R}^n) \$, its Bochner-Riesz approximation $\ B_R^{\alpha } \$ converges uniformly to the source function f, that is

$$\begin{aligned} \lim _{R \rightarrow \infty } ||B_R^{\alpha }[f] - f||_{\infty } = 0. \end{aligned}$$

(5.4)

Proof

The proof is the same as in Theorem 5.1 and may be omitted. $\square$

Remark 5.2

As before, if $f\in C_0(\mathbb {R}^n)$, then

$$\begin{aligned} ||B_R^{\alpha }[f]||_{\infty } \le ||f||_{\infty } + \int _{\mathbb {R}^n} \omega _{\infty }[f](|v|/R) \ |K_{\lambda }(|v|)| \ dv \end{aligned}$$

and moreover

$$\begin{aligned} \forall f \in C_0(\mathbb {R}^n) \ \ \Rightarrow \ \ \sup _{R \ge 1} ||B_R^{\alpha }[f]||_{\infty } \le 3 ||f||_{\infty }. \end{aligned}$$

6 Concluding remarks

A.:

In our opinion, the method described in this paper may be essentially generalized to more operators of convolutions type, linear or not. See some preliminary results [34].

B.:

The study of the maximal Bochner-Riesz operator $B^\alpha _*$, defined by

$$\begin{aligned} B^\alpha _*[f](t)=\sup _{R>0}|B^\alpha _R[f](t)|, \end{aligned}$$

is closely related to the pointwise convergence of the associated Bochner–Riesz operator $B^\alpha _R[f]$ as $R \rightarrow \infty$, for any $f\in L_p(\mathbb {R}^n), \ p \in (1,\infty )$, see e. g. [7, 33, 44].

It is interesting to generalize the estimates obtained in the previous Sections to the maximal operator associated to the Bochner–Riesz one considered here, in the spirit of the works [18, 28], and so one:

$$\begin{aligned} ||\sup _{R \ge 1} B_R^{\alpha }[f]||_p \le M(\alpha ,n;p) \cdot ||f||_p, \ \ \ p \ge p_0, \end{aligned}$$

or

$$\begin{aligned} ||\sup _{R \ge 1} B_R^{\alpha }[f]||_r \le M(\alpha ,n; p,r) \cdot ||f||_p, \ \ p \ge p_0, \ \ \ r = r(p), \end{aligned}$$

in order to obtain the GLS estimate for the Bochner-Riesz maximal operator of the form

$$\begin{aligned} ||\sup _{R \ge 1} B_R^{\alpha }[f]||_{G\zeta } \le L(\alpha ,n; \zeta ,\psi ) \cdot ||f||_{G\psi } \end{aligned}$$

for some generating functions $\ \psi , \ \zeta . \$

C. :

It is not hard to generalize the estimates obtained here in the case when the kernels are of the form

$$\begin{aligned} \phi (t) := \prod _{s=1}^k (1 - t^{\alpha (s)})_+^{m(s)}, \end{aligned}$$

instead of the the kernel $(1 - t^2)^{\alpha }_+$, considered here. Necessary estimates for this purpose for the Fourier transform $\ F[\phi ](y) \$ of these functions may be found in [45, 46].