# Channel Orthogonalization with Reconfigurable Surfaces

Juan Vidal Alegría1, Fredrik Rusek12
<EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of
Electrical and Information Technology, Lund University, Lund, Sweden
2Sony Research Center, Lund, Sweden

###### Abstract

Orthogonal multi-user multiple-input multiple-output (MU-MIMO) channels allow
for optimum performance with simplified precoding/equalization, and they
achieve maximum multiplexing gain which is shared fairly among users.
Reconfigurable intelligent surface (RIS) constitutes a promising cost-
efficient solution to improve the wireless channel, since they consist of
passive reflecting elements able to adjust the phases of the incoming waves.
However, it is still widely unclear how these surfaces can improve spatial-
multiplexing. In fact, the common RIS model cannot achieve perfect
orthogonalization of MU-MIMO channels with a reasonable number of elements.
Furthermore, efficient channel estimation algorithms for RIS, which are key
for taking advantage of its benefits, are still a matter of research. We study
two types of reconfigurable surfaces (RSs), namely amplitude-reconfigurable
intelligent surface (ARIS) and fully-reconfigurable intelligent surface
(FRIS), with extended capabilities over RIS. We show how these RSs allow for
perfect channel orthogonalization, and, by minimizing the applied power, we
show that they can potentially be implemented without the need of
amplification. We also present an efficient channel estimation method for each
of them that allows the base station (BS) to select the desired propagation
channel.

###### Index Terms:

Reconfigurable surface (RS), MU-MIMO, Reconfigurable intelligent surface
(RIS), Amplitude-reconfigurable intelligent surface (ARIS), Fully-
reconfigurable intelligent surface (FRIS), channel orthogonalization.

##  I Introduction

Multi-user multiple-input multiple-output (MU-MIMO) [1], has become a standard
solution for base station (BS) implementation in 5G networks. After the first
commercial deployments of Massive MIMO [2, 3], its large scale counterpart,
MU-MIMO is now a mature technology that allows multiplexing user equipments
(UEs) in the spatial domain. However, the ability to exploit multiplexing
gains with MU-MIMO depends on the conditions of the wireless propagation
channels.

Reconfigurable intelligent surface (RIS) has emerged as a promising enabling
technology towards future generation networks [4, 3, 5]. Also known as
intelligent reflective surface (IRS), this technology works as a passive
reflector which can adjust the propagation environment in a power and cost-
efficient manner. The reflected waves at the RIS can be redirected to create
constructive interference and increase the received signal, leading to
impressive gains in energy efficiency [6, 7].

Previous work has also considered RIS for improving spatial multiplexing in
MIMO settings. For example, [8] uses RIS for improving the rank of a single-
user MIMO channel. RIS has also been considered for maximizing the user rates
in different settings [9, 10]. However, most of the previous results rely on
the availability of channel state information at the BS, while channel
estimation in RIS scenarios becomes extremely challenging [5] due to its
limited capabilities and large number of elements.

In this work, we study two alternatives to RIS, namely amplitude-
reconfigurable surface (ARIS) and fully-reconfigurable surface (FRIS), for
orthogonalizing MU-MIMO channels. To the best of our knowledge, the available
research has not considered the fundamental problem of using reconfigurable
surfaces (RS) for obtaining orthogonal MIMO channels, while this is of
essential interest since it leads to full-multiplexing gain with fair user
sharing (same channel power per UE) [11]. Furthermore, for these channels,
optimal processing at the base station is achieved by simple maximum ratio
combining/transmission (MRC/MRT). We present a channel estimation method for
each RS that allows the BS to select its own channel by computing and sending
the RS configuration with a reduced number of pilots. We also show that these
RSs can be realized without the need for amplification by minimizing the
required power.

The rest of the paper is organized as follows. Section II describes the system
model, and defines the different RSs capabilities. Section III describes how
to achieve perfectly orthogonal channels with RSs. In Section IV, we present
the channel estimation processes for configuring the RSs. Section V the power
minimization setting. Section VI gives numerical results. The paper is
concluded in Section VII.

##  II System model

Let us consider an uplink MU-MIMO scenario where $K$ UEs are transmitting to
an $M$-antenna BS, with $M>K$, through a narrow-band channel with the aid of
an $N$-element RS. The $M\times 1$ received complex vector, $\boldsymbol{y}$,
can be expressed as

$\boldsymbol{y}=\boldsymbol{H}\boldsymbol{s}+\boldsymbol{n},$ (1)

where $\boldsymbol{H}$ is the $M\times K$ channel matrix, $\boldsymbol{s}$ is
the $K\times 1$ vector of symbols transmitted by the UEs, with
$\mathbb{E}(|s_{k}|^{2})=E_{s}$ $\forall k$, and
$\boldsymbol{n}\sim\mathcal{CN}(\boldsymbol{0}_{M\times
1},N_{0}\mathbf{I}_{M})$ is the noise vector. Considering that there exists a
direct channel, as well as a reflected channel through the RS, we can express
the channel matrix as

$\boldsymbol{H}=\boldsymbol{H}_{0}+\boldsymbol{H}_{1}\boldsymbol{\Theta}\boldsymbol{H}_{2},$
(2)

where $\boldsymbol{H}_{0}$ corresponds to the $M\times K$ direct channel
between the BS and the UEs, $\boldsymbol{H}_{1}$ and $\boldsymbol{H}_{2}$
correspond to the $M\times N$ channel between the BS and the RS and the
$N\times K$ channel between the RS and the UEs, respectively, and
$\boldsymbol{\Theta}$ is the reflection matrix applied at the RS.

In the literature, it is common to restrict the RS to have phase shifting
capabilities. This corresponds to the widely known concept of RIS where

$\boldsymbol{\Theta}_{\mathrm{RIS}}=\mathrm{diag}\left(\exp(j\phi_{1}),\dots,\exp(j\phi_{N})\right).$
(3)

In this paper, however, we propose two RS technologies where said restriction
is relaxed, and we compare them in the task of orthogonalizing the channel
matrix.

Let us consider an RS, here referred to as ARIS, whose elements can also
adjust amplitude. The corresponding reflection matrix is then defined by

$\boldsymbol{\Theta}_{\mathrm{ARIS}}=\mathrm{diag}\left(\alpha_{1},\dots,\alpha_{N}\right),\;\;\alpha_{i}\in\mathbb{C}\;\;\forall
i.$ (4)

Note that the restriction of having each $\alpha_{i}$ of amplitude $1$ is here
relaxed. The idea of adding amplification to a RIS has already been considered
in the literature, and some of the hardware implications to realize these
systems are given in [12, 13]. However, one of our goals is to restrict the
power of these surfaces so that they can still be implemented without the need
for active amplification.

We also consider an RS, here referred to as FRIS, whose reflection matrix is a
complete matrix. Thus, we have

$\boldsymbol{\Theta}_{\mathrm{FRIS}}\in\mathbb{C}^{N\times N}.$ (5)

In this work, we will not elaborate on the challenges of realizing such a RS.
However, we can think of architectures based on vector modulators such that
the matrix multiplication can be performed by an analog combiner as in [14],
although, if future technology allows it, fully-digital implementations would
be desirable so that processing is done per sub-carrier.

##  III Channel orthogonalization

The main goal of employing RSs is to adjust the propagation channel to make it
more beneficial in some metric, e.g., array gain, channel capacity,
multiplexing gain, etc. Within the considered framework, orthogonal
channels111A more accurate term would be unitary channels due to the complex
nature of the channel matrix. are channels whose columns are constructed from
unitary matrices, i.e.,
$\boldsymbol{H}=\sqrt{\beta}\widetilde{\boldsymbol{U}}$, where

$\widetilde{\boldsymbol{U}}=\boldsymbol{U}\begin{bmatrix}\mathbf{I}_{K}\\\
\boldsymbol{0}_{(M-K)\times K}\end{bmatrix},$ (6)

and $\boldsymbol{U}\in\mathcal{U}(M)$ ($M\times M$ unitary). Note the slight
abuse of notation so that $\sqrt{\beta}$ corresponds to the singular values of
the orthogonal channel. We then have

$\boldsymbol{H}^{\mathrm{H}}\boldsymbol{H}=\beta\mathbf{I}_{K}.$ (7)

Since the early research on MIMO systems, orthogonal channels were found to be
desirable for several reasons [11]:

  * •

Full multiplexing gain is available since all eigenvalues of the channel
matrix are non-zero.

  * •

Waterfilling algorithms are not required for maximizing capacity since all
eigenvalues of the channel are equal.

  * •

In the case of MU-MIMO, the users are served fairly since the different
spatial streams have equal power.

  * •

Simple linear equalization or precoding, namely MRC or MRT, achieves optimum
performance, since it can exploit the orthogonal paths of the channel without
the need for UE cooperation in MU-MIMO.

We next show how to construct $\boldsymbol{\Theta}$, for the case of ARIS and
FRIS, so that the resulting channel (2) is orthogonal.

###  III-A ARIS

We are interested in finding $\alpha_{1},\dots,\alpha_{N}$ such that

$\boldsymbol{H}_{0}+\boldsymbol{H}_{1}\boldsymbol{\Theta}_{\mathrm{ARIS}}\boldsymbol{H}_{2}=\sqrt{\beta}\widetilde{\boldsymbol{U}},$
(8)

Let us define

$\boldsymbol{H}_{1}=\begin{bmatrix}\boldsymbol{h}_{11},\dots,\boldsymbol{h}_{1N}\end{bmatrix},\;\;\;\;\boldsymbol{H}_{2}=\begin{bmatrix}\boldsymbol{h}_{21},\dots,\boldsymbol{h}_{2N}\end{bmatrix}^{\mathrm{T}},$

where $\boldsymbol{h}_{1i}$ corresponds to column $i$ of $\boldsymbol{H}_{1}$
and $\boldsymbol{h}_{2i}^{\mathrm{T}}$ corresponds to row $i$ of
$\boldsymbol{H}_{2}$. We can then rewrite (8) as

$\sum_{i=1}^{N}\alpha_{i}\boldsymbol{h}_{1i}\boldsymbol{h}_{2i}^{\mathrm{T}}=\sqrt{\beta}\widetilde{\boldsymbol{U}}-\boldsymbol{H}_{0}.$
(9)

By noting that (9) is a linear equation in the vector
$\boldsymbol{\alpha}=\begin{bmatrix}\alpha_{1},\dots,\alpha_{N}\end{bmatrix}^{\mathrm{T}}$,
we can use the vectorization operation to reach

$\boldsymbol{\mathcal{H}}_{12}\boldsymbol{\alpha}=\mathrm{vec}\left(\sqrt{\beta}\widetilde{\boldsymbol{U}}-\boldsymbol{H}_{0}\right),$
(10)

where
$\boldsymbol{\mathcal{H}}_{12}=\begin{bmatrix}\mathrm{vec}(\boldsymbol{h}_{11}\boldsymbol{h}_{21}^{\mathrm{T}})&\dots&\mathrm{vec}(\boldsymbol{h}_{1N}\boldsymbol{h}_{2N}^{\mathrm{T}})\end{bmatrix}$,
which corresponds to an $MK\times N$ matrix. Assuming
$\boldsymbol{\mathcal{H}}_{12}$ is full-rank, (10) leads to an
orthogonalization requirement for ARIS, namely $N\geq MK$. We would then solve
(10) by

$\boldsymbol{\alpha}=\boldsymbol{\mathcal{H}}_{12}^{\dagger}\mathrm{vec}\left(\sqrt{\beta}\widetilde{\boldsymbol{U}}-\boldsymbol{H}_{0}\right),$
(11)

where $\boldsymbol{\mathcal{H}}_{12}^{\dagger}$ is the right pseudo-
inverse222Note that, although we can generate different right pseudo-inverses
by adding matrices in the null-space of $\boldsymbol{\mathcal{H}}_{12}$, the
common expression for right pseudo-inverse
$\boldsymbol{\mathcal{H}}_{12}^{\dagger}=\boldsymbol{\mathcal{H}}_{12}^{\mathrm{H}}(\boldsymbol{\mathcal{H}}_{12}\boldsymbol{\mathcal{H}}_{12}^{\mathrm{H}})^{-1}$
minimizes the norm of $\boldsymbol{\alpha}$ for the given
$\widetilde{\boldsymbol{U}}$ and $\beta$, which is most desirable in this
work. of $\boldsymbol{\mathcal{H}}_{12}$. Note that for obtaining
$\boldsymbol{\alpha}$ we have not used the fact that the desired channel
should be orthogonal. In fact, we could generate any channel matrix if we
substitute $\sqrt{\beta}\widetilde{\boldsymbol{U}}$ in (11) by the desired
channel.

###  III-B FRIS

We are interested in finding a full-matrix
$\boldsymbol{\Theta}_{\mathrm{FRIS}}$ such that

$\boldsymbol{H}_{0}+\boldsymbol{H}_{1}\boldsymbol{\Theta}_{\mathrm{FRIS}}\boldsymbol{H}_{2}=\sqrt{\beta}\widetilde{\boldsymbol{U}}.$
(12)

Assuming $\boldsymbol{H}_{1}$ and $\boldsymbol{H}_{2}$ are full-rank, we can
select the reflection matrix as
$\boldsymbol{\Theta}_{\mathrm{FRIS}}=\boldsymbol{H}_{1}^{\dagger}\boldsymbol{B}\boldsymbol{H}_{2}^{\dagger}$,
where $\boldsymbol{H}_{1}^{\dagger}$ is the right pseudo-inverse of
$\boldsymbol{H}_{1}$, $\boldsymbol{H}_{2}^{\dagger}$ is the left pseudo-
inverse of $\boldsymbol{H}_{2}$, and $\boldsymbol{B}$ is an $M\times K$ matrix
to be selected. This removes, with minimum power, the effect of
$\boldsymbol{H}_{1}$ and $\boldsymbol{H}_{2}$ on the overall channel, and
gives the orthogonalization requirement for FRIS $N\geq\min(M,K)$, which,
given $M>K$, leads to $N\geq M$. We then get

$\boldsymbol{\Theta}_{\mathrm{FRIS}}=\boldsymbol{H}_{1}^{\dagger}\left(\sqrt{\beta}\widetilde{\boldsymbol{U}}-\boldsymbol{H}_{0}\right)\boldsymbol{H}_{2}^{\dagger}.$
(13)

As happened with ARIS, we can also generate a non-orthogonal channel matrix by
substituting $\sqrt{\beta}\widetilde{\boldsymbol{U}}$ in (13) with any other
channel matrix.

###  III-C RIS baseline

Achieving perfect channel orthogonalization is generally not possible if we
consider the widely studied RIS model (3). In case there existed a solution,
it would come from finding a vector $\boldsymbol{\alpha}$ in (11) such that
$|\alpha_{n}|^{2}=1$ $\forall n$. Obtaining said solution would correspond to
finding a combination of $\widetilde{\boldsymbol{U}}$ (from a subspace of the
unitary matrices), $\beta$, and a vector in the null-space of
$\boldsymbol{\mathcal{H}}_{12}$ leading to a solution of (10) with
$|\alpha_{n}|^{2}=1$ $\forall n$. This problem seems analytically intractable,
so we can only restrict ourselves to approximate solutions by numerical
optimization. Since our goal is channel orthogonalization, we can find
approximate solutions by numerical minimization of

$\min_{\phi_{1},\dots,\phi_{N}}\kappa(\boldsymbol{H}_{0}+\boldsymbol{H}_{1}\boldsymbol{\Theta}_{\mathrm{RIS}}\boldsymbol{H}_{2}),$
(14)

where $\kappa(\cdot)$ is the condition number of a matrix, given by the
division between its maximum and minimum singular value, i.e.,
$\kappa(\cdot)=\sigma_{\max}(\cdot)/\sigma_{\min}(\cdot)$. Note that
$\kappa(\cdot)\geq 1$, with equality only for orthogonal matrices. Thus, by
minimizing it we would achieve a channel as close as possible to orthogonal,
which will be used as a baseline approach.

##  IV Channel estimation and RS configuration

In this section, we propose two techniques (one for ARIS and one for FRIS) for
estimating the channel and RS configuration at the BS. The idea is that, since
it is desirable for RSs to have limited energy consumption[15, 16], and thus
limited computation capabilities, we propose to leave most of the task of
channel estimation and RS weight computation to the BS. For the sake of
simplicity, we assume that each channel estimation step works perfectly.
Proposing more specific channel estimation methods and characterizing the
estimation errors is left as future work.

###  IV-A ARIS configuration

We start by proposing a method to configure the ARIS and obtain the desired
channel. The main goal is to estimate the necessary channel information at the
BS to be able to compute $\boldsymbol{\alpha}$ given by (11). Since the
channel matrix can be arbitrarily chosen by selecting
$\sqrt{\beta}\widetilde{\boldsymbol{U}}$ in (11) (recall it needs not be
orthogonal), we can assume that it is the BS itself that selects the desired
channel so that it does not need to further estimate it. The following steps
describe the method for finding the ARIS configuration at the BS:

####  IV-A1 Estimation of $\boldsymbol{H}_{0}$

First, the ARIS fixes $\boldsymbol{\alpha}=\boldsymbol{0}_{N\times 1}$, and
the UEs send $K$ orthogonal pilots. The received symbols over $K$ slots would
be then given by the $M\times K$ matrix

$\boldsymbol{Y}_{1}=\boldsymbol{H}_{0}\boldsymbol{P}+\boldsymbol{N}_{1},$ (15)

where $\boldsymbol{P}$ is the previously known pilot matrix, which can be
fixed to, e.g., $\boldsymbol{P}=\mathbf{I}_{K}$, and $\boldsymbol{N}_{1}$ is
the noise matrix with IID entries $n_{ij}\sim\mathcal{CN}(0,N_{0})$. From (15)
we can directly estimate $\boldsymbol{H}_{0}$ using state-of-the art channel
estimation methods.

####  IV-A2 Estimation of $\boldsymbol{\mathcal{H}}_{12}$

Since the effect of $\boldsymbol{H}_{1}$ and $\boldsymbol{H}_{2}$ for the
selection of $\boldsymbol{\alpha}$ only comes through
$\boldsymbol{\mathcal{H}}_{12}$, it is enough to estimate said matrix, whose
columns are given by

$[\boldsymbol{\mathcal{H}}_{12}]_{:,n}=\mathrm{vec}(\boldsymbol{h}_{1n}\boldsymbol{h}_{2n}^{\mathrm{T}}).$
(16)

Let us assume that the ARIS is configured such that for a given $n$ we have
$\alpha_{n}=1$ and $\alpha_{i}=0$ for $i\neq n$. Transmitting $K$ orthogonal
pilots from the UEs would then lead to

$\boldsymbol{Y}_{2n}=\boldsymbol{H}_{0}\boldsymbol{P}+\boldsymbol{h}_{1n}\boldsymbol{h}_{2n}^{\mathrm{T}}\boldsymbol{P}+\boldsymbol{N}_{2n},$
(17)

where $\boldsymbol{P}$ and $\boldsymbol{N}_{2n}$ are defined as before.
Assuming we already know $\boldsymbol{H}_{0}$ from the previous step, we can
cancel it at the BS, leading to

$\widetilde{\boldsymbol{Y}}_{2n}=\boldsymbol{h}_{1n}\boldsymbol{h}_{2n}^{\mathrm{T}}\boldsymbol{P}+\widetilde{\boldsymbol{N}}_{2n},$
(18)

where $\widetilde{\boldsymbol{N}}_{2n}$ would include the estimation error
from the previous step, which could lead to correlated entries. From (18) we
can estimate $\boldsymbol{h}_{1n}\boldsymbol{h}_{2n}^{\mathrm{T}}$ using
state-of-the-art estimation methods, which, after vectorizing, would give the
estimate of the $n$th column of $\boldsymbol{\mathcal{H}}_{12}$ given in (16).
If we iteratively perform this estimation step for $n=1,\dots,N$, i.e.,
setting to $1$ each element of the ARIS at a time, the BS would construct a
full estimate of $\boldsymbol{\mathcal{H}}_{12}$.

####  IV-A3 Computation and forwarding of $\boldsymbol{\alpha}$

Once the BS has estimated $\boldsymbol{H}_{0}$ and
$\boldsymbol{\mathcal{H}}_{12}$ it can select its desired channel, which in
our case corresponds to $\sqrt{\beta}\widetilde{\boldsymbol{U}}$, and compute
$\boldsymbol{\alpha}$ using (11). The BS should then forward
$\boldsymbol{\alpha}$ to the ARIS so that it can be configured to create the
desired channel, which is already known at the BS.

The proposed method allows to configure the ARIS to generate any channel
matrix by using a total of $(N+1)K$ pilot slots. This corresponds to a notable
decrease with respect to estimating $\boldsymbol{H}_{0}$,
$\boldsymbol{H}_{1}$, and $\boldsymbol{H}_{2}$ independently, which would at
least require $MK+N(M+K)$ slots, or even more for practical methods such as in
[17] for RIS. Moreover, since the BS selects its desired channel, it can
directly use it for equalization/precoding purposes. Also, form the
restriction of the channels to be orthogonal, optimum equalization/precoding
would be achieved through simple MRC/MRT, i.e., multiplying the conjugate
transpose of the channel matrix at the BS.

###  IV-B FRIS configuration

If we inspect (13), we note that we need an estimate of both
$\boldsymbol{H}_{1}$ and $\boldsymbol{H}_{2}$ to be able to compute the
corresponding FRIS configuration, so an efficient method such as the the one
for ARIS may not be available. Let us thus consider that the FRIS can transmit
pilots through each of its elements. Although this might not be desirable in
practice, the concept of FRIS is not yet well-established in contemporary
literature, so we use this assumption as a first step towards defining the
operation of such RSs. Coming up with more elaborate methods to avoid the
requirement of sending pilots from the FRIS will be considered in future work.
The following steps describe the proposed method for FRIS configuration:

####  IV-B1 Estimation of $\boldsymbol{H}_{0}$

In the initial step the FRIS would fix
$\boldsymbol{\Theta}_{ARIS}=\boldsymbol{0}_{N\times N}$, and the UEs would
send pilots to perform the channel estimation of $\boldsymbol{H}_{0}$ as in
the ARIS case.

####  IV-B2 Estimation of $\boldsymbol{H}_{1}$

In the case of FRIS we need to have an estimate of $\boldsymbol{H}_{1}$ and
$\boldsymbol{H}_{2}$ to compute their pseudo-inverses in (13). With the
assumption that the FRIS has the ability to send pilots through each of its
elements, the FRIS would send $N$ orthogonal pilots leading to the received
matrix at the BS

$\boldsymbol{Y}_{2}=\boldsymbol{H}_{1}\boldsymbol{P}_{\mathrm{FRIS{}}}+\boldsymbol{N}_{2},$
(19)

where $\boldsymbol{P}_{\mathrm{FRIS{}}}$ is the $N\times N$ known pilot
matrix, which can be set to $\mathbf{I}_{N}$. From (19), the BS can estimate
$\boldsymbol{H}_{1}$ using state-of-the-art estimation methods.

####  IV-B3 Estimation of $\boldsymbol{H}_{2}$

Let the FRIS fix alternatively each group of $M$ elements to 1, i.e., at
instant $n$ we select

$\boldsymbol{\Theta}_{\mathrm{FRIS}}=\mathrm{diag}(\begin{bmatrix}\boldsymbol{0}_{1\times(n-1)M}&\boldsymbol{1}_{1\times
M}&\boldsymbol{0}_{1\times(N-nM)}\end{bmatrix}),$ (20)

and we send $K$ orthogonal pilots from the UEs. The BS would then receive

$\boldsymbol{Y}_{3n}=\boldsymbol{H}_{0}\boldsymbol{P}+\boldsymbol{H}_{1,\mathrm{sq}}(n)\boldsymbol{H}_{2,\mathrm{sq}}(n)\boldsymbol{P}+\boldsymbol{N}_{3},$
(21)

where $\boldsymbol{H}_{1,sq}(n)$ is the $M\times M$ matrix formed by the
columns $(n-1)M+1$ to $nM$ of $\boldsymbol{H}_{1}$, $\boldsymbol{H}_{2,sq}(n)$
is the $M\times K$ matrix formed by rows $(n-1)M+1$ to $nM$ of
$\boldsymbol{H}_{2}$, and $\boldsymbol{N}_{3}$ and $\boldsymbol{P}$ are the
noise and pilot matrix, respectively. Assuming $\boldsymbol{H}_{1,sq}(n)$ is
full-rank $\forall n$, we can get an estimate of $\boldsymbol{H}_{2,sq}(n)$
applying state-of-the-art estimation methods to

$\widetilde{\boldsymbol{Y}}_{3n}=\widehat{\boldsymbol{H}}_{1,\mathrm{sq}}^{-1}(n)(\boldsymbol{H}_{1,\mathrm{sq}}(n)\boldsymbol{H}_{2,\mathrm{sq}}(n)\boldsymbol{P}+\boldsymbol{N}_{3}),$
(22)

where $\widehat{\boldsymbol{H}}_{1,\mathrm{sq}}(n)$ is the estimate of
$\boldsymbol{H}_{1,\mathrm{sq}}(n)$ from the previous stage. In the last
instant, given by $n=\left\lceil N/M\right\rceil,$ $nM$ might exceed $N$, so
everything should be cropped to $N$ in (22), and we would change the inverse
for the left pseudo-inverse of the cropped $\widehat{\boldsymbol{H}}_{1}$.
Note that the assumption of having $\boldsymbol{H}_{1,\mathrm{sq}}(n)$ of rank
$M$ $\forall n$ is a bit more restrictive than the requirement of solvability
of (12), where only the whole matrix $\boldsymbol{H}_{1}$ should be rank $M$.
However, in case some $\boldsymbol{H}_{1,\mathrm{sq}}(n)$ are ill-conditioned,
which can be known at the BS from the estimate of $\boldsymbol{H}_{1}$, we
could think of alternative solutions, e.g., selecting groups of $M$ linearly
independent rows. In the worst case, we could also fix a smaller number of 1s
in the FRIS and use the pseudo-inverse instead of inverse of the resulting
cropped $\boldsymbol{H}_{1}$, but this would require larger number of pilot
slots.

####  IV-B4 Computation and forwarding of
$\boldsymbol{\Theta}_{\mathrm{FRIS}}$

As a final step, the BS would select the desired channel
($\sqrt{\beta}\widetilde{\boldsymbol{U}}$) and compute the FRIS configuration,
$\boldsymbol{\Theta}_{\mathrm{FRIS}}$, using (13) with the estimates of
$\boldsymbol{H}_{0}$, $\boldsymbol{H}_{1}$, and $\boldsymbol{H}_{2}$. The BS
would then forward $\boldsymbol{\Theta}_{\mathrm{FRIS}}$ to the FRIS, which
would then apply it.

The proposed method allows the BS to configure the FRIS for inducing some
desired channel, in this case orthogonal, by employing a total of
$\left(1+\left\lceil N/M\right\rceil\right)K+N$ pilots, where $N$ of them
would correspond to pilots sent from the FRIS. For a moderate number of users,
this leads to a notable decrease with respect to the ARIS method, which
requires $(N+1)K$ pilots. Furthermore, we should note that the required $N$
for FRIS can also be remarkably smaller than for ARIS. A summary of the
orthogonalization conditions for each RS can be found in Table I.

##  V RS power constraints

In this section we study the problem of reducing the power requirements for
the RS configurations achieving channel orthogonality. As shown in Table I, we
define the power of the different RS settings as the squared Frobenius norm of
the reflection matrix $\boldsymbol{\Theta}$, which corresponds to the sum
power throughout its entries. Let us then assume that each RS can operate
without amplification as long as the average power per RS element is no
greater than 1 (RIS achieves this with equality), which translates to
$\|\boldsymbol{\Theta}\|^{2}_{\mathrm{fro}}\leq N$. Note that, ideally, each
RS element should have power no greater than 1, which will be considered in
the extended version of the paper.

Another factor to consider is the power of the resulting orthogonal sub-
channels of $\boldsymbol{H}=\sqrt{\beta}\widetilde{\boldsymbol{U}}$. Said
power, given by $\beta$ (orthogonal channels have all eigenvalues equal),
would be linearly related to the post-processed SNR per UE (after MRC/MRT),
$\eta=\beta E_{s}/N_{0}$, where we have assumed that the RS does not introduce
extra noise.333RSs with amplification might suffer from noise enhancement
similar to that of zero-forzing (ZF) equalizers. A thorough characterization
of it may be considered in future work. Recall that, from the orthogonality of
the channel, there is no interference between UEs and all UEs have the same
post-processed SNR. Thus, for a limited RS power, we would ideally like to
have a large $\beta$ so as to increase the capacity per UE .

###  V-A ARIS

The ARIS sum power required for having
$\boldsymbol{H}=\sqrt{\beta}\widetilde{\boldsymbol{U}}$ is given by (see Table
I)

$P_{\mathrm{A}}(\beta,\widetilde{\boldsymbol{U}})=\beta
g_{1}(\widetilde{\boldsymbol{U}})-2\sqrt{\beta}f_{1}(\widetilde{\boldsymbol{U}})+c_{1},$
(23)

where we defined
$f_{1}(\widetilde{\boldsymbol{U}})\\!=\\!\mathfrak{Re}\left\\{\mathrm{vec}(\widetilde{\boldsymbol{U}})^{\mathrm{H}}\boldsymbol{G}_{12}^{-1}\mathrm{vec}(\boldsymbol{H}_{0})\right\\}$,
$g_{1}(\widetilde{\boldsymbol{U}})\\!=\\!\mathrm{vec}(\widetilde{\boldsymbol{U}})^{\mathrm{H}}\boldsymbol{G}_{12}^{-1}\mathrm{vec}(\widetilde{\boldsymbol{U}})$,
$c_{1}\\!=\\!\mathrm{vec}(\boldsymbol{H}_{0})^{\mathrm{H}}\boldsymbol{G}_{12}^{-1}\mathrm{vec}(\boldsymbol{H}_{0})$,
with
$\boldsymbol{G}_{12}=\boldsymbol{\mathcal{H}}_{12}\boldsymbol{\mathcal{H}}_{12}^{\mathrm{H}}$.
Equation (23) comes from substituting (11) in the ARIS power expression from
Table I and operating. Let us first focus on obtaining the minimum ARIS power
for achieving an orthogonal channel. We can immediately note that the
existence of the direct channel $\boldsymbol{H}_{0}$ is responsible for
requiring a minimum power to be able to orthogonalize the channel with ARIS.
In the absence of $\boldsymbol{H}_{0}$
($c_{1}=f_{1}(\widetilde{\boldsymbol{U}})=0$),
$P_{\mathrm{A}}(\beta,\widetilde{\boldsymbol{U}})$ can be made arbitrarily
small by lowering $\beta$, i.e., sacrificing SNR; therefore, channel
orthogonalization would be achievable without the need for amplification. Let
us then assume $\boldsymbol{H}_{0}$ is present. Note that the BS has freedom
in selecting $\widetilde{\boldsymbol{U}}$ and $\beta$. We can then obtain the
minimum power required for orthogonalization with ARIS by solving

$\displaystyle P_{\mathrm{A,min}}=$
$\displaystyle\min_{\beta,\widetilde{\boldsymbol{U}}}\;\;P_{\mathrm{A}}(\beta,\widetilde{\boldsymbol{U}})$
(24)
$\displaystyle\;\;\mathrm{s.t.}\;\;\widetilde{\boldsymbol{U}}^{\mathrm{H}}\widetilde{\boldsymbol{U}}=\mathbf{I}_{K}.$

Differentiating $P_{\mathrm{A}}(\beta,\widetilde{\boldsymbol{U}})$ over
$\beta$ and equalling to 0 gives us the minimum $\beta$

$\beta_{\mathrm{o}1}=\left(\frac{f_{1}(\widetilde{\boldsymbol{U}})}{g_{1}(\widetilde{\boldsymbol{U}})}\right)^{2}.$
(25)

We can then substitute $\beta_{\mathrm{o}1}$ in (23) to get
$P_{\mathrm{A}}(\beta_{\mathrm{o}1},\widetilde{\boldsymbol{U}})$, which can
then be minimized using gradient descent within the unitary space. In order to
improve accuracy of the optimization, we consider optimization over the
geodesics of the unitary space as proposed in [18]. Thus, we need to obtain
the Euclidean gradient by differentiating
$P_{\mathrm{A}}(\beta_{*},\widetilde{\boldsymbol{U}})$ over
$\boldsymbol{U}^{*}$ (recall (6)), and use it for algorithm in [18, Table II],
which includes Armijo line-search for better convergence. We get

$\displaystyle\frac{\partial
P_{\mathrm{A}}(\beta_{\mathrm{o}1},\widetilde{\boldsymbol{U}})}{\partial\widetilde{\boldsymbol{U}}^{*}}=\frac{b}{g_{1}^{2}(\widetilde{\boldsymbol{U}})}\mathrm{vec}^{-1}\Big{(}-f_{1}^{2}(\widetilde{\boldsymbol{U}})\boldsymbol{G}_{12}^{-1}\mathrm{vec}(\widetilde{\boldsymbol{U}})$
(26)
$\displaystyle+f_{1}(\widetilde{\boldsymbol{U}})g_{1}(\widetilde{\boldsymbol{U}})\boldsymbol{G}_{12}^{-1}\mathrm{vec}(\boldsymbol{H}_{0})\Big{)}$
,

where $b=1-2\mathrm{sign}\left(f_{1}(\widetilde{\boldsymbol{U}})\right)$. Note
that, for differentiating over $\boldsymbol{U}^{*}$ instead of
$\widetilde{\boldsymbol{U}}^{*}$, we would just complete (26) with zeros,
since the corresponding extra columns of $\boldsymbol{U}$ have no bearing on
$P_{\mathrm{A}}(\beta_{\mathrm{o}1},\widetilde{\boldsymbol{U}})$. Once we have
obtained $P_{\mathrm{A,min}}$, any other ARIS sum power above it can be
achieved from (23) by solving a second order equation over $\sqrt{\beta}$.
Note that for every different $\beta$ there may be a new optimal
$\widetilde{\boldsymbol{U}}$, i.e., different from the one solving (24), which
minimizes the resulting power. Alternatives of (24) will be studied in the
extended version.

###  V-B FRIS

The FRIS sum power giving
$\boldsymbol{H}=\sqrt{\beta}\widetilde{\boldsymbol{U}}$ corresponds to

$\displaystyle P_{\mathrm{F}}=$ $\displaystyle\beta
g_{2}(\widetilde{\boldsymbol{U}})-2\sqrt{\beta}f_{2}(\widetilde{\boldsymbol{U}})+c_{2},$
(27)

where we defined
$f_{2}(\widetilde{\boldsymbol{U}})\\!=\\!\mathfrak{Re}\left\\{\mathrm{tr}(\boldsymbol{G}_{2}^{-1}\widetilde{\boldsymbol{U}}^{\mathrm{H}}\boldsymbol{G}_{1}^{-1}\boldsymbol{H}_{0})\right\\}$,
$g_{2}(\widetilde{\boldsymbol{U}})\\!=\\!\mathrm{tr}(\boldsymbol{G}_{2}^{-1}\widetilde{\boldsymbol{U}}^{\mathrm{H}}\boldsymbol{G}_{1}^{-1}\widetilde{\boldsymbol{U}})$,
$c_{2}\\!=\\!\mathrm{tr}(\boldsymbol{G}_{2}^{-1}\boldsymbol{H}_{0}^{\mathrm{H}}\boldsymbol{G}_{1}^{-1}\boldsymbol{H}_{0})$,
with $\boldsymbol{G}_{1}=\boldsymbol{H}_{1}\boldsymbol{H}_{1}^{\mathrm{H}}$
and $\boldsymbol{G}_{2}=\boldsymbol{H}_{2}^{\mathrm{H}}\boldsymbol{H}_{2}$. We
can use the same reasoning as in the case for ARIS throughout the different
steps. Let us thus focus on solving

$\displaystyle P_{\mathrm{F,min}}=$
$\displaystyle\min_{\beta,\widetilde{\boldsymbol{U}}}\;\;P_{\mathrm{F}}(\beta,\widetilde{\boldsymbol{U}})$
(28)
$\displaystyle\;\;\mathrm{s.t.}\;\;\widetilde{\boldsymbol{U}}^{\mathrm{H}}\widetilde{\boldsymbol{U}}=\mathbf{I}_{K}.$

Proceeding as in the previous case we can get

$\beta_{\mathrm{o}2}=\left(\frac{f_{2}(\widetilde{\boldsymbol{U}})}{g_{2}(\widetilde{\boldsymbol{U}})}\right)^{2},$
(29)

which leads to the euclidean gradient to be used for minimizing over
$\widetilde{\boldsymbol{U}}$ using [18, Table II],

$\displaystyle\frac{\partial
P_{\mathrm{F}}(\beta_{\mathrm{o}1},\widetilde{\boldsymbol{U}})}{\partial\widetilde{\boldsymbol{U}}^{*}}=\frac{b}{g_{2}^{2}(\widetilde{\boldsymbol{U}})}\\!\Big{(}\\!-f_{2}^{2}(\widetilde{\boldsymbol{U}})\boldsymbol{G}_{1}^{-1}\widetilde{\boldsymbol{U}}^{\mathrm{H}}\boldsymbol{G}_{2}^{-1}$
(30)
$\displaystyle+g_{2}(\widetilde{\boldsymbol{U}})f_{2}(\widetilde{\boldsymbol{U}})\boldsymbol{G}_{1}^{-1}\boldsymbol{H}_{0}\boldsymbol{G}_{2}^{-1}\Big{)}$
,

where $b=1-2\mathrm{sign}\left(f_{2}(\widetilde{\boldsymbol{U}})\right)$.

| ARIS | FRIS | RIS
---|---|---|---
Minimum $N$ for orthogonalization | $MK$ | $\min(M,K)$ | -
Number of pilots | $(N+1)K$ | $\left(1+\left\lceil\frac{N}{M}\right\rceil\right)K+N$ |  $>MK+N(M+K)$ [17]
RS sum power | $\|\Theta_{\mathrm{ARIS{}}}\|^{2}_{\mathrm{fro}}=\boldsymbol{\alpha}^{\mathrm{H}}\boldsymbol{\alpha}$ | $\|\Theta_{\mathrm{FRIS{}}}\|^{2}_{\mathrm{fro}}=\mathrm{tr}(\Theta_{\mathrm{FRIS{}}}^{\mathrm{H}}\Theta_{\mathrm{FRIS{}}})$ | $\|\Theta_{\mathrm{RIS}}\|^{2}_{\mathrm{fro}}=N$
TABLE I: Orthogonalization conditions for different RSs.

##  VI Numerical results

For the numerical results, we have tried to solve the optimization problems
defined in (24) and (28). Finding closed form results for said problems is in
general intractable due the constraint in $\widetilde{\boldsymbol{U}}$, which
should live in a subspace of the unitary matrices. However, good local
solutions can be found by using gradient descent along the geodesics, as
proposed in [18]. We cannot assure that the obtained results reach absolute
minima, but, since our main goal is to check if the proposed RS technologies
can be realized without amplification, local minima may be enough for our
purpose. We have thus implemented [18, Table II] with the Euclidean gradients
defined in (26) and (30) to find the minimum power required for perfect
channel orthongonalization using ARIS and FRIS, respectively.

In Fig. 1 we can see the minimum average RS power per element,
$P_{\mathrm{RS,avg}}=P_{\\{\mathrm{A,F}\\}\mathrm{,min}}/N$, and the resulting
channel gain per UE, equal to $\beta$ for all UEs from the orthogonal
restriction, with respect to the normalized power of the direct channel,
$E_{0}$. Since we are most interested in the power relation between the direct
and reflected channels, we have used normalized IID Rayleigh fading channels
with $\|H_{0}\|^{2}_{\mathrm{Fro}}=E_{0}MK$,
$\|H_{1}\|^{2}_{\mathrm{Fro}}=MN$, $\|H_{2}\|^{2}_{\mathrm{Fro}}=NK$. Other
channel models will be considered in future work, but we may note from the
analytical results that ill-conditioned channels are most harmful in the RS-
reflected paths. Fig. 1 (left) shows that in most practical scenarios (direct
links with power below 100 times the reflected one), the minimum average power
for channel orthogonalization, with both ARIS and FRIS, can be smaller than
that of RIS, so these surfaces could potentially be implemented without
amplification. The resulting channel gains for these minimized powers have
analogous linear relation with $E_{0}$, still impressive since they even
outperform RIS, which has been numerically optimized for channel
orthogonalization using (14).444For the RIS, since perfect orthogonality may
not be reachable, we plotted the average channel gain and minimum channel gain
per UE. However, the results for RIS may be far from optimum due to the
difficulty of such task, and the analytical intractability. Finding more
suitable optimization formulations for channel orthogonalization with RIS
should be further studied. A important thing to note is that the channel gains
in Fig. 1 are achieved with RS power dependent on $E_{0}$, and generally below
that of RIS. If we increase the respective gains until all RS powers are equal
to that of RIS, the resulting channel gains, which are plotted in Fig. 2, are
even more impressive, especially for FRIS, which can get 10 times better
channel gains than ARIS with a lower number of elements. In fact, there is
room for improvement by further optimization of $\widetilde{\boldsymbol{U}}$,
as previously discussed.

Figure 1: Minimum average RS power per element (left) and resulting minimum
channel gain per UE with respect to normalized gain of the direct channel.
Figure 2: Channel gain per UE for unit average RS power per element with
respect to normalized gain of the direct channel.

##  VII Conclusions

We have presented the concepts of FRIS and ARIS, two alternative RS
technologies with relaxed restrictions over RIS. We have obtained analytical
results for FRIS and ARIS configurations that achieve perfect channel
orthogonalization. We proposed a channel estimation method for each RS
technology at the BS, which selects the desired channel and forwards the
corresponding RS configuration. We have also showed that these RS can perform
channel orthogonalization without the need of amplification by minimizing over
the unitary space. The achieved channel gains, which are fairly distributed
among users from the orthogonalization, remark the benefits of adding more
processing capabilities at the RSs.

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# A note on Singular integral

Arup Maity  Arup Maity School of Mathematics, Harish-Chandra Research
Institute, Allahabad, 211 019, India<EMAIL_ADDRESS>and  Shyam
Swarup Mondal  Shyam Swarup Mondal Department of Mathematics Indian Institute
of Technology Guwahati, Guwahati-781039, Assam, India.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that for
$\frac{n}{2}+\frac{1}{4}<\alpha\leq\frac{n+1}{2}$, the convolution operator

$S_{\alpha}f(x)=\int_{|y|\geq 1}f(x-y)\left(|y|^{2}-1\right)^{-\alpha}dy$

is bounded from $L^{p}$ to $L^{q}$ for certain values of $p$ and $q$.

###### Key words and phrases:

$L^{p}$-estimate; Convolution; Singular integral; Interpolation, Multiplier

###### 2010 Mathematics Subject Classification:

Primary 42B20, 42B15; Secondary 44A35, 46F12

##  1\. Introduction

In this paper, we consider the $L^{p}-L^{q}$ estimates for the convolution
operators having kernel with singularity along the sphere and at infinity
which is non zero outside the neighbourhood of zero.

For convolutions on Euclidean space with homogeneous kernels, the fractional
integration theorem and the Calderón-Zygmund inequalities give essentially the
best possible results [5]. Such kind of kernels have singularities at the
point $0$ and $\infty$. It was Strichartz who first proved that other than
that kind of singularity, the corresponding operator is still bounded from
$L^{p}$ to other $L^{q}$. Such kind of operators naturally arise in the study
of the wave equation and gives new information about solutions of this type of
equation.

For $0<\alpha<1$, Strichartz considered the operator

$T_{\alpha}f(x)=\int_{|y|\leq 1}f(x-y)\phi_{\alpha}(y)\;dy,$

where the kernel is defined by

$\displaystyle\phi_{\alpha}(y)=\begin{cases}(1-|y|^{2})^{-\alpha},&\text{if}\;|y|<1,\\\
0,&\text{if}\;|y|\geq 1.\end{cases}$ (1.1)

The author investigated $L^{p}-L^{q}$ boundedness of such convolution operator
with kernel having singularities on the sphere and at infinity. In general,
from [3], if we consider the family of generalized functions
$(ax^{2}+bx+c)_{+}^{z}$, where

$(ax^{2}+bx+c)_{+}^{z}=\begin{cases}(ax^{2}+bx+c)^{z},&~{}\text{ if }\quad
ax^{2}+bx+c>0,\\\ 0,&~{}\text{ if }\quad ax^{2}+bx+c\leq 0,\end{cases}$

then it is always possible to perform a linear transformation on $x$ such that
the generalized function $(ax^{2}+bx+c)_{+}^{z}$ is transformed to one of the
forms $\left(1-x^{2}\right)_{+}^{\lambda}$,
$\left(1+x^{2}\right)_{+}^{\lambda}$, $\left(x^{2}-1\right)_{+}^{\lambda}$, or
$x_{+}^{2\lambda}$ respectively, to the four cases in which $ax^{2}+bx+c>0$.
Here in this paper, we mainly interested in the case
$\left(x^{2}-1\right)_{+}^{\lambda}$. Note that, all other cases have becn
treated already. In particular, Strichartz in [16] cosider the first case
$\left(1-x^{2}\right)_{+}^{\lambda}$ and proved the following result.

######  Theorem 1.1.

[16]

  1. (1)

$\left\|T_{\alpha}f\right\|_{q}\leq A_{\alpha}\|f\|_{p}$ provided $1<p\leq
2\leq q<\infty$ and $\frac{1}{p}-\frac{1}{q}\leq\frac{(n+1-2\alpha)}{2n}$ for
$0<\alpha\leq\frac{(n+1)}{2}$.

  2. (2)

$\left\|T_{\alpha}f\right\|_{p^{\prime}}\leq A_{\alpha}\|f\|_{p}$ for
$p=\frac{(n+1)}{(n+1-\alpha)},p^{\prime}=\frac{(n+1)}{\alpha}$ and
$0<\alpha\leq\frac{(n+1)}{2}$.

  3. (3)

$\left\|T_{\alpha}f\right\|_{q}\leq A_{\alpha}\|f\|_{p}$ for
$\frac{1}{2}\leq\alpha\leq\frac{(n+1)}{2}$ provided
$\frac{(n+1)}{(n+1-\alpha)}\leq p\leq 2$ and $\frac{n}{q}=\alpha-\frac{1}{p}$,
or $\frac{n}{\left(n+\frac{1}{2}-\alpha\right)}\leq
p\leq\frac{(n+1)}{(n+1-\alpha)}$ and
$\frac{1}{q}=\alpha-\frac{n}{p^{\prime}}.$

Considerable attention has been devoted to discovering generalizations to new
contexts for the boundedness of convolution operator by several researchers in
different contexts. For example, $L^{p}-L^{q}$ boundedness of convolution
operator was studied by Amri and Gaidi in [2] in the Dunkel setting and
investigated the solutions of wave equations associated to Dunkl Laplacian
operator. Karapetyants studied $L^{p}-L^{q}$ boundedness for convolutions
operator with kernels having singularities on a sphere in [6]. F. Ricci and T.
Giancarlo in [9] studied boundedness of convolution operators defined by
singular measures. Further, $\mathcal{L}$-characteristic of some potential
type convolution operator $A^{\alpha}$ is bounded from $L^{p}$ into $L^{q}$
was given in the works of Karapetyants and Nogin [7]. There are several papers
in the direction of bounds for potential-type convolution operators and close
to them, we refer to [11, 12, 13, 8, 10, 4] and references therein. Such
operators naturally arise in applications to the theory of fractional powers
of differential operators, in particular, classical operators of mathematical
physics: the wave operator, the Klein-Gordon and Schrödinger operators, the
telegraph operator.

The main aim of this paper is to prove $L^{p}$ to $L^{q}$ bounded of the
convolution operator

$\displaystyle S_{\alpha}f(x)=\int_{|y|\geq 1}f(x-y)\xi_{\alpha}(y)\;dy$ (1.2)

for $\alpha>0$, where the kernel is defined by

$\displaystyle\xi_{\alpha}(y)=\begin{cases}(|y|^{2}-1)^{-\alpha},&\text{if}\;|y|>1,\\\
0,&\text{if}\;|y|\leq 1,\end{cases}$ (1.3)

for certain $p$ and $q$. The kernel defined in (1.3) is different form the
kernel (1.1), considered by Strichartz. The main difference in the kernel
defined in (1.3) is that this kernel has singularity along the sphere and in
infinity. Also it is non zero outside the neighbourhood of zero which is not
consider by any author in past and which makes the problem rather slightly
different. However, in the sense of distribution, the range of $\alpha$ can be
extended similarly as in Equation (3) of [16] by integration by parts.

Our main resuls in this article is the following $L^{p}-L^{q}$ boundedness
theorem of convolution operator $S_{\alpha}$.

######  Theorem 1.2.

Let $\frac{n}{2}+\frac{1}{4}<\alpha\leq\frac{n+1}{2}$. Then for $1<p\leq 2\leq
q<\infty$ and

$\frac{1}{p}-\frac{1}{q}\leq\frac{2\alpha-n-\frac{1}{2}}{n},$

the convolution opeartor

$S_{\alpha}f=\xi_{\alpha}*f,\quad f\in\mathcal{S}(\mathbb{R}^{n})$

is bounded from $L^{p}$ to $L^{q}$.

The proof of the theorem is given in Section 3. The main ingridient we use is
the the Hardy-Littlewood multiplier theorem given in Theorem 1.11 of [5], to
prove Theorem 1.2 using a similar technique given in [16].

The presentation of this manuscript is divided into three sections including
the introduction. In section 2, we recall some basics properties and results
from eucledian Fourier analysis which will be used in order to prove our main
results. In Section 3, we prove our main reults.

##  2\. Preliminaries

In this section, we first recall some notation and basic properties of Fourier
analysis on $\mathbb{R}^{n}$. The Fourier transform $\hat{f}$ of a function
$f\in L^{1}\left(\mathbb{R}^{n}\right)$ is defined by

$\mathcal{F}f(\xi)=\widehat{f}(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{ix\cdot\xi}dx,\quad\xi\in\mathbb{R}^{n}.$

The Fourier transform can be extended to $L^{2}\left(\mathbb{R}^{n}\right)$
using the standard density arguments. The inverse Fourier transform is given
by

$f(x)=\int_{\mathbb{R}^{n}}\hat{f}(\xi)e^{-ix\cdot\xi}d\xi,\quad
x\in\mathbb{R}^{n},$

where $f$ belongs to a suitable function space. The space
$S\left(\mathbb{R}^{n}\right)$ is the Schwartz class of rapidly decreasing
smooth functions and $S^{\prime}\left(\mathbb{R}^{n}\right)$ is the space of
tempered distributions. Let $T\in S^{\prime}\left(\mathbb{R}^{n}\right)$, its
fourier transform $\hat{T}$ defined in the sense of distribution by

$\widehat{T}(\phi)=T(\hat{\phi}),\quad\phi\in S(\mathbb{R}^{n}).$

Moreover, $\widehat{T}$ is called multiplier if it is $L^{p}-L^{q}$ bounded.

As usual, the space $L_{p}^{q}$ denotes the Banach space of
$S^{\prime}(\mathbb{R}^{n})$-distributions $T$ such that the closure in
$L^{p}$ of the convolution $T\ast u$ for $u\in S(\mathbb{R}^{n})$, is an
$L^{p}-L^{q}$ bounded translation invariant operator. The spcae $L_{p}{}^{q}$
is thus isomorphic to a closed subspace of the Banach space of all bounded
linear mappings of $L^{p}$ into $L^{q}$ and hence is also a Banach space.
Also, $M_{p}^{q}=\mathcal{F}L_{p}^{q}$ stands for the space of all
$L^{p}-L^{q}$ multipliers type $(p,q)$ [5]. Moreover, we have

$L_{p}^{\infty}=L_{1}^{p^{\prime}}=L^{p^{\prime}},p<\infty~{}\text{and}~{}L_{\infty}^{\infty}=L_{1}^{1}=M.$

From [5], we have the following Hardy-Littlewood multiplier theorem.

######  Theorem 2.1.

[5] Let $0<a<n$ and $m$ be a measurable function such that

$|m(\xi)|\leq\frac{c}{|\xi|^{a}}$

some constant $c>0$. Then the operator
$T_{m}=\mathcal{F}^{-1}\left(m\mathcal{F}\right)$ is bounded from
$L^{p}\left(\mathbb{R}^{n}\right)$ to $L^{q}\left(\mathbb{R}^{n}\right)$,
provided

$1<p\leq 2\leq
q<\infty~{}~{}\text{and}~{}~{}\frac{1}{p}-\frac{1}{q}=\frac{a}{n}.$

By $J_{\left(\alpha\right)}(s)$, we denotes the classical Bessel function of
size $\alpha$. From [15], we have the following estimate:

$\displaystyle\left|t^{-(a+ib)}J_{\left(a+ib\right)}(t)\right|\leq
C_{a}~{}e^{c|b|}~{}(1+t)^{-a-\frac{1}{2}},\quad 0<t<\infty.$ (2.1)

For detailed study and several properties on Bessel functions, we refer to
[17, 15].

##  3\. $L^{p}$-$L^{q}$-boundedness

This section is devoted to study our main results of this paper. The Hardy-
Littlewood multiplier theorem plays a key role in the proof of the theorem.

###  3.1. Proof of Theorem 1.2

###### Proof.

For $\alpha>0$, let us consider

$\xi_{\alpha}(y)=\begin{cases}(|y|^{2}-1)^{-\alpha},&\text{if}\;|y|>1,\\\
0,&\text{if}\;|y|\leq 1.\end{cases}$

Then the Fourier transform of $\xi_{\alpha}$ is not classically defined, as
the function $\xi_{\alpha}$ is not in $L^{1}(\mathbb{R}^{n})$. We define it by
the sense of tempered distribution. For $\alpha\neq n,n\in\mathbb{N}$, we
define

$\widehat{\xi_{\alpha}}(s)=\sqrt{\pi}\;\Gamma(1-\alpha)\frac{|s|}{2}^{\left(\alpha-\frac{n}{2}\right)}\cot\pi\left(\frac{n}{2}-\alpha\right)\left(J_{\left(\alpha-\frac{n}{2}\right)}(|s|)-J_{\left(\frac{n}{2}-\alpha\right)}(|s|)\right)$

for $0\leq\alpha<1$, where $J_{\left(\alpha\right)}(s)$ denotes the classical
Bessel function of size $\alpha$. This is well defined by the analytic
continuation of $\Gamma$ almost everywhere. Since

$\displaystyle S_{\alpha}f=\xi_{\alpha}*f,\quad
f\in\mathcal{S}(\mathbb{R}^{n}),$ (3.1)

then the Fourier transform of $S_{\alpha}f$ is given by

$\displaystyle\widehat{S_{\alpha}f}(s)=\sqrt{\pi}\;\Gamma(1-\alpha)\frac{|s|}{2}^{\left(\alpha-\frac{n}{2}\right)}\cot\pi\left(\frac{n}{2}-\alpha\right)\left(J_{\left(\alpha-\frac{n}{2}\right)}(|s|)-J_{\left(\frac{n}{2}-\alpha\right)}(|s|)\right)\hat{f}(s).$
(3.2)

The right-hand side of the above inequality is a single-valued analytic
function of $\alpha$ in the complex plane except at the positive integers.

Let $\frac{n}{2}+\frac{1}{4}<\alpha\leq\frac{n+1}{2}$ and consider the
function

$\displaystyle
m(s)=\sqrt{\pi}\;\Gamma(1-\alpha)\;\frac{|s|}{2}^{\left(\alpha-\frac{n}{2}\right)}\cot\pi\left(\frac{n}{2}-\alpha\right)\left(J_{\left(\alpha-\frac{n}{2}\right)}(|s|)-J_{\left(\frac{n}{2}-\alpha\right)}(|s|)\right).$
(3.3)

Using the classical estimate (2.1) on the size of Bessel functions, we have

$\displaystyle
J_{\left(\alpha-\frac{n}{2}\right)}(|s|)\leq(1+|s|)^{-\alpha+\frac{n}{2}-\frac{1}{2}}$
(3.4)

and

$\displaystyle
J_{\left(\frac{n}{2}-\alpha\right)}(|s|)\leq(1+|s|)^{\alpha-\frac{n}{2}-\frac{1}{2}}.$
(3.5)

Using the estimates (3.3) and (3.4), from the relation (3.5), we get

$|m(s)|\leq c|s|^{-\left(2\alpha-n-\frac{1}{2}\right)},$

for some costant $c$. Thus by Theorem 2.1, $m$ is a $L^{p}-L^{q}$ multiplier
provided

$1\leq p\leq 2\leq
q<\infty\quad\text{and}\quad~{}\frac{1}{p}-\frac{1}{q}=\frac{2\alpha-n-\frac{1}{2}}{n}.$

Thus, the operator $S_{\alpha}$ is bounded from $L^{p}$ to $L^{q}$ for
$\frac{1}{p}-\frac{1}{q}\leq\frac{2\alpha-n-\frac{1}{2}}{n}$. This completes
the proof of the theorem. ∎

In the next theorem we show that the convolution operator $S_{\alpha}$ is
bounded from $L^{p}$ to $L^{p^{\prime}}$ in one dimentional.

######  Theorem 3.1.

Let $0\leq\alpha\leq 1$. Then for $p=\frac{2}{(2-\alpha)}$ and
$p^{\prime}=\frac{2}{\alpha}$, the convolution opeartor $S_{\alpha}$ is
bounded from $L^{p}$ to $L^{p^{\prime}}$.

###### Proof.

Let $S_{\alpha}$ is an family of operators defined in the strip
$0\leq\operatorname{Re}(\alpha)\leq 1$ in the complex plane in the sense of
Stein [14]. When $\operatorname{Re}(\alpha)=1$, using Plancherel theorem,
(3.1), (3.5), and (3.5), we have

$\left\|S_{\alpha}f\right\|_{2}\leq
c_{1}e^{c\operatorname{Im}(\alpha)}\;\|f\|_{2}.$

Again on the other side, when $\operatorname{Re}(\alpha)=0$, using the fact
that $\left|\xi_{\alpha}(y)\right|\leq 1$ and $|\Gamma(1+ib)|=\left(\frac{\pi
b}{\sinh b}\right)^{\frac{1}{2}}$, from the relation (3.2), we have the
estimate

$\left\|S_{\alpha}f\right\|_{\infty}\leq
c_{1}e^{c\operatorname{Im}(\alpha)}\;\|f\|_{1}.$

Then by Stein’s interpolation theorem theorem [14], we have the result. ∎

An immediate consequence of the above theorem is the follwing $L^{p}-L^{q}$
boundedness of the convolution opeartor $S_{\alpha}$.

######  Theorem 3.2.

Let $\frac{1}{2}\leq\alpha\leq 1$. Then for

$\frac{2}{(2-\alpha)}\leq p\leq
2\quad\text{and}\quad\frac{1}{q}=\alpha-\frac{1}{p}$

or

$\frac{1}{\left(\frac{3}{2}-\alpha\right)}\leq
p\leq\frac{2}{(2-\alpha)}\quad\text{and}\quad\frac{1}{q}=\alpha-\frac{1}{p^{\prime}},$

the convolution opeartor $S_{\alpha}$ is bounded from $L^{p}$ to $L^{q}$.

###### Proof.

If $\alpha>\frac{1}{2},$ then the result follows from Theorem 1.2 and 3.1 by
an application of Riesz interpolation theorem. Now let us consider
$\alpha=\frac{1}{2}$. Since $\xi_{\frac{1}{2}}$ is in weak $L^{2}$, then
$T_{\frac{1}{2}}$ maps $L^{1}$ to weak $L^{2}$. Then the required result
follows by applying the Marcinkiewicz interpolation theorem [18] and a duality
argument [5]. ∎

######  Remark 3.3.

Note that, $\widehat{\xi_{\alpha}}$ (defined in (1.3)) is a multiplier in
terms of Bessel functions.

######  Remark 3.4.

If we define kernels by replacing $|y|^{2}$ by any nondegenerate quadratic
form using the computations of [[3], Chapter $\mathrm{III}]$, then all the
results are also in this case.

## Acknowledgments

The first author gratefully acknowledges the support provided by Harish-
Chandra Research Institute, Government of India. Second author thanks IIT
Guwahati, India, for the support provided during the period of this work. The
authors are deeply indebted to Prof. P. K. Ratnakumar for several fruitful
discussions and generous comments.

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