The aim of this paper is to introduce and to study the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ of the multipliers of the space $\mathcal{S}_ω(\mathbb{R}^N)$ of the $ω$-ultradifferentiable rapidly decreasing functions of Beurling type. We determine various properties of the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$. Moreover, we define and compare some lc-topologies of which $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ can be naturally endowed.
In this paper, we characterize the (left) multiplier algebra of a semidirect product algebra ${\mathcal A}={\mathcal B}\oplus {\mathcal I}$, where ${\mathcal I}$ and ${\mathcal B}$ are closed two-sided ideal and closed subalgebra of ${\mathcal A}$, respectively. As an application of this result we investigate the BSE-property of this class of Banach algebras. We then for two commutative semisimple Banach algebras ${\mathcal A}$ and ${\mathcal B}$ characterize the BSE-functions on the carrier space of ${\mathcal A}\times_ϕ{\mathcal B}$, the $ϕ$-Lau product of ${\mathcal A}$ and ${\mathcal B}$, …
We study the class $\mathcal{M}_p$ of Schur multipliers on the Schatten-von Neumann class $\mathcal{S}_p$ with $1 \leq p \leq \infty$ as well as the class of completely bounded Schur multipliers $\mathcal{M}_p^{cb}$. We first show that for $2 \leq p < q \leq \infty$ there exist $m \in \mathcal{M}_p^{cb}$ with $m \not \in \mathcal{M}_q$, so in particular the following inclusions that follow from interpolation are strict $\mathcal{M}_q \subsetneq \mathcal{M}_p$ and $\mathcal{M}^{cb}_q \subsetneq \mathcal{M}^{cb}_p$. In the remainder of the paper we collect computational evidence that for $p\not …
Let $(Ω_1, \mathcal{F}_1, μ_1)$ and $(Ω_2, \mathcal{F}_2, μ_2)$ be two measure spaces and let $1 \leq p,q \leq +\infty$. We give a definition of Schur multipliers on $\mathcal{B}(L^p(Ω_1), L^q(Ω_2))$ which extends the definition of classical Schur multipliers on $\mathcal{B}(\ell_p,\ell_q)$. Our main result is a characterization of Schur multipliers in the case $1\leq q \leq p \leq +\infty$. When $1 < q \leq p < +\infty$, $ϕ\in L^{\infty}(Ω_1 \times Ω_2)$ is a Schur multiplier on $\mathcal{B}(L^p(Ω_1), L^q(Ω_2))$ if and only if there are a measure space (a probability space when $p\neq q$) $(Ω…
Let ${\mathcal A}$ be a Banach algebra and let $φ$ be a non-zero character on ${\mathcal A}$. Suppose that ${\mathcal A}_M$ is the closure of the faithful Banach algebra ${\mathcal A}$ in the multiplier norm. In this paper, topologically left invariant $φ$-means on ${\mathcal A}_M^*$ are defined and studied. Under some conditions on ${\mathcal A}$, we will show that the set of topologically left invariant $φ$-means on ${\mathcal A}^*$ and on ${\mathcal A}_M^*$ have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. The main application…
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{…
Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L}, -i \mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i \mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator …
In this article, we present some results about Fourier multipliers on Hardy spaces in the product case. We mainly give descriptions of multipliers from the space $H^1(\mathbb{T}\times\mathbb{T})$ into the space $\ell^2$ and from the space $H^1(\mathbb{R}_+^2\times\mathbb{R}_+^2)$ into the space $L^1$.
The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sen…
In this paper we prove that if $Λ\in M_p(\mathbb R^N)$ and has compact support then $Λ$ is a weak summability kernel for $1<p<\infty$, where $M_p(\mathbb R^N)$ is the space of multipliers of $L^p(\mathbb R^N)$.
Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$.…
This article is concerned with the question of whether Marcinkiewicz multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.
Let $Φ_1 , Φ_2 $ and $ Φ_3$ be Young functions and let $L^{Φ_1}(\mathbb{R})$, $L^{Φ_2}(\mathbb{R})$ and $L^{Φ_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(ξ,η)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(Φ_1,Φ_2,Φ_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(ξ) \hat{g}(η)m(ξ,η)e^{2πi (ξ+η) x}dξdη\] defines a bounded bilinear operator from $L^{Φ_1}(\mathbb{R}) \times L^{Φ_2}(\mathbb{R})$ to $L^{Φ_3}(\mathbb{R})$. We denote by $BM_{(Φ_1,Φ_2,Φ_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(Φ_1,Φ_2…
In this paper we prove the bilinear analogue of de Leeuw's result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.
In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator…
In this paper we have characterized the space of summability kernels for the case p=1 and p=2. For other values of p we give a necessary condition for a function $Λ$ to be a summability kernel. For the case p=1, we have studied the properties of measures which are transferred from $M(\mathbb Z)$ to $M(\mathbb R)$ through summability kernels. Further, we have extended every $l_p(\mathbb Z)$ sequences to $L^q(\mathbb R)$ multipliers for certain values of p and q.
In this short note, we prove that the subspace of radial multipliers is contractively complemented in the space of Fourier multipliers on the Bochner space $\mathrm{L}^p(\mathbb{R}^n,X)$ where $X$ is a Banach space and where $1 \leq p <\infty$. Moreover, if $X = \mathbb{C}$, then this complementation preserves the positivity of multipliers.
In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-…
We prove a two-sided transference theorem between $L^{p}$ spherical multipliers on the compact symmetric space $U/K$ and $L^{p}$ multipliers on the vector space $i\mathfrak{p},$ where the Lie algebra of $U$ has Cartan decomposition $\mathfrak{k\oplus }i\mathfrak{p}$. This generalizes the classic theorem transference theorem of deLeeuw relating multipliers on $% L^{p}(\mathbb{T)}$ and $L^{p}(\mathbb{R)}$.
This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$.
Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree $0$ cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group $ \mathbb{H}^n $ and Weyl multipliers on $ \mathbb{C}^n $ acting on Sobolev Spaces.
Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ be the $\mathcal{E}$-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well known Beurling-Lax-Halmos theorem states that every shift invariant subspace of $H^2_{\mathcal{E}}(\mathbb{D})$ other than $\{0\}$ has the form $ΘH^2_{\mathcal{E}_*}(\mathbb{D})$, where $Θ$ is an operator-valued inner multiplier in $H^\infty_{B(\mathcal{E}_*, \mathcal{E})}(\mathbb{D})$ for some Hilbert space $\mathcal{E}_*$. In this paper we identify $H^2(\mathbb{D}^n)$ with $H^2(\mathbb{D}^{n-1})$-valued Hardy space $H^2_{H^…
For $p > 1, γ\in \mathbb{R}$, denote by $H^γ_p(\mathbb{R}^n)$ the Bessel potential space, by $H^γ_{p, unif}(\mathbb{R}^n)$ the corresponding uniformly localized Bessel potential space and by $M[s, -t]$ the space of multipliers from $H^s_2(\mathbb{R}^n)$ into $H^{-t}_2(\mathbb{R}^n)$. Assume that $s, t \geqslant 0, n/2 > \max(s, t) > 0, r: = \min(s, t), p_1: = n/max(s, t)$. Then the following embeddings hold $$ H^{-r}_{p_1, unif}(\mathbb{R}^n) \subset M[s, -t] \subset H^{-r}_{2, unif}(\mathbb{R}^n). $$ The main result of the paper claims the sharpness of the left embedding in the following sens…
We introduce generalized Campanato spaces $\mathcal{L}_{p,ϕ}$ on a probability space $(Ω,\mathcal{F},P)$, where $p\in[1,\infty)$ and $ϕ:(0,1]\to(0,\infty)$. If $p=1$ and $ϕ\equiv1$, then $\mathcal{L}_{p,ϕ}=\mathrm{BMO}$. We give a characterization of the set of all pointwise multipliers on $\mathcal{L}_{p,ϕ}$.
We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(\mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $p\leq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(\mathbb{R}^d)$.
The objective of this paper is to describe the space of multipliers acting from a Bessel potential space $H^s_p(\mathbb R^n)$ into another space $H^{-t}_q(\mathbb R^n)$, provided that the smooth indices of these spaces have different signs, i.e. $s, t \geqslant 0$. This space of multipliers consists of distributions $u$, such that for all $φ\in H^s_p(\mathbb R^n)$ the product $φ\cdot u$ is well-defined and belongs to the space $H^{-t}_q(\mathbb R^n)$. We succeed to describe this space explicitly, provided that $p \leqslant q$ and one of the following conditions $$ s \geqslant t \geqslant 0, \ …
Titchmarsh theorems for Hölder-Lipschitz functions on fundamental domains of lattices in $\mathbb{R}^{d}$ with applications to boundedness of Fourier multipliers
Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F : \mathbb{R} \to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel $\mathcal{K}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of $\mathcal{K}_{F(\mathcal{L})}$ implies continuity of $F$.
For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{α\in R} k(α)$. Denote by $dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x $ the associated measure in $\mathbb R^N$. Let $\mathcal F$ stands for the Dunkl transform. Given a bounded function $m$ on $\mathbb R^N$, we prove that if there is $s>\mathbf N$ such that $m$ satisfies the classical Hörmander condition with the smoothness $s$, then the multiplier operator $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$ is of weak type $(1,1)$, strong type $(p,p)$ for …
We prove in this paper that a sequence $M:\mathbb{Z}^{n}\to\mathcal{L}(E)$ of bounded variation is a Fourier multiplier on the Besov space $B_{p,q}^{s}(\mathbb{T}^{n},E)$ for $s\in\mathbb{R}$, $1<p<\infty$, $1\leq q\leq\infty$ and $E$ a Banach space, if and only if $E$ is a UMD-space. This extends in some sense the Theorem 4.2 in [AB04] to the $n-$dimensional case. The result is used to obtain existence and uniqueness of solution for some Cauchy problems with periodic boundary conditions.
We consider the space $\mathcal{H}(ces_p)$ of all Dirichlet series whose coefficients belong to the Cesàro sequence space $ces_p$, consisting of all complex sequences whose absolute Cesàro means are in $\ell^p$, for $1<p<\infty$. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on $\mathcal{H}(ces_p)$ as the Wiener algebra of Dirichlet series shifted to the vertical half-plane $\mathbb{C}_{1/q}:=\{s\in\mathbb{C}:\Re s>1/q\}$, where $1/p+1/q=1$.
Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an …
We study Fourier multipliers on free group $\mathbb{F}_\infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(\hat{\mathbb{F}}_\infty)$ iff their restriction on $L^p(\hat{\mathbb{F}}_1)=L^p(\mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<\infty$.
Let $\mathcal H$ be the Drury-Arveson or Dirichlet space of the unit ball of $\mathbb C^d$. The weak product $\mathcal H\odot\mathcal H$ of $\mathcal H$ is the collection of all functions $h$ that can be written as $h=\sum_{n=1}^\infty f_n g_n$, where $\sum_{n=1}^\infty \|f_n\|\|g_n\|<\infty$. We show that $\mathcal H\odot\mathcal H$ is contained in the Smirnov class of $\mathcal H$, i.e. every function in $\mathcal H\odot\mathcal H$ is a quotient of two multipliers of $\mathcal H$, where the function in the denominator can be chosen to be cyclic in $\mathcal H$. As a consequence we show that …
We prove a general type description result for the multipliers acting between two periodic Bessel potential spaces, defined on the $n$--dimensional torus, in a case when their smoothness indices are of different signs. This is done through the detailed examination of a periodic analogue of the linear operator $J_s$, which is employed in the definition of the scale of the Bessel potential space defined on the whole space $\mathbb{R}^n$. Our method of defining this periodic analogue of $J_s$ uses the results about an asymptotic behaviour of the generalized Fourier coefficients and existence of a…
The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a Hörmander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that is, half the topological dimension of $\mathbb{S}$. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on $\mathbb{S}$.
Nonexistence of Henkin type projections via a Wiener theorem for multipliers
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
School of Mathematics and Statistics, Hunan First Normal University（湖南第一师范学院数学与统计学院）；School of Mathematics and Statistics, Hunan University of Science and Technology（湖南科技大学数学与统计学院）
We investigate the problem of establishing bilateral continuous embeddings of the uniformly localized Bessel potential spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ into the multiplier spaces between Bessel potential spaces with positive smoothness indices. This problem is considered in the model situation when the natural norms of both of these Bessel potential spaces are generated by some inner product yet the description theorems for the corresponding multiplier space in terms of the spaces $H^γ_{r, \: unif}(\mathbb{R}^n)$ can not be established. The optimal character of the indices figuring in t…
In this paper we study some estimates of norms in variable exponent Lebesgue spaces for maximal multiplier operators.We will consider the case when multiplier is the Fourier transform of a compactly supported Borel measure
In this manuscript we provide necessary and sufficient conditions for the $\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of discrete Fourier multipliers (Fourier multipliers on $\mathbb{Z}^n$). Our main goal is to apply the results obtained to discrete fractional integral operators. Discrete versions of the Calderón-Vaillancourt Theorem and the Gohberg Lemma also are proved.
In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are $\mathcal{R}$-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.
We prove an $L^p$ spectral multiplier theorem for functions of the $K$-invariant sublaplacian $L$ acting on the space of functions of fixed $K$-type on the group $SL(2,\mathbb{R}).$ As an application we compute the joint $L^p(SL(2,\mathbb{R}))$ spectrum of $L$ and the derivative along $K$.
In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.
In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space, and consider the class of (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vector-valued functions.
We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…
In this paper we proof that every Fourier multiplier on homogeneous sobolev space $\dot{W}_1^1(\mathbb{R}^d)$ is a continuous function. Our theorem is generalization of A. Bonami and S. Poornima result for Fourier multipliers, which are homogeneous functions of degree zero.
A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \…
Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $φ$ and the dual left Haar weight $\widehatφ$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehatφ)$ to $L_q(\widehat{\mathbb{G}},\widehatφ)$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},φ)$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\l…
