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International audience; Let IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differen-tiable. Maximal ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime ideal of D(E) is included in a unique maximal ideal and every prime closed ideal of D(E) is a maximal ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a maximal ideal. If IK is locally compact, every maximal ideal of D(E) is of codimension 1. Every maximal ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently , the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required. Introduction and preliminaries

Let Γ denote a lattice in SU(1, p), with p greater than 1. We show that there exists no Zariski dense maximal representation with target SU(m, n) if n > m > 1. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic m-subspaces of a complex vector space V endowed with a Hermitian metric h of signature (m, n) and whose lines correspond to the 2m dimensional subspaces of V on which the restriction of h has signature (m, m).

In this paper, we classify all nonconstant smooth CR maps from a sphere Sn;1 ⊂ Cn with n > 3 to the Shilov boundary Sp;q ⊂ Cp×q of a bounded symmetric domain of Cartan type I under the condition that p-q < 3n-4. We show that they are either linear maps up to automorphisms of Sn;1 and Sp;q or D``Angelo maps. This is the first classification of CR maps into the Shilov boundary of bounded symmetric domains other than sphere that includes nonlinear maps.

Let $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.

We describe the Shilov boundary ideal for a q-analog of the algebra of holomorphic functions on the unit ball in the space of $n\times n$ matrices and show that its $C^*$-envelope is isomorphic to the $C^*$-algebra of continuous functions on the quantum unitary group $U_q(n)$., 27 pages,v.3:accepted for publication in Journal Funct.Anal., crrected som typos, proof of Lemma 10 changed, a reference added, an acknowledgement added

We introduce the concept of an $E$-valued function algebra, a type of Banach algebra that consist of continuous $E$-valued functions on some compact Hausdorff space, where $E$ is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative $E$-valued function algebras. We give some specific examples.

International audience; Let $D=G/K$ be a bounded symmetric domain and $S=K/L$ be its Shilov boundary We consider the action of $G$ with weight $\nu\in\mathbb{Z}$ on functions on $D$ viewed as sections the line bundle and the corresponding eigenspace of $G$-invariant di erential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of Hua operators on the sections of the line bundle. For some special parameter the Poisson transform is of Szegö type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.

It has been shown that any Banach algebra satisfying ‖f2‖ = ‖f‖2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, \(\mathbb{F}\)) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where \(\mathbb{F}\) = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, \(\mathbb{F}\)) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.

We establish general sufficient conditions for maps between function algebras to be composition or weighted composition operators, which extend previous results in [2,4,6,7]. Let $X$ be a locally compact Hausdorff space and $A \subset C(X)$ a dense subalgebra of a function algebra, not necessarily with unit, such that $X = \partial A$ and $p(A)=\delta A$, where $\partial A$ is the Shilov boundary, $\delta A$ -- the Choquet boundary, and $p(A)$ -- the set of $p$-points of $A$. If $T\colon A \to B$ is a surjective map onto a function algebra $B\subset C(Y)$ such that either $\ps(Tf\cdot Tg)\subset\ps(fg)$ for all $f,g \in A$, or, alternatively, $\ps(fg)\subset\ps(Tf\cdot Tg)$ for all $f,g \in A$, then there is a homeomorphism $\psi\colon \delta B\to\delta A$ and a function $\a$ on $\delta B$ so that $(Tf)(y)=\a(y)\,f(\psi(y))$ for all $f \in A$ and $y \in\delta B$. If, instead, $\ps(Tf\cdot Tg)\cap\ps(fg)\neq \varnothing$ for all $f,g\in A$, and either $\ps(f)\subset \ps(Tf)$ for all $f\in A$, or, alternatively, $\ps(Tf)\subset \ps(f)$ for all $f\in A$, then $(Tf)(y)=f(\psi(y))$ for all $f \in A$ and $y \in \delta B$. In particular, if $A$ and $B$ are uniform algebras and $T\colon A \to B$ is a surjective map with $\ps(Tf\cdot Tg)\cap\ps(fg)\neq\varnothing$ for all $f,g \in A$, that has a limit, say $b$, at some $a\in A$ with $a^2=1$, then $(Tf)(y)=b(y)\,a(\psi(y))\, f(\psi(y))$ for every $f\in A$ and $y\in\delta B$.

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality \(\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} \).

Let S be a discrete abelian cancellation semigroup with identity. We consider an algebra of generalized analytic functions defined on the semigroup \( \hat S \) of semicharacters of S that is wider than the Arens-Singer algebra. We show that the strong boundary and the Shilov boundary of this algebra are unions of maximal subgroups of \( \hat S \). If S contains no nontrivial simple ideals, then both boundaries coincide with the character group of S. In this case we calculate the Gelfand spectrum of the algebra under consideration.

A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of the lattice of all subsets of $\N$ into the family of pervasive subalgebras of some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is removed. We discuss further examples, involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures.

Let 𝒟 be the n-dimensional Lie ball and let 𝔅(S) be the space of hyperfunctions on the Shilov boundary S of 𝒟. The aim of this paper is to give a necessary and sufficient condition on the generalized Poisson transform Pl，λf of an element f in the space 𝔅(S) for f to be in Lp(S), 1 < p < 1. Namely, if F is the Poisson transform of some f ∈ 𝔅(S) (i.e., F = Pl，λf ), then for any l ∈ ℤ and ， ∈ C such that , we show that f ∈ Lp(S) if and only if f satisfies the growth condition

International audience; Let $K$ be an ultrametric complete field and let $E$ be an ultrametric space. Let $A$ be the Banach $K$-algebra of bounded continuous functions from $E$ to $K$ and let $B$ be the Banach $K$-algebra of bounded uniformly continuous functions from $E$ to $K$. Maximal ideals and continuous multiplicative semi-norms on $A$ (resp. on $B$) are studied by defining relations of stickness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of $A$ (resp. of $B$) is in bijection with the set of equivalence classes with respect to stickness (resp. to contiguousness). Every prime ideal of $A$ or $B$ is included in a unique maximal ideal and every prime closed ideal of $A$ (resp. of $B$) is a maximal ideal, hence every continuous multiplicative semi-norms on $A$ (resp. on $B$) has a kernel that is a maximal ideal. If $K$ is locally compact, every maximal ideal of $A$, (resp. of $B$) is of codimension $1$. Every maximal ideal of $A$ or $B$ is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on $E$. Consequently, on $A$ as on $B$ the set of continuous multiplicative semi-norms defined by points of $E$ is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of $A , \ Max(A)$ and the Banaschewski compactification of $E$ which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of $A$ (resp. $B$) is equal to the whole set of continuous multiplicative semi-norms.

We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

Let $X=H/L$ be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain $D=G/K$. The intersection $S$ of the Shilov boundary of $D$ with $X$ defines a distinguished subset of the topological boundary of $X$ and is invariant under $H$ and can also be realized as $S=H/P$ for certain parabolic subgroup $P$ of $H$. We study the spherical representations $Ind_P^H(\lam)$ of $H$ induced from $P$. We find formulas for the spherical functions in terms of the Macdonald ${}_2F_1$ hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces $D$. We consider a class of $H$-invariant integral intertwining operators from the representations $Ind_P^H(\lam)$ on $L^2(S)$ to the holomorphic representations of $G$ on $D$ restricted to $H$. We construct a new class of complementary series for the groups $H=SO(n, m)$, $SU(n, m)$ (with $n-m >2$) and $Sp(n, m)$ (with $n-m>1$). We realize them as a discrete component in the branching rule of the analytic continuation of the holomorphic discrete series of $G=SU(n, m)$, $SU(n, m)\times SU(n, m)$ and $SU(2n, 2m)$ respectively.

In this article we study the dominating phenomenon for uniform algebra on pseudoconvex domains in several complex variables. We prove that a subset E of a bounded domain D is a dominating set for A(D) if and only if E contains the Shilov boundary S(D) of A(D).

For any holomorphic function 𝑓 on the unit polydisk we consider its restriction to the diagonal, i.e., the function in the unit disc 𝔻 ⊂ ℂ defined by Diag 𝑓(𝑧) = 𝑓(𝑧, . . . , 𝑧) and prove that the diagonal map Diag maps the space of the polydisk onto the space of the unit disk.

We consider continuity properties of analytic functions in bounded domains in complex space, focusing on the following two questions:Question 1. Given a bounded domain G in complex affine n-space. For which subsets S of the boundary of G does the following hold? If f is a bounded function which is analytic in G and extends continuously to the union of G and S then f extends continously to the closure of G.Question 2. Given a domain G as above and a function f analytic in G and continuous in the closure of G. For which subsets S of the boundary is the rate of continuity of f on the closure of G determined already by its rate of continuity along S?Previously it was known that in some cases, the set S may be taken to be the Shilov boundary of the algebra of analytic functions on the domain which are continuous on the closure of the domain, but examples when this is not sufficient were also known.In Paper I we prove that for a bounded, smoothly bounded pseudoconvex domain G in 2-dimensional complex affine space one may take for S an open neighbourhood of the Shilov boundary (in the boundary of the domain G). This is joint work with B. Joricke.Paper II considers smoothly bounded Reinhardt domains in 2 dimensions and complete Reinhardt domains in complex affine n-space. We prove that in many cases one may let S be the Shilov boundary. Moreover, the respective continuity properties propagate to the closure of the envelope of holomorphy of the domain.Paper III considers pseudoconvex Hartogs domains with connected vertical sections in two dimensions. We prove that in some cases S may be taken to be the Shilov boundary and give examples for when this is not enough.

Let E be the dual of a Frechet nuclear space, then it is well-known that for each open set U in E the space ℋ(U) of all holomorphic functions on U is a nuclear Frechet space. Let be a commutative unital Banach sub-algebra of all bounded holomorphic functions on U which separates points. Applying the nuclearity of ℋ(U) we show that the evaluation on U is given by an integral formula over the Shilov boundary of . We obtain Szego- and Bergman kernels together with some boundary estimates. Moreover, we show that there is a notion of Hardy and Bergman space for ℱ-domains with arbitrary boundary. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

For any n, , such that n ≥ m, let { ; Im − z * z positive definite} be the bounded symmetric domain of type I of rank m and let A′(S) be the space of all hyperfunctions over the Shilov boundary S of . The aim of this article is to give a necessary and sufficient condition on the generalized Poisson transform Pl ,λ f of an element f in the space A′(S) for that f∈L p (S), 1 < p < ∞. More precisely, we establish for any such that that: (i) Let F = Pl ,λ f, f∈L p (S). Then we have (ii) Let f be a hyperfunction on S such that its image F = Pl ,λ f satisfies the growth condition , then necessarily such f is in Lp (S).

We show a sufficient condition for a domain in \({\mathbb{C}^{n}}\) to be a H∞-domain of holomorphy. Furthermore if a domain \({\Omega \subset\subset \mathbb{C}^{n}}\) has the Gleason \({\mathcal{B}}\) property at a point \({p \in \Omega}\) and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then \({\mathcal{M}^{B}}\) is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.

The Shilov boundary of a symmetric domain D = G/K of tube type has the form G/P, where P is a maximal parabolic subgroup of the group G. We prove that the simply connected covering of the Shilov boundary possesses a unique (up to inversion) invariant ordering, which is induced by the continuous invariant ordering on the simply connected covering of G and can readily be described in terms of the corresponding Jordan algebra.

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $ \mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))$ whenever $ A$ is approximately regular.

Let φ: A→ B be a surjective operator between two uniform alge-(1) =1 we Show that if φ satisfies the peripheral multiplicativity bras with φ(1) = 1. We show that if if satisfies the peripheral multiplicativity condition σ π (φ(f)φ(g)) = σ π (fg) for all f,g ∈ A, where σ π (f) is the peripheral spectrum of f, then φis an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.

Let N = V ⊕ Z be a step-two nilpotent group. We study the Radon transform on functions on N defined by integration over the orbits {(z, t)(v, 0); v ∈ V} of the points (z, t) by the action of the subspace V ⊕ 0 of N. For N being of H-type, Siegel-type defined as the Shilov boundary of a complex Siegel domain, or Type F2r, b, a real analogue of the Siegel type, we find an inversion and a Plancherel formula for the Radon transform. We prove certain Lp - Lq boundedness properties for the transform. This generalizes earlier works of Geller–Stein and Strichartz for the Heisenberg groups.

We characterize the L p -range, 1 p + ∞ , of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains Ω. We prove that this range is a Hardy type space for solutions of a Hua system, which are eigenfunctions of all invariant differential operators on Ω.

We study the spectrum of certain Banach algebras of holomorphic functions defined on a domain Ω where ∂−-problems with certain estimates can be solved. We show that the projection of the spectrum onto Cn equals Ω−− and that the fibers over Ω are trivial. This is used to solve a corona problem in the special case where all but one generator are continuous up to the boundary.

Let $\cal D$ be a bounded domain, $G$ its group of biholomorphic diffeomorphisms, and $S$ its Shilov boundary. We define a function $ \iota : S\times S\times S \longrightarrow \Bbb R$ which is invariant under $G$. This invariant generalizes the Maslov index as defined earlier by B. Osrted and the author and the angular invariant constructed by E. Cartan for the unit sphere in $\Bbb C^2$.

Let A1(BX) be the Banach space of all bounded and continuous functions on the closed unit ball BX of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let Au(BX) be the subspace of A1(BX) of those functions which are uniformly continuous on BX. A subset B ⊂ BX is a boundary for A1(BX) if k f k = supx2B |f(x)| for every f ∈ A1(BX). We prove that for X = d(w,1) (the Lorentz sequence space) and X = C1(H), the trace class operators, there is a minimal closed boundary for A1(BX). On the other hand, for X = S, the Schreier space, and X = K(lp, lq) (1 ≤ p ≤ q < ∞), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.

We continue to discuss the example presented in \cite{JarPfl2015}. In particular, we clarify some gaps and complete the description of the Shilov boundary.

We prove that Sp(2d;R), HSp(2d) and pseudo unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non periodic dynamical systems. For Schr\"odinger operators on the strip, we prove a similar result for density of positive Lyapunov exponents.

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if $D$ is a bounded domain in $\mathbb C^n$ having a univalent envelope of holomorphy, say $\widetilde D$, then the Shilov boundary of $D$ with respect to the algebra $\mathcal A(D)$ coincides with the corresponding one for $\widetilde D$.

Let A b ( E ) be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball B E of a complex Banach space E and holomorphic in the interior of B E and let A u ( E ) be the closed subalgebra of those functions which are uniformly continuous on B E . For the case E = M w 0 whose bidual is a Marcinkiewicz sequence space M w , we describe some sufficient conditions for a set to be a boundary of either A b ( E ) or A u ( E ) . Moreover, we consider some analogous problems on M w 0 to those which were studied on the Gowers space G p of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575–586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231–242].

International audience; Let $\Omega$ be a bounded symmetric domain of non-tube type in $\mathbb{C}^n$ with rank $r$ and $S$ its Shilov boundary. We consider the Poisson transform $\mathcal{P}_sf(z)$ for a hyperfunction $f$ on $S$ defined by the Poisson kernel $P_s(z, u)=\left({h(z, z)^{\frac{n}{r}}}/{|h(z, u)^{\frac{n}{r}} |^2}\right)^{s}$, $(z, u)\times \Omega\times S$, $s\in \mathbb{C}$. For all $s$ satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When $\Omega$ is the type $\mathbf{I}$ matrix domain in $M_{n, m}(\mathbb{C})$ ($n\leq m$), we prove that an eigenvalue equation for the second order $M_{n, n}-$valued Hua operator characterizes the image.

Let D={z∈ℂ n /1−2z¯z t+|zz t|2>0 and |zz t| n/2−1, we establish that every function F satisfying the following Hua system of differential equations: is the Poisson transform of an L p -function (1