Marcinkiewicz-type multipliers on products of noncompact symmetric spaces

B{\l}a\.zej Wr\'obel; Stefano Meda

REVIEW 3 major objections 4 minor 32 references

Marcinkiewicz-type multipliers on products of noncompact symmetric spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A product Marcinkiewicz multiplier theorem holds on products of rank-one noncompact symmetric spaces.

desk verdict First real Marcinkiewicz theorem for products of rank-one symmetric spaces; proof is solid but with a load-bearing transference step left as 'straightforward'. read the letter →

arxiv 1908.08831 v1 pith:BP3ULOCY submitted 2019-08-23 math.FA

Stefano Meda , B{l}ażej Wróbel This is my paper · ORCID

classification math.FA MSC 43A8543A32

keywords MarcinkiewiczmultipliersymmetricspacesphericalFouriertransformLpboundednesstransferencerankoneproductnoncompacttype

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that a Marcinkiewicz-type condition on the spherical Fourier transform—one that controls mixed derivatives by products of the one-variable weights $\Theta_p^{(1)}(\lambda_1)\Theta_p^{(2)}(\lambda_2)$ on the tube $T_p^{(1)}\times T_p^{(2)}$—is enough to make a $G$-invariant operator bounded on $L^p(X_1\times X_2)$ for $p\in(1,\infty)\setminus\{2\}$, provided the differentiation orders exceed $(n_j+3)/2$. This fills a gap: Euclidean-type multiplier theorems on the product cannot treat multipliers that are singular along the coordinate walls, such as those associated with operators mixing the two Laplace–Beltrami operators. The proof splits the convolution kernel into local, wall, and infinity pieces, then controls each piece by spherical-function expansions and by transference to the Iwasawa groups. If correct, the result provides the first Marcinkiewicz multiplier theorem for spherical transforms on noncompact symmetric spaces, and it extends to products of any finite number of rank-one factors.

What carries the argument

The central object is the product Marcinkiewicz norm $\|m\|_{M(T_p;N)}$ on the tube $T_p=T_p^{(1)}\times T_p^{(2)}$, defined by the weights $\Theta_p^{(j)}(\lambda_j)=\min(|\lambda_j-i\delta(p)\rho_j|,|\lambda_j+i\delta(p)\rho_j|)$; this is the object that converts derivative estimates into $L^p$ boundedness. The carrying mechanism is a three-piece kernel decomposition: a local piece near the identity uses Bessel-function expansions, a wall piece uses one asymptotic expansion and one local expansion, and the infinity piece uses two asymptotic expansions. Each piece is then handled by a semidirect-product transference theorem that bounds convolution operators on $N_1A_1\times G_2$ (or $N_1A_1\times N_2A_2$) in terms of easier convolution norms on the abelian subgroups $A_1\times A_2$.

What would settle it

Compute, for a nontrivial $K_2$-bi-invariant kernel $\kappa_{v_1}$ on $A_1\times G_2$, both sides of the inequality in Corollary 3.4(ii), using the modular weight $D_1^{1/p}\delta_2$ on $A_1\times A_2$; if the left-hand $\mathrm{Cv}^p(A_1\times G_2)$ norm exceeds the right-hand $\mathrm{Cv}^p(A_1\times A_2)$ norm times the claimed constant, the transference step fails.

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Extended reading notes

Core claim

On the product $X=X_1\times X_2$ of two rank-one noncompact symmetric spaces, the authors establish Theorem 1.1: for $p\in(1,\infty)\setminus\{2\}$ and $N_j>(n_j+3)/2$, every $G$-invariant operator $B$ whose spherical multiplier $m_B$ is holomorphic on $T_p^{(1)}\times T_p^{(2)}$ and satisfies $|\partial_{\lambda_1}^{j_1}\partial_{\lambda_2}^{j_2}m_B(\lambda_1,\lambda_2)|\le C\,\Theta_p^{(1)}(\lambda_1)^{-j_1}\Theta_p^{(2)}(\lambda_2)^{-j_2}$ extends to a bounded operator on $L^p(X)$, with operator norm controlled by the multiplier norm. The point is that the two derivative weights factor: each factor only sees the distance from the corresponding one-variable tube boundary, so the multiplier may be singular along an entire coordinate wall. The proof decomposes the kernel into $B_0+B_1+B_2$, representing contributions near the origin, near one wall, and away from both walls, and analyses each with the local and asymptotic expansions of spherical functions together with a transference theorem that moves estimates from solvable Iwasawa groups to the symmetric space.

Load-bearing premise

The proof depends on an asserted transference step that moves norm estimates from the flat subgroup $A_1\times A_2$ to the larger group $A_1\times G_2$ without increasing them; the step is stated as straightforward but not proved, and the estimates for two of the three kernel pieces collapse if it is wrong.

Editorial extensions

If this is right

Joint spectral multipliers of the pair $(L_1,L_2)$ that satisfy the product Marcinkiewicz estimates are $L^p(X)$ bounded; in particular the theorem covers operators whose multipliers are singular on coordinate walls, a case outside Euclidean-type multiplier theorems on the product.
By interpolation with the trivial $L^2$ case, the $L^p$ bound extends to every $r$ with $|1/r-1/2|\le|1/p-1/2|$.
The statement extends to products $X_1\times\cdots\times X_m$ of $m\ge3$ rank-one noncompact symmetric spaces; the authors indicate that the modifications are straightforward but omit them.
The comparison in Remark 4.2 shows that the product Marcinkiewicz condition is independent of the higher-rank condition used previously, so the theorem genuinely enlarges the class of admissible multipliers near the walls.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural test is to extend the theorem to higher-rank symmetric spaces by using products of one-variable wall distances; the independence example suggests that the correct formulation should weight each root direction separately rather than by a single distance to the polyhedron.
The unproved transference step in Corollary 3.4(ii) should be checked before relying on the wall estimates; writing out the proof or finding a counterexample would settle whether the current route is valid.
The result implies that symmetric-space analogues of operators with multipliers such as $(|\lambda_1|^2+|\lambda_2|^2)^{iu}|\lambda_2|^{iv}$ are $L^p$ bounded, which joint spectral multiplier theorems based on sectors do not deliver.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know about 1908.08831: it proves the first Marcinkiewicz-type multiplier theorem for products of two rank-one noncompact symmetric spaces, and the product condition (4.1) is new and not subsumed by Ionescu's earlier condition. The result is real and not in the literature. The proof follows Ionescu's transference strategy, splitting the kernel into B0, B1, B2; the estimates in Propositions 5.1, 5.3, 6.2 and 7.4 are detailed and mostly written out. The paper handles operators like L1^{it}(L1+L2)^{iu}L2^{iv} that escape Hormander-type theorems, and Remark 4.2 correctly shows (4.1) and (4.3) are independent. I found no internal contradiction.

Soft spots: the paper has several 'details omitted' or 'straightforward to check' passages. The one that matters is Corollary 3.4(ii), the transference from A1xA2 to A1xG2 used in the B1 and B2 estimates. The proof is one sentence citing Coifman-Weiss. I checked the specific worries in the stress-test note: the K2-bi-invariance of kappa_{v1} does follow from kappa being K-bi-invariant, and A1xG2 does admit the Cartan decomposition ({e1}xK2)(A1xA2)({e1}xK2) because A1 is abelian and G2=K2A2K2. So the 'extension is well-defined' objection does not land. What is missing is an explicit verification that [CW, Theorem 8.7] covers this non-semisimple product group, and that the constant is independent of v1. That is a real gap in presentation, not a fatal one; it should be supplied before acceptance. Other omissions are smaller: parts of Lemma 5.2(iii), Lemma 7.1(iv), and the even/odd parity cases in Proposition 5.3 are left as 'similar'; the m>=3 extension is stated as 'straightforward though lengthy' with no proof. The epsilon-regularization is also asserted without proof; for the M-norm it is plausible but deserves a check.

The central claim holds up: I agree with the reader's conditional verdict, with MODERATE confidence because I cannot machine-check the analytic estimates and the omitted transference proof is load-bearing. This is a paper for harmonic analysts working on symmetric spaces and multiplier theorems. It deserves a serious referee. Recommendation: send to peer review; ask the authors to verify the transference hypotheses explicitly and either prove or demote the m>=3 statement.

Referee Report

3 major / 4 minor

Summary. The paper proves a Marcinkiewicz-type multiplier theorem for the spherical Fourier transform on a product X1 times X2 of two rank-one noncompact symmetric spaces. Under separate Marcinkiewicz-type conditions in each variable, with derivative orders N1 > (n1+3)/2 and N2 > (n2+3)/2, any G-invariant operator with such a multiplier is shown to be bounded on Lp(X1 times X2) for p in (1, infinity) other than 2. The proof splits the kernel into a local part B0, a wall part B1, and an infinity part B2, applying local and Harish-Chandra expansions of spherical functions and reducing the estimates to convolutor bounds on abelian groups and semidirect products, with a transference principle from Coifman-Weiss and a previous paper by two of the authors used at a key reduction step.

Significance. If the proof is completed, the result is a substantial extension of Marcinkiewicz multiplier theory to products of rank-one noncompact symmetric spaces, covering operators such as products of imaginary powers of Laplacians that are outside the scope of Hörmander-type theorems. The paper contains a useful comparison with Ionescu's higher-rank condition in Remark 4.2, showing that the two conditions are independent near the walls, and the B0/B1/B2 decomposition is natural and well organized. The analytic estimates in Propositions 5.1, 5.3, 6.2, and 7.4 are extensive and for the most part carefully written. However, the paper is not fully self-contained because a load-bearing transference step is asserted rather than proved, and several supporting lemmas have omitted proofs.

major comments (3)

[Section 3.1, Corollary 3.4(ii)] This corollary is the only step that moves Cv^p bounds from the abelian group A1 times A2 to the non-abelian group A1 times G2, and it is used directly at equation (6.12) in Proposition 6.2(iii) to control the tau^p,3 term for the kernel kappa_{1A2}. The proof is one sentence: the group A1 times G2 is said to admit the Cartan decomposition ({e1} times K2)(A1 times A2)({e1} times K2), and [CW, Theorem 8.7] is applied to the {e1} times K2-bi-invariant extension of kappa_{v1}. The manuscript never verifies that (a) the quoted transference theorem applies to the non-semisimple group A1 times G2, (b) the {e1} times K2-bi-invariant extension of kappa_{v1} is well-defined, which requires checking Weyl invariance on A2, or (c) the extension is norm-preserving with a constant independent of v1. Without (6.12), the estimate for kappa_{1A2} collapses and the boundedness of B1 is not established. Please provide a complete proof of Corollary 3.4(ii), or state precisely the transference theorem used and verify all of its hypotheses.
[Section 5, Lemma 5.2(iii)] The estimate in Lemma 5.2(iii) is stated without proof ('The proof of (iii) follows the lines of the proof of (ii). We omit the details.'), but it is needed in Proposition 5.3(iii) for the case where n1 is even and n2 is odd. Specifically, it provides the Marcinkiewicz condition for the function d_{v1} H that is required to conclude kappa_{1,1} belongs to Cv^p(X). The mixed-parity case is not literally the same as (ii), because after the change of variables only one angular integration remains and the Weyl-invariance argument must be checked separately. The omitted details should be supplied.
[Section 4, Outline after (4.8)] The proof assumes that the multiplier mB is pre-multiplied by ~h_eps(lambda) = exp(-eps(lambda1^2 + lambda2^2)) and states that this is no loss of generality because the final bounds depend on ||mB||_{M(Tp;N)} and the norms of mB ~h_eps converge to that value. No proof of this convergence or of the passage to the limit as eps tends to 0 is given. Since Theorem 1.1 is stated for all multipliers in M(Tp;N) and not only for rapidly decreasing ones, this limiting argument must be supplied, or the theorem must be restricted to the regularized class and a separate argument given for the general case.

minor comments (4)

[Definition 4.1, Eq. (4.2); Definition 4.3, Eq. (4.7)] In both definitions the displayed norm for the 'at infinity' class is written with the same symbol as the non-infinity class: Eq. (4.2) should be ||m||_{M_infty(Tp;N)} and Eq. (4.7) should be ||m||_{M_infty(a*;N)}.
[Section 4, Theorem restatement] The sentence 'We denote by 3 the two dimensional vector (3, 3)' is confusing; use a distinct multi-index notation, for example N0 = (3,3), or write the condition componentwise as N1 > (n1+3)/2 and N2 > (n2+3)/2.
[Section 3.1, Corollary 3.4] The symbol delta2 is used in Corollary 3.4(ii) but is not defined there; it should be defined explicitly (as the density in Cartan coordinates on A2, introduced in Section 2.3) at the point of first use.
[Section 7, Lemma 7.1(iv)] The proof of Lemma 7.1(iv) is omitted with the remark that it is similar to (iii) with roles interchanged. Since the statement is used in Lemma 7.2 to prove that phi^p_{11} lies in Cv^q(A), a short proof should be included for completeness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed multiplier theorem is derived from independent rank-one and transference results, not from its own conclusion.

full rationale

The paper contains no fitted inputs called predictions, no parameter calibrated to a data subset, and no definition of the theorem's conclusion inside its hypotheses. Theorem 1.1 assumes a genuine Marcinkiewicz-type multiplier condition and derives L^p boundedness through a chain of reductions: local and remote kernel estimates (Stanton-Tomas, Ionescu), a reduction to the abelian group A1 x A2, and the classical Marcinkiewicz theorem (Theorem 4.5). The transference results used in Corollary 3.4 and Theorem 3.3 are quoted from Coifman-Weiss [CW] and Celotto-Meda-Wróbel [CMW]; [CMW] is a self-citation, but it is a published, parameter-free theorem about convolution operators on semidirect products and homogeneous trees, not a restatement of Theorem 1.1, and it does not assume the target conclusion. The skeptical point about Corollary 3.4(ii) (applying [CW, Theorem 8.7] to the non-semisimple group A1 x G2 and to a K2-bi-invariant extension) is a possible missing verification, hence a correctness risk, not a circular step: the estimate sought there is not the same as the theorem's hypothesis, and no quantity is fitted. Accordingly, the paper's central claim is not forced by self-citation or by definition.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theorem adds no fitted parameters and no invented entities. It assumes standard spherical analysis and uses two transference results: a semidirect-product transference from [CMW] and a Coifman-Weiss application to A1×G2 in Corollary 3.4(ii). The latter is the most fragile input.

assumptions (6)

standard math Spherical Fourier transform, Plancherel formula, Harish-Chandra c-function and spherical function expansions with derivative estimates (equations (2.1), (2.2), (2.4), (2.5))
Invoked throughout Sections 5-7 to expand the kernel kB near the origin, near walls, and at infinity; sourced to [GV], [ST], [I1, Appendix A].
standard math Classical Marcinkiewicz multiplier theorem on Euclidean spaces (Theorem 4.5)
Used to conclude that inverse Fourier transforms are Cv^p(A) multipliers in Propositions 5.3, 6.2, and 7.4.
standard math Ionescu's rank-one transference and kernel estimates (Lemmas 2.6 and 2.7)
Provides the rank-one building blocks for φp and κ1 estimates; the paper sketches proofs and cites [I1].
standard math Semidirect product transference principle [CMW, Corollary 3.4] (Theorem 3.3)
Reduces Cv^p norms on N1A1×G2 and NA to integrals over N of Cv^p norms on A1×G2 or A; load-bearing for B1 and B2.
ad hoc to paper Coifman-Weiss transference applied to the non-semisimple group A1×G2 with compact subgroup {e1}×K2 (Corollary 3.4(ii))
The paper asserts without proof that Cv^p boundedness on A1×A2 transfers to A1×G2 via the Cartan decomposition ({e1}×K2)(A1×A2)({e1}×K2); this is the most fragile premise.
domain assumption G=G1×G2 with K=K1×K2, rank-one factors and product Weyl chamber structure
Defines the setting of Theorem 1.1; A+ = A1+×A2+ and the tube Tp factorizes as Tp^{(1)}×Tp^{(2)} (Section 2.3).

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Pith. "Pith review of Marcinkiewicz-type multipliers on products of noncompact symmetric spaces." pith.science (2026). https://pith.science/paper/BP3ULOCY

@misc{pith2026190808831,
  author       = {Pith},
  title        = {Pith review of: Marcinkiewicz-type multipliers on products of noncompact symmetric spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BP3ULOCY}},
  note         = {Machine review of arXiv:1908.08831}
}

read the original abstract

In this paper we prove a Marcinkiewicz-type multiplier result for the spherical Fourier transform on products of rank one noncompact symmetric spaces.

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Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages

[1]

Anker, L_p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann

J.-Ph. Anker, L_p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. 132 (1990), 597--628

work page 1990
[2]

Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math

J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257--297

1992
[3]

Anker and L

J.-Ph. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces I, Geom. Funct. Anal. 9 (1999), 1035--1091

work page 1999
[4]

Anker and N

J.-Ph. Anker and N. Lohou\'e, Moltiplicateurs sur certain espaces sym\'etriques, Amer. J. Math 108 (1986), 1303--1354

work page 1986
[5]

Carbonaro, G

A. Carbonaro, G. Mauceri and S. Meda,

work page
[6]

Celotto, S

D. Celotto, S. Meda and B. Wr\'obel, L^p spherical multipliers on homogeneous trees, Studia Math. 247 (2019), 175--190

work page 2019
[7]

Cheeger, M

J. Cheeger, M. Gromov and M. Taylor,

work page
[8]

Clerc and E.M

J.-L. Clerc and E.M. Stein, L^p multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U. S. A. 71 (1974), 3911--3912

work page 1974

Show all 32 references

[9]

Coifman and G

R.R. Coifman and G. Weiss, Extensions of Hardy
[10]

Coifman and G

R.R. Coifman and G. Weiss, Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31, American Mathematical Society, Providence, R.I., 1976

1976
[11]

Cowling, Some applications of Grothendieck's theory of topological tensor products in Harmonic Analysis, Math

M.G. Cowling, Some applications of Grothendieck's theory of topological tensor products in Harmonic Analysis, Math. Ann. 232 (1978), 273--285

1978
[12]

Cowling, S

M.G. Cowling, S. Giulini and S. Meda, Estimates for functions of the Laplace--Beltrami operator on noncompact symmetric spaces. II, J. Lie Th. 5 (1995), 1--14

1995
[13]

Gangolli, On the Plancherel

R. Gangolli, On the Plancherel
[14]

Gangolli and V.S

R. Gangolli and V.S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer-Verlag, 1988

1988
[15]

Giulini, G

S. Giulini, G. Mauceri and S. Meda, L^p multipliers on noncompact symmetric spaces, J. reine angew. Math. 482 (1997), 151--175

1997
[16]

Guivarc'h, L

Y. Guivarc'h, L. Ji and J.C. Taylor,
[17]

Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984

S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984

1984
[18]

Helgason, Differential Geometry, Lie groups, and Symmetric Spaces, Academic Press, New York, 1978

S. Helgason, Differential Geometry, Lie groups, and Symmetric Spaces, Academic Press, New York, 1978

1978
[19]

Helgason, Geometric analysis on symmetric spaces, Math

S. Helgason, Geometric analysis on symmetric spaces, Math. Surveys & Monographs 39, Amer. Math. Soc., 1994

1994
[20]

Hewitt and K.A

E. Hewitt and K.A. Ross, Abstract Harmonic Analysis , A Series of Comprehensive Studies in Mathematics 1 (1979), n. 115, Springer- Verlag

1979
[21]

H\"ormander, Estimates for translation invariant operators in L^p spaces, Acta Math

L. H\"ormander, Estimates for translation invariant operators in L^p spaces, Acta Math. 104 (1960), 93--140

1960
[22]

Ionescu, Fourier integral operators on noncompact symmetric spaces of real rank one, J

A.D. Ionescu, Fourier integral operators on noncompact symmetric spaces of real rank one, J. Funct. Anal. 174 (2000), 274--300

2000
[23]

Ionescu, Singular integrals on symmetric spaces of real rank one, Duke Math

A.D. Ionescu, Singular integrals on symmetric spaces of real rank one, Duke Math. J. 114 (2002), 101--122

2002
[24]

Ionescu, Singular integrals on symmetric spaces, II, Trans

A.D. Ionescu, Singular integrals on symmetric spaces, II, Trans. Amer. Math. Soc. 335 (2003), 3359--3378

2003
[25]

Lebedev, Special Functions and Their Applications, Dover, New York, 1972

N.N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972

1972
[26]

Meda and M

S. Meda and M. Vallarino, Weak type estimates for spherical multipliers on noncompact symmetric spaces, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2993--3026

2010
[27]

Stanton, P.A

R.J. Stanton, P.A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251--276

1978
[28]

Stein, Harmonic Analysis

E.M. Stein, Harmonic Analysis. Real variable methods, orthogonality and oscillatory integrals, Princeton Math. Series No. 43, Princeton N. J., 1993

1993
[29]

Str\"omberg, Weak type L^1 estimates for maximal functions on non-compact symmetric spaces, Ann

J.-O. Str\"omberg, Weak type L^1 estimates for maximal functions on non-compact symmetric spaces, Ann. of Math. 114 (1981), 115--126

1981
[30]

Trombi and V.S

P.C. Trombi and V.S. Varadarajan, Spherical transforms on semisimple Lie groups, Ann. of Math. 94 (1971), 246--303

1971
[31]

Watson, A treatise on the theory of Bessel functions, Cambridge Univ

G.N. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press Cambridge, 2nd edition, 1944

1944
[32]

Fourier Anal

B.\ Wr\'obel, Joint spectral multipliers for mixed systems of operators, J. Fourier Anal. Appl. (2) 23 (2017), 245--287

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Anker, L_p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann

J.-Ph. Anker, L_p Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. 132 (1990), 597--628

work page 1990

[2] [2]

Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math

J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257--297

1992

[3] [3]

Anker and L

J.-Ph. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces I, Geom. Funct. Anal. 9 (1999), 1035--1091

work page 1999

[4] [4]

Anker and N

J.-Ph. Anker and N. Lohou\'e, Moltiplicateurs sur certain espaces sym\'etriques, Amer. J. Math 108 (1986), 1303--1354

work page 1986

[5] [5]

Carbonaro, G

A. Carbonaro, G. Mauceri and S. Meda,

work page

[6] [6]

Celotto, S

D. Celotto, S. Meda and B. Wr\'obel, L^p spherical multipliers on homogeneous trees, Studia Math. 247 (2019), 175--190

work page 2019

[7] [7]

Cheeger, M

J. Cheeger, M. Gromov and M. Taylor,

work page

[8] [8]

Clerc and E.M

J.-L. Clerc and E.M. Stein, L^p multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U. S. A. 71 (1974), 3911--3912

work page 1974