Combinatorial property of all positive measures in dimensions $2$ and $3$

We prove multi-parameter dyadic embedding theorem for Hardy operator on the multi-tree. We also show that for a large class of Dirichlet spaces of holomorphic functions in bi-disc and tri-disc this proves the embedding theorem of those spaces on biand tri-disc. We completely describe the Carleson measures for such embeddings. Funding. We acknowledge the support of the following grants: NSF grant DMS-1900286, Theorem 2 was obtained in the frameworks of the Russian Science Foundation grant 17-11-01064-P, the third author was supported also by Alexander von Humboldt foundation. Manuscript received 10th February 2020, revised 25th June 2020, accepted 26th June 2020. Version française abrégée Un n -arbre T n , n ≥ 1, est un produit cartésien de n arbres dyadiques identiques avec un ordre partiel induit par la structure du produit. Etant donné un point β ∈ T n , nous définissons son successeur en posant S (β) = {α ∈ T n : α≤ β}. Soient w,μ deux fonctions positives sur T n , nous définissons la constante de boîte comme le plus petit nombre [w,μ]Box tel que ES (β)[μ] := ∑ α≤β w(α)(I∗μ(α))2 ≤ [w,μ]Boxμ(S (β)), ∀β ∈ T n . (1) La constante de plongement de Carleson est la plus petite constante [w,μ]C E telle que l’inégalité suivante ait lieu: E [ψμ] ≤ [w,μ]C E ∑ ω∈T n |ψ(ω)|2μ(ω) (4) Le résultat principal de cet article est le théorème suivant: ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 722 Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg and Pavel Zorin Kranich Theorem 1. Soit μ : T n → R+, n = 1,2,3 et soit w : T n → [0,∞) un poids d’une forme tensorielle. Alors l’inégalité suivante a lieu [w,μ]C E . [w,μ]Box . 1. Hardy inequality on the n-tree and energy of measures A (finite) tree T is a finite partially ordered set such that for every ω ∈ T the set {α ∈ T : α ≥ ω} is totally ordered (here we identify the tree with its vertex set). In what follows we consider rooted dyadic trees, i.e. there is a unique maximal element in T , and every element (except for the minimal ones) has exactly two children. An n-tree T n , n ≥ 1 is a Cartesian product of n identical dyadic trees with order induced by the product structure. In what follows no estimate will depend on the depth of the tree. A subset U (resp. D) of a partially ordered set T n is called an up-set (resp. down-set) if, for every α ∈U and β ∈ T with α≤β (resp. β≤α), we also have β ∈U (resp. β ∈D). Given a point β ∈ T n we define its successor set S (β) = {α ∈ T n : α≤β}, clearly it is a down-set. From now on we assume that the weight w : T n → R+ is fixed. The Hardy operator associated with w is defined by Iwφ(γ) := ∑ γ′≥γ w(γ′)φ(γ′) and I∗ψ(γ) = ∑ γ′≤γ ψ(γ′). For a measure (non-negative function) μ on T n we define the (w-) potential to be Vμw (α) := (Iw I∗μ)(α), α ∈ T n , again we usually drop the index w . Let E ⊂ T n andμ be a measure on T n . The E-truncated energy of μ is EE [μ] := ∑ α∈E (I∗μ)2(α)w(α). If E = T n , we write E [μ] instead, and so E [μ] = ∫ T n Vdμ := ∑ α∈T n V(α)μ(α). If E is a δ-level set of V for some δ> 0, i.e. E = {α : V ≤ δ}, then we write Eδ[μ] := EE [μ]. We define the box constant to be the smallest number [w,μ]Box such that ES (β)[μ] := ∑ α≤β w(α)(I∗μ(α))2 ≤ [w,μ]Boxμ(S (β)), ∀β ∈ T n . (1) The Carleson constant is the smallest number [w,μ]C such that ED [μ] ≤ [w,μ]Cμ(D), ∀D ⊂ T n down-set. (2) The hereditary Carleson constant (or restricted energy condition constant, or REC constant) is the smallest constant [w,μ]HC such that E [μ1E ] ≤ [w,μ]HCμ(E), ∀ E ⊂ T n . (3) Finally the Carleson embedding constant is the smallest constant [w,μ]C E such that the adjoint embedding E [ψμ] ≤ [w,μ]C E ∑ ω∈T n |ψ(ω)|2μ(ω) (4) holds for all functions ψ on T n . If [w,μ]C E < +∞, we call (w,μ) the trace pair for the weighted Hardy inequality on T n . For positive numbers A,B , we write A . B if A ≤ C B with an absolute constant C , that in particular does not depend on the pair (w,μ). The inequalities [w,μ]Box ≤ [w,μ]C ≤ [w,μ]HC ≤ [w,μ]C E are obvious. The converse inequalities for 1-trees were proved in [6]. Our main result is the extension to the 2and 3-trees. The reader can see details in preprints [3], [4] and partially in [1] for 2-tree. C. R. Mathématique, 2020, 358, n 6, 721-725 Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg and Pavel Zorin Kranich 723 Theorem 1. Let μ : T n → R+, n = 1,2,3. Let w : T n → [0,∞) be of tensor product form. Then the reverses of the above inequalities also hold: [w,μ]C E . [w,μ]HC . [w,μ]C . [w,μ]Box . The key element in the proof of all these inequalities is the so-called surrogate Maximum Principle. To elaborate let us consider the one-parameter case. Then V μ δ (α) ≤ δ for any α ∈ T , so, in particular, for any measure ρ on T one has trivially ∫ T V μ δ dρ ≤ δ|ρ|, where |ρ| = ρ(T ) is the total mass of ρ. In higher dimensions the situation is different, i.e. V δ can blow up at some points, however one can still measure the size of the blow-up set in potentialtheoretic terms. We conjecture that the following holds for all n. Theorem 2. For n = 1,2,3 and a measure ρ on T n one has ∫ T n V δ dρ. ( δ|ρ|) 2 n+1 (Eδ[μ]E [ρ]) n−1 n+1 . 2. Applications to holomorphic Sobolev spaces on the polydisc The Hardy inequality can be interpreted with regards to its connections to certain problems in the theory of Hilbert spaces of analytic functions on the poly-disc. These connections actually motivated the study of the Hardy operator in [2], [5] and [7]. We start with some additional notation. Given an integer n ≥ 1 and s = (s1, . . . , sn) ∈ Rn we consider a Hilbert space H s (Dn) of analytic functions on the poly-disc Dn with the norm ‖ f ‖H s (Dn ) := ∑ k1,...,kn≥0 | f̂ (k1, . . . ,kn)|(k1 +1)s1 · · · · · (kn +1)sn , where f (z) = ∑k1,...,kn≥0 f̂ (k1, . . . ,kn)z1 1 · · · · · zn n , z = (z1, . . . , zn) ∈ Dn . Observe, that, clearly H s (Dn) = ⊗nj=1 H s j (D). In particular, the choice s = (0, . . . ,0) gives a classical Hardy space on the poly-disc, on the other hand s = (1, . . . ,1) corresponds to the Dirichlet space. Definition 3. A measure ν on Dn is called a Carleson measure for H s , if there exists a constant C such that ∫ Dn | f (z)|2 dν(z) ≤C‖ f ‖H s (Dn ), (5) or, in other words, the embedding I d : H s (Dn) → L2(Dn ,dν) is bounded. The smallest constant C such that (5) holds is denoted by [ν]s . Trace pairs for the weighted Hardy inequality on n-tree and Carleson measures for H s are closely related. Below we give a brief overview of this relationship. We start by assuming that s ∈ (0,1]n (so that H s (Dn) is a weighted Dirichlet space on the poly-disc), and that suppν⊂ rDd for some r < 1 (the latter is just a convenience assumption). It is well known that H s j (D), 1 ≤ j ≤ n, is a reproducing kernel Hilbert space (RKHS) with kernel Ks j satisfying (possibly after a suitable change of norm) |Ks j |(z j ,ζ j ) ≈ |1− zζ̄|1−s j , 0 < s j < 1, |Ks j |(z j ,ζ j ) ≈ log |1− zζ̄|−1, s j = 1. It follows immediately that H s (Dn) is RKHS as well, and Ks (z,ζ) = n ∏ j=1 Ks j (z j ,ζ j ), z,ζ ∈Dn . Going back to the Carleson embedding we see that Id : H s (Dn) → L2(Dn ,dν) is bounded if and only if its adjointΘ is bounded as well. Let us compute its action on a function g ∈ L2(Dn ,dν) (Θg )(z) = 〈Θg ,Ks (z, · )〉H s (Dn ) = 〈g ,Ks (z, · )〉L2(Dn ,dν) = ∫ Dn g (ζ)Ks (z,ζ)dν(ζ). C. R. Mathématique, 2020, 358, n 6, 721-725 724 Pavel Mozolyako, Georgios Psaromiligkos, Alexander Volberg and Pavel Zorin Kranich Hence, forΘ to be bounded it must satisfy ‖g‖L2(Dn ,dν) & ‖Θg‖H s (Dn ) = 〈g ,Θg 〉L2(Dn ,dν) = ∫ D2n g (z)g (ζ)Ks (z,ζ)dν(z)dν(ζ). (6) Observe now that if 1−s j , j = 1, . . . ,n is small enough, we can replace the reproducing kernel with its real part. Lemma 4. For any n ≥ 1 there exists a number εn > 0 such that if sup1≤ j≤n(1− s j ) ≤ εn , then |Ks (z,ζ)| ≤C (n)RKs (z,ζ). (7) In this case that (6) is equivalent to