Random forest methods belong to the class of non-parametric machine learning algorithms. They were first introduced in 2001 by Breiman and they perform with accuracy in high dimensional settings. In this article, we consider, a simplified kernel-based random forest algorithm called simplified directional KeRF (Kernel Random Forest). We establish the asymptotic equivalence between simplified directional KeRF and centered KeRF, with additional numerical experiments supporting our theoretical results.

We present a personal overview of some of the areas in complex analysis that are current subject of intense activities. We discuss a variety of topics, in one and in several variables. However, we make no pretense to be exhaustive in any way. Our intention is to possibly stimulate thoughts and discussions in some areas of the theory of holomorphic function spaces.

Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of unitary, and of self-adjoint operators

Random forests are notable learning algorithms introduced by Breiman in 2001. They are widely used for classification and regression tasks and their mathematical properties are under ongoing research. We consider a specific class of random forest algorithms related to kernel methods, the so-called Kernel Random Forests (KeRF). In particular, we investigate thoroughly two explicit algorithms, designed independently of the data set, the centered KeRF and the uniform KeRF. In the present article, we provide an improvement in the rate of convergence for both algorithms and we explore the related reproducing kernel Hilbert space defined by the explicit kernel of the centered random forest.

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse H\"older inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.

Bi-parameter potential theory and Carleson measures for the Dirichlet space on the bidisc, Discrete Analysis 2023:22, 58 pp. Carleson measures arise naturally when considering harmonic or holomorphic extensions from the boundary of a domain to the interior of the domain. For instance, suppose one has an $L^p$ function $f$ on the real line for some $1 \leq p < \infty$, and let $u$ be its standard harmonic extension to the upper half-plane, given by convolution with the Poisson kernel. The function $u$ will not be expected to be an $L^p$ function on the upper half-plane with respect to the usual Lebesgue measure, but turns out to be an $L^p(\mu)$ function for other measures $\mu$ on the upper half-plane. Indeed, the celebrated Carleson embedding theorem asserts that a measure $\mu$ has this property if and only if it is what is now known as a "Carleson measure", which means that for every interval $[x-r,x+r]$ on the real line, the measure $\mu( [x-r,x+r] \times [0,r])$ assigned to the rectangle $[x-r,x+r] \times [0,r]$ is bounded by a constant times the length of the interval. Analogous results hold when $f$ is an $H^p$ function on the circle (roughly speaking, an $L^p$ function with a holomorphic extension to the unit disc); this theorem has many applications in complex analysis and harmonic analysis, for instance to the corona problem of determining the spectrum of the Hardy space $H^\infty$ (viewed as a Banach algebra), and in the theory of functions of bounded mean oscillation (BMO). They are also instrumental in describing multipliers of Hardy spaces $H^p$: holomorphic functions $g$ with the property that multiplication by $g$ is a bounded operation on $H^p$, though the description is a bit more complicated, in which the intervals $[x-r,x+r]$ have to be replaced with finite unions of intervals, and the notion of length replaced with the more complicated notion of "Bessel capacity" from potential theory. The original Carleson embedding theorem can be extended to higher dimensions without much difficulty (intervals get replaced by balls, and certain exponents get adjusted accordingly). However, when dealing with holomorphic functions of several complex variables, defined on the polydisc (the product of several copies of the unit disc), the situation becomes more delicate, even for functions of two complex variables on the bidisc, basically because intervals get replaced by axis-parallel rectangles of arbitrary eccentricity, which can no longer be interpreted as single-parameter balls in a metric space, but are instead genuinely bi-parameter objects. Many of the classical harmonic analysis techniques that can handle the geometry of single-parameter metric balls will fail in the bi-parameter setting if adapted naively, but over time many authors have come up with ingenious substitutes for the classical theory that can handle bi-parameter or multi-parameter settings. Often, even just stating the right generalizations correctly is a significant part of the problem. In this paper, the authors state and prove the analogue of the Carleson embedding theorem and the multiplier characterization for the bidisc, where in both cases the characterization involves the Bessel capacity of finite unions of rectangles. This is achieved by first discretizing the problem to an analogous problem on the bitree (the product of two infinite dyadic trees), and then by carefully developing a bi-parameter capacity theory first on the bitree, and then on the bidisc. There are many technical subtleties, as some (but not all) of the classical one-parameter techniques are known to fail in the multi-parameter setting.

We characterize Q-dimensional Ahlfors regular spaces among trees' boundaries and show how to construct, for each 0 < alpha < Q, an alpha-regular subspace. As an application, we give an alternative simple proof of the existence of alpha-regular subspaces of a Q-dimensional complete Ahlfors regular metric space (X, rho), which was proved in [8].

We characterize the Carleson measures for the Dirichlet space on the bidisc, hence also its multiplier space. Following Maz'ya and Stegenga, the characterization is given in terms of a capacitary condition. We develop the foundations of a bi-parameter potential theory on the bidisc and prove a Strong Capacitary Inequality. In order to do so, we have to overcome the obstacle that the Maximum Principle fails in the bi-parameter theory.

We study the Riesz (a, p)-capacity of the so called Dobiński set. We characterize the values of the parameters a and p for which the (a, p)-Riesz capacity of the Dobiński set is positive. In particular we show that the Dobiński set has positive logarithmic capacity, thus answering a question of Dayan, Fernandéz and González. We approach the problem by considering the dyadic analogues of the Riesz (a, p)-capacities which seem to be better adapted to the problem.

We give a characterization of equilibrium measures for p-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For p=2 , this provides, in the special case of trees, a converse to a theorem of Benjamini and Schramm, which interpretes the equilibrium measure of a planar graph’s boundary in terms of square tilings of cylinders.

In this work we study what we call Siegel–dissipative vector of commuting operators $$(A_1,\ldots , A_{d+1})$$ ( A 1 , … , A d + 1 ) on a Hilbert space $${{\mathcal {H}}}$$ H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space $${{\mathcal {U}}}$$ U . The operator $$A_{d+1}$$ A d + 1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $$\{e^{-i\tau A_{d+1}}\}_{\tau <0}$$ { e - i τ A d + 1 } τ < 0 . We then study the operator $$e^{-i\tau A_{d+1}}A^{\alpha }$$ e - i τ A d + 1 A α where $$A^{\alpha }=A_1^{\alpha _1}\cdots A^{\alpha _d}_d$$ A α = A 1 α 1 ⋯ A d α d for $$\alpha \in {\mathbb N}_0^d$$ α ∈ N 0 d and prove that can be studied by means of model operators on a weighted $$L^2$$ L 2 space. To prove our results we obtain a Paley–Wiener type theorem for DA and we investigate some multiplier operators on DA as well.

In this note we give a proof-by-formula of certain important embedding inequalities on a dyadic tree. We also consider the case of a bi-tree, where a different approach is explained.

Abstract. Coifman–Meyer multipliers represent a very important class of bilinear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding from the bi-torus to the bi-disc is equivalent to a simple “box” condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the “box” condition does not suffice for the embedding to hold. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.

We give an overview of parts of the theory of Hardy spaces from the viewpoint of signals and systems theory. There are books on this topic, which dates back to Bode, Nyquist, and Wiener, and that eventually led to the developement of $H^{\infty}$ optimal control. Our modest goal here is giving a beginner's dictionary for mathematicians and engineers who know little of either systems or $H^2$ spaces.

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.

N. Arcozzi1, S. Benvenuti1, A. Cattabriga1, D. Gouthier2 1Università di Bologna (ITALY) 2Scienza Express (ITALY)

Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on the bi-tree. In this note we give another proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree in a previous paper of the authors (Arcozzi et al. in Bellman function sitting on a tree, arXiv:1809.03397, 2018), which used the Bellman function technique, the proof here is based on some rather subtle comparisons of energies of measures on the bi-tree.

Lennart Carleson showed in 1974 that the natural generalization, using a box condition, from the one parameter case (disc) to the bi-parameter case (bi-disc) of his theorem does not work. Sun-Yang A. Chang in 1979 found the necessary and sufficient condition for the validity of the Carleson embedding for bi-harmonic extensions into the bi-disc. In both works the underlying measure was the Lebesgue measure on bi-torus, and the embedding measure was a priori arbitrary. In this article we switch the constraints on the two measures involved: the underlying measure on the bi-torus is arbitrary, while the embedding measure has a product structure. For uniform embedding measures several necessary and sufficient conditions of Chang--Carleson type for a Carleson embedding were found in (arXiv:1811.04990, arXiv:1809.03397). In this article we show the unexpected fact that in the bi-parameter case for embedding measures with product structure a simple box condition turns out to be equivalent to Chang--Carleson type conditions. This seems to be a new combinatorial fact about positive measures on the plane.

Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the bi-torus to the bi-disc is equivalent to a simple ``box'' condition, for product weights on the bi-disc and arbitrary weights on the bi-torus. This gives a new simple necessary and sufficient condition for the embedding of the whole scale of weighted Dirichlet spaces of holomorphic functions on the bi-disc. This scale of Dirichlet spaces includes the classical Dirichlet space on the bi-disc. Our result is in contrast to the classical situation on the bi-disc considered by Chang and Fefferman, when a counterexample due to Carleson shows that the ``box'' condition does not suffice for the embedding to hold. But this was the embedding of bi-harmonic functions in bi-harmonic Hardy class. Our result can be viewed as a new and unexpected combinatorial property of all positive finite planar measures.