We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete... more We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete description of the analogous sequences for a discrete model of the Dirichlet space.
We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G... more We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G_y$ that are multiply connected, composed of a discrete abelian component $G_x$ and a compact connected component $G_y$ . We prove sharp $L^p$ estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set $G = G_x \times G_y$ . The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local.
We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most... more We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most of the results are already known but some results are new. Among the new results that we prove there is the characterization of the compactness of the Hardy operator, a reverse Hölder inequality for trace measures and a simple proof of the characterization of trace measures based on a monotonicity argument. Furthermore we give a probabilistic proof of an inequality due to Wolff. We also provide a list of open problems and suggest some possible lines of future research. 2010 Mathematics Subject Classification. 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85.
Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, whic... more 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-d...
We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functi... more We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functions to the closed subspace of slice L 2 functions. The aim of this short note is to study the orthogonal projection Π from the space of quaternion valued L 2 functions to its closed subspace of slice L 2 functions. In particular, to compute the norm of the projection operator we will first show that we can write Π in terms of a quaternionic slice Poisson kernel. Let H = R + Ri + Rj + Rk denote the non commutative 4-dimensional real algebra of quaternions and let B = {q ∈ H : |q| < 1} be the unit ball in H. Its boundary ∂B contains elements of the form q = e It , I ∈ S, t ∈ R, where S = {q ∈ H : q 2 = −1} is the two dimensional sphere of imaginary units in H. We endow ∂B with the measure dΣ e It = dσ(I)dt, which is naturally associated with the Hardy space H 2 (B) of the slice regular functions on B, see [1]. We here normalize the measures so to have Σ(∂B) = σ(S) = 1. The functions we are concerned with here satisfy algebraic conditions and size conditions. A function f : ∂B → H is a slice function if for any I, J ∈ S
We give a direct, combinatorial proof that the logarithmic capacity is essentially invariant unde... more We give a direct, combinatorial proof that the logarithmic capacity is essentially invariant under quasisymmetric maps of the circle.
Transactions of the American Mathematical Society, 2006
We establish a potential theoretic approach to the study of twist points in the boundary of simpl... more We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2006
We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, thi... more We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.
We investigate connections between potential theories on a Ahlfors-regular metric space X, on a g... more We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given. Contents 1. Introduction 1.1. Main Results and Outline of the Contents 1.
We study the smoothness of the distance function from a set in the Heisenberg group, with some ap... more We study the smoothness of the distance function from a set in the Heisenberg group, with some applications to the mean curvature of a surface. Contents 1. Introduction 1 2. Notation and preliminaries 3 3. The metric normal and the distance function for a plane 5 4. The metric normal and the oriented metric normal for a smooth surface 7 5. Regularity of the distance function near a smooth surface 11 6. The Hessian of the distance function for a smooth surface 17 7. Mean Curvature 22 8. Examples 25 8.1. Nonvertical planes 25 8.2. The Carnot-Charathèodory sphere and the Hessian of the distance function 27 References 29
Journal of Mathematical Analysis and Applications, 2008
The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections ... more The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections are solutions of an isoperimetric problem in the Grushin plane.
We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and s... more We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and symmetric MSF wavelets of L 2 (R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L 2 (R) and H 2 (R). We also enumerate all symmetric wavelets of L 2 (R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H 2 (R). Finally, we construct families of MSF wavelets of L 2 (R) whose Fourier transform does not vanish in any neighbourhood of the origin.
We show that the capacity of a class of plane condensers is comparable to the capacity of corresp... more We show that the capacity of a class of plane condensers is comparable to the capacity of corresponding "dyadic condensers". As an application, we show that for plane condensers in that class the capacity blows up as the distance between the plates shrinks, but there can be no asymptotic estimate of the blow-up.
We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined... more We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined by the norm f p B p (s) = |f (0)| p + D |(1 − |z| 2)f (z)| p (1 − |z| 2) s dA(z) (1 − |z| 2) 2 , 1 < p < ∞ and 0 < s < 1, and for the corresponding multiplier spaces M (Bp(s)).
Abstract. In this paper we survey many results on the Dirichlet space of analytic functions. Our ... more Abstract. In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on ...
Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the functio... more Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the CarnotCharathéodory distance from S in terms of the mean curvature of S and of an imaginary curvature which was introduced in [2] in order to find ...
We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete... more We describe a class of "onto interpolating" sequences for the Dirichlet space and give a complete description of the analogous sequences for a discrete model of the Dirichlet space.
We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G... more We study a class of combinations of second order Riesz transforms on Lie groups $G = G_x \times G_y$ that are multiply connected, composed of a discrete abelian component $G_x$ and a compact connected component $G_y$ . We prove sharp $L^p$ estimates for these operators, therefore generalizing previous results [13][4]. The proof uses stochastic integrals with jump components adapted to functions defined on the semi-discrete set $G = G_x \times G_y$ . The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local.
We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most... more We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most of the results are already known but some results are new. Among the new results that we prove there is the characterization of the compactness of the Hardy operator, a reverse Hölder inequality for trace measures and a simple proof of the characterization of trace measures based on a monotonicity argument. Furthermore we give a probabilistic proof of an inequality due to Wolff. We also provide a list of open problems and suggest some possible lines of future research. 2010 Mathematics Subject Classification. 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85.
Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, whic... more 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-d...
We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functi... more We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functions to the closed subspace of slice L 2 functions. The aim of this short note is to study the orthogonal projection Π from the space of quaternion valued L 2 functions to its closed subspace of slice L 2 functions. In particular, to compute the norm of the projection operator we will first show that we can write Π in terms of a quaternionic slice Poisson kernel. Let H = R + Ri + Rj + Rk denote the non commutative 4-dimensional real algebra of quaternions and let B = {q ∈ H : |q| < 1} be the unit ball in H. Its boundary ∂B contains elements of the form q = e It , I ∈ S, t ∈ R, where S = {q ∈ H : q 2 = −1} is the two dimensional sphere of imaginary units in H. We endow ∂B with the measure dΣ e It = dσ(I)dt, which is naturally associated with the Hardy space H 2 (B) of the slice regular functions on B, see [1]. We here normalize the measures so to have Σ(∂B) = σ(S) = 1. The functions we are concerned with here satisfy algebraic conditions and size conditions. A function f : ∂B → H is a slice function if for any I, J ∈ S
We give a direct, combinatorial proof that the logarithmic capacity is essentially invariant unde... more We give a direct, combinatorial proof that the logarithmic capacity is essentially invariant under quasisymmetric maps of the circle.
Transactions of the American Mathematical Society, 2006
We establish a potential theoretic approach to the study of twist points in the boundary of simpl... more We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2006
We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, thi... more We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.
We investigate connections between potential theories on a Ahlfors-regular metric space X, on a g... more We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given. Contents 1. Introduction 1.1. Main Results and Outline of the Contents 1.
We study the smoothness of the distance function from a set in the Heisenberg group, with some ap... more We study the smoothness of the distance function from a set in the Heisenberg group, with some applications to the mean curvature of a surface. Contents 1. Introduction 1 2. Notation and preliminaries 3 3. The metric normal and the distance function for a plane 5 4. The metric normal and the oriented metric normal for a smooth surface 7 5. Regularity of the distance function near a smooth surface 11 6. The Hessian of the distance function for a smooth surface 17 7. Mean Curvature 22 8. Examples 25 8.1. Nonvertical planes 25 8.2. The Carnot-Charathèodory sphere and the Hessian of the distance function 27 References 29
Journal of Mathematical Analysis and Applications, 2008
The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections ... more The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections are solutions of an isoperimetric problem in the Grushin plane.
We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and s... more We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and symmetric MSF wavelets of L 2 (R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L 2 (R) and H 2 (R). We also enumerate all symmetric wavelets of L 2 (R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H 2 (R). Finally, we construct families of MSF wavelets of L 2 (R) whose Fourier transform does not vanish in any neighbourhood of the origin.
We show that the capacity of a class of plane condensers is comparable to the capacity of corresp... more We show that the capacity of a class of plane condensers is comparable to the capacity of corresponding "dyadic condensers". As an application, we show that for plane condensers in that class the capacity blows up as the distance between the plates shrinks, but there can be no asymptotic estimate of the blow-up.
We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined... more We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined by the norm f p B p (s) = |f (0)| p + D |(1 − |z| 2)f (z)| p (1 − |z| 2) s dA(z) (1 − |z| 2) 2 , 1 < p < ∞ and 0 < s < 1, and for the corresponding multiplier spaces M (Bp(s)).
Abstract. In this paper we survey many results on the Dirichlet space of analytic functions. Our ... more Abstract. In this paper we survey many results on the Dirichlet space of analytic functions. Our focus is more on the classical Dirichlet space on the disc and not the potential generalizations to other domains or several variables. Additionally, we focus mainly on ...
Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the functio... more Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the CarnotCharathéodory distance from S in terms of the mean curvature of S and of an imaginary curvature which was introduced in [2] in order to find ...
Uploads
Papers by Nicola Arcozzi