We derive Adams inequalities for potentials on general measure spaces, extending and improving pr... more We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results in [FM1]. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on R n , where the kernel is large, but they behave better where the kernel is small. The new element is a "critical integrability" condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name "Riesz subcritical domains". Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser-Trudinger inequalities on R n , on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincaré inequality holds.
We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams for... more We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CRpluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss. Contents 0. Introduction 1 1. Intertwining operators on the CR sphere 9 2. Adams and Moser-Trudinger inequalities on the CR sphere 25 3. Beckner-Onofri's inequality 32 4. The logarithmic Hardy-Littlewood-Sobolev inequalities 42 Appendix A. Intertwining operators on S 2n+1 44 Appendix B. Proofs of (3.11) and (3.12) 47 References 48
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main ste... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
We derive sharp Adams inequalities for the Riesz and other potentials of functions with arbitrary... more We derive sharp Adams inequalities for the Riesz and other potentials of functions with arbitrary compact support in R n. Up to now such results were only known for a class of functions whose supports have uniformly bounded measure. We obtain several sharp Moser-Trudinger inequalities for the critical Sobolev space W α, n α on R n and on the hyperbolic space H n. The only known results so far are for α = 1, both on R n and H n , and for α = 2 on R n. Other sharp inequalities are obtained for general elliptic operators with constant coefficients and for trace type Borel measures. We introduce critical potential spaces on R n on which our results can be extended to noninteger values of α.
We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in ... more We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L 1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W k,n/k 0 .
Proceedings of the American Mathematical Society, 1998
In this note the Hilbert transform is characterized in terms of function algebras with respect to... more In this note the Hilbert transform is characterized in terms of function algebras with respect to pointwise multiplication.
We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the wh... more We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R n. As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W α, n α (R n), 0 < α < n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.
We study the ∂-Neumann problem using the Sobolev space inner product. We show that the problem ca... more We study the ∂-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev Hodge theory.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2014
We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in ... more We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L 1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W k,n/k 0 .
Memoirs of the American Mathematical Society, 1998
The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a... more The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in R N +1 , and on the half space R N +1 +. The novelty is that the topology used is not an L 2 topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary value problem belonging to a class of problems first introduced by Višik and Eskin, and by Boutet de Monvel.
Journal of Mathematical Analysis and Applications, 2013
We show that the Riemann zeta function ζ has only countably many self-intersections on the critic... more We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z P C the equation ζp 1 2`i tq " z has at most one solution t P R. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R, then either the set tpa, bq P R 2 : a ‰ b and F paq " F pbqu is countable, or the image F pRq is a loop in C.
We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2 (B n... more We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2 (B n) into the exponential class, where B n is the unit ball of R n. The corresponding sharp results for the spaces W d,n/d 0 (Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d = 1 and are due to Chang-Yang, Cianchi and Leckband. Our proof is based on general abstract results recently obtained by the authors in [FM], and on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is the unique solution of the equation ∆u = f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. We will also consider the situation where the exponential class is endowed with more general Borel measures, and obtain corresponding sharp Moser-Trudinger inequalities of trace type.
Thus, in particular, ker∂ is a closed subspace in W s (0,q) (Ω). Sometimes we shall use the notat... more Thus, in particular, ker∂ is a closed subspace in W s (0,q) (Ω). Sometimes we shall use the notation∂ (0,q) to stress the fact that the operator∂ is acting on (0, q) forms. Consider now the W s (Ω)-Hilbert space adjoint∂ * of∂. We want to study the boundary value problem (∂∂ * +∂ * ∂)u = f on Ω u,∂u ∈ dom∂ * , (3) where f is a given (0, q) form. When appropriate, we shall refer to this problem as (3,s) in order to emphasize that the topology is coming from the W s inner product. The condition that u and∂u lie in the domain of∂ * leads to thē ∂-Neumann s-order boundary conditions. We shall refer below to the (∂, s)-Neumann conditions, and the (∂, s)-Neumann problem. Notice that if the Hilbert space under consideration is L 2 (Ω) (that is, s = 0) with respect to the Lebesgue measure, then the problem (3,s) reduces to the classical∂-Neumann problem. J. J. Kohn solved the∂-Neumann (= (∂, 0)-Neumann) problem in a series of papers in 1963-4 (see [FK] and references therein). This work has proved important in the theory of partial differential equations, in geometry, and in function theory. Recent work of Christ [Ch] has shown that the canonical solution-the solution that is minimal in L 2 norm-that arises from Kohn's work in the L 2 topology is not as well behaved as one might have hoped. The program presented in this paper endeavors to seek other canonical solutions that may serve when Kohn's solution will not. This work is also interesting from the point of view of partial differential equations-particularly boundary value problemsand in the study of the energy integral in geometry. We mention that H. Boas [Bo1] and [Bo2] studied properties and regularity of the Hilbert space orthogonal projection of W s (Ω) onto the subspace consisting of the holomorphic functions. The present paper is the first of a series of papers that we devote to the study of the (∂, s)-Neumann problem. We begin by showing that problem (3,s) can
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main ste... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
Journal of Mathematical Analysis and Applications, 2003
We find a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on... more We find a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on finite weighted networks. We also discuss the periodicity in time of the solution in terms of the spectrum of the discrete graph associated with the network and finally we present two significant examples to illustrate and clarify the general analysis.
We derive Adams inequalities for potentials on general measure spaces, extending and improving pr... more We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results in [FM1]. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on R n , where the kernel is large, but they behave better where the kernel is small. The new element is a "critical integrability" condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name "Riesz subcritical domains". Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser-Trudinger inequalities on R n , on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincaré inequality holds.
We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams for... more We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CRpluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss. Contents 0. Introduction 1 1. Intertwining operators on the CR sphere 9 2. Adams and Moser-Trudinger inequalities on the CR sphere 25 3. Beckner-Onofri's inequality 32 4. The logarithmic Hardy-Littlewood-Sobolev inequalities 42 Appendix A. Intertwining operators on S 2n+1 44 Appendix B. Proofs of (3.11) and (3.12) 47 References 48
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main ste... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
We derive sharp Adams inequalities for the Riesz and other potentials of functions with arbitrary... more We derive sharp Adams inequalities for the Riesz and other potentials of functions with arbitrary compact support in R n. Up to now such results were only known for a class of functions whose supports have uniformly bounded measure. We obtain several sharp Moser-Trudinger inequalities for the critical Sobolev space W α, n α on R n and on the hyperbolic space H n. The only known results so far are for α = 1, both on R n and H n , and for α = 2 on R n. Other sharp inequalities are obtained for general elliptic operators with constant coefficients and for trace type Borel measures. We introduce critical potential spaces on R n on which our results can be extended to noninteger values of α.
We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in ... more We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L 1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W k,n/k 0 .
Proceedings of the American Mathematical Society, 1998
In this note the Hilbert transform is characterized in terms of function algebras with respect to... more In this note the Hilbert transform is characterized in terms of function algebras with respect to pointwise multiplication.
We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the wh... more We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R n. As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W α, n α (R n), 0 < α < n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.
We study the ∂-Neumann problem using the Sobolev space inner product. We show that the problem ca... more We study the ∂-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev Hodge theory.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2014
We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in ... more We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L 1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson-Brezis-Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W k,n/k 0 .
Memoirs of the American Mathematical Society, 1998
The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a... more The authors study the Hodge theory of the exterior differential operator d acting on q-forms on a smoothly bounded domain in R N +1 , and on the half space R N +1 +. The novelty is that the topology used is not an L 2 topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary value problem belonging to a class of problems first introduced by Višik and Eskin, and by Boutet de Monvel.
Journal of Mathematical Analysis and Applications, 2013
We show that the Riemann zeta function ζ has only countably many self-intersections on the critic... more We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z P C the equation ζp 1 2`i tq " z has at most one solution t P R. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R, then either the set tpa, bq P R 2 : a ‰ b and F paq " F pbqu is countable, or the image F pRq is a loop in C.
We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2 (B n... more We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2 (B n) into the exponential class, where B n is the unit ball of R n. The corresponding sharp results for the spaces W d,n/d 0 (Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d = 1 and are due to Chang-Yang, Cianchi and Leckband. Our proof is based on general abstract results recently obtained by the authors in [FM], and on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is the unique solution of the equation ∆u = f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. We will also consider the situation where the exponential class is endowed with more general Borel measures, and obtain corresponding sharp Moser-Trudinger inequalities of trace type.
Thus, in particular, ker∂ is a closed subspace in W s (0,q) (Ω). Sometimes we shall use the notat... more Thus, in particular, ker∂ is a closed subspace in W s (0,q) (Ω). Sometimes we shall use the notation∂ (0,q) to stress the fact that the operator∂ is acting on (0, q) forms. Consider now the W s (Ω)-Hilbert space adjoint∂ * of∂. We want to study the boundary value problem (∂∂ * +∂ * ∂)u = f on Ω u,∂u ∈ dom∂ * , (3) where f is a given (0, q) form. When appropriate, we shall refer to this problem as (3,s) in order to emphasize that the topology is coming from the W s inner product. The condition that u and∂u lie in the domain of∂ * leads to thē ∂-Neumann s-order boundary conditions. We shall refer below to the (∂, s)-Neumann conditions, and the (∂, s)-Neumann problem. Notice that if the Hilbert space under consideration is L 2 (Ω) (that is, s = 0) with respect to the Lebesgue measure, then the problem (3,s) reduces to the classical∂-Neumann problem. J. J. Kohn solved the∂-Neumann (= (∂, 0)-Neumann) problem in a series of papers in 1963-4 (see [FK] and references therein). This work has proved important in the theory of partial differential equations, in geometry, and in function theory. Recent work of Christ [Ch] has shown that the canonical solution-the solution that is minimal in L 2 norm-that arises from Kohn's work in the L 2 topology is not as well behaved as one might have hoped. The program presented in this paper endeavors to seek other canonical solutions that may serve when Kohn's solution will not. This work is also interesting from the point of view of partial differential equations-particularly boundary value problemsand in the study of the energy integral in geometry. We mention that H. Boas [Bo1] and [Bo2] studied properties and regularity of the Hilbert space orthogonal projection of W s (Ω) onto the subspace consisting of the holomorphic functions. The present paper is the first of a series of papers that we devote to the study of the (∂, s)-Neumann problem. We begin by showing that problem (3,s) can
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of Rn. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main ste... more In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams' results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
Journal of Mathematical Analysis and Applications, 2003
We find a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on... more We find a d'Alembert type formula for the solution of the Cauchy problem for the wave equation on finite weighted networks. We also discuss the periodicity in time of the solution in terms of the spectrum of the discrete graph associated with the network and finally we present two significant examples to illustrate and clarify the general analysis.
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Papers by Luigi Fontana