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An extension problem, trace Hardy and Hardy’s inequalities for the Ornstein–Uhlenbeck operator
... The higher order local problem was considered in [12], where the diffusion term is given by ∆ m , m ∈ N. Fractional powers of the Ornstein-Uhlenbeck operator, that is, (−∆ + 2x · ∇ + n) s were studied in [19]. ...
Hermite polynomials, which are associated to a Gaussian weight and solve the Laplace equation with a drift term of linear growth, are classical in analysis and well-understood via ODE techniques. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials, which appear as eigenfunctions of a L\'evy Fokker-Planck equation. We will restrict, without loss of generality, to radially symmetric functions. A crucial tool in our analysis is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns weighted derivatives into multipliers. This allows to write the weighted space in the fractional case that replaces the usual . After proving compactness, we obtain a exhaustive description of the spectrum of the L\'evy Fokker--Planck equation and its dual, the fractional Ornstein--Uhlenbeck problem, which forms a basis thanks to the spectral theorem for self-adjoint operators. As a corollary, we obtain a full asymptotic expansion for solutions of the fractional heat equation.
In this paper we present a new approach based on the heat equation and extension problems to some intertwining formulas arising in conformal CR geometry.
In this paper we use the heat equation in a group of Heisenberg type to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators and , . Here, is the fractional power of the horizontal Laplacian, and is the conformal fractional power of the horizontal Laplacian on . One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equations and semigroup methods. When s=1 our results recapture the famous fundamental solution found by Folland and generalised by Kaplan.
We prove Hardy inequalities for the conformally invariant fractional powers
of the sublaplacian on the Heisenberg group . We prove two
versions of such inequalities depending on whether the weights involved are
non-homogeneous or homogeneous. In the first case, the constant arising in the
Hardy inequality turns out to be optimal. In order to get our results, we will
use ground state representations. The key ingredients to obtain the latter are
some explicit integral representations for the fractional powers of the
sublaplacian and a generalized result by M. Cowling and U. Haagerup. The
approach to prove the integral representations is via the language of
semigroups. As a consequence of the Hardy inequalities we also obtain versions
of Heisenberg uncertainty inequality for the fractional sublaplacian.
We study boundary value problems for some differential operators on Euclidean
space and the Heisenberg group which are invariant under the conformal group of
a Euclidean subspace resp. Heisenberg subgroup. These operators are shown to be
self-adjoint in certain Sobolev type spaces and the related boundary value
problems are proven to have unique solutions in these spaces. We further find
the corresponding Poisson transforms explicitly in terms of their integral
kernels and show that they are isometric between Sobolev spaces and extend to
bounded operators between certain -spaces.
The conformal invariance of the differential operators allows us to apply
unitary representation theory of reductive Lie groups, in particular recently
developed methods for restriction problems.
The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (− Δ + |x|2)σ to deduce a Harnack's inequality. A pointwise formula for H σ f(x) and some maximum and comparison principles are derived.
1. vyd. Vydáno pro Matematický ústav AV ČR
The operator square root of the Laplacian (-\lap)^{1/2} can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.
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 CR-pluriharmonic functions on
the sphere, and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev
inequality in the form given by Carlen and Loss.
Authored by a ranking authority in Gaussian harmonic analysis, this book embodies a state-of-the-art entrée at the intersection of two important fields of research: harmonic analysis and probability. The book is intended for a very diverse audience, from graduate students all the way to researchers working in a broad spectrum of areas in analysis. Written with the graduate student in mind, it is assumed that the reader has familiarity with the basics of real analysis as well as with classical harmonic analysis, including Calderón-Zygmund theory; also some knowledge of basic orthogonal polynomials theory would be convenient. The monograph develops the main topics of classical harmonic analysis (semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, spectral multipliers, fractional integrals and fractional derivatives, singular integrals) with respect to the Gaussian measure. The text provide an updated exposition, as self-contained as possible, of all the topics in Gaussian harmonic analysis that up to now are mostly scattered in research papers and sections of books; also an exhaustive bibliography for further reading. Each chapter ends with a section of notes and further results where connections between Gaussian harmonic analysis and other connected fields, points of view and alternative techniques are given. Mathematicians and researchers in several areas will find the breadth and depth of the treatment of the subject highly useful.
We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators. © 2019 American Institute of Mathematical Sciences. All rights reserved.
In this paper we study the extension problem for the sublaplacian on a H-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.
We prove Hardy-type inequalities for a fractional Dunkl--Hermite operator which incidentally give Hardy inequalities for the fractional harmonic oscillator as well. The idea is to use h-harmonic expansions to reduce the problem in the Dunkl--Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by R. L. Frank, E. H. Lieb and R. Seiringer in the Euclidean setting, to get a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a "good" spectral theorem and an integral representation for the fractional operators involved.
Preface.- Smoothness and Function Spaces.- BMO and Carleson Measures.- Singular Integrals of Nonconvolution Type.- Weighted Inequalities.- Boundedness and Convergence of Fourier Integrals.- Time-Frequency Analysis and the Carleson-Hunt Theorem.- Multilinear Harmonic Analysis.- Glossary.- References.- Index.
We show that the conformally invariant fractional powers of the sub-Laplacian
on the Heisenberg group are given in terms of the scattering operator for an
extension problem to the Siegel upper halfspace. Remarkably, this extension
problem is different from the one studied, among others, by Caffarelli and
Silvestre.
Optimal constants are found in Hardy–Rellich inequalities containing derivatives of arbitrary (not necessarily integer) order l. Some new inequalities of this type are also obtained.
The Fourier algebra A(G) of a locally compact group G is the space of matrix coefficients of the regular representation, and is the predual of the yon Neumann algebra VN(G) generated by the regular representation of G on L 2 (G). A multiplier m of A (G) is a bounded operator on A (G) given by pointwise multiplication by a function on G, also denoted m. We say m is a completely bounded multiplier ofA (G) if the transposed operator on VN(G) is completely bounded (definition below). It may be possible to find a net ofA (G)-functions, (m i : ie I) say, such that mi tends to
We derive the sharp constants for the inequalities on the Heisenberg group
H^n whose analogues on Euclidean space R^n are the well known
Hardy-Littlewood-Sobolev inequalities. Only one special case had been known
previously, due to Jerison-Lee more than twenty years ago. From these
inequalities we obtain the sharp constants for their duals, which are the
Sobolev inequalities for the Laplacian and conformally invariant fractional
Laplacians. By considering limiting cases of these inequalities sharp constants
for the analogues of the Onofri and log-Sobolev inequalities on H^n are
obtained. The methodology is completely different from that used to obtain the
R^n inequalities and can be (and has been) used to give a new, rearrangement
free, proof of the HLS inequalities.
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide short proofs of the Frank-Seiringer "Hardy inequalities" where fractional smoothness is measured by a Besov norm. Comment: v.6. Added new results extending estimates for fractional smoothness to the Heisenberg group and product spaces with mixed homogeneity. 25 pages, AMSLaTeX
We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schroedinger-like operators remain true, with possibly different constants, when the critical Hardy-weight is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schroedinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge .
We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.
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Notes on some points in the integral calculus, LI. On Hilbert's double series theorem and some connected theorems concerning the convergence of infinite series and integrals
- G H Hardy
G. H. Hardy, Notes on some points in the integral calculus, LI. On Hilbert's double series theorem
and some connected theorems concerning the convergence of infinite series and integrals, Messenger
of Math. 48(1919),107-112.
Integrals and series
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A. P. Prudnikov, A. Y. Brychkov and O. I. Marichev, Integrals and series, Elementary Functions,
Volume 1 (Gordon and Breach Science Publishers, New York, 1986).
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L. Roncal and S. Thangavelu, Holomorphic extensions of eigenfunctions on NA groups,
arXiv:2005.09894.
On the unreasonable effectiveness of Gutzmer's formula. Harmonic analysis and partial differential equations
- S Thangavelu
S. Thangavelu, On the unreasonable effectiveness of Gutzmer's formula. Harmonic analysis and
partial differential equations, 199-217, Contemp. Math., 505, Amer. Math. Soc., Providence, RI,
2010