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Positive Gaussian Kernels also Have Gaussian Minimizers
We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put
forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give
necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of
multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends
several inverse inequalities.
eBook
Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals
The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces
More precisely, Street (2014) studied the
eBook
Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag
setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function
and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this
paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the
Plancherel–Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels
and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in
terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the
multi-parameter Hardy space with the flag structure.
eBook
Algebras of Singular Integral Operators with Kernels Controlled by Multiple Norms
The authors study algebras of singular integral operators on \\mathbb R^n and nilpotent Lie groups that arise when considering the composition of Calderón-Zygmund operators with different homogeneities, such as operators occuring in sub-elliptic problems and those arising in elliptic problems. These algebras are characterized in a number of different but equivalent ways: in terms of kernel estimates and cancellation conditions, in terms of estimates of the symbol, and in terms of decompositions into dyadic sums of dilates of bump functions. The resulting operators are pseudo-local and bounded on L^p for 1 \\lt p \\lt \\infty . While the usual class of Calderón-Zygmund operators is invariant under a one-parameter family of dilations, the operators studied here fall outside this class, and reflect a multi-parameter structure.
eBook
The Spin L-function on GSp6 Via a non-unique model
We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on ${\\rm PGSp}_6$. We deduce that for a wide class of cuspidal automorphic representations $\\pi$, the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $\\pi$ is an exceptional theta lift from ${\\rm G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $\\pi$ upon which the aforementioned model becomes unique. The other integral can be regarded as a higher rank analogue of the integral of Kohnen-Skoruppa on ${\\rm GSp}_4$.
Journal Article
Multilinear Singular Integral Forms of Christ-Journé Type
We introduce a class of multilinear singular integral forms
eBook
Dyadic-probabilistic methods in bilinear analysis
We demonstrate and develop dyadic–probabilistic methods in connection with non-homogeneous bilinear operators, namely singular
integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view.
The main result is a new global
While proving our bilinear results we also advance and
refine the linear theory of Calderón–Zygmund operators by improving techniques and results. For example, we simplify and make more
efficient some non-homogeneous summing arguments appearing in
eBook
Local fractional integral transforms and their applications
Local Fractional Integral Transforms and Their Applications provides information on how local fractional calculus has been successfully applied to describe the numerous widespread real-world phenomena in the fields of physical sciences and engineering sciences that involve non-differentiable behaviors.
eBook