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Register a new userStrichartz transforms with Riesz potentials and Semyanistyi integrals
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Authors
Yingzhan Wang
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In this paper, we study the general orthogonal Radon transform $R_{p,q}^k$ first studied by R.S Strichartz in cite{Stri}. An sharp existence condition of $R_{p,q}^k f$ on $L^p$-spaces will be given. Then we devote to the relation formulas connecting Strichartz transform $R_{p,q}^k$ and Semyanistyi integrals. We prove the corresponding Fuglede type formulas, through which a number of explicit inversion formulas for $R_{p,q}^k f$ will be given. Different from the inclusion Radon transform and Gonzalez type orthogonal transform, Strichartz transform is more complicated. Our conclusions generalize the corresponding results of the two particular cases above.
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In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the Riesz fractional integrals and compare their properties with the standard Coulomb and Yukawa potentials commonly used. It is demonstrated, that Riesz potentials may serve as a promising extension of standard potentials and may be reckoned as a smooth transition from Coulomb to Yukawa like potentials, depending of the fractional parameter $alpha$. For the macroscopic part of the total energy the Riesz potentials treat the Coulomb-, symmetry- and pairing-contributions from a generalized point of view, since they turn out to be similar realizations of the same fractional integral at distinct $alpha$ values.
The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.
In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.
In this article, we prove a Strichartz type inequality %associated with Schrodinger equation for a system of orthonormal functions associated with the special Hermite operator $mathcal{L}=-Delta+frac{1}{4}|z|^{2}-i sum_{1}^{n}left(x_{j} frac{partial}{partial y_{j}}-y_{j} frac{partial}{partial x_{j}}right), $ where $Delta$ denotes the Laplacian on $mathbb{C}^{n}$.
The Strichartz inequality for the system of orthonormal functions for the Hermite operator $H=-Delta+|x|^2$ on $mathbb{R}^n$ has been proved in cite{lee}, using the classical Strichartz estimates for the free Schrodinger propagator for orthonormal systems cite{frank, frank1} and the link between the Schrodinger kernel and the Mehler kernel associated with the Hermite semigroup cite{SjT}. In this article, we give an alternative proof of the above result in connection with the restriction theorem with respect to the Hermite transform with an optimal behavior of the constant in the limit of a large number of functions. As an application, we show the well-posedness results in Schatten spaces for the nonlinear Hermite-Hartree equation.
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