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Table Of Integrals Series And Products

We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a two-sided estimate on the fractal considered. The results of this paper are among the marvelous consequences of the heat kernel on the fractal.

https://www.minet.uni-jena.de/geometrie/publicat/pdffiles/psof_hz.pdf

Potential spaces on fractals

In this paper, we introduce and study the Sobolev spaces of exponential type associated with the Weinstein operator, via some elements of harmonic analysis related to this operator. In particular, some properties, including completeness and imbedding theorem, are proved. Finally, using the theory of reproducing kernels, some applications are given for these spaces.

Weinstein-Sobolev spaces of exponential type and applications

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L
 p
 -L
 q
 estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L
 p
 -L
 q
 estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L
 p
 -L
 q
 bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Napoli, Drelichman and Duran, and Duoandikoetxea.

Potential operators associated with Hankel and Hankel-Dunkl transforms

/pdf/tu-di-li-yong-duo-gong-neng-xing-yan-jiu-jin-zhan-yu-zhan-32vykl9y0g.pdf

土地利用多功能性研究进展与展望@@@Progress and prospects of multi-functionality of land use research

Sufficient conditions on weights are found that ensure that a multidimensional singular integral operator generated by the generalized shift operator acts boundedly from one weighted Lp-space into another. It is shown that in some cases the conditions are sharp.Bibliography: 9 titles.

https://iopscience.iop.org/article/10.1070/SM1994v077n01ABEH003428/pdf

Weighted estimates of multidimensional singular integrals generated by the generalized shift operator

We introduce a family of wavelet-like transforms associated to certain admissible
semigroups of operators acting on Lp-spaces, and prove the corresponding reproducing formula
of Calderon’s type. The new transforms constitute a unified approach to inversion of a wide class
of integral operators in analysis and applications. We illustrate the general theory by considering
Riesz and Bessel potentials (associated to the ordinary and the generalized translation), and the
k-plane Radon transform on ℝn.

Wavelet-Like Transforms for Admissible Semi-Groups; Inversion Formulas for Potentials and Radon Transforms

Parabolic wavelet transforms associated with the singular heat operators +@=@t andI +@=@t, where = Pn=1 @ 2 =@x 2 +(2 =xn)@=@xn, are introduced. These transforms are dened in terms of the relevant generalized translation operator. An analogue of the Calder on reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

/pdf/parabolic-potentials-and-wavelet-transforms-with-the-1s5h832e0g.pdf

Parabolic potentials and wavelet transforms with the generalized translation

We introduce a new concept of the so-called {\it composite wavelet transforms}. These transforms are generated by two components, namely, a kernel function and a wavelet function (or a measure). The composite wavelet transforms and the relevant Calder\'{o}n-type reproducing formulas constitute a unified approach to explicit inversion of the Riesz, Bessel, Flett, parabolic and some other operators of the potential type generated by ordinary (Euclidean) and generalized (Bessel) translations. This approach is exhibited in the paper. Another concern is application of the composite wavelet transforms to explicit inversion of the k-plane Radon transform on $\bbr^n$. We also discuss in detail a series of open problems arising in wavelet analysis of $L_p$-functions of matrix argument.

Composite Wavelet Transforms: Applications and Perspectives

Inversion of Bessel Potentials with the Aid of Weighted Wavelet Transforms

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