Fractional integrals on radial functions with applications to weighted inequalities

2011 ◽  
Vol 192 (4) ◽  
pp. 553-568 ◽  
Author(s):  
Javier Duoandikoetxea
Keyword(s):  
Fractional Integrals ◽  
Weighted Inequalities ◽  
Radial Functions

scholarly journals On weighted inequalities for fractional integrals of radial functions

2011 ◽  
Vol 55 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Pablo L. De Nápoli ◽  
Irene Drelichman ◽  
Ricardo G. Durán
Keyword(s):  
Fractional Integrals ◽  
Weighted Inequalities ◽  
Radial Functions

Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

2020 ◽  
Vol 23 (4) ◽  
pp. 967-979
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Fractional Integrals ◽  
Radon Transforms ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Compactly Supported ◽  
Radial Case ◽  
Compactly Supported Functions

AbstractWe apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

Weighted inequalities for the Stieltjes transform and the maximal spherical partial sum operator on radial functions

1995 ◽  
Vol 125 (1) ◽  
pp. 195-204
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Fourier Transform ◽  
Lorentz Space ◽  
Sufficient Conditions ◽  
Weighted Inequalities ◽  
Stieltjes Transform ◽  
Radial Functions ◽  
Partial Sum ◽  
Power Weights ◽  
The Fourier Transform ◽  
Spherical Partial Sum

If TRf(x) is the spherical partial sum of the Fourier transform of f and T*f(x) = SUPR > 0 | TRf(x)|, sufficient conditions are given on the non-negative weight function ω(x) which ensure that T* restricted to radial functionsis bounded on the Lorentz space Lp,s(Rn,ω) into Lp,q(Rn, ω) For power weights, these conditions are also necessary. The weight pairs (u,v) for which the generalised Stieltjes transform Sλ is bounded from LP,S(R+, v)into Lp,q(R+, u)are also characterised. These are an essential ingredient for the study of T*.

scholarly journals On Radon transforms between lines and hyperplanes

2017 ◽  
Vol 28 (13) ◽  
pp. 1750093 ◽  
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Radon Transform ◽  
Symmetric Case ◽  
Fractional Integrals ◽  
Radon Transforms ◽  
Full Generality ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Kelvin Transform ◽  
Funk Transform ◽  
Dual Transform

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in [Formula: see text]. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John [Formula: see text]-plane transform in [Formula: see text], the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.

scholarly journals Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sami Aouaoui ◽  
Rahma Jlel
Keyword(s):  
Euclidean Space ◽  
Nonlinear Equations ◽  
Exponential Type ◽  
Growth Conditions ◽  
Concentration Compactness ◽  
Weighted Inequalities ◽  
Radial Functions ◽  
Compactness Arguments ◽  
Logarithmic Type ◽  
Conditions At Infinity

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

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