spherical means
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2021 ◽  
Vol 47 (1) ◽  
pp. 23-37
Author(s):  
Yoshihiro Mizuta ◽  
Tetsu Shimomura

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).


Author(s):  
Antonio Tristán‐Vega ◽  
Guillem París ◽  
Rodrigo Luis‐García ◽  
Santiago Aja‐Fernández

2021 ◽  
Vol 88 (1-2) ◽  
pp. 88
Author(s):  
M. G. Grigoryan ◽  
L. S. Simonyan

In this paper we consider the convergence by measure of Fourier integral spherical means of Riesz at a critical exponent δ = 1/2 after changing the values of the integrable function on the given set of a small measure.


2020 ◽  
Vol 12 ◽  
Author(s):  
Irfan Alam

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof.


2020 ◽  
Vol 42 (5) ◽  
pp. A2910-A2942
Author(s):  
Boxi Xu ◽  
Jin Cheng ◽  
Shingyu Leung ◽  
Jianliang Qian

2019 ◽  
Vol 2019 (3) ◽  
pp. 49-59
Author(s):  
I.A. Ikromov ◽  
A.A. Rahmonov
Keyword(s):  

2019 ◽  
Vol 276 (3) ◽  
pp. 815-866
Author(s):  
Amy Peterson ◽  
Ambar N. Sengupta

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