Search results for "Radial function"
showing 4 items of 4 documents
The variation of the maximal function of a radial function
2017
We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.
Bochner-Riesz means of functions in weak-L p
1993
The Bochner-Riesz means of order delta greater-than-or-equal-to 0 for suitable test functions on R(N) are defined via the Fourier transform by (S(R)(delta)f)(xi) = (1 - \xi\2/R2)+(delta)f(xi). We show that the means of the critical index delta = N/P - N + 1/2, 1 + infinity, to f(x) in norm and for almost every x in R(N). We also observe that the means of the function absolute value of x-N/p, which belongs to L(p,infinity) (R(N)) but not to the closure of test functions, converge for no x
Scattering of Particles by Potentials
2013
The three prototypes of spectra of self-adjoint operators, the discrete spectrum, with or without degeneracy, the continuous spectrum, and the mixed spectrum, as well as the corresponding wave functions, contain important information about the physical systems that they describe
The Variation of the Fractional Maximal Function of a Radial Function
2017
Abstract In this article, we study the regularity of the non-centered fractional maximal operator $M_{\beta}$. As the main result, we prove that there exists $C(n,\beta)$ such that if $q=n/(n-\beta)$ and $f$ is radial function, then $\|DM_{\beta}f\|_{L^{q}({\mathbb{R}^n})}\leq C(n,\beta)\|Df\|_{L^{1}({\mathbb{R}^n})}$. The corresponding result was previously known only if $n=1$ or $\beta=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}({\mathbb{R}^n})$.