Geodesic
Norms of the Positive Powers of the Bessel Operator in the Spaces of Even Schlömilch j-Polynomials
Matematičeskie zametki, Tome 106 (2019) no. 4, pp. 549-564 Cet article a éte moissonné depuis la source Math-Net.Ru

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The definition of a $B$-derivative is based on the notion of generalized Poisson shift; this derivative coincides, up to a constant, with the singular Bessel differential operator. We introduce the fractional powers of a $B$-derivative by analogy with fractional Marchaud and Weyl derivatives. We prove statements on the coincidence of these derivatives for the classes of even smooth integrable functions. We obtain analogs of Bernstein's inequality for $B$-derivatives of integer and fractional order in the space of even Schlömilch j-polynomials with sup-norm and $L_p^\gamma$-norm (the Lebesgue norm with power weight $x^\gamma$, $\gamma>0$). The resulting estimates are sharp and define the norms of powers of the Bessel operator in the spaces of even Schlömilch j-polynomials.
Keywords: Bessel j-function, generalized Poisson shift, and Weyl fractional derivatives, Schlömilch polynomial, Bernstein's inequality, Bernstein–Zygmund inequality, operator norm.
Mots-clés : Liouville, Marchaud, Riesz interpolation formula 
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L. N. Lyakhov; E. Sanina. Norms of the Positive Powers of the Bessel Operator in the Spaces of Even Schlömilch j-Polynomials. Matematičeskie zametki, Tome 106 (2019) no. 4, pp. 549-564. http://geodesic.mathdoc.fr/item/MZM_2019_106_4_a6/

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