Estimates of the norms of Riesz derivatives of multivariate functions

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Estimates of the norms of Riesz derivatives of multivariate functions

Journal Title: Дослідження в математиці і механіці - Year 2017, Vol 22, Issue 1

Abstract

Let Δ = 𝜕2 𝜕𝑥1 + ... + 𝜕2 𝜕𝑥𝑚 be the Laplace operator. We denote by 𝐿𝑠(R𝑚) (1 6 𝑠 6 ∞) the spaces of measurable functions 𝑓 : R𝑚 → R with a finite norm ‖𝑓‖𝑠. Let 𝐹 and 𝐸 be ideal lattices on R𝑚 with finite norms ‖ ・ ‖𝐹 and ‖ ・ ‖𝐸. By 𝐿Δ 𝐹,𝐸(R𝑚) we denote the space of functions 𝑓 ∈ 𝐹 such that Δ𝑓 ∈ 𝐸. If 𝐹 = 𝐿𝑠(R𝑚) (1 6 𝑠 6 ∞), then we use the notation 𝐿Δ 𝑠,𝐸, or 𝐿Δ 𝑠,𝑠 if, in addition, 𝐸 = 𝐿𝑠(R𝑚). We obtain new Kolmogorov-type inequalities for the norms of the Riesz derivatives 𝐷𝛼𝑓 of functions 𝑓 ∈ 𝐿Δ ∞,𝐸(R𝑚). As a corollary, we obtain inequalities for the functions 𝑓 ∈ 𝐿Δ 𝑠,𝑠(R𝑚). The problem on approximation an unbounded operator 𝐷𝛼 by bounded ones on a class of functions 𝑓 such that ‖Δ𝑓‖𝐸 6 1 is solved.

Authors and Affiliations

N. V. Parfinovych

Keywords

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EP ID EP342734
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