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Constructive description of analytic Besov spaces in strictly pseudoconvex domains

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Abstract

We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.

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Notes

  1. For a definition of Smirnov class \(E^p(\varOmega )\) see [1].

  2. Let \(0<s<\infty \) and \(1\le p,q\le \infty ,\) \(f\in L^p(\mathbb {R}^l).\) A classical definition of Besov classes \(B^s_{pq}\) is based on difference operators that can be defined by induction. Let \(t>0,\) \(1\le j\le l,\) and

    $$\begin{aligned} \varDelta ^1_{j,t} f(x) = f(x_1,\ldots ,x_j+t,\dots ,x_l)-f(x),\quad \varDelta ^k_{j,t} f(x) = \varDelta ^1_{j,t}(\varDelta ^{k-1}_{j,t} f)(x),\ k\in \mathbb {N}. \end{aligned}$$

    Then \( \varDelta ^k_t f(x) = \varDelta ^{k}_{1,t}\circ \ldots \circ \varDelta ^{k}_{l,t} f(x) \) is a difference operator of order k.

    We say that f belongs to Besov class \(B^s_{pq}(\mathbb {R}^l)\) if \( \left\Vert \omega ^m(f,h)_p h^{-s-1/q} \right\Vert _{L^q(0,+\infty )}<\infty \) for some \(m>s,\) where \( \omega ^{m} (f,t)_p = \Vert \varDelta _t^m f \Vert _{L^p(\mathbb {R}^l)} \) is a modulus of smoothness of function f.

    Let \(\varOmega \subset {\mathbb {C}}^n\) be a bounded domain with a smooth boundary. We let \(f\in B^s_{pq}( {\partial \varOmega } )\) if for every \(\xi \in {\partial \varOmega } \) there exists a neighbourhood \(V_\xi \) of \(\xi \) with a smooth diffeomorphism \(\varphi : [0,1]^{2n-1}\rightarrow V_\xi \) and a smooth function \(\chi (z)\) supported on \(V_\xi \) and equal to 1 in some neighbourhood of \(\xi \) and such that \((\chi f)\circ \varphi \in B^s_{pq}(\mathbb {R}^{2n-1}).\)

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  1. St. Petersburg University, Saint Petersburg, Russia

    Aleksandr Rotkevich

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  1. Aleksandr Rotkevich
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The work is supported by Russian Science Foundation Grant 19-11-00058.

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Rotkevich, A. Constructive description of analytic Besov spaces in strictly pseudoconvex domains. Anal.Math.Phys. 11, 26 (2021). https://doi.org/10.1007/s13324-020-00466-0

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