Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents. (English) Zbl 1242.46044
Let \(p(x)\), \(q(x)\), \(s(x)\) be real \(\log\)-Hölder continuous functions in \(\mathbb R^n\) with
\[
0<c\leq p(x), ~q(x) \leq C <\infty, \qquad x \in \mathbb R^n.
\]
Let \(B^{s(\cdot)}_{p(\cdot), q(\cdot)} (\mathbb R^n)\) and \(F^{s(\cdot)}_{p(\cdot), q(\cdot)} (\mathbb R^n)\) be the Besov and Triebel-Lizorkin spaces with variable exponents generalizing the nowadays well-known spaces \(B^s_{p,q} (\mathbb R^n)\) and \(F^s_{p,q} (\mathbb R^n)\) with constant exponents. The paper deals with the complex interpolation
\[
\Big( F^{s_0 (\cdot)}_{p_0(\cdot), q_0 (\cdot)} (\mathbb R^n), F^{s_1 (\cdot)}_{p_1 (\cdot), q_1 (\cdot)} (\mathbb R^n) \Big)_\theta = F^{s(\cdot)}_{p(\cdot), q(\cdot)} (\mathbb R^n),
\]
similarly for the \(B\)-spaces, with \(0<\theta<1\),
\[
\frac{1}{p(\cdot)} = \frac{1-\theta}{p_0 (\cdot)} + \frac{\theta}{p_1 (\cdot)}, \quad \frac{1}{q(\cdot)} = \frac{1-\theta}{q_0 (\cdot)} + \frac{\theta}{q_1 (\cdot)}, \quad s(\cdot) = (1-\theta)s_0 (\cdot) + \theta s_1 (\cdot),
\]
extending the method developed in the book [H. Triebel, Theory of Function Spaces. Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser Verlag (1983; Zbl 0546.46027)], Section 2.4.7, from constant exponents to variable exponents.
Reviewer: Hans Triebel (Jena)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42B35 | Function spaces arising in harmonic analysis |
| 46B70 | Interpolation between normed linear spaces |
Keywords:
Besov spaces; Triebel-Lizorkin spaces; atoms; complex interpolation; variable exponent function spacesCitations:
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