On \(L^p\)-\(L^1\) estimates of logarithmic-type in Hardy-Sobolev spaces of the disk and the annulus. (English) Zbl 1293.30073
Summary: The aim of this paper is to prove some optimal estimates of logarithmic-type in Hardy-Sobolev spaces \(H^{1,p}\), \(1\leq p\leq\infty\), of both the unit disk and the annulus. These estimates extend those previously established by L. Baratchart and M. Zerner [J. Comput. Appl. Math. 46, No.1–2, 255–269 (1993; Zbl 0818.65017)] in the Hardy-Sobolev space \(H^{1,2}\) of the unit disk and by S. Chaabane and the first author [C. R., Math., Acad. Sci. Paris 347, No. 17–18, 1001–1006 (2009; Zbl 1181.46023)] in the case of the spaces \(H^{1,\infty}\) of the unit disk and of the annulus.
MSC:
| 30E10 | Approximation in the complex plane |
| 30H99 | Spaces and algebras of analytic functions of one complex variable |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
Keywords:
Hardy-Sobolev spaces; Hardy’s convexity theorem; Kolmogorov-type inequality; logarithmic estimateReferences:
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