Hölder and locally Hölder Continuous Functions, and Open Sets of Class C^k, C^{k,lambda}

1. Front Matter

   Pages i-xi

   PDF

2. Hölder and locally Hölder continuous functions. The linear spaces \(C^{k}(\Omega), C^{k,\lambda}(\Omega),\; \mathrm{and}\;C^{k,\lambda}_{\mathrm {loc}}(\Omega)\)

   • Renato Fiorenza

   Pages 1-35

3. Coordinate changes in \(\mathbb{R}^n\). Rotations. Cones in \(\mathbb{R}^n\)

   • Renato Fiorenza

   Pages 37-75

4. Open sets with boundary of class \(C^{k}\) and of class \(C^{k, \lambda}\). The cone property

   • Renato Fiorenza

   Pages 77-96

5. Open sets of class \(C^{k}\) and of class \(C^{k,\lambda}\)

   • Renato Fiorenza

   Pages 97-123

6. Majorization formulas for functions in \(C^{m,\lambda}_{\mathrm {loc}}(\Omega), C^{m,\lambda}(\Omega),\; \mathrm{and}\;C^{m}(\Omega)\)

   • Renato Fiorenza

   Pages 125-144

7. Back Matter

   Pages 145-152

   PDF

This book offers a systematic treatment of a classic topic in Analysis. It fills a gap in the existing literature by presenting in detail the classic λ-Hölder condition and introducing the notion of locally Hölder-continuous function in an open set Ω in Rⁿ. Further, it provides the essential notions of multidimensional geometry applied to analysis.

Written in an accessible style and with proofs given as clearly as possible, it is a valuable resource for graduate students in Mathematical Analysis and researchers dealing with Hölder-continuous functions and their applications.

• Hölder continuity
• Local Hölder continuity
• Lipschitz continuity
• The Cone property
• Multidimensional geometry
• Regularity of boundary of sets
• Classes of open sets
• Local coordinate systems
• partial differential equations

• Dipartimento di Architettura, Università di Napoli "Federico II" Dipartimento di Architettura, Napoli, Italy

  Renato Fiorenza

Renato Fiorenza has been a professor in mathematical analysis at the University of Naples (Italy) "Federico II" since 1967. He became a professor when Italy was leading the field with Miranda, Caccioppoli, De Giorgi (who solved Hilbert's 19th problem, proving that for a class of elliptic partial differential equations with analytic coefficients the solutions had first derivatives which are Hölder continuous). Renato Fiorenza was an assistant of Caccioppoli. In his career he has established some important results about oblique derivative problems for partial differential equations.

Policies and ethics