Uniform Convexity of Paranormed Generalizations of L p Spaces

Abstract

<jats:p> For a measure space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">(\Omega,\Sigma,\mu )</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:mi>μ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and a bijective increasing function <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi:\left[ 0,\infty \right) \rightarrow \left[0,\infty \right)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mo fence="true">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo fence="true">)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mrow> <mml:mo fence="true">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo fence="true">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L^{p}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -like paranormed ( <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">F</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -normed) function space with the paranorm of the form <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbf{p}_{\varphi }(x)=\varphi ^{-1}\left( \int_{\Omega }\varphi \circ \left\vert x\right\vert d\mu \right)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="bold">p</mml:mi> <mml:mi>φ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>φ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo fence="true">(</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:msub> <mml:mi>φ</mml:mi> <mml:mo>∘</mml:mo> <mml:mrow> <mml:mo fence="true">∣</mml:mo> <mml:mi>x</mml:mi> <mml:mo fence="true">∣</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> <mml:mo fence="true">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L^{p}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -space is generalized. Under some specific assumptions imposed on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> we give not only a proof of the uniform convexity but also show the formula of a modulus of convexity. We establish the uniform convexity of all finite-dimensional paranormed spaces, generated by a strictly convex bijection <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">[0, \infty)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . However, the a contrario proof of this fact provides no information on a modulus of convexity of these spaces. In some cases it can be done, even an exact formula of a modulus can be proved. We show how to make it in the case when <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">S={\mathbb R}^2</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varphi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow>