Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:msubsup><a:mrow><a:mi mathvariant="script">B</a:mi></a:mrow><a:mrow><a:mi>H</a:mi></a:mrow><a:mrow><a:mi>λ</a:mi></a:mrow></a:msubsup><a:mo>,</a:mo><a:mfenced open="(" close=")"><a:mrow><a:mn>0</a:mn><a:mo><</a:mo><a:mi>λ</a:mi><a:mo><</a:mo><a:mrow><a:mo>∞</a:mo></a:mrow></a:mrow></a:mfenced></a:math> to the weighted harmonic Zygmund space <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M2"><f:msubsup><f:mrow><f:mi mathvariant="script">Z</f:mi></f:mrow><f:mrow><f:mi>H</f:mi></f:mrow><f:mrow><f:mi>β</f:mi></f:mrow></f:msubsup><f:mo>,</f:mo><f:mfenced open="(" close=")"><f:mrow><f:mn>0</f:mn><f:mo><</f:mo><f:mi>β</f:mi><f:mo><</f:mo><f:mrow><f:mo>∞</f:mo></f:mrow></f:mrow></f:mfenced></f:math> . The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M3"><k:msubsup><k:mrow><k:mi mathvariant="script">B</k:mi></k:mrow><k:mrow><k:mi>H</k:mi></k:mrow><k:mrow><k:mi>λ</k:mi></k:mrow></k:msubsup></k:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M4"><n:msubsup><n:mrow><n:mi mathvariant="script">Z</n:mi></n:mrow><n:mrow><n:mi>H</n:mi></n:mrow><n:mrow><n:mi>β</n:mi></n:mrow></n:msubsup></n:math> is bounded. Secondly, we will study the compactness case of the composition operator between <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M5"><q:msubsup><q:mrow><q:mi mathvariant="script">B</q:mi></q:mrow><q:mrow><q:mi>H</q:mi></q:mrow><q:mrow><q:mi>λ</q:mi></q:mrow></q:msubsup></q:math> and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M6"><t:msubsup><t:mrow><t:mi mathvariant="script">Z</t:mi></t:mrow><t:mrow><t:mi>H</t:mi></t:mrow><t:mrow><t:mi>β</t:mi></t:mrow></t:msubsup></t:math> . Finally, we will estimate the essential norms of the composition operator between <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M7"><w:msubsup><w:mrow><w:mi mathvariant="script">B</w:mi></w:mrow><w:mrow><w:mi>H</w:mi></w:mrow><w:mrow><w:mi>λ</w:mi></w:mrow></w:msubsup></w:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M8"><z:msubsup><z:mrow><z:mi mathvariant="script">Z</z:mi></z:mrow><z:mrow><z:mi>H</z:mi></z:mrow><z:mrow><z:mi>β</z:mi></z:mrow></z:msubsup></z:math> .