Suppose that <mml:math alttext="$\varphi(z)$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3C6;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is an analytic self-map of the unit disk <mml:math alttext="$\Delta$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x394;</mml:mi></mml:math>. We consider the boundedness of the composition operator <mml:math alttext="$C_{\varphi}$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mi>&#x3C6;</mml:mi></mml:msub></mml:math> from Bloch space <mml:math alttext="$\mathcal{B}$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x212C;</mml:mi></mml:math> into the spaces <mml:math alttext="$Q_T$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math> (<mml:math alttext="$Q_{T,0}$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>) defined by a nonnegative, nondecreasing function <mml:math alttext="$T(r)$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>T</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> on <mml:math alttext="$0\le r &#x3C; \infty$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>r</mml:mi><mml:mo>&#x3C;</mml:mo><mml:mi>&#x221E;</mml:mi></mml:math>.

Composition operators from the Bloch space into the spaces <mml:math alttext="$Q_T$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Q</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math>

Composition operators from the Bloch space into the spaces <svg style="vertical-align:-3.27605pt" height="15.5375" version="1.1" viewBox="0 0 21.725 15.5375" width="21.725" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D444" d="M745 361q0 -134 -83.5 -233t-214.5 -130l16 -11q97 -67 250 -132l-8 -23q-76 3 -131 16q-81 19 -242 125l-20 13q-129 8 -209 91t-80 208q0 160 116 271t289 111q136 0 226.5 -83t90.5 -223zM645 356q0 127 -57.5 201.5t-169.5 74.5q-126 0 -210.5 -104.5t-84.5 -248.5&#xA;q0 -97 46 -166.5t129 -87.5l84 15l29 -19q104 21 169 121.5t65 213.5z" /></g><g transform="matrix(.012,-0,0,-.012,13.113,15.475)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g></svg> 


International Journal of Mathematics and Mathematical Sciences

Composition operators from the Bloch space into the spaces QT

Abstract

Copyright