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• W1567104751 abstract "The Dirichlet-type space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript p Baseline left-parenthesis 1 less-than-or-equal-to p less-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mtext> </mml:mtext> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>D^{p} (1 leq p leq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Superscript p> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>A^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi> <mml:semantics> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:annotation encoding=application/x-tex>Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an analytic self-map of the disc and define <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal upper Phi Baseline left-parenthesis f right-parenthesis equals f ring normal upper Phi> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∘<!-- ∘ --></mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{Phi }(f) = f circ Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f element-of upper D Superscript p> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>f in D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The operator <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal upper Phi Baseline colon upper D Superscript p Baseline right-arrow upper D Superscript p> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>C_{Phi }: D^{p} rightarrow D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded (respectively, compact) if and only if a related measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu Subscript p> <mml:semantics> <mml:msub> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mu _{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Carleson (respectively, compact Carleson). If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal upper Phi> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>C_{Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded (or compact) on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript p> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the same behavior holds on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript q Baseline left-parenthesis 1 less-than-or-equal-to q greater-than p> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mtext> </mml:mtext> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>D^{q} (1 leq q > p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) and on the weighted Dirichlet space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript 2 minus p> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>D_{2-p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Compactness on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript p> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Subscript normal upper Phi> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>C_{Phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript p> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Inner functions which induce bounded composition operators on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Superscript p> <mml:semantics> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>D^{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are discussed briefly." @default.
• W1567104751 created "2016-06-24" @default.
• W1567104751 creator A5069004534 @default.
• W1567104751 date "2000-08-17" @default.
• W1567104751 modified "2023-10-16" @default.
• W1567104751 title "Composition operators on Dirichlet-type spaces" @default.
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