<p>For Hardy spaces and weighted Bergman spaces on the open unit ball in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {C}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, we determine exactly when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^p_\alpha \subset H^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> <mo>⊂</mo> <msup> <mi>H</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^p\subset A^q_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mo>⊂</mo> <msubsup> <mi>A</mi> <mi>α</mi> <mi>q</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0<q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$-\infty<\alpha <\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>∞</mi> <mo><</mo> <mi>α</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called <i>tight fitting</i> and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.</p>

Embedding and Compact Embedding Between Bergman and Hardy Spaces

期刊论文
发表日期：

2026年5月22日

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Guanlong Bao,
Pan Ma,
Fugang Yan,
Kehe Zhu

摘要

For Hardy spaces and weighted Bergman spaces on the open unit ball in ${\mathbb {C}}^n$ $C^{n}$ , we determine exactly when $A^p_\alpha \subset H^q$ $A_{α}^{p} \subset H^{q}$ or $H^p\subset A^q_\alpha $ $H^{p} \subset A_{α}^{q}$ , where $0<q<\infty $ $0 < q < \infty$ , $0<p<\infty $ $0 < p < \infty$ , and $-\infty<\alpha <\infty $ $- \infty < α < \infty$ . For each such inclusion we also determine exactly when it is a compact embedding. Although some special cases were known before, we are able to completely cover all possible cases here. We also introduce a new notion called tight fitting and formulate a conjecture in terms of it, which places several prominent known results about contractive embeddings in the same framework.