| Summary: | The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mi>p</mi></msup></semantics></math></inline-formula> multiplier theorem on the Heisenberg group. If an operator-valued function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfies certain conditions, the right-multiplier operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math></inline-formula> is bounded on the Hardy space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">H</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.
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