| Summary: | We consider analytic functions in tubes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mi>B</mi><mo>⊂</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula> with values in Banach space or Hilbert space. The base of the tube <i>B</i> will be a proper open connected subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open connected cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, an open convex cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, and a regular cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula>, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space <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> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup></semantics></math></inline-formula>. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions <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>. We obtain boundary value results on the distinguished boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mrow><mo>{</mo><mover><mn>0</mn><mo>¯</mo></mover><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and on the topological boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>+</mo><mi>i</mi><mo>∂</mo><mi>B</mi></mrow></semantics></math></inline-formula> of the tube for the analytic functions in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> and vector valued tempered distribution topologies. Suggestions for associated future research are given.
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