A Continuous Characterization of Triebel-Lizorkin Spaces Associated with Hermite Expansions
A Continuous Characterization of Triebel-Lizorkin Spaces Associated with Hermite Expansions
We study the properties of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math>). In addition to the endpoint estimates, we give the continuous characterizations of these spaces via harmonic and thermic extensions from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>into<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>. Based on this result, …