The Hausdorff–Young Inequality and Hermite Based Triebel–Lizorkin Spaces

Abstract In this note, we investigate orthogonal expansions adapted to exponential weights and function spaces endowed with Hermite-frequency decompositions. Our objective is to formulate the Hausdorff–Young inequality within the framework of Hermite based Triebel–Lizorkin type spaces. The results rest on two underlying principles. First, since the Hermite functions $$\{{\mathcal {H}}_m(x)\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , along with their generalized counterparts, are particular instances of generalized Freud functions associated with generalized Freud weights, we rely on the $$L^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> and $$L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> , $$q&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , estimates of Kasuga and Sakai which extend Hille’s estimate $$|{\mathcal {H}}_m(x)|\lesssim m^{-1/12}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> <mml:mo>≲</mml:mo> </mml:mrow> <mml:msup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , $$m=1,2,\ldots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> Second, we apply interpolation techniques involving the Orlicz–Lorentz spaces $$\Lambda (\varphi _X, C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where C is a Young (or concave) function and $$\varphi _X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> denotes the fundamental function of a rearrangement-invariant space X . This approach enables us to derive Hausdorff–Young inequalities in $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> , Lorentz, Orlicz, and $$\Lambda (\varphi _X, C)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> based Hermite Triebel–Lizorkin spaces. Finally, in the Coda we present a general Hausdorff–Young inequality for the generalized Freud coefficients of functions $$f\in L^{p_0}({\mathbb {R}})+L^{p_1}({\mathbb {R}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:msub> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi>