Nonlinear operations and factorizations on a class of affine modulation spaces

Abstract In the first part of the paper, we obtain a necessary and sufficient condition on a complex-valued function 𝐹 on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> \mathbb{R}^{2} such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi mathvariant="bold-italic">M</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> F(f)\in{\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{n}) for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi mathvariant="bold-italic">M</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> f\in{\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>r</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> 1\leq r&lt;\infty , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>s</m:mi> <m:mo>&lt;</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> 1\leq s&lt;2 , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="bold-italic">M</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{n}) is the affine modulation space associated with the fractional Fourier transform. In this context, we obtain that if 𝐹 acts on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="bold-italic">M</m:mi> <m:mi>θ</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) by composition, then 𝐹 is necessarily real analytic and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>F</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo>