The Fourier transform in variable exponent Lebesgue spaces

Abstract

Abstract In this work we define a Fourier transform for each $$f\in L^{p(\cdot )}({{\,\mathrm{\mathbb {R}}\,}})$$ <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:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , for a large class of exponent functions $$p(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , as the distributional derivative of a Hölder continuous function. A norm is defined in the space of such Fourier transforms so that it is isometrically isomorphic to $$L^{p(\cdot )}({{\,\mathrm{\mathbb {R}}\,}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mspace/> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We also prove several properties of this Fourier transform, such as inversion in norm and an exchange theorem.