Martingale Theory of Variable Lorentz-Karamata Spaces and Applications to Walsh-Fourier Analysis

Abstract Let $$p(\cdot ):\Omega \rightarrow (0,\infty )$$ <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:mo>:</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>→</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> be a variable exponent, $$0&lt;q\le \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and b be a slowly varying function. In this paper, we discuss the martingale theory of variable Lorentz-Karamata spaces $$L_{p(\cdot ), q,b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <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:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:math> and apply it to Walsh-Fourier analysis. More precisely, we introduce the generalized BMO martingale spaces, which enable us to characterize the dual spaces of martingale Hardy spaces $$H^s_{p(\cdot ), q,b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> </mml:math> for $$0&lt;p_-\le p_+&lt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> </mml:msub> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$0&lt;q\le \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . The John-Nirenberg theorem for the generalized BMO martingale spaces are presented by the dual results. We also investigate the boundedness of fractional integral operators on martingale Hardy spaces $$H^M_{p(\cdot ), q,b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>M</mml:mi> </mml:msubsup> </mml:math> . As applications in Walsh-Fourier analysis, we consider the Walsh-Fourier series on variable Lorentz-Karamata spaces. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means. The results obtained here generalize the known results in previous literature.