A Counterexample to an Endpoint Mixed Norm Estimate of Calderón-Zygmund Operators

Abstract

It is known that that the endpoint mixed norm estimate $||||Tf(x,y)|{|}_{L_x^p}|{|}_{L_y^{\infty }}\lesssim ||||f(x,y)|{|}_{L_x^p}|{|}_{L_y^{\infty }}$ in general does not hold for Calderón-Zygmund operator $T$. In this article, we show that when $p=2$, even if we make the right hand side of the above estimate larger by replacing it with $||||e^{x^2+y^2}f(x,y)|{|}_{L_y^{\infty }}|{|}_{L_x^{\infty }}$, the estimate does not hold for the double Riesz transform given by the kernel $K(x,y)=\frac{xy}{2\pi (x^2+y^2{)}^2}$. As a consequence we will show that the mixed norm estimate $||||Tf(x,y)|{|}_{L_x^p}|{|}_{L_y^{\infty }}\lesssim ||||f(x,y)|{|}_{L_y^{\infty }}|{|}_{L_x^p}$ does not hold for double Riesz transform and $p\ge 2$.

arXiv comment9 pages