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

It is known that that the endpoint mixed norm estimate $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }Tf(x,y)\mathrm{\mid }{\mathrm{\mid }}_{L_x^p}\mathrm{\mid }{\mathrm{\mid }}_{L_y^{\mathrm{\infty }}}\lesssim \mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }f(x,y)\mathrm{\mid }{\mathrm{\mid }}_{L_x^p}\mathrm{\mid }{\mathrm{\mid }}_{L_y^{\mathrm{\infty }}}$ 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 $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }{e}^{x^2+y^2}f(x,y)\mathrm{\mid }{\mathrm{\mid }}_{L_y^{\mathrm{\infty }}}\mathrm{\mid }{\mathrm{\mid }}_{L_x^{\mathrm{\infty }}}$, 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 $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }Tf(x,y)\mathrm{\mid }{\mathrm{\mid }}_{L_x^p}\mathrm{\mid }{\mathrm{\mid }}_{L_y^{\mathrm{\infty }}}\lesssim \mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }f(x,y)\mathrm{\mid }{\mathrm{\mid }}_{L_y^{\mathrm{\infty }}}\mathrm{\mid }{\mathrm{\mid }}_{L_x^p}$ does not hold for double Riesz transform and $p\ge 2$.